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![xv
List of Figures
2.1 Asymptotic Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Complexity Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Search space for a minimization problem: local and global optima . . . . 18
2.4 Hill Climbing: downhill transition to a local optimum . . . . . . . . . . . 19
2.5 Simulated Annealing: occasional uphill moves and escaping local optima 21
2.6 Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.7 Mutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.8 Partially Matched Crossover (PMX) . . . . . . . . . . . . . . . . . . . . 27
2.9 Foraging behaviour of ants . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.10 The problem of cycling . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1 Vehicle routing and scheduling problems . . . . . . . . . . . . . . . . . . 39
3.2 Taxonomy of the VRP literature [48] . . . . . . . . . . . . . . . . . . . . 41
3.3 The vehicle routing problem with time windows . . . . . . . . . . . . . . 43
3.4 Solution Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5 The savings heuristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.6 The insertion heuristic . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.7 2-exchange move . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.1 The single vehicle pickup and delivery problem with time windows . . . . 73
5.2 Solution Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 77](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-15-2048.jpg)
![xix
List of Algorithms
2.1 Hill Climbing (HC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 The Simulated Annealing (SA) Algorithm . . . . . . . . . . . . . . . . . 23
2.3 The Basic Genetic Algorithm (GA) . . . . . . . . . . . . . . . . . . . . . 25
2.4 The Ant Colony Optimization (ACO) Algorithm . . . . . . . . . . . . . . 30
2.5 The Tabu Search (TS) Algorithm . . . . . . . . . . . . . . . . . . . . . . 33
2.6 The Variable Neighbourhood Search (VNS) Algorithm [70] . . . . . . . 34
2.7 The Variable Neighbourhood Descent (VND) Algorithm [70] . . . . . . 35
5.1 Create Test Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2 Generate Time Windows . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.3 The SV-PDPTW Genetic Algorithm . . . . . . . . . . . . . . . . . . . . 86
6.1 Calculating Annealing Parameters [40] . . . . . . . . . . . . . . . . . . . 98
6.2 The Main SA Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.3 The Main HC Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.1 The HC Routing Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.2 The Sequential Construction . . . . . . . . . . . . . . . . . . . . . . . . 125
7.3 Parallel Construction: First Route . . . . . . . . . . . . . . . . . . . . . 127
7.4 Parallel Construction: Best Route . . . . . . . . . . . . . . . . . . . . . 128
7.5 Parallel Construction: Best Request . . . . . . . . . . . . . . . . . . . . 129
7.6 Basic Construction Algorithm for the VRP [24] . . . . . . . . . . . . . . 138](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-19-2048.jpg)
![xx List of Algorithms
8.1 The Complete Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . 157
8.2 The 2-Stage SA Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 169
9.1 VND(x) Procedure [73] . . . . . . . . . . . . . . . . . . . . . . . . . . 179
9.2 Hybrid GRASP/VND Procedure [73] . . . . . . . . . . . . . . . . . . . 179
9.3 The EPS Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
10.1 Find Initial Solution & Calculate Starting Temperature . . . . . . . . . . 198
10.2 Adaptive VNS-SA (AVNS-SA) Algorithm . . . . . . . . . . . . . . . . . 202
10.3 The VNSSA Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 203](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-20-2048.jpg)
![xxi
Acronyms
1M One Level Exchange Mutation
1-PDP One Commodity Pickup and Delivery Problem
ACO Ant Colony Optimization
ALNS Adaptive Large neighbourhood Search
AVNS-SA Adaptive Variable neighbourhood Search - Simulated Annealing
BF Best Fit
CDARP Capacitated Dial-A-Ride Problem
CEL centre TW - Early TW - Late TW
CKKL Evolutionary Approach to the MV-PDPTW in [33]
CL Number of Closest neighbours
CLE centre TW - Late TW - Early TW
CPP Chinese Postman Problem
CTSPPD Capacitated Traveling Salesman Problem with Pickup and Delivery
CV Capacity Violations
CVRP Capacitated Vehicle Routing Problem
DARP Dial-A-Ride Problem
DLS Descent Local Search
DP Dynamic Programming
ECL Early TW - centre TW - Late TW
ELC Early TW - Late TW - centre TW
EPA Exact Permutation Algorithm
EPS Evolutionary Perturbation Scheme
FCGA Family Competition Genetic Algorithm
GA Genetic Algorithm
GALIB Genetic Algorithm Library
GDP Gross Domestic Product
GGA Grouping Genetic Algorithm
GIS Geographic Information Systems
GLS Guided Local Search](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-21-2048.jpg)
![xxii Acronyms
GPS Geographic Positioning Systems
GRASP Greedy Randomized Adaptive Search Procedure
GRGA Grouping Routing Genetic Algorithm
GRS Greedy Random Sequence
GUI Graphical User Interface
HC Hill Climbing
HPT Handicapped Person Transportation
IBX Insertion-Based Crossover
ILS Iterated Local Search
IMSA Iterated Modified Simulated Annealing
IRNX Insert-within-Route Crossover
LC1 Li & Lim benchmark instances [100]- clustered customers- short TW
LC2 Li & Lim benchmark instances [100]- clustered customers- long TW
LCE Late TW - centre TW - Early TW
LEC Late TW - Early TW -centre TW
LNS Large neighbourhood Search
LR1 Li & Lim benchmark instances [100]- random customers- short TW
LR2 Li & Lim benchmark instances [100]- random customers- long TW
LRC1 Li & Lim benchmark instances [100]- random & clustered customers -
short TW
LRC2 Li & Lim benchmark instances [100] - random & clustered customers -
long TW
LS Local Search
LSM Local Search Mutation
MSA Modified Simulated Annealing
MTSP Multiple Vehicle Traveling Salesman Problem
MV Multiple Vehicle
MV-PDPTW Multiple Vehicle Pickup and Delivery Problem with Time Windows
MX1 Merge Crossover
NN Nearest neighbour
N P Non Deterministic Polynomial Time Problems
OR Operations Research
OR/MS Operations Research/ Management Science
P Polynomial Time Problems
PBQ Parallel Construction Best Request
PBR Parallel Construction Best Route
PD Pickup and Delivery](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-22-2048.jpg)
![1
Chapter 1
Introduction
Efficient transportation and logistics plays a role in the economic wellbeing of society.
Almost everything that we use in our daily lives involves logistic planning, and practically
all sections of industry and society, from manufacturers to home shopping customers,
require the effective and predictable movement of goods. In today’s dynamic business
environment, transportation cost constitutes a significant percentage of the total cost of a
product. In fact, it is not unusual for a company to spend more than 20% of the product’s
value on transportation and logistics [76]. In addition, the transportation sector itself is a
significant industry, and its volume and impact on society as a whole continues to increase
every day. In a 2008 report, the UK Department of Transport estimates that freight and
logistics sector is worth £74.5 billion to the economy and employs 2.3 million people
across 190,000 companies [2].
In the last few decades, advancement in transportation and logistics has greatly improved
people’s lives and influenced the performance of almost all economic sectors. Never-
theless, it also produced negative impacts. Approximately two thirds of our goods are
transported through road transport [2]. However, vehicles moving on our roads contribute
to congestion, noise, pollution, and accidents. Reducing these impacts requires coopera-
tion between the various stakeholders in manufacturing and distribution supply chains for
more efficient planning and resource utilization. Efficient vehicle routing is essential, and
automated route planning and transport management, using optimization tools, can help
reduce transport costs by cutting mileage and improving driver and vehicle usage. In ad-
dition, it can improve customer service, cut carbon emissions, improve strategic decision
making and reduce administration costs.
Research studying effective planning and optimization in the vehicle routing and sche-
duling field has increased tremendously in the last few decades [48] . Advances in tech-
nology and computational power has encouraged researchers to consider various problem
types and real-life constraints, and to experiment with new algorithmic techniques that can
be applied for the automation of vehicle planning activities. A number of these techniques
have been implemented in commercial optimization software and successfully used in](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-25-2048.jpg)
![2 1.1 Research Problem and Motivation
every day transportation and logistics applications. However, due to the large increase in
demand and the growing complexity of this sector, innovations in solution techniques that
can be used to optimize vehicle routing and scheduling are still urgently needed.
Our research is concerned with developing efficient algorithmic techniques that can be
applied in vehicle routing and planning activities. In Section 1.1 we introduce the research
problems that we are trying to tackle and emphasize the motivation behind our selection
of these problems. Section 1.2 highlights the hypothesis and contribution of the thesis.
Section 1.3 gives an overview of the contents of this thesis and indicates the publications
related to each part of the research. Finally, some brief remarks about the computational
experimentation performed in this research are presented in Section 1.4.
1.1 Research Problem and Motivation
The Vehicle Routing Problem (VRP) is at the heart of scientific research involving the
distribution and transportation of people and goods. The problem can be generally des-
cribed as finding a set of minimum cost routes for a fleet of vehicles, located at a central
depot, to serve a number of customers. Each vehicle should depart from and return to the
same depot, while each customer should be visited exactly once. Since the introduction
of this problem by Dantzig and Fulkerson in 1954 [35] and Dantzig and Ramser in 1959
[35], several extensions and variations have been added to the basic VRP model, in or-
der to meet the demand of realistic applications that are complex and dynamic in nature.
For example, restricting the capacity of the vehicle and specifying certain service times
for visiting clients are among the popular constraints that have been considered while
addressing the VRP.
One important variant of the VRP is the Pickup and Delivery Problem (PDP). Unlike the
classical VRP, where all customers require the same service type, a central assumption in
the PDP is that there are two different types of services that can be performed at a customer
location, a pickup or a delivery. Based on this assumption, other variants of the PDP have
also been introduced depending on, for example, whether the origin/destination of the
commodity is the depot or some customer location, whether one or multiple commodities
are transferred, whether origins and destinations are paired, and whether people or goods
are transported. In addition, some important real-life constraints that have been applied to
the basic VRP have also been applied to the PDP, such as the vehicle capacity and service
time constraints.
Applications of the PDP are frequently seen in transportation and logistics. In addition,](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-26-2048.jpg)
![1.2 Thesis Hypothesis and Contribution 3
the problem is likely to become even more important in the future, due to the rapid growth
in parcel transportation as a result of e-commerce. Some applications of this problem
include bus routing, food and beverage distribution, currency collection and delivery bet-
ween banks and ATM machines, Internet-based pickup and delivery, collection and distri-
bution of charitable donations between homes and different organizations, and the trans-
port of medical samples from medical offices to laboratories, just to name a few. Also, an
important PDP variant, known as dial-a-ride, can provide an effective means of transport
for delivering people from door to door. This model is frequently adopted for the disa-
bled, but could provide a greener alternative than the car and taxi to the wider community.
Besides road-based transportation, applications of the PDP can also be seen in maritime
and airlift planning.
As previously mentioned, considerable attention has been paid to the classical VRP and
its related variants by the scientific community. Literally thousands of papers have been
published in this domain (see for example survey papers [142], [96], [48], [147] and the
survey paper in the 50th anniversary of the VRP [97]). Despite this, research on Pickup
and Delivery (PD) problems is relatively scarce [137]. A possible reason is the complexity
of these problems and the difficulty in handling the underlying problem constraints. In-
novations in solution methods that handle different types of PD problems are certainly in
great demand.
We have selected two important variants of PD problems as the focus of this research.
These are: the Pickup and Delivery Problem with Time Windows (PDPTW), and the
One-commodity pickup and delivery problem (1-PDP), with more emphasis on the first
of these two variants and a thorough investigation of both its single and multiple vehicle
cases. The main difference between the two problems is that the PDPTW assumes that
origins and destinations are paired, while the 1-PDP assumes that a single commodity may
be picked up from any pickup location and delivered to any delivery location. In fact, the
problems dealt with in this research are regularly encountered in every day transportation
and logistics applications, as will be demonstrated during the course of this thesis.
1.2 Thesis Hypothesis and Contribution
Similar to the VRP, the PDP belongs to the class of Combinatorial Optimization (CO)
problems, for which there is usually a huge number of possible solution alternatives. For
example, the simplest form of the VRP is the Traveling Salesman Problem (TSP), where
a salesman must visit n cities starting and ending at the same city, and each city should
be visited exactly once. It is required to find a certain route for the salesman such that the](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-27-2048.jpg)
![4 1.2 Thesis Hypothesis and Contribution
total traveling cost is minimized. Although this problem is simple for small values of n,
the complexity of solving the problem increases quickly as n becomes large. For a TSP
problem of size n, the number of possible solutions is n!/2. So, if n = 60 for example,
the number of possible solutions that must be searched is approximately 4.2 × 1081
.
If one considers the VRP with added problem constraints, the solution process becomes
even more complex. Despite the advancement in algorithmic techniques, solving the VRP
and similar highly constrained CO problems to optimality may be prohibitive for very
large problem sizes. Exact algorithms that may be used to provide optimum problem
solutions cannot solve VRP instances with more than 50-100 customers, given the current
resources [72]. Approximation algorithms that make use of heuristic and meta-heuristic
approaches are often used to solve problems of practical magnitudes. Such approaches
provide good solutions to CO problems in a reasonable amount of time, compared to exact
methods, with no guarantee to solve the problem to optimality (see Chapter 2 for more
details about combinatorial optimization, complexity theory and exact and approximation
techniques).
Many heuristic and meta-heuristic algorithms have been applied to solving the different
variants of PD problems. However, most of these approaches are adaptations of algo-
rithms that have been previously developed for the classical VRP. The solution process
for such problems is usually divided into two phases: solution construction and solution
improvement. In the solution construction phase one or more initial problem solutions
is generated, while in the solution improvement phase the initial solution(s) is gradually
improved, in a step-by-step fashion, using a heuristic or a meta-heuristic approach. Ap-
plying classical VRP techniques to the PDP, though, requires careful consideration, since
the PDP is in fact a highly constrained problem that is much harder than other variants of
the VRP.
We noticed during our literature survey that research tackling PD problems tends to exhi-
bit complexity and sophistication in the solution methodology. Probably the main reason
behind this is that handling the difficult, and sometimes conflicting, problem constraints is
often a hard challenge for researchers. Straightforward heuristic and meta-heuristic algo-
rithms cannot normally be applied to PD problems, without augmenting the approach with
problem-specific techniques. In many cases, researchers resort to sophisticated methods
in their search for good quality solutions. For example, they may hybridize several heu-
ristics and meta-heuristics, or utilize heuristics within exact methods. Some researchers
use adaptive and probabilistic search algorithms, which undergo behavioural changes dy-
namically throughout the search, according to current features of the search space or to
recent performance of the algorithm. Dividing the search into several stages, adding a](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-28-2048.jpg)
![8 1.3 Thesis Overview
The survey presented in this chapter is intended to be a concise overview. More details
about state-of-the-art techniques are presented in later chapters, when related work to each
individual problem that we handled is summarized.
Chapter 5 is the first of six chapters that detail the research carried out in this thesis. This
chapter describes the first problem that we tackled, the Single Vehicle Pickup and Deli-
very Problem with Time Windows (SV-PDPTW). A general overview of the problem, and
a summary of related work from the literature is given. In addition, a Genetic Algorithm
(GA) approach to solving this problem is described in detail. A novel solution representa-
tion and intelligent neighbourhood moves that may help the algorithm overcome difficult
problem constraints are tried within the GA approach. The experimental findings of this
part of the research were published, as a late breaking paper, in GECCO2007 conference
[79].
Chapter 6 continues with the research started in Chapter 5 for the SV-PDPTW. Never-
theless, a more extensive investigation of the problem is given here. Several heuristic and
meta-heuristic techniques are tested for solving the problem, employing the same tools
introduced in Chapter 5. Three approaches are applied to solving the problem: genetic
algorithms, simulated annealing and hill climbing. A comparison between the different
approaches and a thorough analysis of the results is provided. The algorithms and the
findings of this part of the research were published in the Journal of Heuristics [84].
Chapter 7 starts the investigation of the more general Multiple-Vehicle case of the Pi-
ckup and Delivery Problem with Time Windows (MV-PDPTW). After summarizing some
related work in this area, new solution construction methods are developed and compa-
red, using the single vehicle case as a sub-problem. The algorithms introduced here act
as a first step towards a complete solution methodology to the problem, which will be
presented in the next chapter. The different construction heuristics are compared, and
conclusions are drawn about the construction algorithm that seems most appropriate for
this problem. This part of the research was published in the MIC2009 [81].
Chapter 8 continues the investigation of the MV-PDPTW, by augmenting solution construc-
tion, introduced in the previous chapter, with solution improvement. In this chapter, we
tried both a GA and an SA for the improvement phase. Several techniques from the first
part of our research (explained in Chapters 5 to Chapter 7) are embedded in both the GA
and the SA. The GA approach is compared with two related GA techniques from the li-
terature, and the results of both the GA and the SA are thoroughly analyzed. Part of the
research carried out in this chapter was published in the MIC2009 [80].
Chapter 9 starts the investigation of another important PD problem which is the One-
commodity Pickup and Delivery Problem (1-PDP). In this chapter, we explain the pro-](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-32-2048.jpg)
![1.4 A Note on Implementation and Computational Experimentation 9
blem, provide a literature review, and highlight some related problems. After that, we try
to solve the 1-PDP using an Evolutionary Perturbation Scheme (EPS) that has been used
for solving other variants of vehicle routing problems. Experimental results on published
benchmark instances are reported and analyzed.
Chapter 10 continues the investigation of the 1-PDP, using another meta-heuristic ap-
proach, which is a Variable Neighbourhood Search (VNS) hybridized with Simulated
Annealing (SA). Also, adaptation of search parameters is applied within this heuristic
for more efficient searching. The approach is thoroughly explained and the experimental
results on published test cases are reported. Two conference papers that discuss this ap-
proach have now been accepted for publication in GECCO2010 (late breaking abstract)
[82], and PPSN2010 [85]. In addition, a third journal paper has been submitted to the
Journal of Heuristics and is currently under review [83].
Chapter 11 looks beyond the current research phase and discusses how a theoretical
scientific research, such as that carried out in this thesis, can be used in real-life com-
mercial applications of vehicle routing problems. We give examples of some commercial
software tools, and we also highlight important industrial aspects of vehicle routing that
the research community should be aware of. An overview of future trends in scientific
research tackling this issue is also provided.
Chapter 12 summarizes the research undertaken in this thesis and elaborates on the thesis
contribution. Some critical analysis of parts of the current research and suggestions for
future work are also presented.
1.4 A Note on Implementation and Computational Expe-
rimentation
All algorithmic implementations presented in this thesis are programmed in C++, using
Microsoft Visual Studio 2005. Basic Genetic Algorithms (GAs) were implemented with
the help of an MIT GA library GALIB [155]. Most of the computational experimentations
were carried out using Intel Pentium (R) CPU, 3.40 GHz and 2 GB RAM, under a Win-
dows XP operating system, unless otherwise indicated. The algorithms developed in this
thesis were tested on published benchmark data for the selected problems, or on problem
instances created in this research. We indicate in the relevant sections of this thesis the
exact source of the data, and provide links where applicable.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-33-2048.jpg)
![2.1 Combinatorial Optimization, Algorithm and Complexity Analysis 13
frequently used notations for algorithm analysis are the Ω notation and the Θ notation.
The Ω notation provides an asymptotic lower bound, i.e., f(n) is of Ω(g(n)) if there exist
positive constants c and n0 such that 0 ≤ cg(n) ≤ f(n), for all n ≥ n0. On the other hand,
the Θ notation provides an asymptotically tight bound for f(n), i.e., f(n) is of Θ(g(n))
if there exist positive constants c1, c2 and n0 such that 0 ≤ c1g(n) ≤ f(n) ≤ c2g(n), for
all n ≥ n0 [32]. Figures 2.1(a), 2.1(b) and 2.1(c) demonstrate a graphic example of the O
notation, the Ω notation, and the Θ notation respectively.1
(a) The O notation (b) The Ω notation
(c) The Θ notation
Figure 2.1: Asymptotic Notations.
In addition to analyzing the efficiency of an algorithm, we sometimes need to know what
types of algorithms exist for solving a particular problem. The field of complexity ana-
lysis analyzes problems rather than algorithms. Two important classes of problems are
usually identified in this context. The first class is called P (polynomial time problems).
1
In this thesis we use the O notation for the analysis of algorithms when needed, since it is the most
commonly used type of asymptotic notations among the three notations introduced in this section.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-37-2048.jpg)
![14 2.1 Combinatorial Optimization, Algorithm and Complexity Analysis
It contains problems that can be solved using algorithms with running times such as O(n),
O(log(n)) and O(nk
) 2
. They are relatively easy problems, sometimes called tractable
problems. Another important class is called N P (non-deterministic polynomial time
problems). This class includes problems for which there exists an algorithm that can
guess a solution and verify whether the guessed solution is correct or not in polynomial
time. If we have an unbounded number of processors that each can be used to guess and
verify a solution to this problem in parallel, the problem can be solved in polynomial time.
One of the big open questions in computer science is whether the class P is equivalent
to the class N P. Most scientists believe that they are not equivalent. This, however, has
never been proven.
Researchers also distinguish a sub class of N P, called the N P-complete class. In a sense,
this class includes the hardest problems in computer science, and is characterized by the
fact that either all problems that are N P-complete are in P, or not in P. Many N P-
complete problems may require finding a subset within a base set (e.g. the SAT problem),
or an arrangement (or permutation) of discrete objects (e.g. the TSP), or partitioning an
underlying base set (e.g. graph coloring). As mentioned above, these problems belong
to the combinatorial optimization class of problems, and sometimes called intractable
problems.
An optimization problem for which the associated decision problem is N P-complete is
called an N P-hard problem. A decision problem is a problem that has only two pos-
sible answers: yes or no. For example, if the problem is a cost minimization problem,
such that it is required to find a solution with the minimum possible cost, the associated
decision problem would be formulated as: “ is there a solution to the problem whose cost
is B, where B is a positive real number?”. Figure 2.2 demonstrates how the different
complexity classes may relate to each other. For more details about algorithm analysis
and complexity theory, the reader is referred to the book by Garey and Johnson [54] and
Cormen et al. [32].
Solving CO problems has been a challenge for many researchers in computer science and
operations research. Exact methods used to solve regular problems cannot be used to
solve CO problems given current resources, since searching among all possible solutions
of a certain intractable problem is usually prohibitive for large problem sizes. The natural
alternative would be to use approximation methods that give good, rather than optimal,
solutions to the problem in a reasonable amount of time.
In the next section we briefly consider some exact methods that can be used to solve
2
Running times of O(n) are usually called linear running times. Also, running time of O(nk
) are often
referred to as polynomial running times, while running times of O(2n
) are called exponential running times.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-38-2048.jpg)
![16 2.3 Heuristic Algorithms
complex calculation into a series of simpler calculations. Only certain CO problems
are amenable to DP.
2. Branch-and-Bound (B&B): is an optimization technique which is based on a sys-
tematic search of all possible solutions, while discarding a large number of non-
promising candidate solutions, using a depth-first strategy. The decision to discard
a certain solution is based on estimating upper and lower bounds of the quantity
to be optimized, such that nodes whose objective function are lower/higher than
the current best are not explored. The algorithm requires a ‘branching’ tool that
can split a given set of candidates into two smaller sets, and a ‘bounding’ tool that
computes upper and lower bounds for the function to be optimized within a given
subset. The process of discarding fruitless solutions is usually called ‘pruning’. The
algorithm stops when all nodes of the search tree are either pruned or solved.
3. Branch-and-Cut (B&C): is a B&B technique with an additional cutting step. The
idea is to try to reduce the search space of feasible candidates by adding new
constraints (cuts). Adding the cutting step may improve the value returned in the
bounding step, and could allow solving subproblems without branching.
2.3 Heuristic Algorithms
Given the limitation of exact methods in solving large CO problems, approximation tech-
niques are often preferred in many practical situations. Approximation algorithms, like
heuristics and meta-heuristics3
, are techniques that solve ‘hard’ CO problems in a rea-
sonable amount of computation time, compared to exact algorithms. However, there is
no guarantee that the solution obtained is an optimal solution, or that the same solution
quality will be obtained every time the algorithm is run.
Heuristic algorithms are usually experience-based, with no specific pre-defined rules to
apply. Simply, they are a ‘common-sense’ approach to problem solving. On the other
hand, an important subclass of heuristics are meta-heuristic algorithms, which are general-
purpose frameworks that can be applied to a wide range of CO problems, with just minor
changes to the basic algorithm definition. Many meta-heuristic techniques try to mimic
biological, physical or natural phenomena drawn from the real-world.
3
In this discussion we use the term approximation/heuristic algorithm to refer to any ‘non-exact’ so-
lution method. More accurately, however, the term approximation algorithm is often used to refer to an
optimization algorithm which provides a solution that is guaranteed to be within a certain distance from the
optimum solution every time it is run, with provable runtime bounds [153]. This may not be necessarily the
case for heuristic algorithms, though.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-40-2048.jpg)
![2.3 Heuristic Algorithms 17
In solving a CO problem using a heuristic or a meta-heuristic algorithm, the search for
a good problem solution is usually divided into two phases: solution construction and
solution improvement. Solution construction refers to the process of creating one or more
initial feasible solutions that will act as a starting point from which the search progresses.
In this phase, a construction heuristic usually starts from an empty solution and gradually
adds solution components to the partially constructed solution until a feasible solution
is generated. On the other hand, solution improvement tries to gradually modify the
starting solution(s), based on some predefined metric, until a certain solution quality is
obtained or a given amount of time has passed.
Within the context of searching for a good quality solution, we usually use the term search
space to refer to the state space of all feasible solutions/states that are reachable from the
current solution/state. To find a good quality solution, a heuristic or a meta-heuristic al-
gorithm moves from one state to another, i.e., from one candidate solution to another,
through a process that is often called Local Search (LS). During LS, the new solution
is usually generated within the neighbourhood of the previous solution. A neighbou-
rhood N(x) of a solution x is a subset of the search space that contains one or more
local optima, the best solutions in this neighbourhood. The transition process from one
candidate solution to another within its neighbourhood requires a neighbourhood move
that changes some attributes of the current solution to transform it to a new solution x′
.
A cost function f(x) is used to evaluate each candidate solution x and determine its cost
compared to other solutions in its neighbourhood. The best solution within the overall
search space is called the globally optimal solution, or simply the optimum solution.
Figure 2.3 shows a search space for a minimization function, i.e., our goal is to obtain the
global minimum for this function. In this figure three points (solutions) are local optima
within their neighbourhoods, A, B and C. The global optimum among the three points is
point C.
Several modifications to the basic local search algorithm have been suggested to solve CO
problems. For example, Baum [10] suggested an Iterated Local Search (ILS) procedure
for the TSP, in which a local search is applied to the neighbouring solution x′
to obtain
another solution x′′
. An acceptance criterion is then applied to possibly replace x′
by x′′
.
Voudouris and Tsang [154] suggest a Guided Local Search (GLS) procedure, in which
penalties are added to the objective function based on the search experience. More spe-
cifically, the search is driven away from previously visited local optima, by penalizing
certain solution features that it considers should not occur in a near-optimal solution.
In addition, some local search methods have been formulated under the meta-heuristic fra-
mework, in which a set of predefined rules or algorithmic techniques help guide a heuristic](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-41-2048.jpg)
![18 2.3 Heuristic Algorithms
Figure 2.3: Search space for a minimization problem: local and global optima.
search towards reaching high quality solutions to complex optimization problems. Among
the most famous meta-heuristic techniques are: Hill Climbing (HC), Simulated Annea-
ling (SA), Genetic Algorithms (GAs), Ant Colony Optimization (ACO), Tabu Search and
Variable Neighbourhood Search (VNS). For more details about local search techniques
and its different variations and applications to CO problems, the reader is referred to [3]
and [125]. In what follows we describe some important meta-heuristic techniques that
have been widely used for solving CO problems.
2.3.1 Hill Climbing (HC)
Hill Climbing is the simplest form of local search, where the new neighbouring solution
always replaces the current solution if it is better in quality. The process can thus be
visualized as a step-by-step movement towards a locally optimum solution. Figure 2.4
shows how an HC algorithm transitions ‘downhill’ from an initial solution to a local mi-
nimum. On the other hand, Algorithm 2.1 shows the steps of an HC procedure applied to
a minimization problem.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-42-2048.jpg)
![20 2.3 Heuristic Algorithms
2.3.2 Simulated Annealing (SA)
Simulated annealing is a well-known meta-heuristic search method that has been used
successfully in solving many combinatorial optimization problems. It is a hill climbing
algorithm with the added ability to escape from local optima in the search space. However,
although it yields excellent solutions, it is very slow compared to a simple hill climbing
procedure.
The term simulated annealing is adopted from the annealing of solids, where we try to
minimize the energy of the system using slow cooling until the atoms reach a stable state.
The slow cooling technique allows atoms of the metal to line themselves up and to form a
regular crystalline structure that has high density and low energy. The initial temperature
and the rate at which the temperature is reduced is called the annealing schedule.
The theoretical foundation of SA was led by Kirkpatrick et al. in 1983 [93], where they
applied the Metropolis algorithm [107] from statistical mechanics to CO problems. The
Metropolis algorithm in statistical mechanics provides a generalization of iterative im-
provement, where controlled uphill moves (moves that do not lower the energy of the
system) are probabilistically accepted in the search for obtaining a better organization and
escaping local optima. In each step of the Metropolis algorithm, an atom is given a small
random displacement. If the displacement results in a decrease in the system energy, the
displacement is accepted and used as a starting point for the next step. If on the other hand
the energy of the system is not lowered, the new displacement is accepted with a certain
probability exp(−E/kbT)
where E is the change in energy resulting from the displacement,
T is the current temperature, and kb is a constant called a Boltzmann constant . Depending
on the value returned by this probability either the new displacement is accepted or the
old state is retained. For any given T, a sufficient number of iterations always leads to
thermal equilibrium. The SA algorithm has also been shown to possess a formal proof of
convergence using the theory of Markov Chains [47].
In solving a CO problem using SA, we start with a certain feasible solution to the pro-
blem. We then try to optimize this solution using a method analogous to the annealing
of solids. A neighbour of this solution is generated using an appropriate method, and the
cost (or the fitness) of the new solution is calculated. If the new solution is better than
the current solution in terms of reducing cost (or increasing fitness), the new solution is
accepted. If the new solution is not better than the current solution, though, the new solu-
tion is accepted with a certain probability. The probability of acceptance is usually set to
exp(−∆/T)
, where ∆ is the change in cost between the old and the new solution and T is
the current temperature. The probability thus decreases exponentially with the badness of](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-44-2048.jpg)
![22 2.3 Heuristic Algorithms
number of iterations N(t) to be executed at each temperature, and a stopping criterion to
terminate the algorithm.
As previously mentioned, the initial temperature T(0) is generally chosen high enough
so that almost any move is accepted regardless of its fitness. This choice is adopted from
the physical analogy and corresponds to heating up a substance until all particles are
randomly arranged in a liquid. The temperature update function is usually a proportional
temperature function T(t + 1) = αT(t), where t is the current iteration. Typical values
of α lie between 0.8 and 0.99. Using such values provides very small decrements of the
temperature, which corresponds to slow cooling of the substance until the temperature
approaches zero.
The number of iterations carried out at each temperature value should be large enough to
bring the system to a stable state equating to thermal equilibrium in the physical analogy.
Some applications may choose N(t) to be constant for each temperature. The stopping
criterion of the algorithm is usually the stagnation of the system when no change in the
result can be obtained for a specified number of iterations or temperature changes.
Implementing SA also requires a set of problem-specific decisions. These include: iden-
tifying the set of feasible solutions to the problem, defining a clear objective function, ge-
nerating an initial solution, and defining a neighbourhood operator that generates moves
using the current solution.
The topology of the neighbourhood structure is also critical to the performance of the SA
algorithm. In general, a smooth topology with shallow local optima is favoured over a
bumpy topology with many deep local minima. A neighbourhood function is also easy to
implement for discrete problems, while implementing a neighbourhood function for conti-
nuous problems is more challenging. Constrained problems also raise some difficulties.
A choice must be made between restricting the solution space to solutions that conform to
the constraints, or allowing solutions that break the constraints at the expense of a suitably
defined penalty function. The generic SA algorithm is described in Algorithm 2.2, where
the underlying optimization problem is again assumed to be a minimization function.
Some modifications to the basic SA algorithm have been suggested. These modifications
are intended to provide an improvement of the quality of the solution and/or processing
time. One attempt is to store the best solution found so far. Since the SA algorithm accepts
solutions probabilistically, it may accept solutions that are worse than the current solution.
A good solution found during the run may be discarded because it was not lucky during
the acceptance attempt. Storing the best solution found so far prevents the SA algorithm
from returning a final solution that is worse than the best solution ever found (e.g. [63]).](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-46-2048.jpg)
![2.3 Heuristic Algorithms 23
Algorithm 2.2: The Simulated Annealing (SA) Algorithm.
1: Generate an initial solution x
2: T ← T0 {initialize current temperature}
3: repeat
4: for (i = 0; i < numIterations; i + +) do
5: Generate a new solution x′ within the neighbourhood of x (x′ ∈ N(x))
6: ∆ ← f(x′) − f(x)
7: if (∆ < 0) then
8: x ← x′
9: else
10: p = Random(0, 1) {generate a random number in the interval (0,1)}
11: if (p < exp(−∆/T)) then
12: x ← x′
13: T ← α × T {reduce current temperature}
14: until (Done){stopping condition is reached}
15: Return solution x
Some researches also modify the basic SA algorithm using problem specific information.
For example [148] suggest a neighbourhood operator that is able to identify promising
areas in the neighbourhood, and give a greater probability to generated moves that fall
in the promising areas. SA has also been hybridized with other optimization techniques.
This can be done by using another technique to generate a good initial solution that SA
can improve, or using SA to generate a good solution that can be used by another search
technique.
To sum up, SA has several attractive features, especially in difficult optimization problems
in which obtaining a good solution, with a reasonable computational effort and proces-
sing time, is preferred to an optimal solution with considerably higher cost. The basic
advantages of SA are the following:
1. It is very easy to implement, since it just requires a method for generating a move in
the neighbourhood of the current solution, and an appropriate annealing schedule.
2. It can be applied to a wide range of problem types. For example, any combinatorial
optimization problem can be tackled using SA, if an appropriate neighbourhood
structure has been devised.
3. High quality solutions can be obtained using SA, if a good neighbourhood structure
and a good annealing schedule have been chosen.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-47-2048.jpg)
![24 2.3 Heuristic Algorithms
These benefits, however, may be at the expense of a longer processing time, compared
to simple hill climbing. In our research we applied SA to solving the SV-PDPTW, as
explained in Chapter 6, and for the MV-PDPTTW, as explained in Chapter 8. We also
hybridized SA with a Variable Neighbourhood Search (VNS) approach for solving 1-PDP,
as will be shown in detail in Chapter 10.
2.3.3 Genetic Algorithms (GAs)
The idea of simulation of biological evolution and the natural selection of organisms dates
back to the 1950’s. One of the early pioneers in this area was Alex Fraser with his research
published in 1957 [50] [51]. Nevertheless, the theoretical foundation of GAs were esta-
blished by John Holland in 1975 [78], after which GAs became popular as an intelligent
optimization technique that may be adopted for solving many difficult problems.
The theme of a GA is to simulate the processes of biological evolution, natural selection
and survival of the fittest in living organisms. In nature, individuals compete for the re-
sources of the environment, and they also compete in selecting mates for reproduction.
Individuals who are better or fitter in terms of their genetic traits survive to breed and
produce offspring. Their offspring carry their parents’ basic genetic material, which leads
to their survival and breeding. Over many generations, this favourable genetic material
propagates to an increasing number of individuals. The combination of good characteris-
tics from different ancestors can sometimes produce ‘super fit’ offspring who out-perform
their parents. In this way, species evolve to become better suited to their environment.
GAs operate in exactly the same manner. They work on a population of individuals re-
presenting possible solutions to a given problem. In traditional GAs, each individual is
usually represented by a string of bits analogous to chromosomes and genes, i.e., the
parameters of the problem are the genes that are joined together in a solution chromo-
some. A fitness value is assigned to each individual in order to judge its ability to survive
and breed. The highly fit individuals are given a chance to breed by being selected for
reproduction. Thus, the selection process usually favours the more fit individuals. Good
individuals may be selected several times in one iteration, while poor ones may not be
selected at all. By selecting the ‘most fit’ individuals, favourable characteristics spread
throughout the population over several generations, and the most promising areas of the
search space are explored. Finally, the population should converge to an optimal or near
optimal solution. Convergence means that the population evolves toward increasing uni-
formity, and the average fitness of the population will be very close to the highest fitness.
During the reproduction phase of a GA, two individuals breed by combining their genes in](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-48-2048.jpg)
![2.3 Heuristic Algorithms 27
Chromosome representation as a bit string is not suitable for many problems types, though.
For example, a permutation of integers representing nodes or locations has been often used
for the TSP and other vehicle routing problems. This kind of representation is called an
order-based representation, since the fitness depends on the order in which the genes ap-
pear. Different crossover and mutation operators must be used in this case, since classical
genetic operators, such as the ones described above, will result in ‘infeasible’ chromo-
somes. Goldberg in [64] described several order-based operators, such as the Partially
Matched Crossover (PMX). In a PMX, it is not the genes that are crossed, but the order
in which they appear, i.e., offspring have genes that inherit order information from each
parent. This avoids the problem of generating offspring that violate problem constraints,
such as having duplicate cities in a chromosome that represents a solution to the TSP, or
having a chromosome with some non-visited cities. Figure 2.8 shows an example of a
PMX, where cities 5,6,7 exchange their positions with 2, 3 and 10 respectively. Other
crossover operators that can be used for order-based representations are cycle crossover
and order crossover (see [64] for details).
Figure 2.8: Partially Matched Crossover (PMX).
To select two individuals in the population for mating, several selection methods may
be applied. For example, in a rank selection, individuals are sorted according to their
objective function values, and each individual is assigned a number of offspring that is
a function of its rank in the population. Roulette wheel selection picks an individual,
with a probability, based on the magnitude of the fitness score relative to the rest of the
population. The higher the score, the more likely an individual will be selected, and the
probability of the individual being chosen is equal to the fitness of the individual divided
by the sum of the fitness values of all individuals in the population. On the other hand, a
tournament selection uses the roulette wheel, or another selection method, to select two](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-51-2048.jpg)
![28 2.3 Heuristic Algorithms
or more individuals, then picks the one with the higher score.
Similarly, many replacement schemes can be used to integrate the new offspring and
produce the next generation. In a Simple GA, the new generation of offspring comple-
tely replace the old population, while a Steady State GA allows generations to overlap,
with a certain percentage of replacement. On the other hand, an Incremental GA only
allows a small degree of overlap between generations, for example by replacing one or
two individuals at each generation.
In summary, GAs represent an intelligent search method, since they operate on a popu-
lation of solutions and allocate trials to promising areas of the search space. GAs do not
depend heavily on information available from the underlying problem. They are easy
to connect to existing simulations and models, and can be easily hybridized to generate
knowledge-augmented GAs. Using the operations of selection of the fittest, mutation,
and crossover, GAs can quickly reach fit individuals (not always the most fit), but who are
usually good enough as solutions to problems of a large magnitude. Crossover is consi-
dered as the main GA operator. Having to combine two solutions rather than one, makes
designing an appropriate crossover operator often more challenging than developing a
mutation operator or a simple neighbourhood move. This usually makes GAs implemen-
tation more difficult compared to SA or simple HC.
In our research we applied GAs to solving the PDPTW, for both the single and multiple
vehicle variants, as will be explained in detail in Chapters 5, 6 and 8. We also applied a
GA within a perturbation scheme for the 1-PDP, as will be detailed in Chapter 9.
2.3.4 Ant Colony Optimization (ACO)
Ant Colony Optimization is a meta-heuristic technique that is inspired by the behaviour of
real ants. Its principles were established by Dorigo et al. in 1991 [42]. Real ants coope-
rate to find food resources by laying a trail of a chemical substance called ‘pheromone’
along the path from the nest to the food source. Depending on the amount of pheromone
available on a path, new ants are encouraged, with a high probability, to follow the same
path, resulting in even more pheromone being placed on this path. Shorter routes to food
sources have higher amounts of pheromone. Thus, over time, the majority of ants are di-
rected to use the shortest path. This type of indirect communication is called ‘stigmergy’
[41], in which the concept of positive feedback is exploited to find the best possible path,
based on the experience of previous ants. Figure 2.9 visualizes the foraging behaviour of
real ants in their search for food sources.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-52-2048.jpg)
![2.3 Heuristic Algorithms 31
cities. Certain parameters are also used in the probabilistic rule to determine the
relative importance of the pheromone trail and the heuristic information. During
the solution construction, individual ants may update the pheromone trail on the
edges they visited in a process called local pheromone update.
2. DaemonActions: in this optional step of the ACO algorithm, some problem-specific
action may be required, which cannot be usually performed by a single ant. For
example, local search may be applied to optimize the set of generated solutions.
The new optimized solutions are then used to decide which pheromone trails to
update.
3. UpdatePheromones: this is a global pheromone update process that is performed
at the end of each iteration, where updating the pheromone values depends on the
quality of the generated solutions in the current iteration. This is usually done by
decreasing the pheromone value for all solutions, in a process called evaporation,
and increasing the pheromone value of good solutions. Evaporation is a tool that
ACO uses to explore new areas of the search space and avoid being trapped in local
optima.
ACO has become very popular in solving CO problems. For example, its versatility and
robustness have been demonstrated in [43] by tailoring the approach to the TSP, the asym-
metric TSP, the Quadratic Assignment Problem (QAP), and the job-shop scheduling. It
has also been applied to variants of vehicle routing problems (e.g. [39] and [90]). The idea
of ‘attractiveness’ and ‘pheromone trails’ were also exploited within other meta-heuristic
techniques. For example, in the crossover operator used by [158] for solving the 1-PDP.
For further information about the various types of Ant systems and their applications, the
reader is referred to the book by Dorigo and Stützle [44].
2.3.5 Tabu Search (TS)
Tabu search is another popular search technique proposed by Glover in 1977 [59]. Since
then, it has been widely used for solving CO problems. Its name is derived from the word
‘taboo’ meaning forbidden or restricted. TS, like SA, allows for exploring the search
space ‘intelligently’ in an attempt to escape the trap of local optima. Nevertheless, there
are three main differences between TS and SA. Firstly, unlike SA, TS only accepts moves
within the vicinity of the current solution that improve the objective function. Secondly,
TS always searches for the best solution in the current neighbourhood before applying
the replacement criterion. Thirdly, the most distinguishing feature of TS is the use of](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-55-2048.jpg)
![2.3 Heuristic Algorithms 33
previously identified tabu moves may be prohibited during certain stages of the search.
Algorithm 2.5 shows the main steps of TS for a cost minimization problem.
Algorithm 2.5: The Tabu Search (TS) Algorithm.
1: Initialize an empty Tabu List T
2: Generate an initial solution x
3: Let x∗ ← x {x∗ is the best so far solution}
4: repeat
5: Generate a subset S of solutions in N(x) {N(x) is the current neighbourhood of x}
6: Select the best neighbourhood move x′ ∈ S, where f(x′) < f(x)
7: if (f(x′) < f(x∗)) then
8: x∗ ← x′ {aspiration condition: if current solution improves best so far, accept it even if
it is in the tabu list}
9: x ← x′ {update current solution}
10: T ← T + x′ {update tabu list}
11: else
12: if (x′ ∈ N(x) T) then
13: x ← x′ {update current solution if the new solution is not tabu}
14: if (f(x′) < f(x∗)) then
15: x∗ ← x′ {update the best so far solution if the new solution is better in quality}
16: T ← T + x′ {update tabu list}
17: until (Done){stopping condition is reached}
18: Return solution x∗
It is also sometimes fruitful in TS to make use of an intensification and/or a diversifica-
tion mechanism. Intensification tries to enhance the search around good solutions, while
diversification tries to force the algorithm to explore new search areas, in order to escape
local optima. For example, intensification can be performed by encouraging solutions that
have some common features with the current solution. On the other hand, diversification
may be enforced by applying a penalty in the objective function, at some stage of the
search, to solutions that are close to the present one [156].
Some variations of TS also exist in the literature, for example Probabilistic TS assigns
a probability to neighbourhood moves, such that some attractive moves that lower the
solution cost are given a higher probability, while moves that result in a repetition of
some previous state are given a lower probability [63]. Also, a Reactive TS was proposed
by Battiti and Tecchiolli [8] in which the size of the tabu list is adapted dynamically during
the search, according to the frequency of repeated moves.
TS has been extensively applied to many CO problems. For example, it has been applied](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-57-2048.jpg)
![34 2.3 Heuristic Algorithms
to scheduling in manufacturing systems in [113], to uniform graph partitioning in [36],
to the Quadratic Assignment Problem (QAP) in [140]. In addition, it was applied to
many variants of vehicle routing problems, for example in [129], [100], [112], [143] and
[99]. For more details about some variations and applications of tabu search, the reader is
referred to [62] and [55].
2.3.6 Variable Neighbourhood Search (VNS)
Variable Neighbourhood Search (VNS) is a relatively new meta-heuristic that has been
suggested by Hansen and Mladenovi´c in [69] and [70]. The main idea is based on explo-
ring the search space by gradually increasing the neighbourhood size, within which a new
solution is generated, until a certain stopping condition is reached. In addition, whenever
a new solution is generated in the current neighbourhood, a local search is applied to this
solution to optimize it before the replacement decision is undertaken. The basic VNS
algorithm, as described in [70], is shown in Algorithm 2.6.
Algorithm 2.6: The Variable Neighbourhood Search (VNS) Algorithm [70] .
1: Initialization: Select the set of neighbourhood structures Nk, (k = 1, ...kmax), that will be
used in the search
2: Generate an initial solution x
3: repeat
4: k ← 1
5: while (k < kmax) do
6: Shaking: Generate a point x′ at random from the kth neighbourhood of x (x′ ∈ Nk(x))
7: Local Search: Apply some local search method with x′ as initial solution; denote with
x′′ the so-obtained local optimum
8: Move or not: if the local optimum x′′ is better than the incumbent, move there (x ← x′′),
and continue the search with N1 (k ← 1); otherwise set k ← k + 1
9: until (Done){stopping condition is reached}
10: Return solution x
In the above VNS algorithm, the neighbourhood size k increases from 1 to a certain maxi-
mum value kmax. The most important step is the Shaking step in which a new point
(solution) is generated within the current neighbourhood Nk(x). It is crucial to choose a
shaking procedure that will allow enough perturbation of the solution, while preserving, at
the same time, the most favourable solution features which should be utilized in obtaining](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-58-2048.jpg)
![2.3 Heuristic Algorithms 35
a near optimum solution as the search progresses. For example, in solving the TSP, sha-
king my be performed by displacing or inverting a sequence of k cities in each iteration,
i.e., the number of cities to be displaced is the variable neighbourhood size that gradually
increases from one iteration to the next. On the other hand, the local search step in the
VNS algorithm intensifies the search to obtain a local optimum, which may replace the
current solution. For example, in a TSP a 2-Opt improvement heuristic may be applied as
a local improvement method (see Section 3.4 for more details about the 2-Opt heuristic).
The basic VNS procedure is a descent algorithm with a first acceptance criterion. Ho-
wever, the algorithm can be easily changed to a descent/ascent procedure, similar to SA,
by accepting bad moves with a certain probability in Step 8 (move or not) of Algorithm
2.6. Also, the first improvement criterion adopted by the basic VNS in the same step,
may be changed to a best improvement by selecting the neighbourhood k∗
, which yields
the best improvement, among all kmax neighbourhoods. Other variations include starting
the neighbourhood size from a certain value kmin rather than 1, and varying the step size,
such that increasing the neighbourhood size allows jumps to far away regions of the space
[70].
Another important variant of the basic VNS algorithm, called Variable Neighbourhood
Descent (VND), was also suggested by Hansen in [70]. The idea is to apply a change of
neighbourhood size within the local search as well. The steps of the VND algorithm as
described in [70] are shown in Algorithm 2.7.
Algorithm 2.7: The Variable Neighbourhood Descent (VND) Algorithm [70] .
1: Initialization: Select the set of neighbourhood structures Nk, (k = 1, ...kmax), that will be
used in the local search
2: Generate an initial solution x
3: repeat
4: Set k ← 1
5: while (k < kmax) do
6: Exploration of the neighbourhood: Find the best neighbour x′ of x (x′ ∈ Nk(x))
7: Move or not: if x′ is better than x, set (x ← x′); otherwise set k ← k + 1.
8: until (Done){no further improvement is obtained}
9: Return solution x
The VNS meta-heuristic offers many attractive features that have encouraged researchers
to use it for CO problems. First, the algorithm is simple, easy to understand and imple-
ment, and, to a large extent, parameter-independent. Second, well-known neighbourhood
moves and local search methods can be easily integrated in several ways and applied wi-](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-59-2048.jpg)
![36 2.4 Chapter Summary
thin the VNS framework. Third, it is a robust technique that can be used in a variety of
problem types with almost no modification to the basic structure of the algorithm. Fi-
nally, it can be easily hybridized with other heuristic and meta-heuristic approaches to
solve hard optimization problems. In fact, VNS has been successfully used in solving
many well-known CO problems. For example, it has been applied to the TSP and the
facility location problem in [70], and to the graph coloring problem in [6]. It has also
been extensively applied to vehicle routing problems, for example in [19], [119], [118]
and [73]. For more details about advances in VNS and its applications, the reader is re-
ferred to the recent work by Hansen et al. [71]. As previously mentioned, our research
applied the VNS approach, hybridized with SA, in solving the 1-PDP, as will be explained
in detail in Chapter 10.
2.4 Chapter Summary
Conventional OR techniques may not be sufficient for solving complex optimization pro-
blems, which model a large number of decision making strategies in real-world applica-
tions. Heuristic and meta-heuristic techniques, which usually perform well in most prac-
tical situations, have become increasingly popular among researchers in the optimization
field.
In this chapter we introduced combinatorial optimization problems and the theory of al-
gorithm and complexity analysis. We then briefly highlighted some exact algorithms that
can be used to solve these problems to optimality, for limited problem sizes only. For
more practical applications, though, a heuristic or a meta-heuristic approach is usually the
preferred option. We reviewed in this chapter some important meta-heuristic techniques
like Local Search, Simulated Annealing, Genetic Algorithms, Ant Colony Optimization,
Tabu Search, and Variable Neighbourhood Search.
In the next chapter, we will introduce an important class of CO problems, which is the
class of vehicle routing and scheduling. A particular variant of vehicle routing problems,
pickup and delivery problems, is introduced in Chapter 4. Since pickup and delivery
problems are the main theme of our research, our investigation of some selected problems
from this category continues in Chapters 5 to 10.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-60-2048.jpg)
![37
Chapter 3
Vehicle Routing Problems: A Literature
Review
Solving the Vehicle Routing Problem (VRP) and its related variants is an important field
in the area of Operations Research (OR), which has attracted a growing interest in recent
years. Effective decision support tools that can be adopted in logistic planning are cur-
rently in great demand, since they can lead to a substantial cost reduction and efficient
resource consumption. In addition, they can help reduce the negative environmental im-
pacts of transportation. An European Commission White Paper [1] states:
A modern transport system must be sustainable from an economic and social as well
as an environmental viewpoint. Plans for the future of the transport sector must
take account of its economic importance. Total expenditure runs to some EUR 1000
billion, which is more than 10% of gross domestic product. The sector employs
more than 10 million people. It involves infrastructure and technologies whose cost
to society is such that there must be no errors of judgment.
The use of automated route planning and scheduling can lead to huge savings in transpor-
tation costs, typically ranging from 5% to 20% [147], which should contribute to boosting
the overall economic system. This great potential, together with the advancement in tech-
nological and computational powers in the last three decades, has encouraged researchers
to experiment with diverse algorithms and address different applications, in order to meet
the increase in demand for effective vehicle routing support tools. This indeed resulted
in a large increase in the literature dealing with the VRP, especially given that its theme
encompasses several intertwined disciplines, such as scientific computing, operations re-
search, and business and management [48].
The rest of this chapter is organized as follows: Section 3.1 gives an idea about the dif-
ferent types of vehicle routing problems existing in the literature and highlights the diver-
sity of published research in this area. Section 3.2 briefly describes one particular VRP va-
riant that is of interest to our research: the Vehicle Routing Problem with Time Windows](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-61-2048.jpg)
![38 3.1 Vehicle Routing Problems (VRPs)
(VRPTW). Section 3.3 emphasizes heuristic approaches for the solution construction of
the VRPTW, while Section 3.4 describes some popular solution improvement heuristics.
Section 3.5 then briefly summarizes some meta-heuristic algorithms that were applied to
the VRPTW. Finally, section 3.6 concludes this chapter with a short summary.
3.1 Vehicle Routing Problems (VRPs)
VRPs generally involve problems that deal with the transportation of goods or people bet-
ween depots and customers, where the objective is to design an optimum schedule for one
or more vehicles to service the clients with minimum possible operational cost and maxi-
mum customer satisfaction. For example, the routing and scheduling plan may try to mi-
nimize the number of vehicles used, the total distance traveled, and the labour force used.
At the same time, it may try to maximize orders and volumes serviced per unit distance.
Moreover, for practical applicability of the problem, a number of constraints are usually
added to the basic model. These may include respecting the capacity of the vehicle, ap-
plying a certain visiting order, or adhering to preferred service times for clients and to the
maximum working hours of drivers. Other practical considerations include service types
for customers (delivery and/or collection), number of depots (one/two/multiple), types of
operating vehicles (homogeneous/heterogeneous), drivers availability (fixed/variable start
times), and whether the information concerning routing requests is known in advance or
revealed in real-time.
As a generalization of the famous TSP, the VRP is an N P-hard problem [54] [147]. This
fact, together with the above mentioned constraints and practical considerations, add to
the difficulty of handling the different variants of vehicle routing problems, and contribute
to the existence of a wealth of models and algorithms that deal with this problem.
Given the large amount of published research tackling the various types of VRPs [48],
it becomes increasingly difficult to keep track of all problem types and underlying so-
lution methods. However, a general classification of the basic routing and scheduling
problems and their interconnections is shown in Figure 3.1. Two main problem catego-
ries are distinguished in this figure, based on the observation that routing and scheduling
problems are usually represented as graphical networks, in which pickup and/or delivery
points are represented as nodes connected with line segments called arcs. The first cate-
gory is called node routing problems, in which the service demand is associated with
nodes (locations). The second category is called arc routing problems, in which the
service demand is associated with the arc connecting two nodes. Arc routing problems
involve applications like refuse collection and winter gritting, and are often referred to as](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-62-2048.jpg)
![3.1 Vehicle Routing Problems (VRPs) 39
the Chinese Postman Problem (CPP) [66]. The CPP, though, is beyond the scope of our
research and will not be addressed further. As Figure 3.1 indicates, the simplest type of
node routing problems is the TSP, in which a single vehicle is required to visit a number
of nodes, with no restriction on the capacity of the vehicle. It is required in the TSP that
the trip starts and ends at the same node, without identifying a particular depot point. The
multiple-vehicle variant of the TSP is called the Multiple Traveling Salesman Problem
(MTSP). If the demands of locations are added to the problem, the capacity of the vehicle
is restricted, and a depot point is identified, it is usually called the Capacitated Vehicle
Routing Problem (CVRP). Requesting two different service types (pickup and delivery)
in the problem instance, transforms it to the Pickup and Delivery Problem (PDP) category.
In addition, a service time window (TW) may be imposed on any of the above variants
to allow for more realistic applications of the problem, in which the visiting time of each
node is restricted between certain pre-defined bounds. For further details about the main
variants of the VRP and both exact and heuristic methods applied to solving it, the reader
is referred to the book edited by Toth and Vigo [147].
Figure 3.1: Vehicle routing and scheduling problems.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-63-2048.jpg)
![40 3.1 Vehicle Routing Problems (VRPs)
Recently, Eksioglu et al. in 2009 carried out a taxonomic review of the VRP [48], in
which they demonstrated that the growth in the literature published from 1955 to 2005 is
almost perfectly exponential, if we eliminate the first two years of data. Despite this large
increase in published VRP literature, other OR/MS (Operations Research/Management
Sience) disciplines have grown at a much higher rate, which indicates that research in the
VRP area is probably more sophisticated and requires high skills in analysis and design.
The authors classified the characteristics of different VRPs, which should help understand
the extent of the available literature and also identify potential areas where more research
seems to be needed. Their taxonomy of the VRP research as it appears in [48] is shown
in Figure 3.2.
This taxonomy classifies the VRP literature under five major categories:
1. Type of Study: which identifies the nature of the study itself, for example: theore-
tical or applied. Some techniques listed under applied literature include exact and
heuristic methods.
2. Scenario Characteristic: which indicates the problem characteristics and the ope-
rating scenario of the vehicle routing process. For example, under this category,
static problems, in which all requests are determined in advance of the solution pro-
cess, are distinguished from dynamic problems in which some requests may arrive
later (in real-time). Dynamic problems should account for some probable changes
in the current progressing routing plan. Other problem characteristics identified
here include the type of service (pickup/delivery), and the precedence and coupling
requirements 1
.
3. Problem Physical Characteristics: which outlines factors that directly affect the
solution, such as number of depots, number of vehicles, capacity and time win-
dows constraints. This category also distinguishes node routing problems from arc
routing problems.
4. Information Characteristics: which identifies the nature of information presen-
ted and accessed by the underlying solution methodology, and is mainly directed
to the study of fuzzy and dynamic routing problems. For example, acting under
uncertainty, when certain information (e.g. delays/vehicle breakdown) are revealed
during an emergency situation, may fall under this category.
5. Data Characteristics: identifies the type of data used to evaluate the solution me-
thod, such as real-world or synthesized data.
1
Precedence means that a pickup location must precede its corresponding delivery, while coupling re-
quires that both the pickup and its delivery must be served by the same vehicle.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-64-2048.jpg)
![3.1 Vehicle Routing Problems (VRPs) 41
Figure 3.2: Taxonomy of the VRP literature [48].](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-65-2048.jpg)
![42 3.2 The Vehicle Routing Problem with Time Windows (VRPTW)
The above taxonomy gives a general idea about how sophisticated and diverse the VRP
literature is. As previously mentioned in the introduction to this thesis (Chapter 1), our
research mainly focuses on one particular variant of vehicle routing problems, which are
problems that deal with two types of services simultaneously, pickup and delivery (PD).
Specifically, we selected two main problems from this category, the Pickup and Delivery
Problem with Time Windows (PDPTW) and the One-commodity Pickup and Delivery
Problem (1-PDP), with more emphasis on the first of these two variants. In the next chap-
ter, we will present a brief summary of some important PD problems, and give examples
of published research in this field. On the other hand, we will devote the rest of this chapter
to an important and closely related problem in the literature, which is the Vehicle Routing
Problem with Time Windows (VRPTW). This problem is significant to our research, be-
cause it bears similarities to our selected PD problems, in terms of the constraints involved
and the solution approaches used, as well as the objective function. Moreover, it is one of
the most well-studied problems in the vehicle routing field.
As previously mentioned in Chapter 2, exact algorithms have their limitations in solving
large scale combinatorial optimization problems. In the vehicle routing domain, Hasle
and Kloster [72] indicate that today’s exact methods cannot consistently solve VRP ins-
tances having more than 50-100 customers, which is generally small for most realistic
applications. Moreover, given the complexity of the problem, it is very unlikely that an
algorithm that solves the problem to optimality in a reasonable time will be developed
in the near future. Hence, our research mainly investigates heuristics and meta-heuristics
as possible solution methods to the problems of interest. This literature review will, the-
refore, focus on the approximation techniques introduced in Chapter 2. In other words,
we will highlight selected published research that deals with heuristic and meta-heuristic
methods only. Information about research that uses exact methods can be found in several
vehicle routing and pickup and delivery problems surveys that we are going to mention
throughout this discussion . In this chapter and the next chapter, a general classification
and overview of the literature is presented, while more details about previous research par-
ticularly related to the individual problems we tackled will be presented in the chapters
dedicated to these problems (i.e., Chapters 5 to 10).
3.2 The Vehicle Routing Problem with Time Windows
(VRPTW)
The VRPTW is an important problem occurring frequently in transportation systems. The
problem deals with a number of customer requests that must be dealt with by a fleet of](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-66-2048.jpg)
![3.2 The Vehicle Routing Problem with Time Windows (VRPTW) 43
vehicles. All customers are assumed to require the same type of service (either pickup
or delivery but not both) 2
. Each vehicle route must start and end at a central depot, and
each customer must be visited exactly once. Two main constraints are strictly enforced
in this problem. The capacity constraint requires that the vehicle capacity should not be
exceeded at any time, while the time window (TW) constraint requires that each customer
must be visited during a pre-specified time window interval, i.e., if the vehicle reaches
the customer before the beginning of its TW, it should wait until the allowed service
time begins. Similarly, arrival after the deadline of the TW means a violation of problem
constraints. The solution objective is usually hierarchical, such that the minimization
of the number of vehicles used is a primary objective, followed by minimizing the total
travel distance of the operating fleet, or the total schedule duration. Figure 3.3 shows a
representation of a small instance of the VRPTW, before solving it (Figure 3.3(a)) and
after solution (Figure 3.3(b)).
(a) VRPTW - before solution (b) VRPTW - after solution
Figure 3.3: The vehicle routing problem with time windows.
The VRPTW is one of the most well-studied problems in the vehicle routing research.
Both exact and heuristic methods have been widely applied to solving this problem. In our
current literature summary, we mostly base our review on the 2-part survey of the VRPTW
by Bräysy and Gendreau [20] and [21]. As previously mentioned, our investigation of the
problem focuses on heuristic and meta-heuristic solution methods, since exact methods
are only limited to small problem sizes. Up-to-date exact algorithms were only capable of
solving some of Solomon’s benchmark instances (developed in [141]) of 100 customers
2
In the current discussion, we use the terms requests and customers interchangeably to refer to the set of
nodes to be visited. Nevertheless, a distinction between the two terms will be introduced within the context
of the PDPTW in Chapters 5 to 8.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-67-2048.jpg)
![44 3.3 Solution Construction for the VRPTW
to optimality. Moreover, most of the solved instances are characterized by having a short
schedule horizon (a short TW width). The remaining (unsolved) instances have a long
schedule horizon, and are much harder for exact algorithms to solve. For further details
about some important exact algorithms in the VRPTW literature, the reader is refereed to
Cordeau et al. [28].
Solving the VRPTW generally requires two types of decisions: 1) the assignment or the
grouping decision, which means to assign to each vehicle a subset of nodes from the
customers set, and 2) the routing and scheduling decision which involves generating a
minimum cost route for each vehicle to visit its assigned requests, such that the generated
route respects the capacity and TW constraints. It is often observed that for vehicle routing
problems of this sort, the assignment decision is usually of more importance than the
routing and the scheduling decision, in determining the final solution quality [137]. In
addition, the solution process often consists of two phases: solution construction, in which
one or more solutions for the problem is generated, and solution improvement, in which
the initial solution is improved using a heuristic or a meta-heuristic approach. Both phases
are discussed in the following two sections.
3.3 Solution Construction for the VRPTW
Solution construction refers to the creation of a set of routes for the vehicles by selecting
nodes (customers) and inserting them in one of the partial routes already created, or in
a new route. The decision to select a particular node for insertion is usually based on a
cost-minimization criterion, and requires that the insertion of the node in a selected route
does not cause violations of problem constraints. Two main types of solution construction
exist in the VRPTW literature: sequential construction, which builds routes one after the
other, and parallel construction which builds several routes at the same time. Sequential
construction does not attempt to allocate an additional vehicle unless no more requests
can be ‘feasibly’ added to the current vehicle. A parallel construction, on the other hand,
initially pre-specifies the number of vehicles that could be used, but more vehicles can
be added as needed if the initial estimate of the number of vehicles does not serve all
requests without violating the constraints of the problem. Figure 3.4 demonstrates these
two variants.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-68-2048.jpg)
![3.3 Solution Construction for the VRPTW 45
(a) Sequential construction (b) Parallel construction
Figure 3.4: Solution Construction.
Several famous construction heuristic are described by Solomon in [141]. These are the
savings heuristics, the time oriented nearest neighbour heuristic, the insertion heuristics,
and the time oriented sweep heuristic. These heuristics are briefly explained below.
3.3.1 Savings Heuristics
This method is based on one of the most successful construction heuristics for the VRP,
the savings heuristic proposed by Clarke and Wright [26]. The idea is to initially assign
each customer individually to one vehicle. Thus, an initial vehicle route will only consist
of a trip from the depot to the customer and then back to the depot. If we have two such
routes, one serving customers i and another serving customer j, combining the two routes
such that the new route will service both i and j will result in a saving of service cost
Sij = ci0 + c0j − cij. This idea is illustrated in Figure 3.5.
The parallel savings construction algorithm for the VRPTW can be summarized as fol-
lows:
1. Step 1: compute saving Sij for every two customers i and j, and sort them from
largest to smallest;
2. Step 2: take saving Sij from top of the saving list. Search for a route containing
(0, i) and another containing (j, 0). If such routes exist, join them to form route
(0, i, j, 0).
Solomon in [141] adapted the Clarke and Wright savings heuristic to the VRPTW by
taking the route orientation into account, as shown in Figure 3.5 and Step 2 above. In
addition the algorithm must check the capacity and time window feasibility at every step](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-69-2048.jpg)
![46 3.3 Solution Construction for the VRPTW
Figure 3.5: The savings heuristic.
in the heuristic process 3
. A limit on the waiting time is also imposed when routes are
joined.
A sequential version of the savings heuristic will consider one route at a time, and then
implement the best saving that can be achieved if another route is joined with the current
route. The sequential savings heuristic can be briefly described as follows:
1. Step 1: compute saving Sij for every two customers i and j, and sort them from
largest to smallest;
2. Step 2: consider the current route (0, i, ..., j, 0). Find routes containing (k, 0) and
(0, l) that have the best savings Ski and Sjl in the saving list, and select the best
one among them for joining with the current route. If no more savings can be
implemented for the current route, move to the next route.
3.3.2 Time Oriented Nearest Neighbour Heuristic
This heuristic is a sequential construction process that starts by initializing the current
route with the depot. Then the customer closest to the depot, in terms of a certain cost
measure c, is inserted next in the route. Subsequently, the customer ‘closest’ to the last
added customer will be inserted next, if its insertion causes no constraint violation. If
3
Solomon also proposed an important heuristic for speeding the TW feasibility checking called the push
forward (PF) heuristic, which is based on calculating the extra time occurring in the schedule due to the
insertion of the new node in the route. For more derails see [141].](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-70-2048.jpg)
![3.3 Solution Construction for the VRPTW 47
all remaining un-routed customers cannot be feasibly inserted in the current route, a new
route is allocated and the process is repeated.
The cost function c used by Solomon in [141] to determine the ‘closeness’ between two
customers, is based on both geographical and temporal separation between the two nodes.
This measure is described by cij = w1 ×dij +w2 ×Tij +w3 ×vij, where dij is the distance
between the two customers4
, Tij is the time difference between the completion of the
service at i and the beginning of the service at j, and vij is the urgency of visiting customer
j, which is calculated as the time remaining until the deadline of servicing customer j is
reached. The weights w1, w2 and w3 satisfy: w1 + w2 + w3 = 1.0, and w1 ≥ 0, w2 ≥ 0
and w3 ≥ 0.
3.3.3 Insertion Heuristics
This is another class of sequential construction heuristics described by Solomon in [141],
and is based on expanding the current route by inserting one un-routed customer at each
iteration. The general idea is demonstrated in Figure 3.6, where the un-routed customer k
(left) was inserted between customers i and j (right) in the progressing route.
Figure 3.6: The insertion heuristic.
More specifically, the insertion heuristic starts by initializing the route under consideration
with a seed customer. Other customers are then selected for addition to the current route,
4
One simplifying assumption that is usually made in the VRP is that the speed of the vehicle is constant
(often = 1). If this assumption is made, the distance separating i and j (dij) and the travel time between i
and j (tij) may be used interchangeably.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-71-2048.jpg)
![48 3.3 Solution Construction for the VRPTW
based on two cost measures to be defined shortly. When no more un-routed customers
can be ‘feasibly’ inserted in the current route, the process is repeated for a new route.
The seed customer could be selected as the customer farthest from the depot, or as the
customer having the earliest allowed starting service time.
Three insertion heuristics have been defined by Solomon, based on the criteria used for
selecting an un-routed customer to be inserted next in the current route. The first and most
successful insertion heuristic (called I1 insertion) proceeds as follows, after initializing the
route with a seed customer:
1. Assume the current partial route created is (i0, i1, i2, ...im), where i0 and im re-
present the depot;
2. For each un-routed customer u, find its best feasible insertion position in the route,
between two adjacent customers i and j. The best insertion position is the one that
minimizes the first cost measure, which we will call c1(u). c1(u) is a measure that
is calculated based on both the extra travel distance and time delay that happens in
the route, due to the insertion of u;
3. Find the best customer u∗
to be inserted in the route, where u∗
is the customer
having the maximum value of a second measure, which we will call c2(u). c2(u) is
calculated based on both c1(u) (the first measure) and the distance from the depot.
The second insertion heuristic (I2) differs from I1 in the definition of the measure c2(u),
since it is now a combined measure of the total route distance and time resulting after the
insertion of u (rather than the extra distance and time, as in I1). On the other hand, the
third sequential insertion heuristic (I3) differs from I1 in the definition of c1(u), since the
urgency of servicing the new customer u is now also considered in this measure5
.
These insertion heuristics are in fact a generalization of the time oriented nearest neigh-
bour heuristic, described in Section 3.3.2, since they allow for inserting a new customer
between any two nodes in the route, rather than only at the end. They are sometimes refer-
red to in the literature as the cheapest insertion heuristics, with I1 being the most popular
among them.
A parallel construction heuristic for the VRPTW was proposed by Potvin and Rousseau
[121], based on the I1 heuristic described above. In this heuristic several routes are first
initialized with seed customers. To determine the number of initial routes, the authors
5
Our presentation of Solomon’s insertion techniques and the cost measures adopted within these heuris-
tics is simplified to a large extent, in order to fit the general overview intended here. For further information,
the reader is referred to [141] and [20].](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-72-2048.jpg)
![3.4 Solution Improvement for the VRPTW 49
first ran the I1 sequential construction algorithm. Further routes are later added as needed
if the initial number of routes does not yield a feasible solution. After determining the
best (feasible and least cost) insertion position for each un-routed customer, the customer
whose insertion in the solution causes the least increase in the overall cost is selected next
for insertion. Besides a weighted sum of the extra travel distance and time delay used
by Solomon to determine the cost of the insertion, this parallel heuristic adds a regret
measure over all routes. The regret measure is a type of ‘lookahead’ estimate of the cost
of not inserting the customer immediately in its current best route, i.e., if its insertion was
postponed to be carried out later in a different route.
Although these construction algorithms have mainly been designed for the VRPTW, re-
searchers have also adapted them to different problem variants, such as the PDPTW. More
discussion about solution construction heuristics from the perspective of the PDPTW will
be presented in Chapter 7.
3.3.4 Time Oriented Sweep Heuristic
In this heuristic Solomon [141] suggests a two phase approach to construct a VRPTW
solution. The first phase is a clustering phase, in which customers are assigned to vehicles
in a way similar to the original sweep heuristic suggested by Gillett and Miller [58]. In
this heuristic, a centre of gravity is computed based on the angle between the depot and
a randomly selected customer. Then, the remaining customers are assigned to vehicles
according to the polar angle separating their locations from the centre of gravity. After
the assignment phase, the scheduling phase starts by inserting the selected customers
in their respective routes, such that the insertion process follows Solomon’s I1 insertion
heuristic, explained in Section 3.3.3.
3.4 Solution Improvement for the VRPTW
Solution improvement within the context of the VRPTW refers to the gradual and repea-
ted modification of the initial solution until a certain stopping condition is satisfied. The
initial solution is usually a feasible solution, obtained using a construction algorithm, such
as the ones described in the previous section. One must then define a neighbourhood move
that can be applied to the initial/current solution to obtain a new solution within its neigh-
bourhood, i.e., the new solution only differs in a few attributes from the current solution.
For example, the new solution could be obtained by modifying some edges connecting
customers in the current solution. Afterwards, the generated solution is evaluated based](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-73-2048.jpg)
![50 3.4 Solution Improvement for the VRPTW
on the objective function, and may replace the previous solution if it is smaller in cost.
Nevertheless, two main acceptance criteria are usually adopted in this context. The first
acceptance criterion selects the first solution found in the neighbourhood of the current
solution that improves the objective function. On the other hand, the best acceptance exa-
mines all solutions in the neighbourhood of the current solution, and selects the best one
among them.
The above process is often referred to as local search, and usually results in a local opti-
mum that may be very far from the optimal solution. In addition, the quality of the local
optimum depends heavily on the quality of the initial solution. Within the vehicle rou-
ting literature, edge exchanges are the most popular solution improvement method, first
described by Lin [103] for the TSP.
Edge exchanges are applied to one route in the current solution, and depend on removing a
number k of edges from the current route, and replacing them with another set of k edges.
Thus, the process is often called k-exchange. A route that cannot be further improved
by a k-exchange is called a k-optimal. Performing all possible k-exchanges on a route
requires O(nk
) time. Thus, moves beyond 2 or 3 exchanges are very time consuming.
The 2-exchange move is illustrated in Figure 3.7.
Figure 3.7: 2-exchange move.
Several other improvement methods have been described for the VRPTW, a few of them
are modifications to the basic edge-exchange moves. We will briefly mention some of the
most important methods, without going into details, since most of them are beyond the](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-74-2048.jpg)
![3.5 Meta-heuristics for the VRPTW 51
scope of our research.
Potvin and Rousseau [122] introduce a modification to the 2-Opt heuristic of [103], called
the 2-Opt* exchange heuristic. 2-Opt* works on two different routes (unlike 2-Opt which
exchanges edges belonging to the same route). 2-Opt* tries to combine the two routes
without changing the orientation of the tours, by appending the last customers of the
second route after the first customers of the first route. Also, Or [114] modified the 3-Opt
operator of [103] and introduced the Or-Opt operator for the TSP. In this operator, three
edges in the original tour are replaced by three new ones without changing the orientation
of the tour.
Inter-route operators for the VRP were introduced by Savelsbergh [136] and used by Pos-
ser and Shaw [123] for the VRPTW. These are the re-locate, exchange and cross opera-
tors. The relocate operator moves a customer from one route to another. The exchange
swaps two visits in different routes and the cross is similar to the 2-Opt* proposed in
[122]. Other neighbourhood operators are the λ-exchange of Osman [115], the CROSS-
exchange of Taillard et al. [143], the GENI-exchange of Gendreau et al. [56], ejection
chains of Glover [60] [61] and cyclic k-transfers of Thompson and Psaraftis [146].
In addition, other improvement heuristics include the Large Neighbourhood Search (LNS)
of Shaw [139]. This algorithm removes a large number of customers from their current
routes, and re-schedules them at optimal cost. The removed customers could be selected at
random, or based on some relatedness measure that takes into account customers demands
or starting times. Schrimpf [138] used a similar approach and calls it ruin and recreate.
Finally, Bräysy [18] introduces a modified ejection chain approach that also considers
re-ordering of the routes. For the interested reader, more details and illustrations of the
different types of solution improvement heuristics are presented in [20].
Similar to solution construction approaches, solution improvement methods have also
been adapted and applied to PD problems. Some examples of their application to these
problems will be presented in the upcoming chapters, when related work to our selected
problems is investigated.
3.5 Meta-heuristics for the VRPTW
Since our research is particularly focused on meta-heuristic techniques, we highlight in
this section some papers that use meta-heuristic approaches for the VRPTW, and explain
briefly some important common features among them. Table 3.1 summarizes research](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-75-2048.jpg)
![52 3.5 Meta-heuristics for the VRPTW
that uses Tabu Search (TS), Simulated Annealing (SA), Ant Colony Optimization (ACO),
Variable Neighbourhood Search (VNS), and Guided Local Search (GLS).
As Table 3.1 indicates, most meta-heuristic techniques apply some cheapest insertion
method for the construction of the initial solution, with Solomon’s insertion being the
most popular among them. On the other hand, solution improvement is usually per-
formed using well-known neighbourhood moves like 2-Opt, 2-Opt*, 3-Opt, Or-Opt, λ-
interchange. . . etc. To accelerate the optimization, the search is sometimes limited to
neighbourhoods having particular features. For example, in [53] only moves involving
arcs close in distance are allowed, while the approach in [22] only considers selected
routes and selected customers for merging during the solution construction process.
Some approaches allow infeasibility of solutions (e.g. [99]) and penalize the violation in
the objective function. In addition, many algorithms apply a specific strategy for reducing
the number of vehicles. For example, the algorithm in [53] removes customers from
routes having a small number of nodes and inserts them into other routes, using Or-Opt
exchanges. Also, in [11] a 2-phase approach is applied, where the first phase is an SA
procedure intended to minimize the number of routes, while the second phase focuses on
minimizing total distance. On the other hand, the ACO of [52] uses two colonies, one for
the minimization of the number of vehicles and the other to minimize total travel distance.
Different diversification mechanisms have also been applied within the various meta-
heuristics to improve solution quality, so that the algorithm is directed towards exploring
new areas of the search space. For example, in [129] the search is diversified using an
adaptive memory, where routes taken from best solutions found during the search are sto-
red. Some of these routes are combined to form solutions that will act as new starting
solutions for the TS. Also, in [143] diversification is applied by penalizing frequently per-
formed exchanges. On the other hand, the approach in [22] adopts a Guided Local Search
(GLS) diversification mechanism that penalizes certain solution features, if they are going
to bring the search close to previously visited solutions.
Intensification of the search around best solutions discovered during the search have also
often been considered. For instance, in [101] the SA algorithm is forced to start from
the current best solution several times, while the ACO in [52] uses global pheromone
updating, to further explore search areas around the best solutions obtained. Also, in [143]
intensifying the search is done by selecting the best routes and re-ordering customers
within these routes, using Solomon’s I1 insertion.
A post-optimization phase is sometimes added to the basic meta-heuristic to further en-
hance best solutions. For example, the approach in [129] performs a post-optimization](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-76-2048.jpg)
![3.5 Meta-heuristics for the VRPTW 53
using an exact set-partitioning algorithm on routes obtained from the adaptive memory.
Also, in [143] the post-optimization technique of [56] is applied to each route of the final
solution obtained.
In many cases, hybridization of more than one meta-heuristic technique is adopted. For
example, in [101] a tabu-embedded SA is used. Also, in [25] the annealing process is
enhanced using a varying size tabu-list, and the authors in [11] use both SA and LNS.
To complete our review of meta-heuristic techniques for the VRPTW, Table 3.2 summa-
rizes some Genetic Algorithm (GA) approaches that have been applied to this problem.
Almost all of these techniques hybridize GAs with some other heuristic, local search, or
another meta-heuristic approach for solution construction and/or solution improvement.
Solution representation is usually based on integer representation, rather than the classical
bit string encoding. Thus, genetic operators are directly applied to problem solutions. The
initial population is often created randomly, or by adapting some well known construction
methods. For example, in [144], customers are grouped randomly, and then the cheapest
insertion method of [65] is used for routing each group of customers.
The fitness function is usually based on solution cost, i.e., number of routes, total dis-
tance and total duration. However, since some approaches allow violations of problem
constraints, the fitness function should take this into account. For example, in [16], the
fitness considers the number of un-serviced customers in an infeasible solution. Also, in
[15] a penalty is added in the objective function to violated constraints. On the other hand,
the selection scheme often applied is roulette-wheel selection. However, other selection
methods have also been used. For instance, a ranking scheme is used in [120].
Regarding recombination, most of the GA techniques apply specialized crossover opera-
tors, since traditional genetic operators do not usually fit the VRPTW. For example, the
authors in [120] propose a Sequence-Based Crossover (SBX) and a Route-Based Cros-
sover (RBX), while the GA in [16] applies special crossover operators that depend on a
global precedence relationship among genes, such as the distance or time windows orde-
ring. On the other hand, mutation is often based on traditional local search operators like
2-Opt, 2-Opt*, Or-Opt and LNS. In addition, mutation is sometimes applied to reduce the
number of routes in the solution, as in [120] and [15].
To sum up, meta-heuristic approaches to the VRPTW often embed solution construction
and improvement methods as described previously. However, they usually obtain bet-
ter quality solutions than algorithms that apply straightforward solution construction and
improvement techniques on their own. This often comes at the expense of greater com-
plexity in implementation and increased computation time, though. Bräysy and Gendreau](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-77-2048.jpg)
![543.5Meta-heuristicsfortheVRPTW
Table 3.1: Meta-heuristics for the VRPTW.
Paper Algorithm Solution Construction Solution Improvement Remarks
Garcia et al. [53] TS Solomon’s insertion 2-Opt*, Or-Opt Parallel implementation - Only allows moves involving arcs close in distance
Rochat and Taillard [129] TS Solomon’s insertion &
2-Opt
2-Opt, relocate Uses adaptive memory containing routes obtained from best solutions visited
during the search, with the purpose of providing new starting solutions
Taillard et al. [143] TS Solomon’s insertion CROSS Decomposes solutions into subsets of routes, based on a polar angle associated
with the centre of gravity of each route - TS applied to each subset separately
- Diversification by penalizing frequently performed exchanges - Intensification
by reordering customers within best routes using I1 insertion
Lau et al. [99] TS Relocate from a holding
list where all customers
are initially stored
Exchange - relocate Allows violation of constraints for a penalty in the objective function - Penalty
parameters adjusted dynamically
Chiang and Russell [25] SA The parallel construction
of Russell [133]
λ-interchange and k-node
interchange
Enhancing of the annealing process via a varying size tabu-list
Li et al. [101] Tabu-
embedded
SA
Insertion and exten-
ded sweep heuristics of
Solomon
Shifting and exchange of
customer segments bet-
ween and within routes
SA restarts from current best solution several times - Reduce routes by reordering
customers and inserting them into other routes - Diversification by random shifts
and exchange of customer segments
Bent and Van Hentenryck
[11]
SA and LNS Not specified 2-Opt, Or-Opt, relocate,
exchange and 2-Opt*
2-phase approach: SA to minimize number of routes and LNS to minimize total
distance
Gambardella et al. [52] ACO Nearest-neighbour heuris-
tic with probabilistic rules
CROSS 2 colonies: first one minimizes number of vehicles, and the second minimizes
total distance - Both cooperate in updating best solution
Bräysy [19] VNS Solomon’s Insertion and
the parallel construction of
Russell [133]
Or-Opt exchanges and
CROSS-exchanges
Reducing number of routes using an ejection-chain phase - Second VNS phase
for improving total distance
Bräysy et al. [22] Threshold
accepting
(TA) &
Guided Lo-
cal Search
(GLS)
Savings heuristic SPLIT (of routes) - limited
CROSS - limited Or-Opt
Construction algorithm only considers selected routes and selected customers for
merging - Initial phase to minimize number of routes, based on simple customer
re-insertion - Further improvement using TA & GLS - TA allows local search
moves that worsen the objective function, provided it is within a certain threshold
limit - GLS penalizes certain solution based on some solution features (e.g. long
edges), not considered part of a near-optimal solution](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-78-2048.jpg)
![3.5Meta-heuristicsfortheVRPTW55
Table 3.2: Genetic Algorithms for the VRPTW.
Paper Initial Population Crossover Mutation Remarks
Thangiah
[144]
A set of K seed angles are planted,
where K is the number of vehicle. Sec-
tor rays are drawn originating from the
depot to each seed angle - Routing done
using cheapest insertion of Golden and
Stewart [65]
Exchange a randomly selected por-
tion of the bit string between chro-
mosomes
Randomly change bit value Cluster first route second approach cal-
led GIDEON - Each chromosome is a
set of possible clustering scheme and
the GA used to improve the clustering
- Routes improved by λ-interchange
Potvin and
Bengio [120]
Chromosome is a problem solution crea-
ted using Solomon’s cheapest insertion
Sequence-Based Crossover (SBX):
two sub-tours from two parents
are linked together - Route-Based
Crossover (RBX) replaces a route
of one parent by another route from
the second parent - Repair operator
applied for infeasible offspring
Reduce number of routes by in-
serting customers from a ran-
domly selected short route into
another - Local search muta-
tion using Or-Opt exchanges
The approach is named GENEROUS
Blanton and
Wainwright
[16]
Solution coded as a permutation of
customers - Insertion heuristic used to
construct solution
Specialized crossover that uses glo-
bal precedence relationship among
customers (TW/distance)
Randomly exchange 2-edges The GA searches for a good ordering of
customers. while construction of fea-
sible solutions is handled by a greedy
heuristic
Berger et al.
[15]
Random insertion heuristic - Chromo-
somes are problem solutions - Two po-
pulations: POP1 used to minimize dis-
tance and POP2 minimizes constraint
violations
Insertion-based crossover (IBX):
combining k routes from parent1
(one at a time) with subsets of
customers from parent2 - Insert-
within-route crossover (IRNX): si-
multaneously combines k routes
from parent1 with subsets of custo-
mers from parent2
Large Neighbourhood Search
(LNS) mutation - Each custo-
mer considered for reinsertion
in an alternate route. eliminate
shortest route - Reorder custo-
mers within a route
Simultaneous evolution of two popula-
tions (minimize distance & constraint
violations) - POP2 has smaller number
of routes than POP1 - When a new better
solution is obtained from POP2, POP1
is replaced by POP2
Repoussis et
al. [128]
Discrete arc-based representation of in-
dividuals combined with a binary vec-
tor of strategy parameters - Individuals
created using greedy insertion from a
Restricted Candidate List (RCL), with a
probabilistic element
Multiple-parent recombination ap-
plied only to strategy parameters
not to problem solutions
Mutation rate determined by
strategy parameters evolved
during the search - Mutation
done using “ruin and recreate”
(λ + µ) Evolution Strategy (ES). Star-
ting from µ individuals, at each iteration
a new intermediate population of λ indi-
viduals is produced by mutation - Evo-
lutionary search is based only on muta-
tion - Each offspring is further optimi-
zed using route elimination and a local
search (based on TS) to improve total
distance](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-79-2048.jpg)
![56 3.6 Chapter Summary
in [21] identified several techniques that seem to give good results when applied within
the different meta-heuristic approaches. Some of these techniques are: 1) saving best
solutions found during the search, 2) varying the neighbourhood structure, with the size
of the neighbourhood being typically small, 3) using a memory structure to facilitate the
search, 4) applying a specific strategy for reducing the number of routes in the solution
and 5) hybridization of different heuristics and meta-heuristics.
3.6 Chapter Summary
Vehicle routing problems have gained considerable attention in the last few decades. This
is mainly due to the increase in complexity in transportation and logistics demands, and
the urgent need for optimizing client services, reducing operational costs, and limiting the
negative environmental impacts that may result from the non-optimal planning of vehicles
and their routes. This chapter provided a general classification of vehicle routing problems
existing in the literature and highlighted the diversity of problem types, solution methods,
and the huge amount of published research that deals with this problem.
We emphasized in this chapter one particular vehicle routing problem that is of interest
to our study, the Vehicle Routing Problem with Time Windows (VRPTW). This problem
is closely related to pickup and delivery problems that we investigated in our research, in
terms of: problem definition, solution methods, objective function and constraint handling
approaches. We presented in this chapter a summary of some important heuristic and
meta-heuristic algorithms that have been applied to solving the VRPTW.
The next chapter is devoted to an overview of pickup and delivery problems, which are
the main focus of our research. We will present a general classification and highlight
some published research in the field, as an introduction to a more in-depth analysis of our
selected pickup and delivery problems, which will follow during the course of this thesis.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-80-2048.jpg)
![57
Chapter 4
Pickup and Delivery Problems: A
Literature Review
We dedicate this chapter to a literature summary of Pickup and Delivery (PD) problems,
since they are the main theme of our research. In addition, there are various types of pi-
ckup and delivery problems and different classifications of these problems that distinguish
them from other variants of vehicle routing problems.
PD problems are widely applicable in areas such as: the transportation of raw materials
from suppliers to factories, Internet-based pickup from sellers and delivery to buyers, food
and beverage collection and delivery, post and parcel delivery, newspaper distribution,
and airline and bus scheduling. In addition, an important related variant is the dial-a-
ride problem and the handicapped persons’ transportation, where people instead of goods
are transported, giving rise to customer inconvenience issues that should be taken into
consideration while constructing a solution.
Similar to other vehicle routing problems, pickup and delivery problems have attracted
the attention of researchers in the last few decades. Nevertheless, published research
tackling these problems is still relatively limited, compared to other variants of vehicle
routing problems [137]. Several important surveys of pickup and delivery problems have
appeared in the literature. The oldest ones are by Solomon and Desrosiers [142], and
Savelsbergh and Sol [137]. Two other surveys appeared recently in [14] and [117]. A
survey dedicated to the dial-a-ride problem is in [29]. Another survey on the one-to-one
pickup an delivery problems (see Section 4.2 for a description of this problem category)
is provided in [31]. In our literature summary, we mostly rely on the information and
classification provided by Parragh et al. in 2008 [117], unless otherwise indicated.
Parragh et al. [117] classify the different types of pickup and delivery problems under
two classes: 1) problems that deal with the transportation of goods from pickup customers
(also called backhaul customers) to the depot, and from the depot to delivery customers
(also called linehaul customers). This class is called the Vehicle Routing Problem with](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-81-2048.jpg)
![58 4.1 The Vehicle Routing Problem with Backhauls (VRPB)
Backhauls (VRPB), and 2) problems that deal with the transportation of goods/people
between two pickup and delivery points. This class is called the Vehicle Routing Pro-
blem with Pickup and Delivery (VRPPD). Sections 4.1 and 4.2 describe in detail these
two categories. Section 4.3 briefly highlights some meta-heuristic techniques that have
been applied to pickup and delivery problems. Finally, Section 4.4 concludes this chapter
and introduces the rest of the thesis.
4.1 The Vehicle Routing Problem with Backhauls (VRPB)
As mentioned above, this class deals with PD problems that involve the transportation of
goods from the depot to customers and vice versa. These problems are further subdivided
into four categories, generally based on the required visiting order of pickups and delive-
ries, and whether a customer can demand both types of services. In all cases the objective
is to minimize the total cost (e.g. total travel distance) of the routing plan, while adhering
to some pre-specified problem constraints. For example, respecting the capacity of the
vehicle at all times, and ensuring that the vehicle starts, if needed, with a load that is equal
to the total load to be delivered.
1. The vehicle Routing Problem with Clustered Backhauls (VRPCB): where all
delivery customers must be visited before all pickup customers. The single vehicle
special case of the same problem is denoted by (TSPCB).
2. The Vehicle Routing Problem with Mixed Linehauls and Backhauls (VRPMB):
where mixed visiting of pickup and delivery customers are allowed. The single
vehicle case of the same problem is denoted by (TSPMB)1
. If the capacity of the
vehicle is greater than or equal to the total sum of demands of pickup and delivery
customers, the problem coincides with a simple TSP.
3. The Vehicle Routing Problem with Divisible Delivery and Pickup (VRPDDP):
where each customer is both a pickup and a delivery customer. In addition, two
visits to the same customer is allowed. The single vehicle variant of the same pro-
blem is denoted by (TSPDDP). A possible scenario in this variant is that the vehicle
could initially visit a few delivery customers to get rid of some of the load it is car-
rying. Afterwards, the vehicle could visit another set of customers to perform both
1
More details about this problem will be presented in Chapter 9, since this problem is closely related to
the One-commodity Pickup and Delivery Problem (1-PDP) that we handled in our research. Nevertheless,
we refer to it in our research by the Traveling Salesman Problem with Pickup and Delivery (TSPPD), the
name most commonly used for this problem in the literature (e.g. [110] and [75]).](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-82-2048.jpg)
![4.1 The Vehicle Routing Problem with Backhauls (VRPB) 59
delivery and collection service at each, and finally a collection service would be
performed at customers that were initially visited for delivery. This type of solution
is called a ‘lasso’ solution. It is also observed in [117] that a solution method of
the VRPMB (the second category) can be applied to the VRPDDP if each custo-
mer demanding both delivery and pickup was modeled by two vertices, one for the
delivery and another for the pickup. Very little research has been applied to this
problem.
4. The Vehicle Routing Problem with Simultaneous Delivery and Pickup (VRPSDP):
where each customer demands both a delivery and a pickup, but, unlike the VRPDDP
(the third category), only one visit is allowed for each customer. The single vehicle
variant is denoted by (TSPSDP).
We summarize in Table 4.1 the most important features of the different variants of the
VRPB described in this section. In this table, we indicate the problem name, whether
the customer is allowed to demand both pickup and delivery, whether delivery customers
must all be visited before pickup customers, or otherwise that mixed sequences of visits
is allowed. The table also shows whether more than one visit is permitted, and if split-
ting the demand of one customer is allowed. Finally, the last column of the table gives
examples of selected published papers that dealt with the underlying problem, together
with the solution method adopted. We used the notations: (LNS) for Large Neighbou-
rhood Search, (TS) for Tabu Search, and (VNS) for Variable Neighbourhood Search. If
the paper handled the single vehicle case, we indicate this by (SV) and if it handled the
multiple vehicle case, we indicated this by (MV).
The authors in [117] concluded from their survey that the LNS algorithm of [132] seems
to be the most flexible and accurate meta-heuristic to date, since it can be applied to seve-
ral problems in the VRPB class, and it provided new best solutions for several benchmark
instances. On the other hand, they selected the heuristic in [111] among the fastest algo-
rithms applied to the VRPB.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-83-2048.jpg)
![604.1TheVehicleRoutingProblemwithBackhauls(VRPB)
Table 4.1: The Vehicle Routing Problem with Backhauls (VRPB).
Problem Customer
both P&D
Deliveries
Visited be-
fore Pickups
Mixed Visi-
ting
One Custo-
mer Visit
Demand
Splitting
Example Papers and Solution Methods
VRPCB
TSPCB
no yes no yes no Gendreau et al. [57]- heuristic (SV)
Thangiah et al. [145]- heuristic (MV)
Mladenovi´c and Hansen [108]- VNS (SV)
Ropke and Pisinger [132]- LNS (MV)
VRPMB
TSPMB
no no yes yes no Mosheiov [110]- heuristic (SV)
Nagy and Salhi [111]- heuristic (MV)
Ropke and Pisinger [132]- LNS (MV)
VRPDDP
TSPDDP
yes no yes no
(at most two)
yes Hoff and Løkketangen [77]- TS (SV)
Salhi and Nagy [134] - heuristic (MV)
VRPSDP
TSPSDP
yes no yes yes no Alshamrani et al. [4]- heuristic (SV)
Nagy and Salhi [111]- heuristic (MV)
Ropke and Pisinger [132]- LNS (MV)](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-84-2048.jpg)
![62 4.2 The Vehicle Routing Problem with Pickups and Deliveries (VRPPD)
We provide in Table 4.2 a summary of the most important features pertaining to the
VRPPD. We indicate in this table the subcategory name, whether the customers are paired
or not, whether a homogeneous commodity or multiple commodities are considered, and
whether the problem is primarily associated with the transportation of people or goods.
In addition, we highlight the most important constraints that are usually taken into consi-
deration while handling the problem. These include: 1) the precedence constraint, in
which a pickup request is required to precede its corresponding delivery, 2) the coupling
constraint, which requires that the pickup and delivery pair must be visited by the same
vehicle (in case of the multiple-vehicle variant), 3) the capacity constraint, which ensures
that the load carried by the vehicle at any given time does not exceed its capacity, 4) the
time window constraint, which requires that a customer has to be served in a pre-specified
time interval, and 5) the maximum ride time constraint, which ensures that when people
are transported, the time they spend in the vehicle does not far exceed the direct travel time
between their origin and destination. Finally, the table gives examples of some published
papers that deal with the problem. Again, we use (SV) for the single vehicle case and
(MV) for the multiple-vehicle variant. (LNS) refers to a Large Neighbourhood Search
algorithm, and (ALNS) refers to an Adaptive version of LNS, (HC) is a Hill Climbing
approach, (GA) is a Genetic Algorithm, (GGA) is a Grouping Genetic Algorithm, (SA) is
a Simulated Annealing algorithm, and (TS) is a Tabu Search algorithm.
In our opinion, the GA in [158], the LNS in [12], the ALNS in [131], and the GGA in
[126] seem to be among the state-of-the-art algorithms applied to the VRPPD, in terms of
their solution quality and/or robustness or flexibility. More details about these approaches
and other important research in the field will be described in the following chapters, where
we address specific pickup and delivery problems in our research. Before we conclude
our review of pickup and delivery problems, though, we present in the next section a brief
summary of some meta-heuristic techniques that have been applied to this problem class.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-86-2048.jpg)
![4.2TheVehicleRoutingProblemwithPickupsandDeliveries(VRPPD)63
Table 4.2: The Vehicle Routing Problem with Pickups and Deliveries (VRPPD).
Problem Paired
Custo-
mers
One
Com-
modity
Type of
Commo-
dity
Precedence Coupling Capacity TW Max Ride
Time
Example Papers and Solution Methods
PDVRP
PDTSP
no yes goods no no yes no no Hernández-Pérez and Salazar-González [75]-
heuristic (SV)
Zhao et al. [158] - GA (SV)
Dror et al. [45]- heuristic (MV)
PDP
SPDP
yes no goods yes yes yes yes no Jih and Hsu [89]- GA (SV)
Hosny and Mumford [84] - HC & SA & GA
(SV)
Li and Lim [100]- TS (MV)
Bent and Van Hentenryck [12] - LNS (MV)
Ropke and Pisinger [131] - ALNS (MV)
Hosny and Mumford [80]- GA (MV)
DARP
SDARP
yes yes people yes yes yes yes yes Psaraftis [124]- heuristic (SV)
Jaw et al. [87]- heuristic (MV)
Baugh et al. [9]- SA (MV)
Rekiek et al. [126]- GGA (MV)
Jørgensen et al. [91]- GA (MV)](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-87-2048.jpg)
![64 4.3 Meta-heuristic Algorithms for Pickup and Delivery Problems
4.3 Meta-heuristic Algorithms for Pickup and Delivery
Problems
Similar to all routing and scheduling problems, pickup and delivery problems have at-
tracted the interest of researchers who deal with meta-heuristic techniques, since exact
algorithms can only be used for small problem sizes and may not be useful for large prac-
tical applications. For example, the approach in [46], for solving the multiple-vehicle
PDPTW, was only capable of solving problem sizes of up to 55 customers using an exact
algorithm. Also, the exact algorithm in [74], solved PDTSP instances of up to 60 cus-
tomers. On the other hand, heuristic and meta-heuristic approaches are able to solve
problems having several hundreds of nodes, which is common in every day routing and
scheduling demands for some industrial applications. For example, Paragon software sys-
tems recently reported, in their October 2009 newsletter2
, that their optimization software
helped Sainsbury’s to manage its transport requirement, involving daily deliveries from
19 distribution centers to 527 supermarkets and 276 smaller stores.
Many heuristic and meta-heuristic techniques that have been successfully applied to the
VRP have been adapted to fit the PDP variant. Table 4.3 summarizes some meta-heuristic
techniques for pickup and delivery problems. In addition to the notations used for the
meta-heuristic techniques in Table 4.2, we use (GRASP) for a Greedy Randomized Adap-
tive Search Procedure, and (VND) for a Variable Neighbourhood Descent algorithm. On
the other hand, Table 4.4 summarizes some GA research on pickup and delivery problems.
It is important to emphasize again that the review presented here is just intended as a quick
introduction to the literature that deals with pickup and delivery problems. As previously
mentioned, further investigation of specific work that is closely connected to our research
will be presented in later chapters, when we address certain variants of pickup and deli-
very problems.
2
http://www.paragonrouting.com/cms/assets/pdf/directionsissue22.pdf](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-88-2048.jpg)
![4.3Meta-heuristicAlgorithmsforPickupandDeliveryProblems65
Table 4.3: Meta-heuristics for Pickup and Delivery Problems.
Paper Problem Algorithm Solution Construction Solution Improvement Remarks
Hernández-
Pérez and
Salazar-
González
[73]
PDTSP
(SV)
Hybrid
GRASP
& VND
Selecting the next ele-
ment for insertion from a
Restricted Candidate List
(RCL), with a probabilis-
tic element in the choice of
the next node
2-Opt and 3-Opt in the VND phase - Move for-
ward and move backward in a post optimization
phase
Combining two optimization heuristics: 1) GRASP (Greedy Ran-
domized Adaptive Search Procedure), which is based on a repe-
tition of a construction phase and a local search phase, 2) VND
(Variable Neighbourhood Descent), which is a variant of the VNS,
where the local optimum found acts as the new starting point for
the local search
Landrieu et al.
[95]
PDPTW
(SV)
TS Simple insertion heuristic A swap move and an insertion move Two tabu search methods compared: a regular deterministic Tabu
Search (TS), and a Probabilistic Tabu Search (PTS)
Nanry and
Barns [112]
PDPTW
(MV)
Reactive
TS
Cheapest insertion Single Paired Insertion (SPI): move all predeces-
sor nodes to better feasible locations, then in-
sert successor nodes in the best possible positions
- Swapping Pairs Between Routes (SBR): ex-
change predecessor nodes between vehicles then
successor nodes - Within Route Insertion (WRI):
re-order nodes within the same route
Reactive tabu search - Allows tuning of search parameters, such
as a short-term memory length, based on an assessment of visited
solutions during the search
Li and Lim
[100]
PDPTW
(MV)
Tabu-
embedded
SA
Modified Solomon’s inser-
tion heuristic - Initializes
each route with a P&D
pair that satisfies a set of
criteria, based on combi-
ned TW intervals and dis-
tance from the depot
PD-Shift: moves a P&D pair from route1 to
route2, and moves another pair from route2 to
route1 - PD-Swap: removes a P&D pair from
each route, and then reinserts each pair in the
other route - PD-Rearrange: removes and then
reinserts a P&D pair in the same route
Main meta-heuristic is a tabu-embedded SA with K-restarts, i.e.,
the algorithm stops when the number of iterations without impro-
vement reaches a pre-defined value K - To prevent cycling, the SA
records the accepted solutions in a tabu list
Urban [149] PDPTW
(MV)
Guided
SA
Assigning only one pickup
and delivery pair to each
route. A guided SA is then
applied to improve this ini-
tial solution by trying to
bundle requests into the
available routes
Removing a selected request from its current
route and inserting it into another - Exchange of
requests served on different routes - Repositio-
ning of a pickup and delivery pair in its same
route
The cost function takes into consideration both wage and non-wage
related traveling and operating costs of the vehicles - The selection
of neighbourhood move depends on a certain probabilistic criterion
- Calculated measurements control the selection of customers and
routes during the generation of the new solution
Cordeau and
Laporte [30]
DARP
(MV)
TS Assigning requests to ve-
hicles randomly and inser-
ting source location follo-
wed by destination at the
end of the route
Moving a request from one route to another Re-insertion of the removed request in its original route is forbid-
den by for a number of iterations - Diversification penalizes fre-
quently occurring moves in the objective function - Intensification
performed by removing all requests from their respective routes,
and re-inserting them in the best possible positions](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-89-2048.jpg)
![664.3Meta-heuristicAlgorithmsforPickupandDeliveryProblems
Table 4.4: Genetic Algorithms for Pickup and Delivery Problems.
Paper Problem Initial Population Crossover Mutation Remarks
Zhao et al. [158] PDTSP
(SV)
Nearest-neighbour construction
heuristic
A new pheromone-based cros-
sover - Edges between last
node and potential new node
that have proved successful in
the past are favoured
3-exchange procedure Initial population is optimized using a 2-Opt move - Phe-
romone trails are updated each generation - Offspring is
optimized using a 2-Opt local search.
Jih and Hsu [89] PDPTW
(SV)
A solution is encoded as a permuta-
tion of locations
Two traditional order-based
crossover operators - Two
merge crossover operators,
MX1 and MX2, that use a
global precedence vector to
guide the inheritance process,
as in [16]
Swap two genes selected at
random - Select two random
sites in the chromosome and
reverses the sub-route between
them - Shuffle genes that pre-
cede a location for which a vio-
lation of constraints is obser-
ved
A Family Competition Genetic Algorithm (FCGA) - Each
individual of the population plays the role of a family fa-
ther in turn - Another randomly selected individual plays
the role of mate for the family father
Jung and Haghani
[92]
PDPTW
(MV)
Chromosome representation is ba-
sed on assigning a 4 digit code to
each request - First digit represents
the number of the vehicle - Remai-
ning 3 digits are used to sort re-
quests according to the visiting or-
der of the vehicle - A pickup request
always has a sorting code that is less
than its corresponding delivery
Two crossover points genera-
ted, and the resulting segments
swapped between the two pa-
rents
Change the first digit of a pi-
ckup and delivery pair to ano-
ther digit, i.e., assign the re-
quest to another vehicle - Ve-
hicle merging also used to re-
duce the number of vehicles
Customers inconvenience is taken into consideration by
assigning a penalty in the objective function to the arrival
of the vehicle before or after the specified TW
Pankratz [116] PDPTW
(MV)
Each gene represents a group of re-
quests that are assigned to one ve-
hicle - Routes are represented by an
independent data structure associa-
ted with each gene
Vehicles with their assigned re-
quests removed from one pa-
rent and inserted in the other
parent
Removes a random cluster
from a chromosome, and re-
assigns its requests to other
clusters
A Grouping Genetic Algorithm (GGA), where the GA is
only responsible for the grouping (assignment) problem
- Routing is handled using an adaptation of Solomon’s
cheapest insertion - Chromosome ‘clean up’ needed after
the genetic operators to remove duplicate vehicles and re-
peated assignment of requests, in addition to re-assigning
requests that are no longer assigned
Rekiek et al.
[126]
DARP
(MV)
Chromosome representation is
group oriented, where each group
represents a vehicle to which a set
of requests is assigned
traditional GGA crossover as
in [49]
Create a new group - Remove
an existing group - Move items
between groups - An inversion
operator is also used to change
the location of groups in the
chromosome
A Grouping Genetic Algorithm (GGA) applied to the
Handicapped Person Transportation (HPT) problem,
which focuses on minimizing clients’ inconvenience - The
insertion heuristic used to insert requests in routes is the
best-fit (BF) heuristic - An unserved client is inserted in
the vehicle that will grant him the best service without vio-
lating problem constraints](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-90-2048.jpg)
![69
Chapter 5
The Single Vehicle Pickup and Delivery
Problem with Time Windows:
Introduction and a Genetic Algorithm
Approach
The Single Vehicle Pickup and Delivery Problem with Time Windows (SV-PDPTW) is
a frequently encountered problem in public and goods transport systems. However, only
a few researchers seem to have tackled it, possibly due to the difficulty in managing the
different underlying problem constraints. As previously mentioned in the introduction to
this thesis (Chapter 1), our research concentrates on representational issues and neigh-
bourhood moves within a simple meta-heuristic framework. For this problem, we adopt
a solution representation that depends on a duplicate code for both the pickup location
and its corresponding delivery. This simple representation will guarantee the satisfaction
of the precedence constraint, among the pickup and delivery pair, throughout the search.
We also present intelligent neighbourhood moves, that are guided by the time window, ai-
ming to overcome the difficult timing constraint efficiently and produce feasible and good
quality solutions in a reasonable amount of time.
This chapter covers two aspects: 1) a detailed analysis of the problem and research chal-
lenges and motivations, as well as a summary of some related work on the SV-PDPTW
and 2) the initial development of our key ideas using a Genetic Algorithm (GA) as a can-
didate solution methodology. The next chapter presents a comparison of three different
approaches to the SV-PDPTW: a genetic algorithm approach, a simulated annealing ap-
proach, and a simple hill climbing heuristic, which all employ the same representation and
neighbourhood move that our initial GA approach adopts. The work related to the current
chapter, i.e., the initial GA technique, was presented in the GECCO’07 conference, as a
late breaking paper [79]. Also, the research tackling other heuristics and meta-heuristics
applied to the SV-PDPTW, as described in the next chapter, was published in the Journal](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-93-2048.jpg)
![70 5.1 Problem and Motivation
of heuristics [84].
The rest of this chapter is organized as follows: Section 5.1 explains the problem tackled
in this part of the research and the motivation behind our selection of the problem and
the underlying solution approaches. Then, Section 5.2 formally defines the SV-PDPTW,
while Section 5.3 provides a brief summary of some related work from the literature.
Section 5.4 highlights the contribution of our research, and focuses on the solution repre-
sentation and neighbourhood move suggested to facilitate dealing with the SV-PDPTW.
Sections 5.5 presents the details of the proposed GA approach to solving this problem,
together with the experimental findings of this part of our research. Finally, Section 5.6
concludes the chapter with our intended future work.
5.1 Problem and Motivation
The Pickup and Delivery Problem with Time Windows (PDPTW) is an important variant
of vehicle routing problems that is likely to assume even greater prominence in the fu-
ture. Current concerns over global warming, resource depletion and the social impact of
traffic congestion and pollution, are driving companies, government organizations and re-
searchers to improve the efficiency of logistics and distribution operations. In addition,
the rapid growth in parcel transportation as a result of e-commerce is likely to have an
increasing impact. As previously mentioned in Chapter 4, an important related variant
of the PDPTW is the dial-a-ride problem, which is concerned with the transportation of
people, especially the elderly and the disabled, from their origins to their destinations,
while minimizing customer inconvenience.
The SV-PDPTW is a special case of the PDPTW dealing with a number of customer
requests that must be satisfied by one vehicle with a known capacity. The route of the
vehicle usually starts and ends at a central depot. A request must be collected from a
pickup location before being dropped off at a corresponding delivery location, and every
pickup and delivery location is associated with a specific time window during which it
must be served. If the vehicle arrives earlier than the beginning of the designated time
window interval, it must wait until the requested service time begins. All requests must
be served in a way that minimizes the total travel cost of the vehicle, without violating
precedence, capacity and time windows constraints [137].
In addition to being a sub-problem in the PDPTW, the SV-PDPTW has also practical ap-
plications, where small scale companies or individuals may operate one vehicle to serve a
set of clients, as for example in a dial-a-ride service. In other applications, the underlying](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-94-2048.jpg)
![5.1 Problem and Motivation 71
vehicle could be a helicopter or a small ship. Also, in some multiple vehicle applications,
the assignment of requests to vehicles may be restricted by some commodity, vehicle,
driver or client conditions. In such cases, certain requests must be assigned to a speci-
fic vehicle, making the optimization of a single vehicle route necessary for reducing the
overall cost of the logistic operation. Moreover, the analysis of the SV-PDPTW is essen-
tial in developing an insight into the more general multiple-vehicle case, and establishing
comprehensive solution algorithms.
As a constrained version of the TSP, the SV-PDPTW is known to be N P-hard [95], with
the presence of time windows making the problem particularly complicated. Since exact
algorithms are too slow for large problem sizes, heuristic and meta-heuristic approaches
seem to be reasonable alternatives. Several heuristics and meta-heuristics are potentially
suitable for this problem. Genetic Algorithms (GAs) are known for their robustness, pa-
rallelism, and their ability to perform reasonably well on a wide variety of problems,
including ordering and grouping problems, as well as highly constrained problems [64]
[105]. Thus, exploring GAs to solve the SV-PDPTW would seem to be a justified option.
Another alternative meta-heuristic approach that could be suitable for this problem is Si-
mulated Annealing (SA), which is analogous to the annealing of solids. As previously
discussed in Chapter 2, this approach has been widely applied to many optimization pro-
blems, successfully transforming ‘random’ low quality solutions to stable high quality
optimized solutions. One appealing feature of simulated annealing is that it is very easy
to implement, since it only requires a method for generating a move in the neighbourhood
of the current solution, and an appropriate annealing schedule. A third solution alterna-
tive is simple Hill Climbing (HC) that first creates a candidate solution and then iteratively
tries to perturb this solution to improve it.
In our research, we mainly focus on developing a meta-heuristic framework for the SV-
PDPTW rather than on a specific algorithmic paradigm. Within the context of the meta-
heuristic, we employ a problem-specific solution representation and guided neighbou-
rhood moves, aiming to overcome and deal efficiently with the hard problem constraints.
Our key ideas were adopted in all three approaches highlighted above, the GA, SA and
HC. In all cases, our view was to design an intelligent and robust solution approach
that can handle all problem constraints efficiently, while keeping the overall algorithm
as simple as possible, a feature often overlooked in most up to date solution methodolo-
gies. Ultimately, we aim to obtain feasible and high quality solutions in an acceptable
amount of time.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-95-2048.jpg)
![72 5.2 The SV-PDPTW
5.2 The SV-PDPTW
Let G = (N, A) be a digraph. The node set is N = {ni ∈ N|i = 0, 1, 2, ..., m},
such that m is an even index. The node n0 denotes the depot, and each ni, i = 1, 2, ...m
denotes a customer location. It is assumed that a customer’s request rk, k = {1, 2, ...m/2}
consists of a pickup and delivery (P&D) pair. We can consider, without loss of generality,
that the set N+
= {ni ∈ N|i = 1, 2, ..., m/2} represents pickup locations, and the set
N−
= {ni ∈ N|i = (m/2) + 1, ..., m} represents delivery locations, such that the pickup
location ni has the corresponding delivery location ni+(m/2). Thus, N = N+
∪ N−
and
|N+
| = |N−
| = m/2. Each location ni is associated with:
• A customer demand qi, such that qi > 0 for a pickup location, qi < 0 for a delivery
location and qi + qj = 0 for the same customer’s pickup and delivery locations
(q0 = 0);
• A service time si (s0 = 0), which is the time needed to load or unload a customer
demand;
• A Time Window (TW) [ei, li] during which the location must be served, and li ≥ ei.
For each pair of nodes < ni, nj > a travel time tij and/or a travel distance dij are specified.
We assume here a symmetric case, i.e., tij = tji and dij = dji. In addition, only edges
satisfying the TW are allowed. Thus the arc set is A = {< ni, nj > |ni, nj ∈ N, ni =
nj, t0i + si + tij < lj}. The vehicle has a limited capacity C. The capacity constraint
ensures that the total load carried by the vehicle at any given time does not exceed its
capacity. The vehicle’s journey should start and end at the depot, while each location is
to be visited exactly once. The time window constraint requires that a location must be
serviced within the specified TW, i.e., if the vehicle reaches the location before the earliest
service time ei, it must wait until ei. The precedence constraint requires that each pickup
location must precede the corresponding delivery location in the visiting order.
The objective function varies depending on the application. In general, one or more of
the following objectives are minimized as far as possible: the total traveling distance,
the total route duration, or the drivers’ total waiting time. Figure 5.1 shows a simplified
representation of a small instance of this problem before and after solving it.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-96-2048.jpg)
![5.3 Related Work 73
(a) SV-PDPTW - before solution (b) SV-PDPTW - after solution
Figure 5.1: The single vehicle pickup and delivery problem with time windows.
5.3 Related Work
As indicated above, there are various approaches to handle pickup and delivery problems.
Some approaches are exact and guarantee to solve the problem to optimality, while others
are approximations and attempt to find an acceptable solution in a reasonable amount
of time. For the SV-PDPTW, an exact algorithm is the dynamic programming approach
of [124]. However, this technique has a time complexity of O(n2
3n
) (where n is the
number of locations) and for this reason is normally limited to solving small problems of
up to about 10 requests (20 locations). Also, the authors in [38] are able to provide exact
solutions to the single vehicle dial-a-ride problem, with precedence, capacity and time
windows constraints, using dynamic programming. Their algorithm uses a set of states
(S, i), such that S is a subset of nodes from the node set, and i is a selected node. State
(S, i) is defined only if there is a feasible route that visits all nodes in S and terminates at i.
To reduce the state space, their dynamic programming algorithm uses sophisticated state
elimination criteria, based on the state (S, i). Computational experimentation indicated
that the elimination criteria work best when the time windows are tight and the vehicle
capacity is small, such that narrowing the constraints in their approach helped to reduce
the usual exponential running time of dynamic programming to a linear running time.
This algorithm can solve instances of up to 40 requests (80 locations).
Approximation algorithms, in which a heuristic or a meta-heuristic is developed to deal](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-97-2048.jpg)
![74 5.3 Related Work
with the problem, make it possible to cope with much larger instances. Among the heu-
ristics for solving the SV-PDPTW is the one suggested in [23]. This heuristic involves
a 2-phase algorithm. In the initial phase a feasible solution is constructed, and in the
second phase this solution is improved. In both phases a variable depth arc exchange
procedure is performed, as a neighbourhood move, in which the number of arcs to be
exchanged is not determined in advance, but calculated dynamically during the search.
In the route construction phase, the time constraint may be violated as long as the pre-
cedence and capacity constraints are satisfied. In the improvement phase, however, only
feasible solutions are permitted and route duration is used as an objective function. Du-
ring the improvement phase, the feasibility of a tour (in terms of precedence and capacity)
is verified using a set of global variables and an algorithm of time complexity O(n2
). To
check the feasibility of the time constraint and determine the promising arc exchanges, a
forward time slack is computed at each node to identify the permissible shift in departure
time that can be introduced without causing violations of time windows for other nodes in
the route. For problem sizes up to 38 customers, a near optimal solution can be reached
by this algorithm in less than 150 seconds. Unfortunately, success is not guaranteed, and
low quality or even infeasible solutions can result, if the 2-phase approach gets trapped
in a poor local optimum. To handle this possible situation, the authors present an alterna-
tive approach, which uses SA. This algorithm accepts time window violations in the early
stages, but penalizes them more severely as the search progresses. The SA approach is
able to obtain good quality solutions, albeit with a relatively high processing time.
The work reported in [88] also deals with the SV-PDPTW. This time, however, a hybrid
strategy is proposed which combines an exact method with a genetic algorithm and both
static and dynamic cases are considered. The static case assumes that all requests are
determined in advance of the route construction process, while the dynamic case allows
some requests to arrive during the construction of the route. The approach in this work
consists of three consecutive stages: a pre-planned module, a dynamic programming mo-
dule, and a GA. The pre-planned module arranges requests and prepares information for
the dynamic programming module. The role of the dynamic programming module is to
create a set of sub-routes, which it will eventually pass on to a temporary result pool,
where the GA module will pick them up, installing these unfinished sub-routes to esta-
blish its initial population. In the GA module, a solution is encoded as a permutation of
locations, and four crossover operators are compared: two traditional order-based cros-
sover operators, and two merge crossover operators, MX1 and MX2, that use a global
precedence vector to guide the inheritance process, as explained in [16] and later in this
chapter. The mutation operator is applied only when the offspring is identical to one of its
parents. Three mutation operators are used: one swaps two genes selected at random, the](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-98-2048.jpg)
![5.3 Related Work 75
second selects two random sites in the chromosome and reverses the sub-route between
the selected sites, finally the third mutation shuffles the genes that precede a location for
which a violation of constraints is observed. If an infeasible solution, violating the prece-
dence constraint, is generated as a result of any genetic operator, this solution is repaired
by swapping the corresponding pickup and delivery pair. The experimental results on
data sets ranging from 10 to 50 requests indicate the merge crossover operator MX1 ge-
nerally performed better than the other crossover operators tested. The first two mutation
operators also achieved better results than the third mutation operator.
In a more recent work [89], the same authors use similar genetic operators to the ones used
in [88], but in the context of a Family Competition Genetic Algorithm (FCGA). Again,
an order-based representation is adopted. The idea is to allow each individual of the po-
pulation to play the role of a family father in turn. Another randomly selected individual
plays the role of mate for the family father. The two individuals are combined to produce
an offspring in a regular GA fashion. The selection of the mate and the reproduction is
repeated for a chosen number of iterations to produce a family of offspring. Only the best
offspring in the family survives and is added to a temporary population of champions. The
new generation is chosen from among the best individuals in both the original population
and the champions of the families. Comparing the performance of the FCGA and a tradi-
tional GA on data sets created by the authors (ranging from 10 to 100 requests), the results
indicate that the merge crossover operators MX1 and MX2 generally worked better in the
context of a traditional GA than a FCGA, possibly due to premature convergence in the
latter case. On the other hand, a traditional uniform order-based crossover (UOX) worked
better within the framework of a FCGA than a traditional GA, possibly due to its uniform,
non greedy, nature of exploring the search space.
The authors in [95] present a heuristic based on tabu search to solve the SV-PDPTW.
The algorithm first creates a route respecting precedence and capacity constraints, using
a simple insertion heuristic. The generated route may be infeasible in terms of the time
window constraint, though. Then, two tabu search methods are compared in transforming
the initial route to a feasible route, with minimum total distance. The two tabu search
methods investigated are a regular deterministic Tabu Search (TS), and a Probabilistic
Tabu Search (PTS), which is based on the same principles as the deterministic one with
the addition of a buffered memory of potential moves and introducing some probabilistic
criteria for the selection of the next move. The creation of a neighbouring solution in
both tabu searches is based on classical neighbourhood moves, a swap operation and an
insertion operation, which respect the precedence and the capacity constraints. Test results
on problem instances created by the authors (from 10 to 40 customers) demonstrate the
superiority of TS compared to the PTS in both run time and solution quality. Possible](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-99-2048.jpg)
![5.4 SV-PDPTW: Research Contribution 77
its corresponding pickup location. Nevertheless, the precedence order may not be main-
tained following the application of a neighbourhood move to a solution. For example, a
simple swap of locations could disturb the precedence order and result in an infeasible so-
lution. Consequently, a repair method would be needed to restore the solution feasibility,
as done, for example, in [88] and [89]. Inevitably, such an approach would increase the
processing time and complicate the algorithm.
We have developed a solution representation which avoids the precedence issue: we sim-
ply assign the same code to both the pickup and its associated delivery location, and rely
on a simple decoder to identify its first occurrence as the pickup and the second as the deli-
very. This straightforward representation eliminates the problem of backtracking to repair
an infeasible solution, and solves the precedence constraint issue in an effective way. As a
result, more effort can be directed towards harder constraints such as the vehicle capacity
and time windows. Figure 5.2 shows an example of a solution with 3 requests1
following
this representation.
Figure 5.2: Solution Representation.
5.4.2 The Neighbourhood Move
As previously mentioned, while creating a neighbouring solution during the search pro-
cess, we are faced with a major challenge: the possible violation of one or more of the
problem constraints that may follow such moves. A neighbourhood move should be in-
telligent enough to direct the search towards high quality and feasible solutions, and thus
avoid valuable time being wasted evaluating a large number of infeasible solutions. When
designing such moves, all problem constraints should be considered. Nevertheless, in our
research we found the time window constraint the most difficult to deal with. Recall that
our duplicate encoding scheme renders the precedence constraint trivial. Furthermore, the
capacity constraint tends to be easy to satisfy in most problem instances, because half of
1
Recall from Section 5.2 that a request refers to a pair of locations (pickup and delivery).](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-101-2048.jpg)
![78 5.5 A Genetic Algorithm Approach for Solving the SV-PDPTW
the locations visited in any route are delivery locations whose demands are removed from
the vehicle.
In our research we adopt neighbourhood search operators that apply ‘randomness’, yet at
the same time take account of the time windows, and bring the more ‘urgent’ locations
earlier in the visiting order, where this is possible. This neighbourhood idea is adopted
in all our search algorithms, with slight variations depending on the context in which it is
applied.
The rest of this chapter and the next chapter are dedicated to our suggested solution me-
thodologies for handling the SV-PDPTW. The following section addresses our problem-
specific genetic algorithm that we adopted for solving this problem. On the other hand,
our investigation of the SV-PDPTW continues in Chapter 6, which presents the proposed
operators within the framework of three different heuristic and meta-heuristic techniques
to deal with this problem.
5.5 A Genetic Algorithm Approach for Solving the SV-
PDPTW
The focus of this part of our research is to investigate the potential of using genetic al-
gorithms to solve the SV-PDPTW. In particular, herein we experiment with a duplicate
gene encoding that guarantees the satisfaction of the precedence constraint, between the
pickup and the delivery, throughout the search. Thus, our chromosome encoding simply
follows the representation depicted in Figure 5.2. In addition, several problem-specific
genetic operators are tested and compared on a number of data sets with various sizes.
The rest of this section presents the details of our GA approach. Section 5.5.1 describes
the fitness function that the GA relies on during the evolutionary process. Section 5.5.2
introduces our suggested genetic operators. Section 5.5.3 explains the procedure adopted
to create test data for the problem. Finally, Section 5.5.4 details the experimental findings
of this part of the research,
5.5.1 The Fitness Function
Following [89], the fitness function treats the constraints as soft constraints, meaning that
an infeasible solution that violates either the capacity and/or the time window constraint](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-102-2048.jpg)
![5.5 A Genetic Algorithm Approach for Solving the SV-PDPTW 79
will be penalized by an added term in the fitness function. The fitness function of a route
r is:
F(r) = w1 × D(r) + w2 × TWV (r) + w3 × CV (r), (5.1)
where D(r) is the total route duration including the waiting time and the service time at
each location, TWV (r) is the total number of time window violations in the route, and
CV (r) is the total number of capacity violations. w1, w2, and w3 are weights in the range
[0, 1] assigned to each term in the fitness function, and w1 + w2 + w3 = 1.0.
The fitness function will thus try to minimize infeasibility as well as the total route du-
ration. Also, the choice of appropriate weights depends on the importance of each term
in the fitness function. Specifically, the highest penalty must be imposed on the number
of time window violations, since it is the most difficult constraint to satisfy in the SV-
PDPTW. We used the following values for the weights of the fitness function: w1 = 0.001,
w2 = 0.7, and w3 = 0.299. Our experimentation with different values in the range [0, 1]
indicated that if a smaller weight value was assigned to the TWV term, the best individual
resulting at the end of the evolutionary process is often infeasible, in terms of violating
the time window constraint.
5.5.2 The Operators
Two crossover operators have been used in our GA, the first crossover operator is a Merge
Crossover (MX1), first suggested in [16]. The second crossover operator is a new pro-
blem specific crossover that we called the Pickup and Delivery Crossover (PDPX).
Also, two mutation operators were used in our research. A regular gene swap mutation
and another constraint oriented directed mutation. These four operators are explained
below.
Crossover
MX1 Crossover: Several crossover operators were considered potentially suitable for
this kind of problem. The first crossover operator we tried follows the merge crosso-
ver operators suggested in [16] for the VRPTW2
, and used in [88] and [89] for the SV-
PDPTW. Unlike traditional crossover operators for ordering problems, which depend only
on the order of genes in a chromosome, the merge crossover operators depend on a glo-
bal precedence among genes, such as the time window or distance ordering. Traditional
2
Recall from Chapter 3 that in the VRPTW, the same service type applies to all locations, either pickup
or delivery.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-103-2048.jpg)
![80 5.5 A Genetic Algorithm Approach for Solving the SV-PDPTW
order-based crossover operators are not very effective for highly constrained problems
like the SV-PDPTW, since they frequently produce infeasible solutions. Merge crossover
operators, however, were shown to be superior to the traditional ones for these types of
problems [16] [88] [89].
In the current research we have slightly modified the MX1 operator used by [88] and [89].
Instead of creating just one child, giving priority to a parent’s gene having an earlier time
window lower bound, we have created two children: the first child favouring genes that
have an earlier lower bound, while the second child favours genes that have an earlier
upper bound. The idea is that visiting a location just before its deadline could be more
beneficial than visiting it as early as possible in its allowed interval. This may help reduce
the waiting time that would result if the vehicle arrives too early at a location, and may
consequently decrease the total route duration. Creating two children instead of one was
suggested in [16], and may help improve the quality of the new generation and accelerate
the optimization process.
To illustrate how the MX1 operator works, assume that the vector shown in Figure 5.3
defines the precedence order, in terms of the lower bound of the time window, among all
pickup and delivery locations, where a (+) following a request number indicates that this
is its pickup location and a (-) indicates the delivery location for the same request.
Figure 5.3: Time window precedence vector.
Figure 5.4 illustrates the steps of the MX1 operator. Consider the two parent solutions,
P1 and P2, shown in Figure 5.4(a). The MX1 crossover starts by decoding the parent
solutions to differentiate the pickup locations from the delivery locations, as illustrated in
Figure 5.4(b). Then, the MX1 proceeds as follows:
• Step 1: since 2+ has a higher precedence than 3+ (as indicated by the precedence
vector depicted in Figure 5.3), the child will inherit 2+ as the first gene, as shown in
Figure 5.4(c);
• Step 2: to maintain feasibility, 2+ in P2 will be swapped to the first location. Now
the second gene in both parents is identical, so it is copied to the child, as shown in](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-104-2048.jpg)
![84 5.5 A Genetic Algorithm Approach for Solving the SV-PDPTW
5.5.3 Test Data
Since there are no standard benchmark test cases available for the SV-PDPTW, our GA
was tested on a data set obtained from the authors of [89]. This data set has a number of
customer requests ranging from 10 to 100 (20 to 200 locations). In addition, for a more
extensive and thorough testing of the operators, we have created a new data set with larger
numbers of customer requests ranging from 100 to 200 (200 to 400 locations).
To create test data for the SV-PDPTW, it is essential to ensure the existence of at least
one solution that satisfies all problem constraints. Similar to [89], our algorithm first
creates a route that respects precedence and capacity constraints, then a time window
interval is generated for each location based on the arrival time realized in the created
route. Unlike [89], however, we rely on our GA to create the initial feasible solution, while
they create their starting feasible solution randomly. More specifically, the following steps
were followed to create the test data:
1. Generate a random vehicle capacity within a certain predetermined range;
2. Generate random x and y coordinates for the depot;
3. For all pickup and delivery locations:
(a) Generate random x and y coordinates;
(b) Generate a random demand (load) within a certain allowable range, such that
the demand of a delivery location is the same as the demand of its correspon-
ding pickup location, but with a negative sign;
(c) Assume a very large time window interval that could not possibly be violated;
4. Run the GA to obtain a feasible solution. Note that, the precedence constraint
is always satisfied in any generated solution, thanks to our duplicate gene enco-
ding. Moreover, due to the nature of the problem, the satisfaction of the capa-
city constraint can be easily accomplished by our genetic operators. The timing
constraint can also be easily satisfied because the TW intervals are very large at this
point;
5. Calculate the arrival time at each location in the feasible route obtained;
6. Create a random time window interval for each location such that the arrival time
falls within the created time window. The width of the permitted time window
interval should be determined in advance.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-108-2048.jpg)
![5.5 A Genetic Algorithm Approach for Solving the SV-PDPTW 85
Algorithm 5.1 describes the initial steps carried out for the creation of test data for the
problem without the final time window intervals, i.e., Steps 1 to 3 above. In this algorithm,
the allowable ranges for the random values were determined empirically. Note that all
ranges were scaled according to the number of requests currently generated, i.e., the larger
the number of requests, the larger the allowable range for the vehicle capacity and the
demand of the location. Algorithm 5.2 describes the procedure for determining the TW
intervals, i.e., Step 6 above.
Algorithm 5.1: Create Test Data.
1: Choose values for constants MinCapacity, MaxCapacity, MinCoord and MaxCoord
{We used MinCapacity = 1, MaxCapacity = 10, MinCoord = 0 and MaxCoord =
200}
2: Choose value for constant MinTW {A very small value (we used 0)}
3: Choose value for constant MaxTW {A very large value (we used 100000)}
4: Let nrequests = n/2 be the number of requests in the set, where n is the total number of
nodes (locations) in the data set, excluding the depot
5: V ehicleCap = Random(MinCapacity×nrequests, MaxCapacity×nrequests){Func-
tion Random generates a random value between its two parameters}
6: AverageDemand = V ehicleCap/nrequests {The average demand of a request}
{Generate coordinates, demand, and TW for the depot and the pickup locations}
7: for (pickloc = 0; pickloc <= nrequests; pickloc + +) do
8: x[pickloc] = Random(MinCoord, MaxCoord){The x-coordinate}
9: y[pickloc] = Random(MinCoord, MaxCoord){The y-coordinate}
10: if (pickloc = 0) then
11: demand[pickloc] = 0 {The demand of the depot is 0}
12: else
13: demand[pickloc] = Random(AverageDemand×m1, AverageDemand×m2) {The
choice of m1 and m2 depends on how big we want the demand of the location}
14: ETW[pickloc] = MinTW{The lower bound of the TW}
15: LTW[pickloc] = MaxTW{The upper bound of the TW}
{Generate coordinates, demand, TW for delivery locations}
16: for (delloc = nrequests + 1; delloc <= n; delloc + +) do
17: x[delloc] = Random(MinCoord, MaxCoord)
18: y[delloc] = Random(MinCoord, MaxCoord)
19: demand[delloc] = −demand[delloc − nrequests]{Same as its pickup demand but -ve}
20: ETW[delloc] = MinTW{The lower bound of the TW}
21: LTW[delloc] = MaxTW{The upper bound of the TW}](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-109-2048.jpg)
![86 5.5 A Genetic Algorithm Approach for Solving the SV-PDPTW
Algorithm 5.2: Generate Time Windows.
1: Choose value for constant CurrentWidth {We used 10}
2: Calculate the arrival time of the vehicle at each location in the feasible route obtained by the
GA
3: for (loc = 1; loc <= n; loc + +) do
4: ETW[loc] = Arrival[loc] − Random(CurrentWidth × m1, CurrentWidth × m2)
{The choice of m1 and m2 depends on whether we want the time window interval to be
wide or narrow}
5: if (ETW[loc] < 0) then
6: ETW[loc] = Arrival[loc] − Random(1, Arrival[loc])
7: LTW[loc] = Arrival[loc] + Random(CurrentWidth × m1, CurrentWidth × m2)
5.5.4 Experimental Results
We implemented a steady state GA with a replacement percentage of 100% and a popu-
lation size of 1000. Algorithm 5.3 outlines the GA used in our implementation.
Algorithm 5.3: The SV-PDPTW Genetic Algorithm.
1: Initialize a random population POP of candidate solutions to the SV-PDPTW, using the
duplicate encoding of pickup and delivery locations
2: while (stopping condition is not reached) do
3: for (i=0; i<NumCrossovers;i++) do
4: Select parents P1 and P2 from POP, using roulette wheel selection
5: Apply crossover to parents P1 and P2 to produce two children C1 and C2
6: With some probability, apply mutation to C1 and C2
7: Update POP by integrating the new generation and eliminating some worst individuals
{i.e., steady state GA with overlapping populations}
To test our GA, different combinations of crossover and mutation operators were compa-
red on two data set samples. The first sample, which we will call SET 1, is the sample
obtained from [89] and includes 30, 80, 90 and 100 customer requests3
. It should be no-
ted that while solving the problem for this data set, the authors of [89] assumed that the
vehicle’s journey is open path. Thus, although the vehicle should start its journey from
the depot, it could end at any of the delivery locations. In order to compare our results
with their results we also make this assumption in our GA approach. An alternative, and
more ‘standard’, assumption is for the vehicle to start and end at the depot.
3
Source: http://wrjih.wordpress.com/2006/12/09/pdptw-test-data/](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-110-2048.jpg)
![5.5 A Genetic Algorithm Approach for Solving the SV-PDPTW 87
The second data set used to test our algorithm, which we will call SET 2, is obtained from
the instances created by us, and includes 130, 170, and 200 customer requests 4
. Note that
the number of locations is always double the number of requests. The algorithm was run
10 times on each test case, with a crossover rate of 1.0 and a mutation rate of 0.45
.
The following combinations of crossover and mutation operators were compared:
1. MX1 crossover and random swap mutation.
2. MX1 crossover and directed mutation.
3. PDPX crossover and random swap mutation.
4. PDPX crossover and directed mutation.
5. Directed mutation without any crossover6
.
6. MX1 crossover without any mutation.
The genetic algorithm was run to convergence or for a maximum of 3000 generations if no
convergence can be reached. Since robustness is one of our main goals in this research, the
GA was run with the same parameters on all problem instances, i.e., it was not specifically
tuned or optimized for each instance separately.
The results for the 10 runs are recorded in Table 5.1 as follows, for each test case under
each GA version: the best score found (in terms of total route duration only), and the
percentage of feasible solutions obtained during the 10 runs. The last column of the table
shows the previous best result. For SET 1, the previous best results are the ones reported
in [89], during their computational experimentation with these problem instance7
, while
for SET 2, the previous best results are the best results achieved in the current experiment.
A score followed by a (*) indicates that this solution is infeasible in terms of time window
constraint violation. When the obtained result is the same as or better than the previous
best result, it is highlighted in boldface.
In terms of the objective function values, Table 5.2 shows the average, best, and worst
objective function values, as calculated by Equation 5.1, for all versions of our GA. the
4
Source: http://users.cs.cf.ac.uk/M.I.Hosny/PDP.zip
5
A higher than usual mutation rate was found necessary to avoid being trapped in a local optimum. One
reason could be the duplicate encoding, a side effect of which is that mutation may swap identical genes
producing the same offspring.
6
Mutation rate was set to 1 and crossover rate to 0
7
The results shown are truncated as they appear in [89].](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-111-2048.jpg)
![90 5.5 A Genetic Algorithm Approach for Solving the SV-PDPTW
Standard Deviation (SD) value of the objective function for the 10 runs is also shown in
this table.
As can be observed in Table 5.1, the best results are obtained from the MX1 crossover
and random swap mutation. Together they achieved a 100% feasibility rate in all test
cases except for the largest task (200 requests) in which the feasibility rate was 80%. In
two cases the results obtained were even better than the previous best results reported in
[89]. These are the results for test cases 30 and 80. Table 5.2 also shows clearly that the
overall average objective function results of the MX1 crossover and swap mutation were
better than those obtained by all other GA versions, for all categories demonstrated in the
table.
Table 5.1 shows that the worst results are obtained by the PDPX crossover and swap mu-
tation. These two operators together failed to achieve any feasible results, in terms of the
time window constraint, even for the small sized tasks. Also, the overall average objective
function results for these two operators were considerably worse than the results obtained
by all other GA versions, as Table 5.2 indicates. These inferior results may be explained
by noticing that both PDPX crossover and swap mutation lacked any guidance towards
the time window ordering. Although they were able to satisfy the capacity constraint in all
test cases, they failed to satisfy the time window constraint. Clearly, they worked blindly
with respect to the timing order, and consequently were not able to find an ordering that
services all requests in their desired intervals.
When directed mutation replaced the random swap mutation, however, the results drama-
tically improved for the PDPX. Table 5.1 shows that PDPX together with directed mu-
tation were able to obtain feasible solutions in all test cases. The feasibility ratio ranged
from 40% for the largest task to 100% for the smallest task. The overall results, though,
were not able to beat the results obtained from the MX1 crossover and swap mutation,
which can be seen in Table 5.2. Figure 5.7 sustains the above observation, by showing
how directed mutation was able to guide the search and helped the PDPX crossover to
find much better solutions for the largest task tested (200 requests), when used in place of
the random swap mutation.
The situation is different, however, when directed mutation replaced random swap mu-
tation with the MX1 crossover. As shown in Tables 5.1 and 5.2, MX1 crossover and
directed mutation produced lower quality solutions than those obtained by MX1 cros-
sover with the random swap mutation. Possibly, the introduction of directed mutation
caused the algorithm to rapidly converge to a local optimum. Figure 5.8 bears this out,
and shows the rapid convergence of the MX1 crossover with directed mutation compared
to the gradual convergence of the MX1 crossover with the random swap mutation.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-114-2048.jpg)
![95
Chapter 6
Several Heuristics and Meta-heuristics
Applied to the SV-PDPTW
Taking into consideration the challenges of developing an appropriate solution represen-
tation and effective neighbourhood moves to facilitate dealing with the constraints of the
SV-PDPTW, in this part of our research we investigate and compare three approaches to
the problem. The first approach is a GA approach, similar to that introduced in Chapter 5.
As previously mentioned, this approach adopts a solution representation having the same
code for both the pickup and its associated delivery, and applies some problem-oriented
genetic operators. The second solution methodology we investigated is Simulated An-
nealing (SA), which adopts the same solution representation as the one used in the GA.
Also inspired by the GA mutations, two SA neighbourhood strategies are tested: a ran-
dom blind move, and an intelligent move that is directed by the time window. The third
approach is a simple Hill Climbing (HC) heuristic that creates a starting solution and then
tries to improve it, replacing the current solution with better solutions generated during
the search. The same solution representation and the directed neighbourhood move used
in both the GA and the SA are also adopted in the HC. As previously indicated in Chap-
ter 5, the details of the three approaches and the results of this part of our research were
published in the Journal of Heuristics [84].
Before we start the discussion of our solution methodologies, we present, in the following
section, the objective function used in this part of the research. The three sections that
introduce our selected approaches to the problem are Sections 6.2 (GA), 6.3 (SA) and
6.4 (HC). Section 6.5 then details our findings when the three approaches were analyzed
and compared, and Section 6.6 elaborates on some further analysis of the SA algorithm,
which is the most promising approach among those tested. Finally, Section 6.7 concludes
this chapter with some suggested future work.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-119-2048.jpg)
![96 6.3 The 3-Stage Simulated Annealing (SA) Approach
6.1 The Objective Function
In this part of the research, we slightly modified the objective function used by [89] and
also by our GA in Chapter 5 (i.e., Equation 5.1). The current objective function additio-
nally penalizes the amount of time delay, together with the number of time window and
capacity violations. The idea is that adding a penalty for the extent of total tardiness in
the route could possibly direct the search towards better quality and feasible solutions,
especially if the problem size increases. Thus, the objective function of a route r is now
described by the following equation:
F(r) = w1 × D(r) + w2 × TWV (r) + w3 × CV (r) + w4 × TD(r), (6.1)
where D(r) is the total route duration including the waiting time, if the vehicles arrives at a
location before the start of its service time. TWV (r) is the total number of time window
violations in the route. CV (r) is the total number of capacity violations, and TD(r)
is the total amount of delay if the vehicle arrives at a location later than the specified
deadline. w1, w2, w3 and w4 are weights in the range [0, 1], and w1 + w2 + w3 + w4 = 1.0.
While experimenting with this objective function, we found that in order to eliminate
infeasibility, a higher penalty should be imposed on the time window violations and the
total delay than the capacity violations or the total route duration. We used the following
values for the penalty weights in our objective function: w1 = 0.001, w2 = 0.6, w3 =
0.099, and w4 = 0.3.
6.2 The Genetic Algorithm (GA) Approach
The genetic algorithm used in the current research is identical to the GA used in the first
part of the research, as presented in Chapter 5. Nevertheless, here we only applied the
two most promising genetic versions. Namely, the GA with MX1 crossover and random
swap mutation, and the GA with directed mutation only. Also, the GA adopted here uses
the new objective function described in Equation 6.1, instead of the previous objective
function (Equation 5.1).
6.3 The 3-Stage Simulated Annealing (SA) Approach
As previously explained in Section 2.3.2, the theoretical foundation of simulated annea-
ling was established in [93]. Since then, the algorithm has been successfully used in](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-120-2048.jpg)
![6.3 The 3-Stage Simulated Annealing (SA) Approach 97
solving many combinatorial optimization problems, including vehicle routing and sche-
duling problems.
To solve the SV-PDPTW using SA, we first need to construct an initial solution, and then
try to optimize this solution using a method analogous to the annealing of solids. Some
researchers choose to apply different heuristic techniques to create an initial solution with
minimum constraint violations, for example [23] and [95]. In our research, we construct
an initial solution s by simply generating a large number of random routes and selec-
ting the route with minimum cost (according to Equation 6.1) to be our starting solution1
.
This technique is straightforward to implement and also computationally inexpensive. Al-
though the quality of the initial solution may be poor, the main focus of the algorithm is
on the route improvement phase, accomplished during the SA search. A good neighbou-
rhood move should be capable of directing the SA algorithm to reach a final high quality
solution, irrespective of the quality of the starting solution. The solution representation
adopted in the SA approach is the same as the solution representation of the GA, i.e.,
assigning the same code to the pickup and its delivery as shown in Figure 5.2.
The choice of an appropriate annealing schedule is critical to the performance of SA.
Ideally it is desirable to devise a scheme that is adaptable for all test cases and problem
sizes, eliminating the need for ‘arbitrary’ parameter tuning, which can be very time consu-
ming. Following [40], we created the annealing parameters for each test case individually
as shown in Algorithm 6.1. To generate a neighbouring solution s′
in this algorithm, a
simple random swap between two different locations in s is performed.
The main SA algorithm is shown in Algorithm 6.2. This algorithm starts with the same
solution s used to calculate the annealing parameters in Algorithm 6.1. This time, ho-
wever, to get a new state s′
from the current state s, an intelligent neighbourhood move,
that depends on the upper bound of the time window, is used. First two locations in the
current solution are selected at random. Then, these two locations are swapped, only if
their deadlines are out of order, i.e., if the latter has a more urgent deadline than the earlier
one. This is exactly the same idea adopted in the directed mutation used in the GA (see
Section 5.5.2 and Figure 5.6), and its purpose is to arrange requests in a way that will best
satisfy the timing constraint.
1
We generated 10000 random solutions and selected the best one among them.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-121-2048.jpg)
![98 6.3 The 3-Stage Simulated Annealing (SA) Approach
Algorithm 6.1: Calculating Annealing Parameters [40].
1: Given an initial solution s
{Initialize Pstart the starting acceptance probability and Pend the final acceptance probabi-
lity}
2: Let Pstart = a large value {We used 0.999}
3: Let Pend = a small value {We used 0.001}
4: ∆avg ← 0
{Generate n neighbouring solutions of s}
5: for (i = 0; i < n; i + +) do
6: Select two random locations in s
7: Swap the current two locations in s to get a new solution s′
8: ∆ ← |cost(s′) − cost(s)|
9: ∆avg ← ∆avg + ∆
10: ∆avg ← ∆avg/n
11: T0 ← −∆avg/ log(Pstart) {T0 is the initial temperature}
12: TN ← −∆avg/ log(Pend) {TN is the final temperature}
13: α ← exp(log(TN )−log(T0))/N {α is the temperature reduction factor, and N is the number of
iterations desired}
14: Return T0, TN and α
Algorithm 6.2: The Main SA Algorithm.
1: Given an initial solution s, a starting temperature T0, and a temperature reduction factor α
2: T ← T0 {Initialize the current temperature}
3: repeat
4: Select two random locations in s
5: if (The latter location has a smaller value of the upper bound of the time window) then
6: Swap the current two locations in s to get a new solution s′
7: ∆ ← cost(s′) − cost(s)
8: if (∆ < 0) then
9: s ← s′
10: T ← α × T {Reduce the current temperature}
11: else
12: p = Random(0, 1) {generate a random number in the interval (0,1)}
13: if (p < exp(−∆/T)) then
14: s ← s′
15: until (Frozen){Stop when no improvement is achieved for a pre-specified number of iterations
(we used 5000 iterations)}
16: Return s](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-122-2048.jpg)
![6.5 Experimental Results 101
bounds of the time window) are out of order. On the other hand, if the deadlines are in
sequence, the current solution remains intact, and two new locations are considered in the
next iteration. When no further improvement is possible using this neighbourhood move,
the second stage starts from the final solution achieved in the first stage and repeats the
same HC procedure, but with a neighbourhood move that adopts the lower bound of the
time window to decide the swapping. Finally, the third stage starts with the final solution
obtained in the second stage, but with a neighbourhood move that swaps locations if they
are out of order in terms of the centre of the time window interval.
6.5 Experimental Results
To test our algorithms, we used the same data set samples on which we tested our GA, as
previously explained in Section 5.5.4. Each of the following algorithms was run 10 times
on each test case:
1. The GA with MX1 crossover and random swap mutation (GA1).
2. The GA with directed mutation only (GA2).
3. The 3-stage SA (SA1).
4. The simple random-move SA (SA2).
5. The Hill Climbing (HC) algorithm .
All algorithms were run on all problem instances using the same set of parameters (e.g.
GA parameters, simulated annealing schedule, and stopping conditions), i.e., no particular
tuning of parameters was performed for each test case separately. The results for the 10
runs are recorded in Table 6.1 as follows: for each test case under each algorithm, the
best result found (in terms of total route duration only), and the percentage of feasible
solutions obtained during the 10 runs. The last column of the table shows the previous
best results, attributed to [89] and our GA approach reported in Chapter 5. Whenever
the result achieved in the current experiment is better than the previous best result, it is
highlighted in boldface.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-125-2048.jpg)
![6.5 Experimental Results 103
As shown in Table 6.1, the best route duration results were achieved by the 3-stage SA
(SA1). In all test cases the best results achieved by this version of SA were better than
the previous results achieved in [89] and by our GA explained in Chapter 5. The results
were also superior to the results achieved by all the other algorithms tested in the current
research. A very high feasibility rate and good quality solutions were obtained in all
tasks tested, with an improvement of approximately 11%, over the previous best2
, for the
smallest task and approximately 3% for the largest task.
In terms of the overall objective function values, Table 6.2 shows the average, best, and
worst objective function values, as calculated by Equation 6.1, for the GA1 and GA2
algorithms. the Standard Deviation (SD) value of the objective function for the 10 runs is
also shown in this table. Table 6.3 shows the same objective function information for the
remaining three algorithms (SA1, SA2, and HC).
Table 6.2: Results Summary for GA1 and GA2 (Objective Function Values).
Task
GA1 GA2
Avg Best Worst SD Avg Best Worst SD
30 3.74 3.74 3.74 0 3.81 3.75 3.89 0.05
80 7.86 7.84 7.87 0.01 7.92 7.87 7.97 0.04
90 8.62 8.62 8.62 0 8.69 8.62 8.77 0.05
100 10.60 10.60 10.60 0 11.08 10.60 14.69 1.27
130 14.51 14.04 15.28 0.35 14.85 14.34 15.16 0.25
170 20.94 20.62 21.25 0.21 20.67 20.33 20.88 0.16
200 26.20 25.80 26.45 0.21 25.04 24.73 25.33 0.21
Avg 13.21 13.04 13.4 0.11 13.15 12.89 13.81 0.29
Tables 6.2 and 6.3 also support the conclusions drawn from Table 6.1. Observing the
overall average results in the last row of Table 6.2 and 6.3, we can clearly see that the
SA1 algorithm outperformed all other heuristics in the average, best and worst objective
function values. On the other hand, its overall average SD value was only slightly larger
than the overall average SD of the GA1 algorithm. The GA1 algorithm obtained a smaller
SD, due to the fact that the population converged to a sub-optimal solution in 3 out of the
7 test cases (30, 90 and 100 requests), as shown in Table 6.2.
2
The improvement percentage is calculated using the formula: ((R∗
− R)/R∗
) × 100, where R is the
current result and R∗
is the previous best result.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-127-2048.jpg)
![108 6.5 Experimental Results
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
0
0.5
1
1.5
2
2.5
3
3.5
4
x 10
6
Iteration
ObjectiveValue
SA2
SA1
Figure 6.3: Random-move SA (SA2) against 3-Stage SA (SA1) for task 200.
Table 6.6 shows the average processing time in seconds for all the algorithms tested.
The table shows that the HC algorithm has the fastest average processing time among all
algorithms, in all test cases. The table also indicates that the overall average processing
time of the 3-stage SA was slightly longer than the overall average processing time of the
random-move SA. In addition, the table clearly indicates that both GA versions were very
slow compared to all other algorithms, which is expected due to the need to maintain a
large population of individuals throughout the search. The SA and HC algorithms, on the
other hand, have lower computational costs due to focusing on a single solution.
Finally, comparing the results obtained by the 3-stage SA with the results obtained by pre-
vious researchers for the same problem, our algorithm seems impressive, in terms of run
time as well as solution quality. We appreciate the difficulty in making such comparisons
when different platforms have been used. Nevertheless, our run times are to a large extent
acceptable. For example, as shown in Table 6.6, the average run time of the 3-stage SA
for the 100-requests task was 44.5 seconds, while the best algorithm tried in [89] appears
to have an average CPU time of approximately 300 seconds for the same task.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-132-2048.jpg)
![118 7.1 Solution Construction for the MV-PDPTW
The MV-PDPTW is known to be N P-hard [131], and the presence of many constraints
makes the problem particularly complicated. Indeed, generating feasible and good quality
solutions to the problem in a reasonable amount of time is often a hard challenge for re-
searchers. The MV-PDPTW is both a grouping problem (assigning requests to vehicles),
and a routing problem (finding the best route for each vehicle). Thus, an intelligent solu-
tion methodology should be able to handle these two aspects efficiently. Similar to other
vehicle routing problems, researchers usually try to solve the problem in two stages: the
first stage is solution construction, while the second is solution improvement. This chapter
is dedicated to the solution construction phase of the MV-PDPTW, while our investigation
of the solution improvement phase is detailed in the next chapter. The findings of this part
of our research were published in the MIC2009 conference [81].
The rest of this chapter is organized as follows: Section 7.1 gives a general idea about
solution construction for the MV-PDPTW and introduces our upcoming research in this
area. Section 7.2 summarizes some related work. Section 7.3 emphasizes the motivation
and the objectives of this part of our research. Section 7.4 explains the routing algorithm
embedded within the different construction heuristics suggested. Section 7.5 details the
construction heuristics implemented in this research. Section 7.6 reports the experimental
results of the algorithms tested. Section 7.7 sheds light on some implementation issues
and complexity analysis of the suggested algorithms. Finally, Section 7.8 concludes this
chapter with our future plans.
7.1 Solution Construction for the MV-PDPTW
Like the construction heuristics for the VRPTW, a solution construction algorithm for the
MV-PDPTW usually selects at each iteration an un-assigned customer whose insertion is
predicted to cause the least increase in the overall cost of the solution. The selected cus-
tomer is then inserted in its best (least cost) feasible insertion position found among all
available routes. This kind of insertion may require multiple calculations to estimate the
effect of the insertion, in terms of the increase in travel distance and time delay, on all cus-
tomers already existing in that particular route who could be affected by the insertion (see
Section 3.3 for more details about construction heuristics). Additional decisions during
the construction of the solution include whether to build routes sequentially or in parallel,
and possibly the selection of seed customers to initialize the routes. Some construction
algorithms order customers before the insertion, and the initial order is also an important
factor that may affect the quality of the generated solution. Common approaches include
sorting customers according to the distance from the depot, or according to the time win-](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-142-2048.jpg)
![7.1 Solution Construction for the MV-PDPTW 119
dow. Finally, the selection of a cost function to assess the quality of the whole solution1
during the construction is sometimes needed, so that insertions which greatly affect the
solution cost can be identified and appropriately handled.
While these considerations also apply to the general VRPTW, where all requests are of
the same type (either pickups or deliveries), the MV-PDPTW in itself entails additional
considerations. This is due to the presence of a pair of related locations for each indivi-
dual request and the precedence and coupling issues resulting thereof. For example, the
decision regarding the best insertion position for a certain request should ideally take both
the pickup and the delivery into account. The sorting criteria for requests may likewise
be based on either the pickup or the delivery location, or perhaps combine both. It is also
frequently the case with the MV-PDPTW that the initial solution is drastically changed
during the improvement phase. For example, both the algorithms in [12] and [131] produ-
ced very good results using an approach that is based on a Large Neighbourhood Search
(LNS). The algorithm removes and then relocates a large number of requests (30% - 40%)
in each iteration. This could possibly indicate that sophisticated construction algorithms,
that are usually time consuming, parameter dependent, and hard to implement, may not
actually merit their cost, compared to more straightforward and faster algorithms.
In an attempt to overcome the difficulties inherent in the construction of a feasible so-
lution, which are mainly due to the hard problem constraints and the complex problem-
specific decisions, we propose, in this part of our research, four different construction
heuristics that aim to build initial feasible solutions to the MV-PDPTW. To the best of
our knowledge, our research is the first attempt in the literature to analyze and compare
different initial solution construction methods for the MV-PDPTW. The aim of the re-
search is to decide which construction algorithm has more potential as a preliminary step
towards a complete solution methodology to the problem. A promising construction al-
gorithm should demonstrate a suitable balance between quality of the generated solution,
processing speed and simplicity of implementation.
All suggested algorithms make use of a simple and efficient routing algorithm to generate
feasible individual vehicle routes. This routing heuristic was based on our investigation
of the SV-PDPTW, as previously detailed in Chapters 5 and 6. These algorithms, never-
theless, differ in whether the construction of vehicle routes is performed sequentially or
in parallel. They also differ in the criteria according to which the next un-routed request
is selected for insertion in a particular route.
1
For example, the cost function may combine the number of routes and the total cost of each route.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-143-2048.jpg)
![120 7.2 Related Work
7.2 Related Work
As previously mentioned in Section 3.3, solution construction, for vehicle routing pro-
blems in general, can be done either sequentially or in parallel. A sequential construction
builds routes one after another, while a parallel construction builds a number of routes
simultaneously. Figure 7.2(a) shows a typical solution, consisting of 8 locations, that is
constructed sequentially, while Figure 7.2(b) shows a solution with the same number of
locations that is constructed in parallel.
(a) Sequential construction (b) Parallel Construction
Figure 7.2: Solution Construction.
To construct initial solutions for the MV-PDPTW sequentially, researchers usually adap-
ted Solomon’s sequential insertion heuristics of the VRPTW [141]. A weighted sum of
the extra travel distance and total time delay resulting from the insertion is often used
to estimate the cost of the insertion (see Section 3.3.3 for more details about Solomon’s
insertion heuristics). This type of construction was used by [100] for the MV-PDPTW,
and was followed by a solution improvement phase called a tabu-embedded simulated
annealing.
As previously mentioned in Section 3.3.3, a parallel construction heuristic was first intro-
duced in [121] for the VRPTW. In a parallel construction, several routes are initialized
with seed customers and requests are subsequently inserted into any of the initialized
routes. Accordingly, the algorithm needs an initial estimate of the number of vehicles to
be used. Routes are later added as needed if the initial estimate does not yield a feasible
solution. The authors also introduced an additional complex measure in the cost function,
which is called “a generalized regret value”, in order to compare the difference between
the cost of an immediate insertion versus a postponed insertion. Requests with a large
regret value are given priority of insertion. The regret value is a kind of a ‘look-ahead’
measure to estimate the difference between inserting the request in its best route and in-
serting it in its second best route. The measure may be generalized by comparing the](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-144-2048.jpg)
![7.3 Research Motivation and Objectives 121
insertion costs of the best k routes of a particular request, rather than only the best two
routes. In a sense, we choose the insertion (of a request) that we will ‘regret’ most if it
was not performed now. This regret measure was also used by [131] for the MV-PDPTW,
and was embedded within an Adaptive Large Neighbourhood Search (ALNS) technique
to improve the solution quality.
The work in [102] presents a sequential construction algorithm for the MV-PDPTW. The
algorithm repeats a cycle of three components. The first component is a constructor, which
uses a sequential greedy algorithm to add pairs of customers in the order they appear in a
priority sequence that is initially random. The analyzer afterwards analyzes the solution
and assigns a certain ‘blame’ value for each customer based on its contribution to the total
solution cost. Finally, the prioritizer reorders the customers, such that customers with a
high blame value are moved forward in the priority sequence.
A parallel construction heuristic that solves the MV-PDPTW is presented in [104]. The
algorithm starts by finding the largest set of customers, where it is impossible to serve
any two customers with the same vehicle due to constraint violation. These customers
are then used as seed customers in the initial set of routes. To insert the remaining custo-
mers afterwards, the algorithm takes into consideration the effect of insertion on both the
classical increase in distance measure, and also the remaining time window slack in the
route, i.e., priority is given to insertions that do not use much of the available time slack,
allowing for more feasible latter insertions. The authors also use a non-standard measure
of the visual attractiveness of the route to select the most desired insertions, by trying to
minimize the number of crossings (intersection points) between the generated routes.
7.3 Research Motivation and Objectives
We noticed during our literature survey of the MV-PDPTW that researchers who adopt
a 2-phase approach (i.e., construction of an initial solution, followed by an improvement
phase) to solving the problem often pay more attention to the solution improvement phase,
such that the results of the initial solution construction phase are seldom reported, if at all.
This makes it difficult to assess the contribution of the construction method to the success
or the failure of the overall algorithm. It is also important to note that the role of the
construction algorithm is not only limited to the initialization phase. The construction
algorithm is often applied at various stages during the improvement phase to create or
modify new or partial solutions, as done for example in [12] and [116]. Therefore, a good
choice of the construction algorithm is vitally important.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-145-2048.jpg)
![7.4 The Routing Algorithm 123
For the purpose of the MV-PDPTW, we selected as a route improvement heuristic, the
simple Hill Climbing (HC) algorithm which tries to gradually modify the current route
until no further improvement is possible. In fact, we chose the HC algorithm to improve
individual routes, because it was very fast compared to the other heuristics investigated for
the SV-PDPTW, and it also gave good quality results. As such, it seemed more suitable
for repeated application within the various construction algorithm, as will be explained
in Section 7.5. Also, only one stage of time window improvement was applied here to
accelerate the generation of multiple routes. The neighbourhood move used by the HC
algorithm in this part of our research depends only on the upper bound of the time window
interval to decide the swapping of locations. Algorithm 7.1 describes this simple heuristic.
Algorithm 7.1: The HC Routing Algorithm.
1: Given a route r
2: repeat
3: for (Each possible pair of locations in r) do
4: if (The latter location is more urgent in its upper time window bound) then
5: Swap the current two locations in r to get a new route r′
6: ∆ ← cost(r′) − cost(r)
7: if (∆ < 0) then
8: r ← r′
9: until (Done){Stop if no improvement has been achieved in the previous pass}
The cost function to evaluate the quality of a route r, in Step 6 of the HC algorithm, is
described by the following equation:
F(r) = w1 × D(r) + w2 × TWV (r) + w3 × CV (r) , (7.1)
where D(r) is the total route duration, including the waiting time and the service time
at each location. TWV (r) is the total number of time window violations in the route,
and CV (r) is the total number of capacity violations. The constants w1, w2, and w3 are
weights in the range [0,1], and w1 + w2 + w3 = 1.0. As previously mentioned, the largest
penalty should be imposed on the time window violations, in order to direct the search
towards more feasible routes. We used the following weights for the route cost function:
w1 = 0.201, w2 = 0.7 and w3 = 0.099 2
.
2
Comparing the current weights to those assigned in Equation 5.1, the penalty on the capacity violations
was reduced, while the weight assigned to the total route duration was increased. This is due to the fact
that routes in the multiple vehicle case are usually shorter than routes in the single vehicle case, and the
satisfaction of the capacity constraint tends to be easier as a result.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-147-2048.jpg)
![7.5 Solution Construction Heuristics 125
Algorithm 7.2: The Sequential Construction.
1: Let M ← 0 {M is the number of vehicles used}
2: repeat
3: Initialize an empty route r
4: M = M + 1
5: for (All unassigned requests) do
6: Get the next unassigned request i
7: Insert request i at the end of the current route r
8: Call the HC routing heuristic (Algorithm 7.1) to improve r
9: if (r is a feasible route) then
10: Mark i as inserted
11: else
12: Remove i from r
13: until (All requests have been inserted)
7.5.2 The Parallel Construction Algorithms
As mentioned previously, for a parallel construction, several routes are considered simul-
taneously for inserting a new request, and an initial estimate of the number of vehicles
is required. Potvin and Rousseau in their parallel construction algorithm for solving the
VRPTW [121], estimate the initial number of vehicles by first running Solomon’s sequen-
tial construction [141]. The number of vehicles in the resulting solution is then used as an
estimate of the initial number of vehicles for the parallel heuristic.
In our research, we adapted the parallel construction heuristic of the VRPTW in [121]
to the MV-PDPTW. However, to avoid extra processing time, we estimated the initial
number of vehicles using a simple formula that divides the total demand of the pickup
requests in the problem instance by the capacity of the vehicle, as shown in Equation 7.2.
M = ⌊(
i∈N+
qi)/C⌋ , (7.2)
where M is the estimated initial number of vehicles, N+
is the set of pickup customers, qi
is the demand (load) of a pickup request, and C is the capacity of the vehicle. However,
this estimate seems to be more suitable for instances with a critical schedule horizon, i.e.,
those having a short time window width. Instances with more flexible (long) time window
intervals, on the other hand, may require fewer vehicles to start with. More specifically,
when the width of the time window interval is short, there will be a limited number of
alternative feasible locations for scheduling the request. Accordingly, instances with such](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-149-2048.jpg)
![126 7.5 Solution Construction Heuristics
a property usually need a large number of vehicles to be able to serve all requests without
any constraint violation, as opposed to instances having a long time window width. Thus,
to be able to handle the different types of test cases, we introduced a small modification
to this formula for some problem instances, as will be explained in Section 7.6.
Similar to the parallel approach for the VRPTW in [121], which initializes each route with
a seed customer, our parallel algorithms initialize each route with a seed request (pickup
and delivery pair) from the sorted list of requests. We then take the remaining requests
in order and attempt to insert the next request in one of the partial routes created. If the
next request cannot be feasibly inserted in any of the already created routes, a new route is
added to accommodate this request. This process is repeated until all requests have been
inserted.
As previously explained, traditional parallel construction algorithms for both the VRPTW
and the PDPTW, usually select the customer who has the current minimum insertion cost
among all remaining un-routed customers to be inserted next. This cost is often a measure
of the extra travel time and distance, which would result from inserting the customer in the
best possible (feasible and minimum cost) insertion position found in all available routes.
Our parallel algorithms, on the other hand, differ among each other in how they select
the next request to be inserted, and also in selecting the route in which this request will
be inserted. Following is an explanation of the different parallel construction algorithms
proposed in our research.
Parallel Construction - First Route:
In our first parallel construction algorithm, the next request in order is inserted in the first
route in which a feasible insertion of this request is found, i.e., no attempt is made to find
the best route for the current request. Thus, our first parallel construction uses a fast first
acceptance criterion for insertion. Algorithm 7.3 describes this procedure.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-150-2048.jpg)
![7.5 Solution Construction Heuristics 127
Algorithm 7.3: Parallel Construction: First Route.
1: Calculate M (the initial estimate of the number of vehicles)
2: Initialize M routes with seed customer pairs from the sorted list of customers
3: for (All remaining unassigned requests) do
4: Get the next unassigned request i
5: r = 0 {start with the first route}
6: while ((r < M) and (i not yet inserted) ) do
7: Insert the request i at the end of the current route r
8: Call the HC routing heuristic (Algorithm 7.1) to improve r
9: if (r is a feasible route) then
10: Mark i as inserted
11: else
12: Remove i from r
13: r = r + 1
14: if (i was not inserted) then
15: Initialize a new route r′
16: M = M + 1 {increase the number of vehicles}
17: Insert the request i in the new route r′
18: Mark i as inserted
Parallel Construction - Best Route:
In our second parallel construction algorithm, the next request in order is inserted in the
best route in which a feasible insertion of this request is found. The best route for each
request is the route that causes the least increase in the overall cost of the solution (the
routing schedule) due to the insertion process. To calculate the overall cost of the solution,
we used an objective function that is suggested by Bent and Hentenryck in [12]. The
objective function consists of three components: the first component tries to minimize
the number of vehicles used in the solution, the second component tries to minimize the
total distance traveled, while the third component is a measure that tries to maximize the
square of the number of nodes visited by each vehicle. This last component is intended
to favour routes that are rather full and those that are rather empty, as opposed to an even
distribution of nodes among routes. The idea is to try to get rid of some vehicles that are
under-utilized during subsequent route improvement phases. The objective function of a
solution S is described by Equation 7.3.
O(S) = α × M + β ×
r∈S
Dist(r) − γ ×
r∈S
|r|2
, (7.3)](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-151-2048.jpg)
![128 7.5 Solution Construction Heuristics
where M is the number of vehicles used in the current solution, Dist(r) is the total dis-
tance traveled by each vehicle, and |r| is the number of nodes visited by each vehicle. The
constants α, β, and γ are weights in the range [0, 1] assigned to each term in the objective
function, and α + β + γ = 1.0 . In our research we try to minimizing the number of ve-
hicles as our primary objective followed by the total distance, thus we chose α > β > γ.
We used the following weight values for Equation 7.3: α = 0.7, β = 0.29 and γ = 0.01.
Algorithm 7.4 describes the second parallel construction algorithm.
It is important to note, in Step 10 of Algorithm 7.4, that since the insertion process only
affects one route, the calculation of the new solution cost does not require evaluating all
routes in the current solution. The calculation is simply done by removing the old cost
of the current route (before the insertion), and adding the new cost resulting from the
insertion.
Algorithm 7.4: Parallel Construction: Best Route.
1: Calculate M (the initial estimate of the number of vehicles)
2: Initialize M routes with seed customer pairs from the sorted list of customers
3: for (All remaining unassigned requests) do
4: Initialize LocalMin to an arbitrary large value
5: for (r = 0; r < M; r + +) do
6: Get the next unassigned request i
7: Insert the request i at the end of the current route r
8: Call the HC routing heuristic (Algorithm 7.1) to improve r
9: if (r is a feasible route) then
10: calculate ∆cost {∆cost is the change in solution cost due to the insertion (where cost
is estimated using Equation 7.3) }
11: if (∆cost < LocalMin) then
12: LocalMin = ∆cost
13: r∗ = r { r∗ is the current best vehicle for request i}
14: Remove i from r {temporarily remove i until the insertion cost of the current request in
all routes is calculated}
15: if (r∗ is found) then
16: Insert i in r∗
17: Mark i as inserted
18: else
19: Initialize a new route r′ {Since no feasible insertion is found for i in any of the available
routes, allocate a new route}
20: M = M + 1 {increase the number of vehicles}
21: Insert the request i in the new route r′
22: Mark i as inserted](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-152-2048.jpg)
![130 7.6 Computational Experimentation
7.6 Computational Experimentation
7.6.1 Characteristics of the Data Set
To test our algorithms, we used several instances from the benchmark data of the MV-
PDPTW created by Li and Lim in [100]. The authors of [100] created this data set based
on Solomon’s test cases of the VRPTW in [141]. There are 6 different categories of
problem instances in this data set: LR1, LR2, LC1, LC2, LRC1, and LRC2. Problems
in the LR category have randomly distributed customers, problems in the LC category
have clustered customers, and problems in the LRC category have partially random and
partially clustered customers. On the other hand, problems identified with the number
‘1’ have a short scheduling horizon (tight time window width), while problems iden-
tified with the number ‘2’ have a long scheduling horizon (large time window width).
Each category has 6 different problem sizes: 100, 200, 400, 600, 800, and 1000 cus-
tomers4
. There are between 56-60 files from each problem size. The total number of
files in the data set is 354. The data and the best known results can be downloaded from
http://www.sintef.no/Projectweb/TOP/Problems/PDPTW/Li--Lim-benchmark/.
For the purpose of testing our algorithms we selected the first 6 files from each category
for each problem size. The total number of files used to test our algorithms is 216. The
files used for testing our algorithms are summarized in Table 7.1.
As mentioned in Section 7.5.2, we used a simple formula (Equation 7.2) to estimate the
initial number of vehicles needed for the parallel construction heuristics. However, during
our preliminary experimentation, we found that this estimate does not suit the different
types of problem instances. Apparently, problems that have a long schedule horizon al-
low for a more flexible visiting schedule, and generally require fewer vehicles. We also
found during our experimentation that an underestimate of the initial number of vehicles
is usually preferred to an overestimate, since reducing the total number of vehicles used is
our primary concern. As a result, to estimate the initial number of vehicles for problems
with a long schedule horizon (problems of category ‘2’) we reduced our initial estimate
by 50%. Thus, Equation 7.4 was used instead of Equation 7.2.
M = ⌊(1/2)(
i∈N+
qi)/C⌋ . (7.4)
However, this estimate is to some extent arbitrary and remains under consideration for
future reassessment.
4
The original data set in [100] contained only 56 100-customers problems. Larger problem sizes were
later added to the original data set.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-154-2048.jpg)
![7.7 The SEQ Algorithm: Complexity Analysis and Implementation Issues 137
problem size in each benchmark category. It appears in this table that problems involving
random customers with a long schedule horizon, LR2 and LRC2, generally require a
longer processing time than the other problem categories, which sustains our previous
observation regarding the large solution space for these problems.
Table 7.5: Average Processing Time (seconds) of the SEQ Algorithm for all Tasks.
Category
Problem Size
100 200 400 600 800 1000
LR1 0.02 0.06 0.22 0.53 0.95 1.54
LC1 0.02 0.05 0.22 0.48 0.92 1.43
LRC1 0.02 0.06 0.21 0.50 0.93 1.44
LR2 0.03 0.13 0.43 1.04 1.71 2.72
LC2 0.03 0.06 0.39 0.61 1.06 1.57
LRC2 0.02 0.12 0.47 1.12 1.84 2.53
7.7 The SEQ Algorithm: Complexity Analysis and Im-
plementation Issues
Before we conclude this part of our research, we present in this section some remarks
concerning the complexity and feasibility checking of the SEQ algorithm in relation to the
common construction methods. Analyzing our SEQ algorithm we find that: the routing
heuristic in Algorithm 7.1 needs O(n2
) time for accessing each pair of locations in the
route, where n is the number of requests in the problem instance. Also, the cost function
of Equation 7.1, which checks the feasibility of the whole route as well, needs O(n) time.
The SEQ algorithm (Algorithm 7.2) needs O(n) since its major iteration processes all
requests in order. This will make the run-time complexity of the whole algorithm O(n4
).
For the sake of comparison, Algorithm 7.6 describes a basic construction method for
vehicle routing problems in general. The algorithm appears in [24].](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-161-2048.jpg)
![138 7.7 The SEQ Algorithm: Complexity Analysis and Implementation Issues
Algorithm 7.6: Basic Construction Algorithm for the VRP [24].
1: N = set of unassigned customers
2: R = set of routes {initially contains one route}
3: while N = ∅ do
4: p∗ = −∞
5: for j ∈ N do
6: for r ∈ R do
7: for (i − 1, i) ∈ r do
8: if (Feasible(i, j) and Profit(i, j) > p∗) then
9: r∗ = r
10: i∗ = i
11: j∗ = j
12: p∗ = Profit(i, j)
13: Insert(i∗, j∗) {insert j∗ between (i∗ − 1) and i∗}
14: N = N j∗
15: Update(r∗)
According to [24], Algorithm 7.6 is of O(n3
), provided that the feasibility and profitability
can be performed in constant time. Feasibility makes sure that the current position
adheres to all problem constraints, while Profitability is usually measured as a weighted
combination of extra travel distance and time delay resulting from the insertion. When
it is more profitable to insert a customer in a new route, a new route will be allocated.
As explained in [24], a special algorithm can be applied to reduce the TW feasibility
test of the VRPTW from a linear time to a constant time. The same algorithm can also
be applied to the PDPTW. Nevertheless, this algorithm requires that extra information is
kept for each customer already existing in the route. In general, two quantities have to be
maintained: the earliest and the latest possible times the service can take place, relative
to the customer’s current location in the route. This information is not fixed and subject
to change after each insertion, which accounts for the existence of an update function to
maintain the desired quantities (Step 15 of Algorithm 7.6).
Unlike the VRPTW, however, checking the capacity feasibility for the PDPTW can only
be done in linear time [86], due to the presence of two different types of customer services
in the route. As a result, the basic construction algorithm when applied to the PDPTW also
results in O(n4
) complexity. The algorithm can also include a selection of seed customers
for route initialization, which usually does not change the complexity of the algorithm.
As mentioned above, it is possible to reduce the TW feasibility check for the PDPTW
from a linear time to a constant time, by maintaining and frequently updating extra route](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-162-2048.jpg)
![7.8 Summary and Future Work 139
information, as done for example in [37]. However, since our route cost function (Equa-
tion 7.1) tests the feasibility of both the TW and the capacity concurrently, it would be
redundant to calculate and store additional service timing information to accelerate the
TW feasibility test, since an O(n) testing would still be needed for the capacity feasibi-
lity.
In addition, since our SEQ algorithm accepts any feasible insertion, it does not have to
check the feasibility nor estimate the profitability of each and every possible insertion
position, as done in Algorithm 7.6. In our algorithm, the cost of the route as a whole will
be calculated, if at all, only if the route has been changed. This is due to the restriction
imposed by the TW condition in Step 4 of Algorithm 7.1.
Finally, during the insertion process, i.e., Step 7 of the SEQ algorithm (Algorithm 7.2),
two locations (a pickup and delivery pair) are simultaneously inserted, then Algorithm 7.1
handles the feasibility checking and the improvement of the underlying route altogether.
Besides overcoming the precedence and the coupling issues, this insertion has the added
advantage of accelerating the solution construction process, since only half the number of
locations is processed in the main iteration of Algorithm 7.2.
On the other hand, the parallel construction algorithms implemented in this research seem
to be one order of magnitude higher than the SEQ algorithm, due to the presence of an
extra loop that passes through all available vehicles, although the number of vehicles is
always less than n (the number of nodes in the data set).
7.8 Summary and Future Work
In this research we investigated several initial solution construction heuristics for the MV-
PDPTW, aiming to identify the best heuristic that can be used as part of a comprehensive
solution methodology. In our opinion, existing approaches in the literature often overlook
and perhaps underestimate this vital component of the overall solution algorithm.
The experimental results on a large number of benchmark instances indicate that the se-
quential construction heuristic (SEQ) seems to be the most favourable solution construc-
tion method, which can be by easily embedded in a heuristic or a meta-heuristic technique
to reach final good quality solutions. With just a few simple lines of code, and without
a pre-determined number of vehicles or a solution evaluation mechanism, this algorithm
produced good quality results, that are sometimes even better than the results obtained by
the most sophisticated parallel algorithm tested in our research (the PBQ algorithm). The
SEQ algorithm also had an impressive speed, with a processing time that is at most 6% of](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-163-2048.jpg)
![141
Chapter 8
Two Approaches for Solving the
Multiple Vehicle Pickup and Delivery
Problem with Time Windows: A
Genetic Algorithm and a Simulated
Annealing
In this chapter we continue our investigation of the MV-PDPTW, introduced in Chapter
7. This part of our research will augment the solution construction heuristic developed in
the previous chapter with an improvement method, using both a Genetic Algorithm (GA)
approach and a Simulated Annealing (SA) heuristic.
This chapter is organized as follows: Section 8.1 highlights the motivation behind the re-
search carried out in this part of the thesis. Section 8.2 provides a brief summary of some
related work from the literature. Section 8.3 discusses genetic algorithms as a solution
approach form the perspective of the MV-PDPTW, and introduces our specific GA sug-
gested for the improvement phase of this problem. Within this section, we describe the
solution representation and the objective function used in Section 8.3.1, how the initial
population is created in Section 8.3.2, the genetic operators in Section 8.3.3, the overall
GA in Section 8.3.4, some attempts to improve the results of the GA in Section 8.3.5, and
finally Section 8.3.6 reports the experimental results of the GA when tested on published
benchmark data and compared with previous GA attempts from the literature. After that,
Section 8.4 explains the SA approach adopted for solving the problem, together with its
experimental results compared to the results of the GA approach. Finally, Section 8.5
gives some concluding remarks and possible future work. Our findings, related to the GA
approach, in this part of our research were published in the MIC2009 conference [80].](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-165-2048.jpg)
![142 8.2 Related Work
8.1 Research Motivation
As previously mentioned in Chapter 2, Genetic Algorithms (GAs) are intelligent search
methods that have been successfully used for solving many hard combinatorial optimiza-
tion problems. In the first part of our research, though, the 3-stage SA approach performed
better than the GA for the SV-PDPTW, especially in terms of processing time. Despite
this, using a GA for the improvement phase of the MV-PDPTW still appeared to be pro-
mising for several reasons:
Firstly, GAs have been used for solving the related VRPTW, producing good results in
many cases, for example [144], [120], [16] and [15]. Secondly, the parallel nature of
GAs and its population based mechanism can probably make it more appropriate than
other meta-heuristic approaches (that focus on improving only one solution), for solving
very hard optimization problems. The multiple-vehicle case of the PDPTW is in fact
considerably harder than the single vehicle case [137], which makes applying GAs an
attractive option for solving this problem. In addition, our simple sequential construction
algorithm (SEQ), developed in Chapter 7, seems to be most appropriate for population-
based meta-heuristics, due to its simplicity, speed and its potential for creating a diverse
population by randomizing the initial order of requests.
Finally, as previously mentioned in the introduction to this thesis (Chapter 1), the main
focus of our research is on developing appropriate representations and neighbourhood
moves that can help guide the search towards good quality solutions and manage infea-
sibility throughout the search. If the appropriate techniques are designed, they should be
widely applicable, and may be used within other heuristics or meta-heuristics, for solving
the underlying problem. Accordingly, we tried to follow the same approach used in our
research for solving the SV-PDPTW, by first developing the representation and neigh-
bourhood moves (represented as genetic operators here) that we believe can handle the
difficult problem constraints. These operators are first tried within a GA approach for sol-
ving the MV-PDPTW, but they were also adapted and employed within an SA approach,
as previously done for the single vehicle case.
8.2 Related Work
When dealing with the MV-PDPTW, some researchers use exact methods to solve the
problem to optimality, but these are limited to small size problems. For example, the
work presented in [46] is an optimization techniques which formulates the problem as a
set partitioning problem, and then employs a column generation method to solve a linear](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-166-2048.jpg)
![8.2 Related Work 143
relaxation of this problem to optimality. Problem sizes of up to 55 customers and 22 ve-
hicles were solved using this approach. Also, the work in [130] presents a new mixed
integer programming formulation for both the MV-PDPTW and the dial-a-ride problem,
and the formulations are solved using two branch-and-cut algorithms. New problem ins-
tances have been created, and the results of the algorithms were compared with upper
bound solutions obtained by applying the heuristic in [131] to the same problem instances.
Problem sizes up to 194 nodes were solved to optimality.
Other approaches adopt approximation techniques to deal with large size problems. As ex-
plained in the previous chapters, the solution process, in this case, often starts by construc-
ting one or more initial solutions to the problem, usually using techniques adopted from
the VRPTW construction algorithms, and then these solutions are improved using heuris-
tics or meta-heuristics. In general, Simulated Annealing (SA) and Tabu Search (TS) have
been the most popular approaches for solving the MV-PDPTW.
The work in [112] is one of the first attempts to solve the MV-PDPTW. The technique is
based on a reactive tabu search which allows the tuning of the search parameters, such as a
short-term memory length, based on an assessment of visited solutions during the search.
The algorithm also tries to detect and escape possible local optima. The solution repre-
sentation is a vector that includes customer nodes separated by vehicle nodes to which
customers are assigned. The objective function consists of three components: the total
schedule duration, the number of capacity violations, and the number of time window
violations. Thus, during the search process, overloads and tardiness in the solutions are
treated as soft constraints, while precedence and coupling constraints are strictly enfor-
ced. To construct an initial feasible solution, a predecessor-successor pair (PS) is inserted
in the best feasible insertion position in the current vehicle’s route. If no feasible insertion
can be found, a new vehicle route is added to the solution. Three neighbourhood moves
are used in this algorithm. The first move is the “Single Paired Insertion” (SPI), which
tries to move all predecessor nodes to better feasible locations in other routes. Successor
nodes are inserted after their predecessors in the best possible positions, although vehicle
capacity or time windows may still be violated. The second move is the “Swapping Pairs
Between Routes” (SBR), which first tries to exchange a pair of predecessor nodes bet-
ween two different vehicles, and then exchanges their successor nodes. The last move is
the “Within Route Insertion” (WRI), which tries to reorder nodes in the same route to re-
duce infeasibility in the solution and improve its quality. The sequence of neighbourhood
moves is selected dynamically during the search, based on a calculated relationship bet-
ween the average time window length (the tightness of the time window) and the average
route duration.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-167-2048.jpg)
![144 8.2 Related Work
The authors in [100] present a tabu-embedded simulated annealing approach to solve the
MV-PDPTW. To construct an initial feasible solution, they modified Solomon’s Insertion
heuristic [141] by initializing each route with a pickup and delivery (PD) pair that satisfies
a set of criteria, based on combined time window intervals and distance from the depot.
Each un-routed PD pair is then inserted into the partial route in the best possible feasible
and minimal cost insertion position. The objective function tries to minimize, in priority
order, the number of vehicles used, the total travel distance, the total service duration,
and the drivers’ total waiting time. To create a neighbouring solution, three local search
methods are presented. The first is a “PD-Shift” operator. In this operator, a PD pair is
first removed from route 1 and then it is inserted in route 2. After that, another PD pair
is removed from route 2 to be inserted in route 1. The second operator is a “PD-Swap”
operator, which simultaneously removes a PD pair from each route, and then reinserts
each pair in the other route. The third is a “PD-Rearrange” operator which first removes
and then reinserts a PD pair in the same route. In all three moves, only feasible insertions
are allowed. For a more thorough neighbourhood search, they extend their local search
method to a descent local search (DLS), which tries to improve the current solution for
a number of iterations. When no further improvement is possible, the DLS algorithm
returns the best current solution. The main meta-heuristic algorithm is a tabu-embedded
simulated annealing procedure with K-restarts, i.e., the algorithm stops when the number
of iterations without improvement reaches a pre-defined value K. To prevent cycling, their
simulated annealing procedure records the accepted solutions in a tabu list. The authors
generated test data for the problem based on Solomon’s test cases for the VRPTW [141].
The authors consider their approach as the first efficient attempt to solve practical size
MV-PDPTW instances. Their generated data set also became the standard benchmark test
data for the MV-PDPTW.
The work in [98] presents a two-phase method to solve the MV-PDPTW. In the first phase
the algorithm creates an initial feasible solution using a strategy that combines the benefits
of Solomon’s classical insertion heuristic [141], and a sweep heuristic, first suggested in
[58], which was particularly adapted to fit the MV-PDPTW. The authors call their insertion
heuristic “Partitioned Insertion Heuristic”. The objective functions is based on the number
of vehicles used and the total travel distance. Their local search is based on three moves,
similar to the moves described in [112], as explained above. The second phase of the
algorithm is a tabu search that tries to improve the current initial solution using a set of
composite neighbourhood moves.
In a relatively recent work [12], the authors present a two-stage algorithm to deal with the
MV-PDPTW. The first stage uses a simple SA approach, whose focus is to minimize the
number of vehicles used in the solution. The second stage uses a Large Neighbourhood](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-168-2048.jpg)
![8.2 Related Work 145
Search (LNS) strategy, first suggested in [139], to minimize the total travel cost of the
entire solution. The primary objective function (used in the LNS stage) minimizes the
number of vehicles first, then the total travel cost. However, according to the authors,
using this objective function is more effective in reducing the total travel cost than mini-
mizing the number of routes in the solution. Therefore, they added an SA phase to boost
the performance of the overall algorithm by reducing the number of routes in the solution
first. To this end, they used a different objective function in the SA stage than the one
used by the LNS. The SA objective function included three components: the first com-
ponent tries to minimize the number of routes. The second component tries to maximize
the number of routes that are rather full and those that are rather empty, with a view to eli-
minating routes with few customers during the search. Finally, the third component of the
SA evaluation function reduces the total travel cost1
. The neighbourhood move adopted
in their SA is a simple pickup and delivery pair relocation operator, which was also used
by previous researchers (e.g. [98], [100] and [112]). The LNS of their algorithm is ac-
tually a sequence of local searches, where a group of customers are selected for relocation
based on a certain relatedness measure between pickup customers. The neighbourhood
generated after the relocation of customers is then exhaustively explored, using a branch
and bound algorithm, to find its best solution. The best solution found replaces the current
solution, if it has a better objective value. The algorithm was tested on benchmark data
obtained from [100], where the results show the effectiveness of the approach since it was
able to produce many new best solutions for instances with 100, 200, and 600 customers.
An interesting solution methodology is presented in [131], where a variant of the MV-
PDPTW is considered. It is assumed here that the starting and ending depots of a vehicle’s
journey need not be the same, and the depots could be different for different vehicles. It
is also assumed that the number of vehicles is limited, so it may not be possible to serve
all requests, in which case the unserved requests are placed in a request bank. Thus, mi-
nimizing the number of unserved requests in the request bank is one of the objectives of
the algorithm, added to minimizing the number of vehicles used and the total travel dis-
tance. It is also assumed that there could be some special requests that cannot be served
by some vehicles, for example if transferring the request requires a special characteris-
tic that is not available in some vehicles (e.g. transferring frozen food or some medical
samples). The basic solution methodology in this work is a Large Neighbourhood Search
(LNS) technique, used in both [139] and [12], where a number of requests are first remo-
ved then re-inserted into the solution. Unlike previous researchers, however, the authors
apply a number of different heuristics for both the removal and the insertion methods.
1
Recall that this 3-component evaluation function is the same as the cost function used by our PBR and
PBQ construction heuristics (Equation 7.3), as explained in Section 7.5.2](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-169-2048.jpg)
![146 8.2 Related Work
They present three different removal heuristics. First, the “Shaw Removal” removes re-
quests according to a certain relatedness measure, which is based on differences between
distances, arrival times, loads and the numbers of vehicles that can serve these requests.
Second, the “Random Removal” heuristic removes randomly selected requests. Finally,
the “Worst Removal” heuristic removes requests with high costs, i.e., requests that appear
to be placed in unfavourable positions in the solution. They also present two different
insertion techniques to insert the removed requests in the partial routes. The basic greedy
heuristic inserts the request with minimum insertion cost in the best possible insertion
positions. On the other hand, the regret insertion heuristic gives priority for insertion
to requests whose insertion would seem to be more costly if it was delayed2
. To select
which heuristic to use, the authors assign weights to the different heuristics and use a
roulette wheel selection method. These weights are adaptively adjusted during the search
based on the performance of the different heuristics employed. Accordingly, the authors
call their algorithm an Adaptive Large Neighbourhood Search (ALNS), as opposed to the
basic LNS search method which only applies one removal and one insertion heuristic.
During the search, a newly generated solution replaces the current solution using an SA
acceptance criterion. Also, similar to [12], the authors first apply a preliminary stage to
minimize the number of vehicles used in the solution before applying the basic ALNS.
During this stage, they create an initial solution using a sequential insertion heuristic.
They then try to remove one or more routs from this solution, using the LNS algorithm,
until the minimum possible number of routes is reached. The algorithm was tested on
the benchmark data of [100]. In addition, new problem instances were created, which
takes into account their modified variant of the MV-PDPTW. When compared with the
LNS, which uses only one removal and one insertion heuristic, the ALNS seems to be
superior. In addition, their ALNS was able to outperform previous heuristics applied to
the MV-PDPTW. To the best of our knowledge, the LNS technique of [12] and the ALNS
of [131] are the current state-of-the-art, since they have both produced best known results
for many benchmark problem instances.
Besides SA and Tabu Search, GAs have been applied by some researchers for solving
the MV-PDPTW and the related dial-a-ride problem. In the early GA approach by [92],
the chromosome representation is based on assigning a four digit code to each location.
The first digit of this code represents the number of the vehicle to which the location is
assigned. Thus, both the pickup location and its associated delivery will be assigned the
same first digit. The other three digits are used to sort locations according to the visiting
order of the vehicle, and a pickup location always has a sorting code that is less than its
corresponding delivery. The crossover operator works by generating two crossover points,
2
See Section 7.2 for more details about the regret measure of the parallel construction heuristics.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-170-2048.jpg)
![8.2 Related Work 147
Figure 8.1: A chromosome in a GA that handles both grouping and routing for the
MV-PDPTW.
and swapping the resulting segments between the two parents. The mutation operator,
simply changes the first digit of a pickup and delivery pair to another digit, i.e., assigns the
request to another vehicle. Another operator called vehicle merging is also used to reduce
the number of vehicles. The algorithm was tested on 24 randomly generated problems
with numbers of requests ranging from 5 to 30.
The authors in [33] introduce an attempt to handle the MV-PDPTW using an evolutionary
algorithm, for both the grouping and the routing aspects of the problem. The solution
representation is a list of vehicles routes, where each route consists of a sequence of
pickup and delivery locations, and all problem constraints are enforced in the solutions
throughout the search. Figure 8.1 shows how a typical chromosome may look in their
suggested evolutionary approach. The objective function tries to minimize the number of
vehicles in the solution, the total travel distance and the total travel time. Two crossover
operators are tried, one exchanges fragments of routes between parents, while the other
exchanges complete routes. Infeasible solutions that may be created following crossover
are repaired by removing repeated requests or adding requests that are not served. If this
is not possible, the offspring is discarded. Two mutation operators are presented, the first
tries to reduce the number of vehicles in a solution by selecting a route and moving all
its requests to other routes in the same solution. The second mutation tries to improve a
route by rearranging its requests.
The work in [116] presents what seems to be a first attempt to apply a Grouping Genetic
Algorithm (GGA) to the MV-PDPTW. In a GGA, the genetic representation is based on
a set of genes, where each gene represents a group of objects rather than a single object.
For the MV-PDPTW, it is assumed that each gene represents a group of requests that are
assigned to one vehicle. Thus an individual solution only covers the grouping aspect of
the problem. The routing aspect, on the other hand, is handled by an independent data](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-171-2048.jpg)
![148 8.2 Related Work
structure associated with each gene. Figure 8.2 is a visual representation of how a chro-
mosome in their GGA approach may look. The objective function is to minimize the total
travel distance, irrespective of the number of vehicles used. The crossover is an adapta-
tion of the original crossover mechanism for the GGA, presented in [49], where clusters
(vehicles with their assigned requests) are removed from one parent and inserted into the
other parent. This is then followed by a chromosome clean up, to remove duplicate ve-
hicles and the repeated assignment of requests, in addition to re-assigning requests that are
no longer assigned. The mutation operator removes a cluster from a chromosome, and re-
assigns its requests to other clusters, creating additional clusters if necessary to maintain
feasibility. The embedded insertion heuristic applied in several stages of this algorithm
is based on inserting a request in the best feasible and minimum cost position, among all
possible insertion positions in the chromosome 3
. In a recent work [126], another GGA
was applied to the Handicapped Person Transportation problem (HPT), which focuses on
minimizing clients inconvenience.
Figure 8.2: A chromosome in a GA that handles only grouping for the MV-PDPTW.
The authors in [91] deal with the dial-a-ride problem using a cluster-first route-second
approach. The clustering is handled using a GA, and the routing is handled using a space-
time nearest neighbour heuristic. In the clustering phase, the chromosome representation
is a two-dimensional array where rows represent routes and columns represent customers.
A cell values of 1 or 0 indicates whether or not a customer is assigned to a particular
route. Crossover is performed by selecting two random routes from parent solutions and
creating a new child route using a traditional uniform crossover. The remaining routes
in the child are inherited from the first parent without change. The newly created child
may be infeasible, though. Accordingly, a repair method is followed to assign unassigned
customers or remove duplicate assignments of customers. The mutation operator just
3
More details about [33] and [116] will be given in Section 8.3.6 when our algorithm is compared with
their approach.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-172-2048.jpg)
![8.2 Related Work 149
moves a random customer to another cluster (route). On the other hand, the work in [34]
tries to use a GA for the whole dial-a-ride problem, i.e., for both the grouping and the
routing phases. In their representation a gene consists of a vehicle-passenger pair. They
used a one-point crossover and a Partial Match Crossover (PMX). For mutation they use
a bit-level mutation and a 2-Opt operator.
A recent paper [37] presents an indirect search method for the MV-PDPTW. The idea
is to separate the meta-heuristic search strategy from the feasibility checking routines,
to simplify and accelerate the optimization process. This technique facilitates the appli-
cation of simple problem-independent moves during the meta-heuristic search, while a
problem-specific greedy decoder handles the construction of the corresponding feasible
solution using the information (encoding) given from the meta-heuristic search engine.
For example, in the PDPTW, the encoding is a permutation of transportation requests,
which determines the sequence in which these requests will be scheduled. Given a certain
permutation, the greedy decoder creates a feasible solution (schedule) using a sequential
construction algorithm and a cheapest insertion rule. To create a neighbouring solution
during the meta-heuristic search, a simple 2-exchange move is applied to the current per-
mutation and the new permutation is supplied again to the greedy decoder to create the
corresponding new schedule. A comparison of the indirect search technique with two
special-purpose algorithms for the MV-PDPTW shows that the technique is competitive
in both solution quality and speed. This may indicate the potential of applying the tech-
nique to other rich combinatorial optimization problems.
The above literature summary shows that the MV-PDPTW is in fact a hard problem, for
which most solution methods tend to be rather complex and hard to explain and repli-
cate. Developing an appropriate solution technique is indeed a challenge for researchers.
An intelligent solution methodology should be able to handle both the grouping and the
routing aspects of the problem efficiently. In addition, in both the construction and the im-
provement phases of the problem, the researcher is faced with many decisions that should
be made. These decisions include, among others, the components of the objective func-
tion, the permissibility of infeasible solutions, how to adhere with the constraints of the
problem, how to generate new solutions, and what acceptance criteria should be applied
to replace the current solution during the improvement phase. It is thus often difficult to
discover good quality solutions that do not violate any problem constraint, in a reasonable
processing time.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-173-2048.jpg)
![150 8.3 A Genetic Algorithm (GA) for the MV-PDPTW
8.3 A Genetic Algorithm (GA) for the MV-PDPTW
Although Genetic Algorithms (GAs) have been successfully used for solving many rou-
ting and scheduling problems, research using GAs for solving the MV-PDPTW is gene-
rally scarce. In addition, the results reported by most GA techniques attempted are often
disappointing in some respects. As previously explained, the MV-PDPTW consists of two
related problems: the grouping or the clustering problem tries to find the best allocation of
requests to vehicles, while the routing problem is concerned with finding the best feasible
route for each vehicle, given the requests assigned to it. When trying to solve this problem
using a GA, it is often hard to tackle these two aspects simultaneously. Moreover, a major
issue is finding a suitable genetic encoding and designing intelligent genetic operators that
are capable of handling all the difficult problem constraints and may encourage the forma-
tion of meaningful genetic building blocks, which may help in generating individuals of
better fitness [116]4
. In the MV-PDPTW, the genetic operators should be smart enough to
transfer the favourable genetic traits from parent solutions to their offspring, while trying
to avoid the frequent generation and evaluation of infeasible problem solutions. As pre-
viously mentioned, infeasible solutions are handled in many solution algorithms using
a repair method to fix infeasibility during the search, which will inevitably increase the
processing time and complicate the algorithm.
Most previous GA research, for example [116] for the MV-PDPTW and [91] for the dial-
a-ride, tried to tackle the difficulties encountered in the GA encoding and operators by
allowing the GA to handle only the grouping aspect. The routing aspect, on the other
hand, was handled by an independent routing algorithm that is hidden from the GA and
is called when a chromosome is decoded. The genetic operators in this case are usually
general-purpose and do not apply any problem-specific knowledge. Attempts to use a GA
for both the grouping and the routing aspects, for example [33] for the MV-PDPTW and
[34] for the dial-a-ride problem, generally produced discouraging results.
We present in this part of our research our experimentation with a new GA for solving
the MV-PDPTW. Our GA tries to face the challenge of handling both the routing and the
grouping aspects of the problem simultaneously. Unlike the most popular approaches in
the literature, in which the GA is only aware of how requests are clustered, but is not
aware of how they are routed, our chromosome representation more naturally accommo-
dates each group of requests together with suggested routes. By explicitly monitoring
and manipulating all the solution information, we aim to preserve the distinctive cha-
racteristic of GAs in identifying the desirable genetic material and transferring it from
4
For details about the “GA schema theorem” and “the building block hypothesis”, the reader is referred
to [64].](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-174-2048.jpg)
![152 8.3 A Genetic Algorithm (GA) for the MV-PDPTW
Figure 8.3: Our chromosome representation for the MV-PDPTW.
Most solution methods from the literature, for example [100], try to minimize the number
of vehicles used in the solution as a primary objective, followed by either or both the total
distance traveled and the total service duration. We used the following objective function
of a solution S to achieve this goal:
O(S) = N(S)2
× TotDist(S) × TotDur(S) , (8.1)
where N(S) is the number of vehicles used in the solution, TotDist(S) is the total dis-
tance traveled by all vehicles, and TotDur(S) is the total schedule duration, which in-
cludes the total travel time, the waiting time of the vehicles, and the service time at each
location. The number of vehicles is squared in this objective function, so that solutions
that use more vehicles will have a considerably higher cost than solutions that use less
vehicles. Initial experimentation with this objective function indicated that it adequately
serves the purpose of giving priority to minimizing the number of vehicles. In addition,
no fine tuning or adjustment of different weights is needed in this objective function.
8.3.2 The Initial Population
To create a solution for our GA, requests are first placed in a relocation pool in a random
order, before being inserted into the solution. Based on our investigation of several solu-
tion construction methods, as explained in Chapter 7, we chose the sequential construction
algorithm (SEQ), to create each solution in the initial GA population. The same algorithm
was also used to create or modify solutions during the evolutionary process, as will be ex-
plained later when we address the genetic operators. The underlying Hill Climbing (HC)
routing heuristic and the overall sequential construction algorithm are explained in detail
in Sections 7.4 and 7.5.1, respectively, and outlined in Algorithms 7.1 and 7.2.
The cost function, in Step 6 of the HC algorithm (Algorithm 7.1), is used to evaluate
the quality of each route, and is different from the objective function (8.1) used to eva-](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-176-2048.jpg)
![8.3 A Genetic Algorithm (GA) for the MV-PDPTW 153
luate the overall solution (chromosome). This cost function tries to minimize the total
route duration as well as the degree of infeasibility in capacity and TW constraints. The
cost function of a route r is described by the following equation (same as Equation 7.1,
repeated for convenience):
F(r) = w1 × D(r) + w2 × TWV (r) + w3 × CV (r) , (8.2)
where D(r) is the total route duration, TWV (r) is the total number of time window
violations in the route, and CV (r) is the total number of capacity violations, while w1, w2
and w3 are assigned weights in the range [0, 1], and w1 + w2 + w3 = 1.0. For this route
cost function, we used the same weights used in Equation 7.1, i.e., w1 = 0.201, w2 = 0.7
and w3 = 0.099.
Our routing algorithm allows routes to dynamically change during the search, i.e., pre-
vious routing decisions (locations) for some requests already existing in the route may
be altered as new requests are added to the route. The new routing information is copied
back to the chromosome whenever a change in the route occurs.
8.3.3 The Genetic Operators
As previously mentioned in the introduction to this thesis, we try to focus on neighbou-
rhood moves (represented by genetic operators here) as a tool for satisfying the hard pro-
blem constraints, and maintaining the feasibility of solutions throughout the search. Two
crossover operators and one mutation operator have been designed to achieve this purpose.
In what follows we explain in detail our proposed genetic operators.
Mutation
The mutation operator, which we will call the Vehicle Merge Mutation (VMM), simply
tries to merge requests from two randomly selected vehicles. The idea is to try to reduce
the number of vehicles by distributing the requests among already existing vehicles, or
possibly combining two vehicles into one.
Figure 8.4 demonstrates the steps of the VMM. Figure 8.4(a) shows that vehicles v2 and
v3 were randomly selected for merging from the current solution. The requests belonging
to them will then be placed in a relocation pool in a random order. The remaining requests
in the solution, i.e., vehicles v1 and v4 will be copied to the new solution to form a partial
solution. Figure 8.4(b) shows that the requests in the relocation pool are then re-inserted
into the partial solution using the SEQ construction, i.e., Algorithm 7.2, and the final](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-177-2048.jpg)
![8.3 A Genetic Algorithm (GA) for the MV-PDPTW 157
8.3.4 The Complete GA
Algorithm 8.1 shows the outline of the complete evolutionary algorithm that we used to
solve the MV-PDPTW.
Algorithm 8.1: The Complete Genetic Algorithm.
1: Initialize a population POP of candidate solutions to the MV-PDPTW, using the SEQ
construction algorithm (Algorithm 7.2)
2: for (a pre-specified number of generations) do
3: for (i=0; i<NumCrossovers;i++) do
4: Select parents P1 and P2 from POP, using roulette wheel selection
5: Randomly select crossover type (VCX or VMX)
6: Apply the selected crossover to parents (P1, P2) to produce child C1
7: Apply the selected crossover to parents (P2, P1) to produce child C2
8: With some probability, apply mutation (VMM) to C1 and C2
9: Update POP by integrating the new generation and eliminating some worst individuals
{i.e., steady state GA with overlapping populations}
8.3.5 Attempts to Improve the Results
Before we report the experimental findings of this part of our research, we describe in
this section some other operators and modifications to the solution approach that we ex-
perimented with, in an attempt to reach the best possible results. These attempts are
briefly explained below, in addition to some concluding remarks about their performance.
However, since the benefits realized from adopting these approaches did not meet our an-
ticipations, they were abandoned in favour of the simpler approach described in Sections
8.3.1 to 8.3.4.
Vehicle Removal Mutation
This mutation operator is based on the GGA of [116]. One vehicle is removed and its
requests are placed in the relocation pool in a random order, where the SEQ algorithm tries
to re-insert them into the solution. This may result in reducing the number of vehicles, if
all the requests from the pool were accommodated into other vehicles. The approach in
[116] selected a random vehicle for removal, while in our research we selected the vehicle
with the minimum number of nodes.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-181-2048.jpg)
![158 8.3 A Genetic Algorithm (GA) for the MV-PDPTW
Large Neighbourhood Search (LNS) Mutation
This mutation operator is based on the Large Neighbourhood Search (LNS) move used in
the work by [131], which is an adaptation of the same operator used by Shaw in [139] for
the VRPTW. The idea is to remove a number of requests from the solution and then try to
re-insert these requests back. Usually a large number of nodes (30% to 40%) are removed.
However, since in our research this operator is used as a mutation operator, it would be
more appropriate to reduce the number of requests removed. In our experimentation we
found that 5% to 10% of the total number of nodes produce reasonable results.
Two variations of this operator were tried in our research, as advised by [131]. One ver-
sion of this mutation selects the nodes to be removed randomly (random removal), while
the other version selects the nodes that seem to be located in the wrong positions in the
solution (worst removal). To find these requests, we calculate the cost of the solution (the
objective function), with and without each request in turn, i.e., the request is temporarily
removed from the solution to calculate its effect on the overall cost. The requests are then
sorted in decreasing order of the change in cost. The requests to be removed are taken,
in order, starting from the top of the list and then placed in a relocation pool in a random
order. The removed requests are then re-inserted in the solution using the SEQ algorithm.
The ‘worst removal’ version of the LNS algorithm gave slightly better results than the
‘random removal’ version.
A Memetic Algorithm
We also tried to improve the performance of the GA by introducing local search at various
stages of the evolutionary process. This idea is based on the memetic algorithm described
in [94]. We tried both the vehicle removal mutation and the LNS mutation (explained
above), as neighborhood moves to create a new solution within the local search. We also
used an SA acceptance for solution replacement.
More specifically, in this memetic algorithm, local search is performed on some indivi-
duals in the population (according to a certain probability), for example at the beginning
of each generation. The local search repeatedly applies a selected ‘mutation’ operator on
the individual for a pre-specified number of trials. Within the local search, a new solu-
tion replaces the current solution using an SA acceptance criterion, i.e., a better solution
always replaces the current solution, while a worse solution may replace the current solu-
tion with some small probability. The best solution found during the local search is finally
the one that replaces the initial solution (the individual that we started with).](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-182-2048.jpg)
![8.3 A Genetic Algorithm (GA) for the MV-PDPTW 159
Our experimentation indicated that the more suitable operator to be used within the local
search seems to be the LNS operator. It should also be noted that a small population
size (e.g. 30 individuals) would be most appropriate for this memetic algorithm, due to
the large increase in processing time that result from the repeated application of the local
search.
Allowing Infeasible Solutions
In our attempt to improve the results, we also tried to add some infeasible solutions to the
set of allowable solutions. The idea is that sometimes good solutions lie in the vicinity
of infeasible solutions, so by allowing infeasibility we may be able to climb out of local
optima. In this version of our algorithm, whenever a solution construction is needed
during the initialization phase of the population, or during the course of the evolutionary
operators, there was a 50% chance that the construction process will allow infeasibility.
In order to restrict the amount of infeasibility and reduce the search space, we only allo-
wed violations in the time window but not in the capacity constraint. To do that, several
modifications to the algorithm were needed.
Firstly, we had to restrict the number of nodes allowed in each individual vehicle. Other-
wise, we could end up with a solution in which all requests are served by one vehicle.
This is due to the fact that even if violating the capacity is not permitted, the algorithm
would still be able to find a solution in which all requests are served, without any capacity
violations, using only one vehicle. For example, a solution in which each pickup is fol-
lowed by its delivery will always satisfy the capacity constraint. Specifically, the allowed
number of nodes, for each vehicle individually, was set to a random number ranging from
1 to m, where m was arbitrarily chosen to be 20% of the total number of nodes in the
problem instance. Thus, nodes are added to the current vehicle until the predetermined
limit on the number of nodes for this vehicle is reached, at which stage a new vehicle has
to be allocated.
Secondly, the objective function has to be changed in order to penalize infeasibility in the
solution. The following equation was used
O(S) = w1 × N(S)2
+ w2 × TotDist(S) + w3 × TWV (S). (8.3)
Where N(S) is the number of vehicles in the solution, TotDist(S) is the total distance
traveled by all vehicles in the solution, and TWV (S) is the number of time window
violations in the solution. w1, w2 and w3 are weights in the range [0,1] and w1 +w2 +w3 =](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-183-2048.jpg)
![8.3 A Genetic Algorithm (GA) for the MV-PDPTW 161
8.3.6 GA Experimental Results
To test our algorithm, we implemented a steady state GA with a 95% replacement. The
following parameters were used: population size= 500, crossover probability= 0.6, mu-
tation probability= 0.05, and the number of generations= 300. In cases where crossover
is performed either VCX or VMX is selected at random. We used the standard 56 (100-
customers) benchmark instances, created by Li & Lim in [100]. For the different types of
problem instances in the data set, the reader is referred to Section 7.6.1. The algorithm
was run 10 times on each problem instance.
We compared our algorithm, which we will call the Grouping-Routing GA (GRGA),
with the GA in [33], denoted by CKKL5
, and the grouping GA in [116], denoted by GGA.
To the best of our knowledge, they are the only GA approaches that have been attempted
in the literature for the MV-PDPTW and applied to the published benchmark data of [100].
The results in [116] are also close to the best known results, so we found that a comparison
with their results will be sufficient for the purpose of this part of our research6
. Before
we report our experimental findings, though, we present in Table 8.1 a comparison of the
most distinctive features of the three algorithms under consideration.
Figures 8.6 and 8.7 show the best results achieved by the three algorithms in terms of the
number of vehicles and the total distance traveled. The two figures show that our GA
clearly achieves better results than the CKKL algorithm in almost all test cases. There
are only 5 cases in which our algorithm produced one more vehicle than the number of
vehicles produced by the CKKL. Moreover, all our total distance results were better than
the results of the CKKL. On average, the improvement of our results compared to the
results of the CKKL in the number of vehicles is approximately 16%, while the average
improvement in the total travel distance is approximately 36%. Also, our results are
also close to the results of the GGA in the number of vehicles produced, with only few
exceptions. Nevertheless, the resulting total distance is larger than the resulting total
distance of the GGA in most test cases. This is even more noticeable in instances with a
long time window width, i.e., instances of category ‘2’.
5
We thank the authors of [33] for providing us with the data files containing their detailed results.
6
The detailed results of the GRGA approach and a comparison with best known results will be presented
in Section 8.4](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-185-2048.jpg)
![8.3 A Genetic Algorithm (GA) for the MV-PDPTW 165
When we try to analyze the computational results in the light of the differences between
the three algorithms summarized in Table 8.1, we will realize the following. First, since
both the GRGA and the CKKL algorithms have the same chromosome structure and the
same components of the objective function, then the obvious success of the GRGA com-
pared to the CKKL algorithm must be due to the routing algorithm and/or the genetic
operators used in the former. The success of the GRGA to produce solutions with less
vehicles in most test cases could be attributed to the presence of two genetic operators the
are specifically designed for this purpose and heavily applied during the search, namely
the VMM mutation and the VMX crossover. On the other hand, the 1M mutation, used
for reducing the number of vehicles in the CKKL, is only performed occasionally during
the search. Also, the noticeable success of the GRGA in terms of reducing the total travel
distance, could be attributed to the routing algorithm that is called whenever a route is
created or modified to try to improve the quality of the route by reducing its overall cost.
This again is in contrast to the LSM mutation of the CKKL that tries to improve the route,
but is only called occasionally during the search. The VCX crossover, used in the GRGA,
also seems to do a better job than the RBX crossover used in the CKKL. The RBX merely
exchanges one route between parents, while the VCX tries to be selective when transfer-
ring routes from the parent to the child, by choosing routes that serve a large number of
nodes with the smallest possible distance. It also seems that the SBX crossover, used in
the CKKL, may not be suitable for the genetic representation used. Since the gene is ac-
tually a complete route, it would seem more appropriate to transfer a collection of routes
rather than route fragments between parents.
We will now try to analyze the reasons behind the sub-optimal results achieved by the
GRGA compared to the GGA, specially in terms of the total travel distance7
. First of all,
one of the major differences between the two algorithms is the objective function. The
objective function of the GRGA includes the number of vehicles, the total distance, and
the total duration, while the objective function of the GGA only includes the total dis-
tance. Thus, the best solution as far as the GRGA is concerned must balance all three
components, i.e., it takes into consideration other parameters involved in the routing sche-
dule like the total waiting time of the vehicle at each location and the total service time.
The second main difference is the routing heuristic used in both algorithms. The routing
(insertion) algorithm of the GGA tries to find the best insertion position for each newly
inserted request in all available routes, while we use a simple and fast algorithm that ac-
cepts any feasible insertion. The attempt to improve the route, using the HC algorithm,
is only local to each route and does not involve comparing the insertion cost with other
7
It should also be noted that our algorithm was run only 10 times on each test case, while the GGA of
[116] was run 30 times and the best result was selected.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-189-2048.jpg)
![166 8.3 A Genetic Algorithm (GA) for the MV-PDPTW
routes. It seems that our routing algorithm could probably use the help of an additional
local search operator (e.g. 2-Opt), to try to improve the routes, which should in turn help
reduce the number of vehicles.
In our opinion, these two issues are the main reasons for the inability of our approach to
reach the solution quality of the GGA. On the other hand, the genetic operators in both
seem to be comparable, since they either try to reduce the number of vehicles, or copy
complete routes from parents to children. However, we think that our VCX crossover may
be more suitable than the crossover of the GGA for this problem type, because transferring
a sequence of routes is not really meaningful, since the order of routes in the chromosome
is irrelevant.
Also, as mentioned previously, our algorithm seems not able to cope with instances of
type ‘2’, as evident by the large gap between our results and the results of the GGA in the
total travel distance. This could be explained if we recall that our routing algorithm had
a neighbourhood move, which was guided by the TW. It seems that the routing algorithm
was thus capable of dealing better with instances in which the TW constraint is hard to
satisfy, i.e., those with a tight TW width. This neighbourhood move may not be sufficient
to improve the route in problems with a large TW width, because of the availability of
many different feasible orders of nodes. An alternative neighbourhood move may be
needed in that case. For example, the neighbourhood move could take into account not
only the bound(s) of the time windows of the swapped locations, but also whether the
swapping can improve other route characteristics, such as the total waiting time, the total
travel time, or the extra travel distance. The results obtained for instances of type ‘2’ also
sustain our previous observation in Section 7.6.3, regarding the large solution space for
this specific type of problem instances.
Finally, the average processing time needed by our algorithm, over all 56 problem ins-
tances, was 176.9 seconds, which is comparable to the processing time of the grouping
GA in [116] having an average of 167.1 seconds. Nevertheless, the separation of the data
structure used for individual routes in [116] from the actual chromosome, seems to be
favourable than our representation which includes all vehicle routes in the chromosome,
as far as processing time is concerned, since the transfer of complete routes during the
recombination and mutation operators is definitely time consuming. The authors in [33],
on the other hand, do not report their processing time.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-190-2048.jpg)
![8.4 Simulated Annealing (SA) for the MV-PDPTW 167
8.4 Simulated Annealing (SA) for the MV-PDPTW
As previously mentioned in Section 8.1, we will try in this part of our research to adopt
some of the neighbourhood operators used in the GA approach for the MV-PDPTW within
an SA approach. As expected, the solution representation used in the SA is similar to
the chromosome representation used in the GA (explained in Section 8.3.1). The same
objective function (Equation 8.1) is also used here.
The initial solution from which the SA will progress is created by generating a number of
random solutions8
, in a manner similar to the creation of an initial genetic population (as
explained in Section 8.3.2), and the best solution among them, in terms of the objective
function value, is selected.
To allow for an adaptive calculation of the SA parameters for each problem instance in-
dividually, we again used the approach proposed by [40], as previously done for the SV-
PDPTW (see Section 6.3 and Algorithm 6.1). Thus, the annealing parameters are calcu-
lated based on the average value of ∆cost, where ∆cost is the difference in the objective
function value between some randomly generated solutions for the current problem ins-
tance. In our approach, the same set of random solutions created for the selection of the
starting solution, were also used for the purpose of calculating the annealing parameters.
A crucial part of any SA algorithm is the neighbourhood move that will be used to generate
a new solution. In this part of the research, we experimented with different neighbourhood
moves from the ones tried in our GA approach. Two neighbourhood moves were found to
be the most appropriate for the SA approach:
1. The Large Neighbourhood Search (LNS) Move: the LNS move is similar to
the LNS mutation operator explained in Section 8.3.5, which is inspired from the
Adaptive Large Neighbourhood Search (ALNS) approach of [131]. As previously
explained, the idea is to remove and then re-insert a large number of requests in each
application of the move. The authors in [131] recommend that 30% to 40% of the
total number of nodes is removed in each iteration. In our LNS move, we slightly
increased the allowed range for the number of removed requests, in order to explore
a wider area of the search space. The number of removed requests in our LNS move
ranges between 20% to 50% of the total number of requests. In addition, we adopted
here the “Worst Removal” variant from the three types of removal heuristics applied
in [131] (see Section 8.2 for more details about the different removal heuristics of
[131]). In our approach, the requests removed are those that are estimated to cause
8
100 random solutions were created in this experiment.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-191-2048.jpg)
![8.4 Simulated Annealing (SA) for the MV-PDPTW 169
Algorithm 8.2: The 2-Stage SA Algorithm.
1: Find an initial solution (InitSol) and calculate the annealing parameters
2: BestSol ← InitSol {Initialize the best so far solution}
3: Initialize MaxAttempts to a small number {We used 10}
4: NoImprovement ← 0
5: repeat
6: OldCost ← Objective(BestSol) {Calculate the cost of the solution using Equation 8.1}
7: BestSol ← SALNS(BestSol) {Perform SA on the current best solution, using the LNS
move, and return the best found solution}
8: BestSol ← SAV M (BestSol) {Perform SA on the current best solution, using the VM
move, and return the best found solution}
9: NewCost ← Objective(BestSol) {Calculate the cost of the new solution using Equation
8.1}
10: if (NewCost is not better than OldCost) then
11: NoImprovement + +
12: else
13: NoImprovement ← 0
14: until (NoImprovement reaches MaxAttempts)
SA Experimental Results
Our SA algorithm was tested on the same 100-customers test cases used for testing our
GA, as explained in Section 8.3.6. Similar to the GA, the SA algorithm was run 10
times on each test case. Table 8.2 shows the best result, in the 10 runs, achieved by both
the GA and the SA for each test case. The best result is the one having the minimum
number of vehicles, and for the same number of vehicles, the one having the minimum
total travel distance. The better obtained result between the two algorithms is highlighted
in boldface. In addition, the processing time in seconds is also shown in the table. The last
two columns of the table show the current best known results for these problem instances,
which are published in:
http://www.sintef.no/Projectweb/TOP/Problems/PDPTW/Li--Lim-benchmark/
The best known results are attributed to [100], [12], and [135].
The last two rows of the table show, respectively, the overall average results of the corres-
ponding column, and the percent difference (gap) between our average and the average of
the best known results, in terms of both the number of vehicles and the total distance.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-193-2048.jpg)
![8.5 Summary and Future Work 173
8.5 Summary and Future Work
In this chapter we investigated both a GA and an SA for solving the MV-PDPTW. Our
GA approach tried to face the challenge of allowing the GA and its operators to be aware
of and manipulate both the grouping and the routing aspects of the problem. A challenge
that most previous GA research on this problem has achieved little success with, or has
been avoided completely by allowing the GA to tackle only one problem aspect. We first
tried a simple representation and an intelligent neighbourhood move to handle the routing
part of the problem. For the grouping part, on the other hand, we designed new genetic
operators that try to exploit problem-specific information and create new solutions from
existing ones, while maintaining the feasibility of solutions throughout the search. Our
operators overcome the difficult problem constraints, and avoid at the same time the need
for a repair method to fix infeasible solutions, a technique that previous GAs and most
other heuristic and meta-heuristic techniques have been relying on to maintain feasibility.
Overall, though, our algorithm is a simple and straightforward GA technique, and the
genetic operators developed here are applicable to other related routing problems.
We compared our results with two previous GA attempts to solve the problem, the CKKL
algorithm of [33] and the GGA algorithm of [116]. The experimental results show that
our algorithm was able to greatly improve upon the results of the CKKL algorithm, using
just a few simple modifications in the routing algorithm and the genetic operators, and
applying problem-specific information. The improvement was evident in both main ob-
jectives, the number of vehicles and the total travel distance. However, our results are still
behind the results of the GGA in most test cases. The three algorithms have been tho-
roughly analyzed and possible reasons behind the differences in performance have been
identified.
The GA was also compared with a 2-stage SA approach that uses two different neigh-
bourhood moves repeatedly. The neighbourhood moves used within this SA approach are
analogous to the genetic operators used previously in the GA, since they rely on removing
and then re-inserting requests or merging selected vehicles.
Both the GA and the SA obtained comparable results when tested on published benchmark
instances, which is mostly due to their reliance on similar neighbourhood operators. Ho-
wever, neither algorithm was capable of competing favourably with best known results. In
general, it appears that our representation and neighbourhood operators in both algorithms
are doing a fair job in guiding the search towards better solutions. However, to cope with
the difficulty of the problem and the different types of problem instances, both approaches
still need further improvement. For this purpose, different neighbourhood moves could be](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-197-2048.jpg)
![175
Chapter 9
The One-Commodity Pickup and
Delivery Problem: Introduction and an
Evolutionary Perturbation Scheme
The One-Commodity Pickup and Delivery Problem (1-PDP) is another important problem
in transportation and logistics systems. The 1-PDP deals with supplying and collecting
one type of commodity from a number of customers, some of them are designated as
pickup customers and the others as delivery customers. Each pickup customer provides
a certain amount of the commodity, while each delivery customer consumes a certain
amount of the same commodity, i.e., goods collected from pickup customers can be de-
livered to any delivery customer. All customers are served by one vehicle with a limited
capacity, and the journey of the vehicle should start and end at a central depot. The de-
pot can supply or consume any additional amount of the commodity that is not supplied
or consumed by the customers. Our goal is to find a feasible and minimum cost route
for the vehicle, such that all customers are served without violating the vehicle capacity
constraint.
This problem has attracted our interest for several reasons. First, there are many applica-
tions for this problem in practice. For example, the commodity could be milk that should
be collected from farms and delivered to factories with no restriction on the origin and
the destination of the product, or it could be money that should be distributed between
the branches of a bank [74]. It can also model any logistic situation in which some wa-
rehouses have an extra supply of some commodity, while others are in short of the same
commodity. A typical situation is when some hospitals need to transfer a certain medicine
to other hospitals, who are in short of this medicine. For example, an H1N1 vaccination
or treatment could be transferred in urgent epidemic circumstances [106].
Second, this problem has not been adequately explored in the literature. Since the in-
troduction of the 1-PDP in [74], only very few papers seem to have handled it. Finally,
there are other important problems in the literature that are closely related to the (1-PDP).](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-199-2048.jpg)
![176 9.1 The 1-PDP
For example, the Traveling Salesman Problem with Pickup and Delivery (TSPPD) (see
Section 9.3 for more details about this and other related problems).
Our initial attempt to handle the 1-PDP was based on a simple Evolutionary Perturba-
tion Scheme (EPS) that has proved successful for other routing problems, such as the
TSP and the Capacitated Vehicle Routing Problem (CVRP). We made some preliminary
experimentation with this technique for the 1-PDP and tested it on published benchmark
data. The details of the approach and its experimental results are reported in this chapter.
The outcome of the approach, though, did not meet the anticipated standard. Some ana-
lysis of possible shortcomings of the algorithm and future directions are also discussed
here. Nevertheless, A more successful heuristic for solving the 1-PDP, will be detailed in
the next chapter.
The rest of this chapter is organized as follows. Section 9.1 formally defines the 1-PDP,
while Section 9.2 is a brief summary of related research in this area. Some problems that
are related to the 1-PDP are presented in Section 9.3. The EPS technique attempted for
solving this problem is explained in Section 9.4, together with its experimental findings.
Finally, Section 9.5 concludes with a summary and some thoughts for future work.
9.1 The 1-PDP
Based on the definition provided in [73], the 1-PDP is characterized by having a set of
customers i(i = 1, 2, ..., n), where customer 1 is the depot. V = 1, 2...n is the vertex
set, and E is the edge set. For each pair of locations (i, j) the travel cost cij is known in
advance. For each customer there is an associated demand qi, such that qi > 0 for a pickup
customer and qi < 0 for a delivery customer. The depot is considered as a customer that
supplies or consumes any amount of the product that is not supplied or consumed by the
set of customers, and the demand of the depot will be calculated as q1 = − n
i=2 qi. All
customers are to be served by one vehicle with a limited capacity Q > 0, and the capacity
of the vehicle should at least be equal to the maximum customer demand (whether a
pickup or a delivery), i.e., Q ≥ maxi∈V {|qi|}.
A feasible path for the vehicle is a path that travels from the first customer to the last
customer, and visits each customer exactly once without exceeding its capacity. The au-
thors in [73] more formally describe the feasibility of a path P through the sequence of
customers i1, ..., ik, with k ≤ n. If we assume that lj(P) is the load of the vehicle after
visiting the jth
customer, and l0(P) = 0, then P is feasible if and only if
max
j=0,...k
{lj(P)} − min
j=0,...k
{lj(P)} − Q ≤ 0. (9.1)](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-200-2048.jpg)
![9.2 Related Work 177
In [158], more explanation of Equation 9.1 is given by indicating that minj=0,...k{lj(P)}
could be either 0 or negative, given the existence of some negative customers’ demands.
Accordingly, there are two possible conditions for a feasible route:
(i) if minj=0,...k{lj(P)} = 0, Equation 9.1 will simply become maxj=0,...k{lj(P)} ≤ Q.
(ii) if minj=0,...k{lj(P)} < 0 (i.e., there is a shortage in the required delivery demands), we
can assume that | minj=0,...k{lj(P)}| will be supplied by the depot when the vehicle starts
its journey. Thus, | minj=0,...k{lj(P)}| will be added to the demands of all nodes, making
minj=0,...k{lj(P)} = 0 again, and we return to the first case, i.e., maxj=0,...k{lj(P)} ≤ Q.
Although the 1-PDP is N P-hard (indeed it coincides with the TSP when the vehicle ca-
pacity is large enough), checking the feasibility of a path can be done in a linear time. It is
also important to note that the feasibility of the path is independent of its orientation, such
that a path that is feasible/infeasible if traversed forward, will be also feasible/infeasible
if traversed backward [75]. This feature allows a solution construction algorithm to start
a tour from any customer node, visit each node, including the depot, exactly once and
return to the first node. This closed circular tour will correspond to starting and ending at
the depot (see Section 9.4.3 for an example of a construction algorithm).
9.2 Related Work
Since this problem is N P-hard, exact algorithms are only suitable for small problem
sizes. For example, [74] presented a branch-and-cut exact algorithm to solve instances
of up to 60 customers. To deal with large size problems, the same authors tried heuristic
approaches in [75]. Two heuristic approaches have been presented in their work. The
first approach starts with a construction heuristic that is based on an adaptation of a TSP
Nearest-Neighbour (NN) insertion heuristic. However, travel distances were modified
using a special formula that intends to penalize the use of edges connecting customers
of the same type, i.e., edges connecting pickup customers together or delivery customers
together. It was thought that this approach would be more likely to lead the construction
algorithm to find feasible solutions. An improvement phase then follows, using 2-Opt and
3-Opt edge exchanges, to try to improve the feasibility and/or the total travel distance.
The process of optimization is repeated several times, each time a new initial solution is
constructed using a different starting node from the node list. The second approach is an
incomplete optimization procedure, based on the branch-and-cut approach presented in
[74] to find the best solution in a restricted feasible region.
In [73] a heuristic approach, named hybrid GRASP/VND, is proposed. The approach
is based on combining two optimization heuristics that have been successfully applied](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-201-2048.jpg)
![178 9.2 Related Work
to combinatorial optimization problems. The first is called GRASP (Greedy Randomi-
zed Adaptive Search Procedure), and is based on a repetition of a construction phase
and a local search phase. The construction process works by selecting the next element
for insertion from a Restricted Candidate List (RCL), which has been previously created
according to the benefit of inserting each node, and depends on the current state of the
solution. There is also a probabilistic element in the choice of the next node from the
RCL. On the other hand, the Variable Neighbourhood Search (VNS) approach is based on
systematically changing the neighbourhood move, during the optimization process, each
time a local optimum is reached. The new neighbourhood move is usually of higher order
than the previous move, and it is applied to the same starting solution. Variable Neigh-
bourhood Descent (VND) is a variant of VNS, where the local optimum found acts as the
new starting point for the local search (see Section 2.3.6 for more details about the VNS
and the VND approaches).
For the 1-PDP, the hybrid GRASP/VND approach in [73] is basically a GRASP, where
the local search is performed using a VND procedure. In this heuristic, two VND al-
gorithms have been applied with different neighbourhood moves in each algorithm. The
first VND (called VND1) is applied in the improvement phase after the construction of
the initial solution. The neighbourhood move used in VND1 is a classical 2-Opt, which
is then followed by a 3-Opt move whenever a local optimum is reached. After the basic
GRASP/VND, a further post-optimization phase is performed, using a second VND (cal-
led VND2). This time, however, the neighbourhood moves applied are a move forward
and a move backward operators respectively. In the move forward operator, a customer
is moved from its current position to a further position in the route. So for example, if a
customer in position i is moved to position j with j > i, this requires that all customers
in positions i + 1, . . . , j must be shifted backwards one position. The locations of other
customers in the route do not change. The move backward operator works similarly, but j
is now a position that precedes i in the tour, and intermediate customers have to be shifted
forward one position. The VND algorithm used in this heuristic is shown in Algorithm
9.1 and the hybrid GRASP/VND algorithm is shown in Algorithm 9.2, both adopted from
[73].](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-202-2048.jpg)
![9.2 Related Work 179
Algorithm 9.1: VND(x) Procedure [73].
1: for (k = 1; k < max; k + +) do
2: x′ ← LocalSearch(x, Nk(x))
3: if (x′ is better than x) then
4: x ← x′
5: Return x
Algorithm 9.2: Hybrid GRASP/VND Procedure [73].
1: while stopping condition is not satisfied do
2: x ← GreedyRandomizedInitSol() {construction phase}
{improvement phase}
3: x ← V ND1(x) {edge-exchange neighbourhoods}
4: if x is feasible and improves the best solution x′ then
5: x′ ← x
{post-optimization}
6: x′ ← V ND2(x′) {vertex-exchange neighbourhood}
7: Return x′
A GA approach was introduced in [158] to solve the problem. The algorithm first starts
by creating a population of feasible solutions using a new nearest-neighbour construction
heuristic. The initial population is then optimized using a 2-Opt neighbourhood move.
The most distinguishing feature of the algorithm is a new pheromone-based crossover
operator, inspired from the Ant Colony Optimization (ACO) technique, where pheromone
trails are updated each generation. During crossover, the selection of the next node to be
inserted in the child is based on a probabilistic rule that takes into account the pheromone
trail of the edge connecting the last inserted node and the potential new node, such that
edges that have proved successful in the past are favoured. The crossover operator also
considers the distance between the two nodes, as well as the demand of the new node
and the current maximum and minimum loads carried by the vehicle. The offspring is
further optimized using a 2-Opt local search, in which only feasible solutions are accepted,
and only the closest neighbours to the current node are considered for edge exchanges.
The mutation operator is based on a 3-exchange procedure. Since there are 5 legal ways
in which 3 nodes could be exchanged, the move that yields the best result is the one
performed in the mutation operator. The algorithm was tested on the benchmark data that
are created by [75], producing the best so far results in most test cases.
The work done in [106] presents a simulation environment to solve the problem. The algo-
rithm starts with a Modified Simulated Annealing (MSA) algorithm, followed by a manual](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-203-2048.jpg)
![180 9.3 Related Problems
improvement phase. The SA approach is called Iterative Modified Simulated Annealing
(IMSA). The idea is to create multiple neighbourhood feasible solutions Y from the cur-
rent feasible solution X by performing a number of changes on X. The changes are done
iteratively on X, starting from 1 change to a maximum of nChangesmax. For each change
value, a fixed number of Y solutions is created, and this is done at the same SA tempera-
ture. This whole process is then repeated in an outer loop for a number nCountsmax of
iterations. The initial solution X is created based on a Greedy Random Sequence (GRS)
algorithm, which is a nearest-neighbour heuristic with some randomness in selecting the
nearest customer or the second nearest customer. A final manual optimization is then
performed to improve the solution. Using a Graphical User Interface (GUI), the user se-
lects a subgraph (of less than 15 nodes) from the optimal result obtained, upon which an
improvement algorithm is applied. This is achieved by performing an Exact Permutation
Algorithm (EPA) for small size problems, or improving a selected sub-route for large size
problems. The algorithm was tested on some instances created in [7], and was found to
be useful for practical applications.
9.3 Related Problems
Hernández-Pérez and Salazar-González in [75] presented an extensive survey of routing
problems that are closely connected to the 1-PDP. Based on this survey, some important
problems that are related to the 1-PDP are:
• The Capacitated Traveling Salesman Problem with Pickup and Delivery (CTSPPD):
which is a special case of the 1-PDP, in which the demand of each pickup or delivery
customer is restricted to one unit (e.g. [5]).
• The Capacitated Dial-a-Ride Problem (CDARP): in which the vehicle should
move one-unit of commodity (for example a person) between pairs of customers
(e.g. [124]).
• The Pickup and Delivery Traveling Salesman Problem (PDTSP): which is the
same as the CDARP, but there is no restriction on the vehicle capacity (e.g. [127]).
• The Traveling Salesman Problem with Backhauls (TSPB): in which all delivery
customers must be visited before all pickup customers (e.g. [57]).
• The Traveling Salesman Problem with Pickup and Delivery (TSPPD): due to
the importance of this problem and its close connection to the 1-PDP, the following
discussion provides more details about this problem.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-204-2048.jpg)
![9.4 Solving the 1-PDP Using an Evolutionary Perturbation Scheme (EPS) 181
The Traveling Salesman Problem with Pickup and Delivery (TSPPD)1
This problem was introduced in [110], and it is assumed here the commodity collected
from pickup customers is different from the commodity delivered to delivery customers.
The depot supplies all the demand of the delivery customers and collects all the supplies
from the pickup customers, and both products must be accommodated in the vehicle wi-
thout exceeding its capacity. An example of an application of this problem is when empty
soft drink bottles are to be collected from homes or shops and delivered to the depot, and
at the same time full bottles are supplied by the depot to be delivered to some customers
[75]. For a TSPPD to be feasible, the vehicle capacity should at least be the maximum
among the sum of the pickup demands and the delivery demands. This condition is not
required for the 1-PDP, though, since the vehicle capacity could even be as small as the
largest customer demand [75].
In mathematical terms, if we let K = max{ i∈V :qi>0 qi, − i∈V :qi<0 qi}, then for a
TSPPD to be feasible Q must be ≥ K. However, we can always assume, as mentio-
ned in [110], that the TSPPD is in ‘standard form’, meaning that K = Q. Accordingly,
we can also assume that in a feasible TSPPD route, the vehicle starts at the depot with a
full load, delivers all the demand needed by the delivery customers, and at the same time
collects the goods provided by the pickup customers. Finally, the vehicle will arrive at
the depot fully loaded again, having collected all the goods from the pickup customers.
This condition does not occur in the 1-PDP, since the product can be collected from the
pickup customers and delivered to the delivery customers, with the depot only supplying
or absorbing any additional amount [75]. In [74] it is observed that a TSPPD can be
solved using the same algorithm as the 1-PDP if the TSPPD instance is transformed by
duplicating the depot into two dummy 1-PDP customers, one collecting all the quantity
supplied by the pickup customers, and the other providing all the quantity needed by the
delivery customers.
9.4 Solving the 1-PDP Using an Evolutionary Perturba-
tion Scheme (EPS)
In this part of our research we are trying to handle the 1-PDP using a technique based
on ‘perturbation’ of the problem instance, first introduced in Codenotti et al. [27] for
solving large instances of the TSP. The idea is to introduce a small perturbation to the
1
This problem was briefly mentioned in our literature survey of pickup and delivery problems in Section
4.1 under the name: The Traveling Salesman Problem with Mixed Linehauls and Backhauls (TSPMB).](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-205-2048.jpg)
![182 9.4 Solving the 1-PDP Using an Evolutionary Perturbation Scheme (EPS)
original problem instance P to transform it to a new instance P′
, for example by making a
small change in the coordinates of the cities to be visited in the TSP. The new coordinates,
which in turn will result in new separating distances between the nodes, will then be
used to construct a solution for the new instance P′
. The new solution is then evaluated,
possibly after applying some local search method to improve it, relative to the original
problem instance P to potentially replace the initial solution2
. Figure 9.1(a) shows a
typical product of a nearest-neighbour construction algorithm to an original small TSP
instance, while Figure 9.1(b) shows an improved solution obtained after perturbing the
coordinates of the original instance.
This idea was extended in [151], [152], [17] and [150], by applying a search algorithm to
find a perturbation that will give a better solution. For example, a GA can be applied to
optimize the set of perturbed coordinates. A simple greedy algorithm could then be used
to create a solution for the TSP, using the perturbed coordinates rather than the original
coordinates.
For the 1-PDP, we tried to apply the same idea of optimizing the set of perturbed coor-
dinates, as done in [150] for the TSP, and in the PhD thesis by Matthew Morgan3
for the
Capacitated Vehicle Routing Problem (CVRP) [109]. We were motivated by the success
of this simple algorithm on these routing problems, as evident by the experimental fin-
dings reported. For both problems, it was demonstrated that the technique was able to
produce high quality solutions compared to simple solution construction heuristics wi-
thout perturbation. The results obtained also compared favourably to published results
from the literature in terms of both solution quality and processing time. Its applicability
for the 1-PDP seemed to be viable, since a simple construction heuristic, such as the one
described in [158], can be easily embedded within the evolutionary perturbation scheme,
in which nodes’ coordinates could be perturbed and optimized using a simple GA tech-
nique.
More specifically, in our perturbation scheme, we use a straightforward GA technique
with a chromosome representing a list of customer locations, with each gene consisting
of the x and y coordinates of the location. Traditional crossover and mutation operators
are then applied to optimize the set of perturbed coordinates. The perturbation of each
coordinate is done by applying a small change to the coordinate within a pre-specified
radius, as suggested in [109]. As mentioned above, the underlying construction heuristic,
2
A local search heuristic, applied to a solution for the new instance P′
could be ‘fooled’ into finding
better solutions for the original instance, because a local optimum for the original instance is not necessarily
a local optimum for the new transformed instance.
3
Matthew Morgan is a member of our research group (Scientific Computing and Optimization) in Cardiff
School of Computer Science & Informatics, UK.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-206-2048.jpg)
![9.4 Solving the 1-PDP Using an Evolutionary Perturbation Scheme (EPS) 183
5 10 15 20 25
5
10
15
20
25
X
Y
(a) TSP solution - original instance
5 10 15 20 25
5
10
15
20
25
X
Y
Original Insstance
Perturbed Instance
(b) TSP solution after perturbation
Figure 9.1: TSP solution before and after perturbation.
used to create a solution using the perturbed coordinates, is a simple nearest-neighbour
insertion heuristic, similar to the heuristic introduced in [158]. The created solution is
then evaluated relative to the original coordinate set, and the cost of the solution is used
as an objective function for the perturbed coordinates that were used to create the solution.
Algorithm 9.3 describes the basic steps of our Evolutionary Perturbation Scheme (EPS).](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-207-2048.jpg)
![9.4 Solving the 1-PDP Using an Evolutionary Perturbation Scheme (EPS) 185
Figure 9.2: EPS chromosome representation.
9.4.2 The Initial Population
A chromosome is created by applying a small perturbation on the coordinates of each
customer location, and storing the perturbed coordinates in the corresponding gene. The
process is repeated for all individuals in the population. The perturbation scheme follows
the recommendation of [109], such that each coordinate is shifted a small distance within
a pre-specified radius. This radius is calculated as a function of the distance between the
current node and its nearest neighbour. More specifically the new coordinates are given
by the following equation:
newx = oldx + (D × S) × r, newy = oldy + (D × S) × r, (9.2)
where r is a random number between −1 and 1, D is the distance between the current
node and its nearest neighbour, and S is a scaling factor used to control the permitted
shift.
9.4.3 The Nearest Neighbour Construction (NN-Construction)
The construction algorithm is called whenever the objective function of the chromosome
has to be calculated. The chromosome is first used to create a distance matrix reflecting
the current perturbed coordinates that are stored in the chromosome. This distance matrix
is then passed to a simple greedy nearest-neighbour construction (NN-Construction) al-
gorithm to create a corresponding solution. The construction algorithm we adopted in our
research is the one described in [158] to create their initial GA population, with a slight
modification. This algorithm works as follows.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-209-2048.jpg)
![186 9.4 Solving the 1-PDP Using an Evolutionary Perturbation Scheme (EPS)
1. Given CL 4
, a user defined parameter, generate the list of CL closest neighbours to
each node in the customer list, based on the current distance matrix;
2. Let m = 1, insert the depot node as the first customer T(m) in the tour, and initialize
both MaxLoad and MinLoad of the vehicle to be equal the demand of the depot;
3. If m = n, where n is the total number of customers including the depot, then stop;
otherwise go to Step 4;
4. Search the CL closet neighbours of T(m) for feasible candidates that have not
been inserted before and can be added to the end of the tour without violating the
vehicle capacity. If there are such customers, select the customer i having the largest
demand (in absolute value) among them, and insert it as T(m + 1), and update
MaxLoad and the MinLoad of the vehicle, go to Step 5; otherwise go to Step 6;
5. m = m + 1, go to Step 3;
6. Search all customers that have not appeared in T for feasible candidates for inser-
tion. If there are such customers, select the closest customer to the last inserted
customer and insert it as T(m + 1) and update MaxLoad and the MinLoad of the
vehicle, go to Step 5; otherwise stop and declare an infeasible solution;
7. Return m ≤ n, the number of nodes that have been successfully inserted in the
solution before infeasibility (if any) is encountered.
The difference between our algorithm and the construction algorithm described in [158],
and used to create a genetic population of random solutions to the 1-PDP, is that they
start their tour from a randomly selected customer, while we always start the tour from
the depot. We chose to fix the starting node to try to make the quality of the constructed
solution only a function of the perturbed coordinates used to create the solution. Any
randomness in the creation of the solution will interfere in the quality of the solution and
will not reflect the true fitness of the perturbed coordinates. Thus, the learning process
of the GA from generation to another may be obstructed. Similarly, some degree of
randomness in Step 6 of the algorithm that was adopted in [158] has been eliminated in
our algorithm. This is in fact essential in our scheme, because the solution constructed
from the perturbed coordinates is not saved during the evolutionary process. Making
the construction algorithm deterministic, by eliminating all randomness in the selection
of nodes, will enable our algorithm to use the best perturbed coordinates (individual) to
re-construct the best solution, when the evolutionary process terminates. In addition to
4
CL stands for a constant that determines a selected number of closet neighbours to a certain node.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-210-2048.jpg)
![9.4 Solving the 1-PDP Using an Evolutionary Perturbation Scheme (EPS) 187
removing randomness, our algorithm stops once the solution becomes infeasible, while
in [158], they repeat the construction algorithm starting from a new node, if the previous
attempt resulted in an infeasible solution.
The selection of the customer with the largest demand in Step 4 of the construction algo-
rithm is justified in [158] by giving priority in the insertion order to nodes that may be
difficult to insert, because of their large demand. Also, as mentioned in Steps 4 and 6,
the minimum load and the maximum load carried by the vehicle are updated each time
a new node is inserted. This step is needed for the feasibility check of each candidate
node. More specifically, according to Equation 9.1, to check the feasibility of insertion,
the difference between MaxLoad and MinLoad after serving the candidate node, should
not exceed its capacity, otherwise the potential candidate cannot be feasibly inserted at
the end of the tour.
9.4.4 The Objective Function
To calculate the fitness of each chromosome in the population, the following steps are per-
formed. First, a new distance matrix is calculated based on the set of perturbed coordinates
stored in the chromosome. Then, this new distance matrix is passed to the construction
algorithm, described in Section 9.4.3, to create a solution S that takes into account the
new distances between the nodes. Finally, the objective function of a chromosome (C) is
calculated using the following equation
F(C) = ((n + 1) − m) × Dist(S) (9.3)
where n is the total number of nodes, m is the number of nodes that have been success-
fully inserted in the solution before infeasibility (if any) is encountered, and Dist(S) is
the ‘true’ total distance traveled when the solution S is decoded relative to the original
coordinate set.
This objective function combines both the total travel distance and the degree of infeasi-
bility in the generated solution. If the created solution is feasible, all nodes would have
been inserted successfully, i.e. m = n. Accordingly, the fitness of this solution will be
Dist(S). On the other hand, if the solution is infeasible, Dist(S) will be multiplied by a
factor that increases with the degree of infeasibility. The smaller the number of nodes that
have been successfully inserted, the larger the multiplication factor. Note also that in case
of an infeasible solution, the distance value will be smaller than the distance of a feasible
solution, because not all nodes have been successfully inserted.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-211-2048.jpg)
![188 9.4 Solving the 1-PDP Using an Evolutionary Perturbation Scheme (EPS)
9.4.5 The Operators
Mutation
The mutation operator first selects the number of nodes (genes) to be mutated. This num-
ber was chosen to be a random number between 1 and 20% of the total number of nodes in
the chromosome. Then, the mutation operator applies Equation 9.2 to each gene selected
at random.
Crossover
Different traditional crossover operators have been tried in our research. The most ef-
fective crossover operator seems to be the EvenOdd crossover operator, which works by
selecting alternative genes from each parent and inserting them in the corresponding lo-
cations in the child. The roles of the parents are reversed for the second child.
9.4.6 Computational Experimentation
We implemented a steady state GA, with an 80% replacement, and the following parame-
ters: population size= 200, number of generations=1000 for large problems (more than 60
customers) and 200 for small problems (from 20 to 60 customers), probability of muta-
tion= 0.3, probability of crossover=0.8, and the perturbation scaling factor S in Equation
9.2 was set to 0.6, after some initial experimentation with different values. The neighbou-
rhood size CL was chosen to be 2, i.e., only the closest two neighbours are checked for
feasibility of insertion before all other nodes are checked. We found, experimentally, that
this ‘tight’ neighbourhood size helped to reduce the runtime, since the neighbourhood list,
which has to be created for all nodes whenever the construction algorithm is called, will
be very small in size. Also, it appeared from trying several other values that this small
value gave better results even for large size problems.
The Data Set: The algorithm was tested on instances created by [75]. There are 2 types
of problem instances. Small instances have a number of customers n in {20, 30, 40, 50,
60}. For these instances, the optimum is known and was obtained using the exact method
proposed in [74]. There are also large instances with n in {100, 200, 300, 400, 500}.
For each combination of n and a different vehicle capacity Q in {10, 15, 20, 25, 30, 35,
40, 45, 1000}, 10 problem instances have been created and given the letters {‘A’ to ‘J’}.
So for example, a problem instance named N100q20A, means the first instance (A) in](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-212-2048.jpg)
![9.4 Solving the 1-PDP Using an Evolutionary Perturbation Scheme (EPS) 189
the 100-customers category (N100) with vehicle capacity 20 (q20). The data set and the
results obtained in [75] and [73] can be downloaded from the Pickup and Delivery Site of
Hernández-Pérez 5
: http://webpages.ull.es/users/hhperez/PDsite/index.html
Experimental Results: We briefly summarize here the experimental results of the EPS
on test cases with the tightest vehicle capacity. These are the problems with Q = 10,
and considered to be the hardest problems in the data set. We summarize the results on
all problem sizes from 20 to 100. Our perturbation heuristic was run 20 times on each
of these instances and the best result was recorded. Larger problem size, however, were
found to be very time consuming and their testing was dealt with differently as will be
explained later.
It should also be noted that there is a slight difference between the final testing version of
the heuristic between small instances and large instances. This was based on preliminary
experimentations and was intended to give the best possible results. For small instances
(from 20 to 60 customers), the solution obtained from the NN-construction heuristic, ex-
plained in Section 9.4.3, was optimized using a Hill Climbing (HC) approach that uses a
simple node swap neighbourhood move. The objective function (Equation 9.3) then uses
the quality of the optimized solution as a fitness of the perturbed coordinates from which
the solution was created. This is the same approach applied in the perturbation scheme of
the CVRP in [109]. For larger problems, however, this technique could not be used due
to the great increase in processing time. Therefore, the objective function was only based
on the quality of the constructed solution without optimization. An optimization phase,
though, was performed only on the best individual after the termination of the whole GA
process. Thus, the set of perturbed coordinates that are represented in the best individual
in the final generation, were used to construct the best solution obtained, using the NN-
construction algorithm. This solution is then further optimized using an HC approach that
uses a simple move forward operator, which was suggested in [73] (see Section 9.2 for
more details about the move forward operator) . The results summarized here are the final
results obtained after this optimization phase.
Results on Problem Sizes 20-100 Customers: Experimental results on problem sizes
ranging from 20-100 customers indicated that the perturbation heuristic was able to achieve
the optimal result in only 16 out of the 50 test instances, where the optimum is known (i.e.,
for problem sizes 20-60). Also, the optimum was achieved only for the smallest size pro-
5
New best results were obtained by the GA in [158], but they do not appear in the pickup and delivery
site yet.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-213-2048.jpg)
![190 9.4 Solving the 1-PDP Using an Evolutionary Perturbation Scheme (EPS)
blems of 20 and 30 customers. The heuristic, on the other hand, was unable to compete
with previous heuristics from the literature for all larger size problems. The average re-
lative difference between our results and the optimal or best known results ranged from
2% for 40 customers to 12% for 100 customers. The heuristic was also slower than the
heuristic in [158]. For example the average run time for 100-customers problems was
43.8 seconds compared to 21.12 seconds reported by [158] for the same problem types.
Thus, the experimental results clearly indicate that the EPS achieved a limited success in
dealing with the 1-PDP problem instances, especially as the problem size increases. The-
refore, testing the algorithm on even larger size problems did not seem worthwhile. The
perturbation scheme in all problem instances, though, always produced feasible solutions.
This indicated to us that the heuristic may be used in cases where a quick and feasible
solution is needed irrespective of its quality.
Results on Problem Sizes 200-500 Customers: For these problem sizes, we shifted
our attention to testing the potential of the EPS in transforming infeasibility to feasibi-
lity, rather than competing with best known results. Hence, we only ran the algorithm
for one generation and compared the result obtained, in terms of the quality of the solu-
tion constructed from the best individual in the population, with the result obtained when
the NN-construction heuristic was run only once to produce an initial solution (using the
original problem coordinates without perturbation). Remember that the NN-construction
heuristic is deterministic in our scheme, since we removed all the randomness involved in
the construction of such solution. Accordingly, there is only one initial solution that can
be constructed using the original coordinates. If the NN-construction managed to insert
all nodes in the solution whilst maintaining feasibility, the created solution is feasible.
Otherwise, the quality of the created solution is determined by the number of nodes that
have been successfully inserted before infeasibility is encountered. In the current expe-
riment, the algorithm was again run 20 times on each problem instance, and we counted
the number of times the EPS heuristic was able to generate a feasible solution after one
generation.
The results of this experiment showed that the NN-construction heuristic failed to produce
feasible solutions in all cases except one, which is the instance N400q10F. On the other
hand, the EPS heuristic was able to find feasible solutions for all problem instances after
one generation. In most test cases, the EPS was able to find a feasible solution in all 20
runs. In 9 out of the 40 problem instances the EPS heuristic did not produce a feasible
solution in all runs. The minimum number of feasible solutions obtained were 3 out of
the 20 runs.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-214-2048.jpg)
![9.5 Summary and Future Work 191
These results, in general, seem to indicate that the EPS heuristic was in fact successful in
transforming the frequent infeasibility problem of the NN-construction heuristic to feasi-
bility. The processing time needed was also reasonable with an average of 1.13 seconds
for 200-customers problems to 7.18 seconds for 500-customers problems.
9.5 Summary and Future Work
In this part of the research we investigated a new heuristic for the 1-PDP. The heuristic is a
perturbation scheme that transforms the problem instance into a new one by performing a
small shift in the coordinates of the nodes in the problem instance. A simple construction
heuristic is used to construct a solution to the problem from the perturbed coordinates
rather then the original coordinates, hoping to produce a better quality solution. The
perturbed coordinates were also optimized using a simple GA technique.
The experimental results on a large number of test cases of different sizes indicated that the
heuristic was in most cases able to improve the quality of the initial constructed solution to
a large extent, removing infeasibility in all test cases. However, the final results obtained
were in general of lesser quality than the best known or optimal results. The algorithm
was also relatively slow, which is mostly due to the nature of the GA involved.
It should be noted that the EPS algorithm is highly sensitive to several underlying factors.
The most important element seems to be the embedded construction heuristic. Previous
attempts in the literature that used a similar technique, for example [109], made a consi-
derable effort in testing several construction methods for the CVRP, to identify the best
construction heuristic that can be used within the EPS. Nevertheless, the literature of the
CVRP is relatively well-established and very rich with such heuristics. The literature on
the 1-PDP, on the other hand, is scarce and only a small number of construction heuris-
tics have been previously attempted. We chose from them the construction algorithm that
seems to be relatively simple and fast, and also gave reasonable results within the context
of other meta-heuristic techniques. This construction heuristic, however, apparently pro-
duce frequent infeasible solutions, as illustrated by the results obtained on large problem
sizes. Having a large number of infeasible solutions will lower the quality of the overall
population of perturbed coordinates, and make identifying the set of perturbed coordinates
that lead to feasible and good quality solutions a difficult task for the EPS.
In addition, the EPS is also highly sensitive to the perturbation parameters and the selected
perturbation neighbourhood shape and size around the coordinates. Fine tuning of these
parameters was also a considerable part of the research done in [109]. In our research, we](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-215-2048.jpg)
![192 9.5 Summary and Future Work
simply relied on the final recommendations of [109] for their CVRP. However, obtaining
the best perturbation parameters for the 1-PDP could allow the heuristic to achieve better
results, but will, nevertheless, be very time consuming.
Another possibility for improvement is to modify the EPS, such that only feasible solution
are allowed in the evolutionary process. Alternatively, the construction algorithm could
be slightly adjusted, such that it will continue inserting nodes in the solution even after
infeasibility is encountered. The objective function, then, should take into account the
amount of infeasibility in the solution. This approach could allow the perturbation scheme
to more easily transform the infeasibility to feasibility and improve the overall quality
of the generated solutions. The viability of this ‘modified’ construction technique was
demonstrated within the context of our VNS heuristic that was applied to the 1-PDP, as
will be detailed in the next chapter.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-216-2048.jpg)
![193
Chapter 10
Solving the One-Commodity Pickup
and Delivery Problem Using an
Adaptive Hybrid VNS/SA Heuristic
In our attempt to investigate possible heuristics to solve the 1-PDP, and perhaps achieve
better results than those achieved by the EPS heuristic, Variable Neighbourhood Search
(VNS) seemed a reasonable choice that may have some potential. First, the idea is simple
and can be easily adapted to the problem in hand. Second, it has been successfully ap-
plied to many routing and scheduling problems, for example to the TSP in [70] and the
VRPTW in [19] and [119], and recently to the Periodic Vehicle Routing Problem with
Time Windows (PVRPTW) in [118]. In this chapter, we explain our investigation of VNS
for solving the 1-PDP. The VNS algorithm introduced here is hybridized with Simula-
ted Annealing (SA) to escape local optima. We also employ adaptation of some search
parameters for more efficient searching.
Section 10.1 briefly describes the VNS meta-heuristic in general, and highlights previous
1-PDP research that uses this approach. Section 10.2 explains in detail our proposed
heuristic, which we will call an Adaptive Hybrid VNS/SA (AVNS-SA) technique for
solving the 1-PDP. An outline of the complete algorithm is presented in Section 10.3.
Experimental results of the algorithm are presented in Section 10.4, and a summary and
some future directions will be presented in Section 10.5.
10.1 Variable Neighbourhood Search (VNS) and its Ap-
plication to the 1-PDP
As previously mentioned in Section 2.3.6, VNS is a relatively new meta-heuristic that
has been introduced by Hansen and Mladenovi´c in [69] and [70]. The idea is to generate](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-217-2048.jpg)
![194 10.1 Variable Neighbourhood Search (VNS) and its Application to the 1-PDP
new solutions that are distant from the incumbent solution, by systematically increasing
the neighbourhood size within which the search is performed. The new solution replaces
the current solution if it is better in quality. This way, many favourable characteristics
of the incumbent solution will be preserved in the new generated solution. In addition, a
local search is performed on the new solution to reach a local optimum within the current
neighbourhood. For convenience, we repeat here the basic steps of the VNS algorithm as
described in [70]:
• Initialization: Select the set of neighbourhood structures Nk, (k = 1, ...kmax), that
will be used in the search; find an initial solution x; choose a stoping condition;
• Repeat the following until the stopping condition is met:
1. Set k ← 1;
2. Repeat the following steps until k = kmax:
(a) Shaking: Generate a point x′
at random from the kth
neighbourhood of
x (x′
∈ Nk(x));
(b) Local Search: Apply some local search method with x′
as initial solution;
denote with x′′
the so obtained local optimum;
(c) Move or not: if the local optimum x′′
is better than the incumbent, move
there (x ← x′′
), and continue the search with N1 (k ← 1); otherwise set
k ← k + 1.
A variant of VNS, called Variable Neighbourhood Descent (VND)1
, has been tried for
the 1-PDP as part of the heuristic proposed by Hernández-Pérez and Salazar-González in
[73]. In their approach, the VND did not include a shaking phase, but only a local search
that changes the neighbourhood move from 2-Opt to 3-Opt, whenever a local optimum is
reached. The VND was embedded within another heuristic called GRASP (Greedy Ran-
domized Adaptive Search Procedure) that repeatedly generates an initial solution upon
which the VND is applied. Their approach is described in Algorithms 9.1 and 9.2 in the
previous chapter. The heuristic in [73] achieved promising results that were better than
the results obtained by the same authors using their previous heuristics suggested in [75].
However, their algorithm apparently was not fully capable of escaping the trap of local
optima. This is evident by the fact that they had to use a post-optimization phase to im-
prove the final result. According to the authors, this post-optimization often made the
difference between beating the results obtained by their previous heuristic in [75] or not.
1
See section 2.3.6 for more details about the VND algorithm.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-218-2048.jpg)
![10.2 The AVNS-SA Heuristic 195
The hybrid GRASP/VND heuristic was also outperformed by the GA of [158], in most
test cases.
A possible shortcoming of the hybrid GRASP/VND heuristic is the absence of a shaking
phase, which should help the diversification of the incumbent solution and allow the suc-
ceeding local search method to escape local optima. In our proposed approach, we try to
apply the basic VNS, with both the shaking and the local search, hoping to overcome the
limitation of this previous VND attempt on the 1-PDP.
10.2 The AVNS-SA Heuristic
To apply VNS to the 1-PDP, several choices should be made. These are outlined and
explained below:
1. The construction heuristic used to create the initial solution: The construction
algorithm we used is the same as the algorithm proposed by [158], which is the
nearest-neighbour construction (NN-construction) heuristic, explained in Section
9.4.3.
2. The set of neighbourhood structures used for shaking: This is in fact the core
of the VNS, and is the tool for diversification of the search. It is essential that the
shaking procedure provides a balance between enough perturbation of the current
solution, and also maintaining the most favourable characteristics of this solution.
The shaking procedure should also allow the systematic change of the neighbou-
rhood size. In our algorithm, we chose a simple move that displaces a sequence of
nodes with or without inversion2
, where the number of nodes to be displaced could
act as the variable neighbourhood size parameter that changes during the search.
3. The local search procedure used to optimize the current solution: This proce-
dure is the tool that the VNS uses for intensification of the search. In our algorithm,
we chose as a local search a simple 2-Opt edge exchange algorithm, based on the
famous procedure proposed by Lin in [103]. We adopted the implementation des-
cribed in [157] for the 2-Opt edge exchanges, having a computational complexity
of O(n2
), where n is the number of nodes in the problem instance. In addition, our
local search is based on exhaustively testing all possible edge exchanges, and using
best improvement as a replacement strategy.
2
Inversion means to reverse the sequence of nodes in a manner similar to the inversion mutation operator
shown in Figure 2.7(b)](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-219-2048.jpg)
![196 10.2 The AVNS-SA Heuristic
4. The acceptance criterion used by the VNS to accept a new solution as a replace-
ment of the incumbent solution: A descent only criterion will only accept a better
solution, while a descent-ascent criterion will also accept a worse solution with
some probability. Both techniques can be used within the VNS procedure as explai-
ned in [70]. In our approach we chose the second criterion, such that the acceptance
is based on an SA acceptance probability, given a certain current temperature value.
In what follows we describe in detail how our proposed AVNS-SA approach was planned
and developed.
10.2.1 The Initial Solution
To choose the initial solution that will be subject to the VNS, a number of solutions were
created using the NN-construction heuristic (adopted from the algorithm in [158] and
explained in Section 9.4.3), and the best solution was selected. However, in order to create
different solutions, we re-introduced the randomness that has been removed in the EPS
heuristic, for reasons previously explained. Specifically, the starting node was chosen at
random, and also there was a small probability of choosing a random customer, rather than
the nearest customer, in Step 6 of this algorithm (refer to the steps of the NN-construction
algorithm in Section 9.4.3).
Also, rather than stopping the construction algorithm once infeasibility is encountered,
we changed the algorithm so that if all remaining nodes are infeasible for insertion, the
closest one to the last inserted node is selected, even if it is infeasible, hoping to fix the
infeasibility later during the search. Thus, all nodes will be inserted in the solution and the
degree of infeasibility is taken into consideration when calculating the cost of the solution.
The value of CL, which is the neighbourhood size used to create the matrix of nearest
neighbours to each node, was chosen to be 15 in the construction algorithm of [158]. In
our approach, though, we tried all values of CL from 2 to 15 on selected test cases from
both the small and the large size instances3
. The value of CL that worked best for all
problem instances tried was 7. Also, unlike the EPS heuristic, where a new neighbourhood
matrix was created with each set of perturbed coordinates, the current approach requires
only one static matrix to be created at the beginning of the run. Accordingly, a very ‘tight’
neighbourhood size was no longer needed to improve the run time4
.
For convenience, we show in detail below the steps of our construction algorithm:
3
We selected test cases: N20q10A, N60q10A, N100q10A, and N500q10A.
4
Recall that the value chosen for CL in the construction algorithm of our EPS heuristic, explained in
Chapter 9, was 2.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-220-2048.jpg)
![198 10.2 The AVNS-SA Heuristic
capacity violations, compared to feasible solutions, whose cost will only be measured
by the total distance traveled. In addition, no arbitrary fine tuning of penalty weights is
needed for this objective function.
10.2.3 The Initial SA Temperature
As previously mentioned, we used an SA acceptance criterion within the VNS approach.
A critical part of any SA algorithm is the SA schedule, especially the starting temperature
and the temperature reduction factor. These parameters significantly affect the perfor-
mance of the SA algorithm, and their adjustment to fit a wide range of problem instances
and sizes is a daunting task and very time consuming. To calculate the SA starting tem-
perature for each problem instance individually, we again adopted the approach proposed
by [40]. Recall that this approach was used in both our SA algorithms that handled the
SV-PDPTW and the MV-PDPTW, as explained in Chapters 5 and 8. The procedure for
creating the initial solution and calculating the SA temperature is shown in Algorithm
10.1.
Algorithm 10.1: Find Initial Solution & Calculate Starting Temperature.
1: Create a starting solution s using the NN-construction heuristic
2: s∗ ← s {s∗ is the best so far solution}
{Initialize Pstart, the starting acceptance probability}
3: Let Pstart = a large value {We used 0.9}
4: ∆avg ← 0
{Generate n solutions using the NN-construction (We used n = 1000)}
5: for (i = 0; i < n; i + +) do
6: generate a new solution s′ using the NN-construction heuristic
7: if (Objective(s′) < Objective(s∗)) then
8: s∗ ← s′
9: ∆ ← |Objective(s′) − Objective(s)|
10: ∆avg ← ∆avg + ∆
11: ∆avg ← ∆avg/n
12: T0 ← −∆avg/ log(Pstart)
13: Return s∗, the initial solution from which the AVNS-SA algorithm will progress, and the
initial SA temperature T0](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-222-2048.jpg)
![10.2 The AVNS-SA Heuristic 199
10.2.4 The Shaking Procedure
As mentioned previously, we chose as a shaking procedure a displacement of a sequence
of nodes with some probability of inverting this sequence (a 50% chance of inversion was
adopted in our algorithm). This move is popular in the VRP literature, and especially
for solving the TSP, but to the best of our knowledge it has not been tried before for the
1-PDP.
Our VNS algorithm passes the current neighbourhood size (NhSize) as a parameter to
the shaking procedure, which will in turn use this parameter as the maximum possible
number of nodes that will be displaced. Specifically, the number of nodes to be displaced
is a random number between 1 and NhSize. So even for large values of NhSize, small
sequences of nodes could still be displaced, and in fact there is a ‘bias’ toward such small
moves, since they have a chance of being executed in all neighbourhood sizes. This is
intended to prevent a large disruption of the current solution, and is recommended by
some VNS implementations, as in [119] for the VRPTW. Both the starting position of the
selected sequence and its new position within the route are chosen at random.
10.2.5 The Maximum Neighbourhood Size
Our VNS algorithm repeats the shaking followed by the local search for the current solu-
tion for all NhSize = 1, 2, 3...NhSizemax, where NhSizemax is the maximum sequence
of nodes that could be displaced. NhSizemax should be chosen in a way that allows
enough perturbation of the solution without disturbing its favourable characteristics. Na-
turally, the value for NhSizemax in our shaking procedure must be smaller than n, where
n is the total number of nodes in the current problem instance. Also, to make NhSizemax
adaptable for any problem instance, it must be calculated relative to n, and not fixed for
all instances. However, large values of NhSizemax will increase the computational cost
and slow down the optimization process. So, in order to reduce the computational cost,
we tried different fractions of n (n/2, n/3, n/4), on selected problem instances from both
the small and large problem instances5
. These values of NhSizemax, though, still resul-
ted in a very long processing time for large problem instances. We then tried the value
2 ×
√
n, which was recommended by [106], for the number of changes (nChangesmax)
in their Iterated Modified Simulated Annealing (IMSA) approach to the 1-PDP, as pre-
viously explained in Section 9.2. NhSizemax = 2 ×
√
n gave the best results among all
values tested, in terms of the balance between processing time and solution quality, for all
instances tried in this experiment.
5
Again we used test cases: N20q10A, N60q10A, N100q10A, and N500q10A.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-223-2048.jpg)
![204 10.4 Experimental Results
10.4 Experimental Results
To test the performance of our AVNS-SA algorithm, we used the same test cases described
in Section 9.4.6. We ran the algorithm 5 times on each test case from 20-300 customers.
On the other hand, only one run was performed on test cases of 400 and 500 customers,
due to time limitation. In this experiment a number of computers with different specifica-
tions were used to run the algorithm. For this reason, the run times we quote in this section
will vary according to the platform. Nevertheless, our timings give useful estimates of the
time requirements of the algorithm.
Table 10.1 shows the results achieved by the AVNS-SA algorithm for small size problems
of 20-60 customers. As previously explained in Section 9.4.6, a problem instance named
N20q10A, for example, means the first instance (A) in the 20-customers category (N20),
with vehicle capacity 10 (q10). Table 10.1 shows the best result obtained in the 5 runs,
the number of times the best result appeared in the 5 runs (Num Seen), and the average
result of the 5 runs. Finally the table also shows the optimum result found by the exact
algorithm in [74].
Table 10.1: AVNS-SA Results (20-60 Customers)
Name Best Num Seen Average Optimum
N20q10A 4963 4 4974.6 4963
N20q10B 4976 5 4976 4976
N20q10C 6333 3 6390.4 6333
N20q10D 6280 4 6341 6280
N20q10E 6415 5 6415 6415
N20q10F 4805 3 4808.6 4805
N20q10G 5119 5 5119 5119
N20q10H 5594 5 5594 5594
N20q10I 5157 2 5195.2 5130
N20q10J 4410 5 4410 4410
N30q10A 6403 1 6455 6403
N30q10B 6603 5 6603 6603
N30q10C 6486 1 6576.2 6486
N30q10D 6652 2 6746 6652
N30q10E 6070 5 6070 6070
N30q10F 5737 4 5817.4 5737
N30q10G 9371 1 9388.2 9371
Continued on Next Page. . .](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-228-2048.jpg)
![206 10.4 Experimental Results
Table 10.1 shows that the algorithm was able to achieve the optimum results at least once
in the 5 runs for 39 out of the 50 test cases. These are shown in boldface in the table. The
maximum relative difference to the optimum was less than 2% among the 11 cases where
the optimum was not found, which was for test case N50q10D. The processing time was
reasonable with an average ranging from 0.66 seconds for 20 customers problems to 47.79
seconds for 60 customers problems.
Table 10.2 shows the results of the AVNS-SA algorithm on large size problems, from 100
to 500 customers. The table shows the best result achieved and the average result of the
5 runs. Note that this average is replaced by the best result for problems of size 400 and
500, since the algorithm was run only once on these problems. Finally, the previous best
known results are also shown in the last column. Only 6 out of the previous best known
results are attributed to [73], which are the results for test cases (N300Q10C, N400Q10A,
N500Q10A, N500Q10D, N500Q10E, N500Q10H). All remaining previous best results
were found by the GA in [158]. The best result achieved by the AVNS-SA is shown in
boldface if it was better than the previous best known result.
Table 10.2: AVNS-SA Results (100-500 customers)
Name Best Average Previous Best
N100q10A 11741 12173.8 11828
N100q10B 13066 13410.6 13114
N100q10C 13893 14073.8 13977
N100q10D 14328 14567.2 14253
N100q10E 11430 11823.6 11411
N100q10F 11813 11947 11644
N100q10G 12025 12118 12038
N100q10H 12821 12844 12818
N100q10I 14025 14278.6 14032
N100q10J 13476 13642.8 13297
N200q10A 17690 17849.2 17686
N200q10B 17618 17887.8 17798
N200q10C 16535 16626.6 16466
N200q10D 21228 21594.2 21306
N200q10E 19220 19485.2 19299
N200q10F 21627 21677.4 21910
N200q10G 17361 17634 17712
N200q10H 20953 21191.4 21276
Continued on Next Page. . .](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-230-2048.jpg)
![208 10.4 Experimental Results
Table 10.2 shows that the AVNS-SA algorithm was able to improve the previous best
known results for 50% of the 100 test cases, 70% of the 200 test cases, 100% of the 300
test cases, 80% of the 400 test cases, and finally 100% of the 500 test cases.
The overall average of the results in the 5 runs for all test cases of size 100 is 13087.94,
which is only 1% worse than the average result of the GA in [158], having a value of
12954.16. Moreover, our overall average for the 200 test cases is 19151.44, while the
overall average of the heuristic in [158] for the same test cases in 10 runs is 19339.48,
i.e., our results account for an improvement of approximately 1%. On the other hand, the
overall average of our results for the 300 test cases was 23683.12, with an improvement of
more than 2% compared the overall average of their results for the same test cases, which
is 24224.28.
Also, our average for the results of the 10 test cases of size 400 was 26059.3, which
accounts for an approximately 2% improvement over the average of the best results of
[158], having the value 26490.4. In addition, our average result of the 10 test cases of size
500 was 29579.4. This is an improvement of approximately 3% over the average of the
best results of [158], having the value 30377.1 for the same instances. These results also
indicate that our algorithm performs even better on larger size problems. The average
processing time in this experiment ranged from approximately 542.22 seconds for 100
customers instances to 151103.04 seconds for 500 customers instances.
To further test the robustness of the AVNS-SA algorithm, we performed an additional
experiment by running the algorithm on 100-customers problems with a vehicle capacity
of 20 and 40. The algorithm was run 10 times on each test case. The results of this
experiment is shown in table 10.3. The table shows the best result achieved among the
10 runs, the average result of the 10 runs, and the best known results published in [73].
Results of the algorithm that are the same or better than the best known results are shown
in boldface.
The results in the table show that the algorithm was able to achieve better than the previous
best known results in 4 out of 10 test cases, for vehicle capacity Q = 20. It was also able
to achieve the same result as the best known result for 6 out of 10 cases, for vehicle
capacity Q = 40. The later results might as well be the optimum results, since they have
also appeared as best results in the heuristic of [75], and each one appeared more than
once among our 10 runs experiment. The average processing time for Q = 20 instances
was 155.76 seconds, and for Q = 40 instances was 146.17 seconds.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-232-2048.jpg)
![210 10.5 Summary and Future Work
0 10 20 30 40 50 60 70 80 90 100 110
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
x 10
4
Iteration
Score
SA
HC
Figure 10.1: SA against HC for N100q10A.
Contrary to the exceptional results achieved by our AVNS-SA algorithm, its processing
time in general was to a large extent disappointing. For example, the average processing
time for 100 customers problems was 542.22 seconds, while the processing time reported
by [158] was 21.12 seconds for the same problem category.
10.5 Summary and Future Work
In this part of the research, we investigated a VNS approach to the 1-PDP problem. The
algorithm is distinguished by performing the VNS repeatedly, each time starting from the
final solution obtained in the previous run. Also, the algorithm is adaptive, in the sense
that the maximum neighbourhood size allowed in each VNS run is not fixed and depends
on the current stage of the run. Early runs are allowed to perform wider jumps in the
solution space from the current solution, using large neighbourhood sizes. Later runs, on
the other hand, are only allowed smaller explorations of the search space, in the vicinity
of the current solution, to maintain the solution quality.
The stopping criterion for each VNS run is also adaptive and depends on the improve-
ment realized in the current solution. The VNS is terminated when a further increase the
neighbourhood size seems unhelpful. During each VNS run, an SA acceptance criterion
is used to allow the algorithm to escape local optima, and explore a wider area of the
search space. In addition, we used a new neighbourhood move in the shaking part of](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-234-2048.jpg)
![213
Chapter 11
The Research and Real-Life
Applications
Efficient transportation has become vital for today’s dynamic society. In the European
Union, the transportation sector constitutes more than 10% of Gross Domestic Product
(GDP) and employs more than 10 million people [1]. Transportation volume continues to
increase rapidly every day, as a result of economic growth and globalization, compared to
a limited expansion in roads and networks capacities.
Transportation demand is not always geographically balanced. In addition, the lack of
coordination between manufacturers, shippers and carriers in supply chains can lead to
inefficient usage of natural and human resources. In many cases, transportation plan-
ning is done manually. However, the advent of today’s technology- including high speed
computers, digital cellular phones, Geographic Information Systems (GIS), Geographic
Positioning Systems (GPS), navigation and tracking technologies, wireless data commu-
nication, digital mapping and web-based services- has increased the demand for more
efficient commercial software for route planning. If applied on a large scale, commercial
software can lead to enormous savings, both economically and environmentally. Vendors
of software tools claim that the reduction in cost may range from 5% - 30% [76]. Given
the huge volume of today’s transportation, such cost reduction is in fact very significant.
Besides cost reduction, efficient routing can greatly reduce the environmental impact of
transportation. For example, in a recent survey of UK brewery, Paragon Software Sys-
tems, Inc.1
identified savings of more than 2.5 million miles- corresponding to 3,700 tons
of CO2- a year, as a result of more efficient routing [68].
To meet this demand, research in vehicle routing and scheduling has grown substantially
in the last few decades [48]. As previously discussed in Chapter 3, a huge number of
problem variants, different problem constraints, and operating scenarios have been inves-
tigated. In fact, research in this field is central to the development of efficient decision
1
http://www.paragonrouting.com/](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-237-2048.jpg)
![214 11.1 Industrial Aspects of Vehicle Routing
support tools that can be adopted in the transportation industry. Despite this, research in
vehicle routing is often accused of being too idealistic. The majority of published research
tackles simplified problems, based on, for example, Euclidean distances, homogeneous
fleets, hard constraints, fixed service times. . . etc. Unfortunately, most of these assump-
tions do not hold in reality. Industrial aspects of vehicle routing have recently started to
gain the attention of researchers, and are increasingly being incorporated into models and
solution methods that address these problems. Thus, the current trend is towards ‘holistic’
approaches that are capable of solving richer and more realistic VRP models.
It is important from a realistic perspective to understand the relationship between theoreti-
cal research and commercial applicability in every day business requirements. To this end,
we discuss in this chapter some industrial aspects that may be considered when addressing
the VRP and its related variants in Section 11.1. Section 11.2 describes the basic compo-
nents of commercial software that are adopted in the transportation industry, and explains
how a theoretical research, like ours, may be integrated within its framework. Examples
of commercial software products and applications in the transportation sector are provided
in Section 11.3. Section 11.4 summarizes some future trends in vehicle routing research.
Finally, Section 11.5 concludes this discussion with a summary and some brief remarks.
Most of the information presented in this chapter is based on the research by SINTEF
research organization in Norway2
, published in [72] and [76], unless otherwise indicated.
11.1 Industrial Aspects of Vehicle Routing
As mentioned above, research in vehicle routing is now shifting towards solving non-
standard and rich VRP models that will facilitate decision making in real-life situations.
The advancement in computational power in the last few decades has encouraged resear-
chers to consider industrial aspects of vehicle routing, in order to meet the demands of
transportation service providers and fleet management companies. Rich VRP models al-
low general and more realistic features to be incorporated, as opposed to conventional
OR models which are simplistic in nature. Some industrial aspects of vehicle routing are
summarized below:
• Heterogeneous Fleet: most of the VRP research makes the assumption that the
operating fleet is homogeneous, with identical characteristics and operating costs.
In reality, though, this is often not the case, since companies usually benefit from
2
http://www.sintef.no/Home/](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-238-2048.jpg)
![216 11.2 Commercial Transportation Software
specific loading sequence may be enforced in order to facilitate un-loading, or to
protect fragile items.
• Uncertainty and Dynamic Situations: dynamic vehicle routing refers to the situa-
tion where routing decisions are affected by input data arriving in real time. Infor-
mation about orders, travel times, service times, vehicle breakdown. . . etc, arriving
while the routing plan is being executed should be taken into consideration in most
real-life applications, and necessitates an immediate response time. Uncertainty
and stochastic variables, for example while planning a certain emergency situation,
also add to the complexity of rich VRP models.
11.2 Commercial Transportation Software
A commercial software package for decision support in the transportation industry usually
integrates an underlying algorithm with an efficient user friendly interface for optimum
usage. A survey of commercial vehicle routing software [67] identifies the basic software
capabilities as:
1. Geocoding addresses using a digital map database, i.e., determining the coordinates
of a location using its address or postal code;
2. Determining the best driving route between pairs of geocoded pints;
3. Solving the VRP, i.e., assigning stops to vehicles and routing vehicles between
stops, and
4. Displaying the results in both graphical and tabular forms, such that the dispatcher
can communicate the solution to the drivers, and edit these solutions, with a ‘drag-
and-drop’ feature if necessary.
Step 3 in the above list is where theoretical research, such that presented in this thesis,
takes its part. It is in fact the ‘core’ or the ‘engine’ of the software tool, sometimes
referred to as aVRP solver. Hasle and Kloster in [72] define a VRP solver as:
A software component with functionality for modeling instances of targeted variants
of the VRP and finding feasible, optimized solutions to a given instance. The effect
of a given routing tool is highly dependent on its VRP solver.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-240-2048.jpg)
![11.3 Examples of Commercial Vehicle Routing Tools 217
The efficiency of a VRP solver can be generally attributed to the richness of the VRP
model (problem) it is trying to solve, in addition to its algorithmic capabilities measured
in terms of the quality of the objective and the processing time needed. To the best of our
knowledge, most VRP solvers of today’s commercial software do not adequately handle
pickup and delivery problems. Improvements in this area are obviously in demand.
11.3 Examples of Commercial Vehicle Routing Tools
Some examples of commercial vehicle routing software are: ILOG Dispatcher, Paragon
Routing and Scheduling System, Direct Route, DISC, and JOpt.SDK. Notable installa-
tions of these and other commercial software include companies like: Sainsbury’s, Argos,
Tesco, Royal Mail, the Home Depot, Samsung, Kraft, Dunkin Donuts, Coca-Cola, BP,
TNT, Fujitsu, and many others [68].
One example of a rich generic VRP solver is SPIDER [72], which was developed by
SINTEF Applied Mathematics research institution in Norway3
. SPIDER VRP solver is
capable of solving a number of vehicle routing problems and its variants, such as the
VRPTW, the PDPTW and the multiple-depot variants of these problems. In addition,
it takes into consideration many industrial aspects, such as the ones described above.
For example, it allows a heterogeneous fleet, multiple time windows, alternative service
locations, variable service times, and travel times that vary according to road network
topology and information available from Geographic Information Systems (GIS).
The algorithmic approach in SPIDER is a unified approach to all problem types and ins-
tances. This technique has advantages in terms of simplifying the code and its mainte-
nance, but may sometimes suffer in terms of computation time, since some operations
will still be performed for problem instances that may not need them.
The SPIDER VRP solver is basically a meta-heuristic approach that integrates several
features from successful academic research in the VRP field. The algorithm consists of
three main components: Construction of Initial Solutions, Tour Depletion and Iterative
Improvement. These are briefly explained below:
• Construction of Initial Solutions: the construction phase is based on extensions
of classical construction heuristics like the Clark and Wright savings heuristic [26],
Solomon’s I1 insertion [141], and the regret-based insertion [121]. In addition, an
3
http://www.spidersolutions.no/](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-241-2048.jpg)
![218 11.4 Future Trends
instance analysis is performed in order to determine if the instance has a heteroge-
neous fleet, in which case a special construction heuristic called SPIDER construc-
tor is used to build the initial solution.
• Tour Depletion: this phase is intended for reducing the number of routes in the
initial solution, but is also used as a local search operator during the iterative impro-
vement phase. Each route is depleted in turn, and an attempt is made to insert all
its requests into other routes. If the attempt was successful, the new solution is ac-
cepted. The routes that have been changed by this operation may also be optimized
using 2-Opt or 3-Opt improvement heuristics of [103].
• Iterative Improvement: the iterative improvement phase is based on Variable
Neighbourhood Descend (VND) [70], using a number of well-known operators wi-
thin each route and in-between routes. For example, 2-Opt, 3-Opt, Or-Opt, EX-
CHANGE, and CROSS (see Section 3.4 for more details about some of these ope-
rators). However, these heuristics have been extended to accommodate the SPIDER
rich VRP model which allows heterogeneous fleets and multiple time windows. In
addition, several neighbourhood filters have been applied to accelerate the opti-
mization, for example by analyzing the current solution and exploring promising
moves only. When the VND reaches a local minimum, a diversification mechanism
is applied using Large Neighbourhood Search (LNS) [131]. The overall process is
a hybrid of VND and Iterated Local Search (ILS).
To evaluate its performance, SPIDER was tested on published benchmark data available
from the literature and compared favourably with state-of-the-art solution methods (see
[72] for some experimental results of SPIDER on published benchmark instances of the
PDPTW).
11.4 Future Trends
Scientific research in transportation optimization is an indispensable part of any commer-
cial software tool, and there is, and will continue to be, a great demand for innovations
in research methodologies. Researchers in this field, however, should also be aware of
new demands in the industrial sector and try to develop richer models in their research.
Issues like heterogeneous fleets, multiple tours, split deliveries, variable service times,
soft time windows, special vehicle equipment or driver certificates, dynamic route plan-
ning, and many other real-life requirements and constraints should motivate researchers
to invest more effort in developing efficient solution techniques that comply with today’s](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-242-2048.jpg)
![11.5 Summary and Conclusions 219
ever changing domain of industrial vehicle routing. The complexity of these models, ne-
vertheless, means that solving the problem to optimality is not an option in most practical
situations. In fact, heuristics, meta-heuristics and hybrid approaches dominate scientific
research in this field.
There is a trend in today’s research towards optimization tools that integrate the whole
supply chain rather than individual components. In addition, current concerns over global
warming has increased the demand towards rewarding lower carbon emissions and green
logistics. More robust planning in dynamic and stochastic situations is also gaining more
attention every day. Decision support tools that enable the service provider to choose bet-
ween more than one good solution is also gaining more popularity in the current research
environment.
Another important factor that needs urgent attention is the development of better bench-
mark test cases. Most of the benchmark cases available to the research community for
the VRP and its related variants are created randomly. In addition, they are often overly
simplified and do not reflect real-world cases. Using such test cases increases the risk
of over fitting, i.e., a great effort may be invested in developing solution methods that
produce good results on published benchmark instances only. Applying these methods
on a larger scale, though, may reveal their shortcomings. The research in [76] identifies
several features that test cases for the vehicle routing research should have. These are: 1)
they should be based on real-world data, 2) they should be as rich as possible, i.e., contain
sufficient details, 3) they should have a common format (e.g. XML), 4) solutions (and not
just their objective functions) should be published, and 5) test cases and their solutions
should be published in the Web.
11.5 Summary and Conclusions
This chapter reviewed some industrial aspects of vehicle routing that are currently un-
der consideration by the research community to meet the increase in demand for more
efficient transportation. The current trend is towards rich VRP models that can be used
in commercial decision support tools, in order to achieve better customer service, cost
reduction, and efficient resource management in transportation systems.
Yet, although scientific research is a major and important step towards a complete real-
life applicable solution to routing and scheduling problems, there is often a considerable
distance between theoretical research and practical applicability. Integrating theoretical
scientific research within a commercially applicable tool is usually done by consulting](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-243-2048.jpg)
![12.1 Research Summary and Contribution 223
1. We used our solution representation and neighbourhood moves to develop new
simple routing heuristics for the SV-PDPTW. The results obtained in this part of
the research were impressive, both in terms of the solution quality and processing
time. One particular approach appears to have the best potential for the SV-PDPTW,
the 3-stage SA routing heuristic. This approach was able to obtain results better
than those previously published in all test cases and also performed significantly
better than all other algorithms implemented in this part of our research. The ex-
perimental results and comparison with other heuristics indicate that the success of
this approach was mostly due to the guided neighbourhood moves that were adop-
ted to overcome the difficult time window constraint. Another heuristic which also
showed potential, especially in terms of processing time, is the Hill Climbing (HC)
routing heuristic. The HC algorithm also employed the same time window guided
neighbourhood moves during the search. The routing heuristics developed for the
SV-PDPTW can be easily integrated within real-world optimization tools that deal
with this problem. Two publications were made out of the results obtained, one late
breaking conference paper [79] and another journal paper [84].
2. Since there are no standard benchmark instances for the SV-PDPTW, we were able
in this research to create test cases that can be used as benchmark data and used by
researchers for testing their algorithms.
3. Based on the simple routing heuristics developed for the SV-PDPTW, we designed
and compared several solution construction methods for the MV-PDPTW. These
construction methods, especially the sequential construction approach (SEQ), can
be used in any heuristic or meta-heuristic that deals with the PDPTW and in the re-
lated dial-a-ride problem as well. The construction heuristics are distinguished by
their simplicity and ease in coding and application, compared to classical construc-
tion methods from the literature. This part of the research was published in [81].
4. We developed new problem-specific genetic operators and neighbourhood moves
for the MV-PDPTW. These operators use techniques developed in the first parts
of our research, i.e., for the SV-PDPTW and for the solution construction of the
MV-PDPTW. Our operators are characterized by the ability to create feasible solu-
tions throughout the search. The operators developed here can be easily adopted by
other GA approaches for different vehicle routing problems. For example, a cros-
sover that ranks routes to guide inheritance is applicable to any routing problem.
These operators can also be adapted and employed within different heuristics and
meta-heuristics for solving the problem, as done in this research by using similar
operators in both the GA and the SA for solving the MV-PDPTW. The research](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-247-2048.jpg)
![224 12.1 Research Summary and Contribution
dealing with the GA approach in this part of the thesis was published in [80].
5. We developed an interesting adaptive hybrid VNS-SA approach for the 1-PDP. The
algorithm is distinguished by adopting, for the first time, the traditional displace-
ment and inversion move from previous vehicle routing research as a ‘shaking’ pro-
cedure within the VNS. In addition, adaptation is introduced in a novel way within
the VNS meta-heuristic. Traditional VNS approaches usually increase the neigh-
bourhood size up to a certain pre-defined limit. In our approach, we adapted this
requirement, such that a further increase in the neighbourhood size is only applied
when it seems beneficial from a search perspective. The algorithm introduced here
proved its potential by beating previously best published results for 90% of the large
problem instances solved. Two conference papers covering this part of the research
have now been accepted for publication in GECCO2010 conference (as a late brea-
king abstract) [82], and the PPSN2010 conference [85]. In addition, a third journal
paper has been submitted to the Journal of Heuristics and is currently under review
[83].
The above achievements indicate that the techniques we developed to guide our heuristic
and meta-heuristic approaches were successful to a large extent in accomplishing the ob-
jectives of the research. Nevertheless, we can also identify a few parts of the research that
did not meet the anticipated standards. Three main areas have shown some shortcomings
and need further investigation. These are:
1. The improvement heuristics of the MV-PDPTW (explained in Chapter 8), since the
results obtained by both the GA and the SA algorithms were in general of lesser
quality than the best known results.
2. The evolutionary perturbation heuristic of the 1-PDP (explained in Chapter 9). Des-
pite its success on other VRPs, this technique did not obtain high quality solutions
for the 1-PDP. However, the approach did show promise in removing infeasibility.
3. The processing time needed by the VNS approach to the 1-PDP (explained in Chap-
ter 10), since the final solution obtained, albeit having a very good quality, needed
quite a long processing time.
The possible reasons behind these shortcomings have been addressed in their respective
chapters. Nevertheless, we will further elaborate on some future research directions that
may be pursued to remedy these shortcomings. To this end, Section 12.2 includes a brief
critical analysis of some parts of the current research and suggestions of additional work
that may be performed to complement the research carried out in this thesis.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-248-2048.jpg)
![12.2 Critical Analysis and Future Work 225
12.2 Critical Analysis and Future Work
Some aspects of this thesis where further work can be done are summarized below.
12.2.1 The SV-PDPTW
To complement the research done so far for this problem, more problem instances of
different sizes and different characteristics need to be created. For example, similar to
the benchmark instances of the VRPTW and the MV-PDPTW, a certain distribution of
nodes may be enforced, such as having some instances with clustered or partially clustered
locations. Also, problem instances may differ according to the width of the time window,
by having short or long schedule horizons.
In addition, a more thorough investigation of the approaches developed in this part of the
research may be undertaken by testing the algorithms on a larger number of test cases and
performing an in-depth statistical analysis of the results.
12.2.2 Solution Construction for the MV-PDPTW
Further investigation of the solution construction heuristics developed in this part of the
research may be carried out by implementing one or more ‘traditional’ construction me-
thods, such as Solomon’s I1 insertion heuristic [141], after adapting it for the PDPTW.
This should then be followed by comparing the performance of the traditional heuristic(s)
with the new construction algorithms developed in this research, on benchmark instances.
12.2.3 Solution Improvement for the MV-PDPTW
Our selection of GAs as a candidate solution improvement method for the MV-PDPTW
was based on the general attractive features that GAs possess, such as simplicity and ro-
bustness. Moreover, we were also encouraged by the success of GAs in solving closely
related vehicle routing problems, like the VRPTW. As previously discussed during the
course of this thesis, though, GAs usually suffer when dealing the MV-PDPTW. This is
mostly due to the number of underlying constraints that must be dealt with when desi-
gning problem-specific genetic operators. Another important factor is the difficulty in
handling the grouping and the routing aspects of the problem simultaneously, in the solu-
tion encoding as well as the genetic operators. Similar difficulties also seem to apply to](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-249-2048.jpg)
![12.3 Final Remarks 227
annealing or a hill climbing algorithm may be used instead of a GA to improve the per-
turbed problem coordinates. Together, with an appropriate and fast construction heuristic
the overall scheme may be considerably enhanced.
12.2.5 The AVNS-SA Approach to the 1-PDP
The AVNS-SA approach was in fact very successful in dealing with the 1-PDP, especially
for large problem sizes. As previously mentioned, the algorithm was able to obtain new
best results for 90% of the large problem instances. The only drawback of the algorithm
seems to be the long processing time. As previously discussed in Chapter 10, an improve-
ment in this area can be achieved if we direct our attention to the local search component
of the algorithm. The 2-Opt heuristic used in the local search phase is the most time
consuming part of the algorithm. This heuristic adopts a best improvement strategy and
is invoked to optimize every new generated solution. Several suggestions may help in this
respect. One option is to reduce the number of calls to the 2-Opt algorithm based on some
solution quality. For example, the 2-Opt procedure can be restricted only to solutions that
look promising from a search perspective (depending on the objective function value or
the number of constraint violations, for instance). In addition, the 2-Opt algorithm may
be applied to selected edges of the solution and not to the whole solution. Some faster
implementations of the 2-Opt algorithm may also be tried to reduce the time requirement
of this component (e.g. [13]).
Finally, the stopping condition of the overall AVNS-SA algorithm also needs further in-
vestigation. To improve the processing time of the algorithm, it is necessary to find an
appropriate criterion to stop repeating the AVNS-SA procedure, without sacrificing the
quality of the final solution returned. However, obtaining a balance between solution
quality and processing time may not be very straightforward for such hard problem.
12.3 Final Remarks
To sum up, we believe that the work done in this thesis contributes positively to scien-
tific research on vehicle routing and scheduling. This study provided simple ideas on
constraint handling mechanisms that can be used in designing effective and robust heuris-
tic and meta-heuristic algorithms. The techniques developed in this research can be easily
integrated within a larger framework and used in optimization tools to help improve trans-
portation and logistic planing.](https://image.slidesharecdn.com/10-171221184752/75/10-1-1-866-373-251-2048.jpg)
![229
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- 1. Investigating Heuristic and Meta-HeuristicAlgorithms for Solving Pickup and Delivery Problems A thesis submitted in partial fulfilment of the requirement for the degree of Doctor of Philosophy Manar Ibrahim Hosny March 2010 Cardiff University School of Computer Science & Informatics
- 3. iii Declaration This work hasnot previously been accepted in substance for any degree and is not concur- rently submitted in candidature for any degree. Signed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (candidate) Date . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statement 1 This thesis is being submitted in partial fulfillment of the requirements for the degree of PhD. Signed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (candidate) Date . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statement 2 This thesis is the result of my own independent work/investigation, except where other- wise stated. Other sources are acknowledged by explicit references. Signed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (candidate) Date . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statement 3 I hereby give consent for my thesis, if accepted, to be available for photocopying and for inter-library loan, and for the title and summary to be made available to outside organisa- tions. Signed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (candidate) Date . . . . . . . . . . . . . . . . . . . . . . . . . . . .
- 5. v Abstract The development ofeffective decision support tools that can be adopted in the trans- portation industry is vital in the world we live in today, since it can lead to substantial cost reduction and efficient resource consumption. Solving the Vehicle Routing Problem (VRP) and its related variants is at the heart of scientific research for optimizing logistic planning. One important variant of the VRP is the Pickup and Delivery Problem (PDP). In the PDP, it is generally required to find one or more minimum cost routes to serve a number of customers, where two types of services may be performed at a customer loca- tion, a pickup or a delivery. Applications of the PDP are frequently encountered in every day transportation and logistic services, and the problem is likely to assume even greater prominence in the future, due to the increase in e-commerce and Internet shopping. In this research we considered two particular variants of the PDP, the Pickup and Delivery Problem with Time Windows (PDPTW), and the One-commodity Pickup and Delivery Problem (1-PDP). In both problems, the total transportation cost should be minimized, without violating a number of pre-specified problem constraints. In our research, we investigate heuristic and meta-heuristic approaches for solving the selected PDP variants. Unlike previous research in this area, though, we try to focus on handling the difficult problem constraints in a simple and effective way, without compli- cating the overall solution methodology. Two main aspects of the solution algorithm are directed to achieve this goal, the solution representation and the neighbourhood moves. Based on this perception, we tailored a number of heuristic and meta-heuristic algorithms for solving our problems. Among these algorithms are: Genetic Algorithms, Simulated Annealing, Hill Climbing and Variable Neighbourhood Search. In general, the findings of the research indicate the success of our approach in handling the difficult problem constraints and devising simple and robust solution mechanisms that can be integrated with vehicle routing optimization tools and used in a variety of real world applications.
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- 7. vii Acknowledgements This thesis wouldnot have been possible without the help and support of the kind people around me. I would like to extend my thanks to all individuals who stood by me throu- ghout the years of my study, but I wish to particulary mention a number of people who have had a profound impact on my work. Above all, I am heartily thankful to my supervisor, Dr. Christine Mumford, whose en- couragement, guidance and support from start to end enabled me to develop my research skills and understanding of the subject. Her wealth of knowledge, sincere advice and friendship with her students made this research fruitful as well as enjoyable. It is an honour for me to extend my thanks to my employer King Saud University (KSU), Riyadh, Saudi Arabia, for offering my postgraduate scholarship, and providing all finan- cial support needed to complete my degree. I am also grateful to members of the IT De- partment of the College of Computers and Information Sciences at KSU, for their support and encouragement during the years of my PhD program. I would like to acknowledge the academic and technical support of the School of Compu- ter Science & Informatics at Cardiff University, UK and all its staff members. I am also grateful for the kindness, help and support of all my colleagues at Cardiff University. Last but by no means least, I owe my deepest gratitude to my parents, my husband, and my children for their unconditional support and patience from the beginning to the end of my study.
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- 9. ix Contents Abstract v Acknowledgements vii Contentsix List of Figures xv List of Tables xvii List of Algorithms xix Acronyms xxi 1 Introduction 1 1.1 Research Problem and Motivation . . . . . . . . . . . . . . . . . . . . . 2 1.2 Thesis Hypothesis and Contribution . . . . . . . . . . . . . . . . . . . . 3 1.3 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 A Note on Implementation and Computational Experimentation . . . . . 9 2 Combinatorial Optimization, Heuristics, and Meta-heuristics 11 2.1 Combinatorial Optimization, Algorithm and Complexity Analysis . . . . 11 2.2 Exact Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Heuristic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
- 10. x Contents 2.3.1 HillClimbing (HC) . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.2 Simulated Annealing (SA) . . . . . . . . . . . . . . . . . . . . . 20 2.3.3 Genetic Algorithms (GAs) . . . . . . . . . . . . . . . . . . . . . 24 2.3.4 Ant Colony Optimization (ACO) . . . . . . . . . . . . . . . . . . 28 2.3.5 Tabu Search (TS) . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3.6 Variable Neighbourhood Search (VNS) . . . . . . . . . . . . . . 34 2.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3 Vehicle Routing Problems: A Literature Review 37 3.1 Vehicle Routing Problems (VRPs) . . . . . . . . . . . . . . . . . . . . . 38 3.2 The Vehicle Routing Problem with Time Windows (VRPTW) . . . . . . 42 3.3 Solution Construction for the VRPTW . . . . . . . . . . . . . . . . . . . 44 3.3.1 Savings Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3.2 Time Oriented Nearest Neighbour Heuristic . . . . . . . . . . . . 46 3.3.3 Insertion Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.3.4 Time Oriented Sweep Heuristic . . . . . . . . . . . . . . . . . . 49 3.4 Solution Improvement for the VRPTW . . . . . . . . . . . . . . . . . . . 49 3.5 Meta-heuristics for the VRPTW . . . . . . . . . . . . . . . . . . . . . . 51 3.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4 Pickup and Delivery Problems: A Literature Review 57 4.1 The Vehicle Routing Problem with Backhauls (VRPB) . . . . . . . . . . 58 4.2 The Vehicle Routing Problem with Pickups and Deliveries (VRPPD) . . . 61 4.3 Meta-heuristic Algorithms for Pickup and Delivery Problems . . . . . . . 64 4.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
- 11. Contents xi 5 TheSV-PDPTW: Introduction and a GA Approach 69 5.1 Problem and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.2 The SV-PDPTW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.3 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.4 SV-PDPTW: Research Contribution . . . . . . . . . . . . . . . . . . . . 76 5.4.1 The Solution Representation . . . . . . . . . . . . . . . . . . . . 76 5.4.2 The Neighbourhood Move . . . . . . . . . . . . . . . . . . . . . 77 5.5 A Genetic Algorithm Approach for Solving the SV-PDPTW . . . . . . . 78 5.5.1 The Fitness Function . . . . . . . . . . . . . . . . . . . . . . . . 78 5.5.2 The Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.5.3 Test Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.5.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 86 5.6 Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 93 6 Several Heuristics and Meta-heuristics Applied to the SV-PDPTW 95 6.1 The Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.2 The Genetic Algorithm (GA) Approach . . . . . . . . . . . . . . . . . . 96 6.3 The 3-Stage Simulated Annealing (SA) Approach . . . . . . . . . . . . . 96 6.4 The Hill Climbing (HC) Approach . . . . . . . . . . . . . . . . . . . . . 100 6.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.6 The 3-Stage SA: Further Investigation . . . . . . . . . . . . . . . . . . . 109 6.7 Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 114 7 New Solution Construction Heuristics for the MV-PDPTW 117 7.1 Solution Construction for the MV-PDPTW . . . . . . . . . . . . . . . . . 118 7.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 7.3 Research Motivation and Objectives . . . . . . . . . . . . . . . . . . . . 121 7.4 The Routing Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
- 12. xii Contents 7.5 SolutionConstruction Heuristics . . . . . . . . . . . . . . . . . . . . . . 124 7.5.1 The Sequential Construction Algorithm . . . . . . . . . . . . . . 124 7.5.2 The Parallel Construction Algorithms . . . . . . . . . . . . . . . 125 7.6 Computational Experimentation . . . . . . . . . . . . . . . . . . . . . . 130 7.6.1 Characteristics of the Data Set . . . . . . . . . . . . . . . . . . . 130 7.6.2 Comparing the Construction Heuristics . . . . . . . . . . . . . . 131 7.6.3 Comparing with Previous Best Known . . . . . . . . . . . . . . . 134 7.7 The SEQ Algorithm: Complexity Analysis and Implementation Issues . . 137 7.8 Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 139 8 Two Approaches for Solving the MV-PDPTW: GA and SA 141 8.1 Research Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 8.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 8.3 A Genetic Algorithm (GA) for the MV-PDPTW . . . . . . . . . . . . . . 150 8.3.1 The Solution Representation and the Objective Function . . . . . 151 8.3.2 The Initial Population . . . . . . . . . . . . . . . . . . . . . . . 152 8.3.3 The Genetic Operators . . . . . . . . . . . . . . . . . . . . . . . 153 8.3.4 The Complete GA . . . . . . . . . . . . . . . . . . . . . . . . . 157 8.3.5 Attempts to Improve the Results . . . . . . . . . . . . . . . . . . 157 8.3.6 GA Experimental Results . . . . . . . . . . . . . . . . . . . . . 161 8.4 Simulated Annealing (SA) for the MV-PDPTW . . . . . . . . . . . . . . 167 8.5 Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 173 9 The 1-PDP: Introduction and a Perturbation Scheme 175 9.1 The 1-PDP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 9.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 9.3 Related Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 9.4 Solving the 1-PDP Using an Evolutionary Perturbation Scheme (EPS) . . 181
- 13. Contents xiii 9.4.1 TheEncoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 9.4.2 The Initial Population . . . . . . . . . . . . . . . . . . . . . . . 185 9.4.3 The Nearest Neighbour Construction (NN-Construction) . . . . . 185 9.4.4 The Objective Function . . . . . . . . . . . . . . . . . . . . . . . 187 9.4.5 The Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 9.4.6 Computational Experimentation . . . . . . . . . . . . . . . . . . 188 9.5 Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 191 10 Solving the 1-PDP Using an Adaptive VNS-SA Heuristic 193 10.1 Variable Neighbourhood Search (VNS) and its Application to the 1-PDP . 193 10.2 The AVNS-SA Heuristic . . . . . . . . . . . . . . . . . . . . . . . . . . 195 10.2.1 The Initial Solution . . . . . . . . . . . . . . . . . . . . . . . . . 196 10.2.2 The Objective Function . . . . . . . . . . . . . . . . . . . . . . . 197 10.2.3 The Initial SA Temperature . . . . . . . . . . . . . . . . . . . . 198 10.2.4 The Shaking Procedure . . . . . . . . . . . . . . . . . . . . . . . 199 10.2.5 The Maximum Neighbourhood Size . . . . . . . . . . . . . . . . 199 10.2.6 A Sequence of VNS Runs . . . . . . . . . . . . . . . . . . . . . 200 10.2.7 Stopping and Replacement Criteria for Individual VNS Runs . . . 200 10.2.8 Updating the SA Starting Temperature . . . . . . . . . . . . . . . 201 10.3 The Complete AVNS-SA Algorithm . . . . . . . . . . . . . . . . . . . . 202 10.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 10.5 Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 210 11 The Research and Real-Life Applications 213 11.1 Industrial Aspects of Vehicle Routing . . . . . . . . . . . . . . . . . . . 214 11.2 Commercial Transportation Software . . . . . . . . . . . . . . . . . . . . 216 11.3 Examples of Commercial Vehicle Routing Tools . . . . . . . . . . . . . . 217 11.4 Future Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
- 14. xiv Contents 11.5 Summaryand Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 219 12 Conclusions and Future Directions 221 12.1 Research Summary and Contribution . . . . . . . . . . . . . . . . . . . . 221 12.2 Critical Analysis and Future Work . . . . . . . . . . . . . . . . . . . . . 225 12.2.1 The SV-PDPTW . . . . . . . . . . . . . . . . . . . . . . . . . . 225 12.2.2 Solution Construction for the MV-PDPTW . . . . . . . . . . . . 225 12.2.3 Solution Improvement for the MV-PDPTW . . . . . . . . . . . . 225 12.2.4 The Evolutionary Perturbation Heuristic for the 1-PDP . . . . . . 226 12.2.5 The AVNS-SA Approach to the 1-PDP . . . . . . . . . . . . . . 227 12.3 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Bibliography 229
- 15. xv List of Figures 2.1Asymptotic Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Complexity Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Search space for a minimization problem: local and global optima . . . . 18 2.4 Hill Climbing: downhill transition to a local optimum . . . . . . . . . . . 19 2.5 Simulated Annealing: occasional uphill moves and escaping local optima 21 2.6 Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.7 Mutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.8 Partially Matched Crossover (PMX) . . . . . . . . . . . . . . . . . . . . 27 2.9 Foraging behaviour of ants . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.10 The problem of cycling . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.1 Vehicle routing and scheduling problems . . . . . . . . . . . . . . . . . . 39 3.2 Taxonomy of the VRP literature [48] . . . . . . . . . . . . . . . . . . . . 41 3.3 The vehicle routing problem with time windows . . . . . . . . . . . . . . 43 3.4 Solution Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.5 The savings heuristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.6 The insertion heuristic . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.7 2-exchange move . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.1 The single vehicle pickup and delivery problem with time windows . . . . 73 5.2 Solution Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
- 16. xvi List ofFigures 5.3 Time window precedence vector . . . . . . . . . . . . . . . . . . . . . . 80 5.4 Merge Crossover (MX1) . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.5 Pickup and Delivery Crossover (PDPX) . . . . . . . . . . . . . . . . . . 82 5.6 Directed Mutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.7 PDPX & Swap against PDPX & Directed - Task 200 . . . . . . . . . . . 91 5.8 MX1 & Swap against MX1 & Directed- Task 200 . . . . . . . . . . . . . 91 6.1 Average objective value for all algorithms . . . . . . . . . . . . . . . . . 106 6.2 HC against 3-Stage SA (SA1) for task 200 . . . . . . . . . . . . . . . . . 107 6.3 Random-move SA (SA2) against 3-Stage SA (SA1) for task 200 . . . . . 108 6.4 Average objective value for all sequences . . . . . . . . . . . . . . . . . 111 7.1 MV-PDPTW before solution (left) and after solution (right) . . . . . . . . 117 7.2 Solution Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 7.3 SEQ algorithm - average vehicle gap for all problem categories . . . . . . 136 7.4 SEQ algorithm - average distance gap for all problem categories . . . . . 136 8.1 Chromosome - grouping & routing . . . . . . . . . . . . . . . . . . . . . 147 8.2 Chromosome - grouping only . . . . . . . . . . . . . . . . . . . . . . . . 148 8.3 MV-PDPTW chromosome representation . . . . . . . . . . . . . . . . . 152 8.4 Vehicle Merge Mutation (VMM) . . . . . . . . . . . . . . . . . . . . . . 154 8.5 Vehicle Copy Crossover (VCX) . . . . . . . . . . . . . . . . . . . . . . 156 8.6 Total number of vehicles produced by all algorithms . . . . . . . . . . . . 164 8.7 Total travel distance produced by all algorithms . . . . . . . . . . . . . . 164 8.8 Average objective value for the GA and the SA algorithms . . . . . . . . 172 9.1 TSP solution before and after perturbation . . . . . . . . . . . . . . . . . 183 9.2 EPS chromosome representation . . . . . . . . . . . . . . . . . . . . . . 185 10.1 SA against HC for N100q10A . . . . . . . . . . . . . . . . . . . . . . . 210
- 17. xvii List of Tables 3.1Meta-heuristics for the VRPTW . . . . . . . . . . . . . . . . . . . . . . 54 3.2 Genetic Algorithms for the VRPTW . . . . . . . . . . . . . . . . . . . . 55 4.1 The Vehicle Routing Problem with Backhauls (VRPB) . . . . . . . . . . 60 4.2 The Vehicle Routing Problem with Pickups and Deliveries (VRPPD) . . . 63 4.3 Meta-heuristics for Pickup and Delivery Problems . . . . . . . . . . . . . 65 4.4 Genetic Algorithms for Pickup and Delivery Problems . . . . . . . . . . 66 5.1 GA Results Summary (Total Route Duration) . . . . . . . . . . . . . . . 88 5.2 GA Results Summary (Objective Function Values) . . . . . . . . . . . . 89 5.3 Average Number of Generations to Reach the Best Individual . . . . . . . 93 6.1 Results Summary for all Algorithms (Total Route Duration) . . . . . . . . 102 6.2 Results Summary for GA1 and GA2 (Objective Function Values) . . . . . 103 6.3 Results Summary for SA1, SA2 and HC (Objective Function Values) . . . 104 6.4 3-stage SA Progress for Task 200 . . . . . . . . . . . . . . . . . . . . . . 105 6.5 HC Progress for Task 200 . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6.6 Average Processing Time in Seconds . . . . . . . . . . . . . . . . . . . . 109 6.7 Average Objective Value for all Sequences . . . . . . . . . . . . . . . . . 110 6.8 Best and Worst Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.10 Average Processing Time in Seconds for all Sequences . . . . . . . . . . 112 6.9 Solution Progress through Stages . . . . . . . . . . . . . . . . . . . . . . 113
- 18. xviii List ofTables 7.1 Test Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 7.2 Frequency of Generated Best Solutions . . . . . . . . . . . . . . . . . . 132 7.3 Average Results for all Algorithms . . . . . . . . . . . . . . . . . . . . . 133 7.4 Average Relative Distance to Best Known . . . . . . . . . . . . . . . . . 135 7.5 Average Processing Time (seconds) of the SEQ Algorithm for all Tasks . 137 8.1 Comparison between the GRGA, GGA and CKKL Algorithms . . . . . . 162 8.2 GA & SA Best Results (100-Customers) . . . . . . . . . . . . . . . . . . 170 10.1 AVNS-SA Results (20-60 Customers) . . . . . . . . . . . . . . . . . . . 204 10.2 AVNS-SA Results (100-500 customers) . . . . . . . . . . . . . . . . . . 206 10.3 AVNS-SA Results (100 Customers with Q=20 and Q=40) . . . . . . . . . 209
- 19. xix List of Algorithms 2.1Hill Climbing (HC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 The Simulated Annealing (SA) Algorithm . . . . . . . . . . . . . . . . . 23 2.3 The Basic Genetic Algorithm (GA) . . . . . . . . . . . . . . . . . . . . . 25 2.4 The Ant Colony Optimization (ACO) Algorithm . . . . . . . . . . . . . . 30 2.5 The Tabu Search (TS) Algorithm . . . . . . . . . . . . . . . . . . . . . . 33 2.6 The Variable Neighbourhood Search (VNS) Algorithm [70] . . . . . . . 34 2.7 The Variable Neighbourhood Descent (VND) Algorithm [70] . . . . . . 35 5.1 Create Test Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2 Generate Time Windows . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.3 The SV-PDPTW Genetic Algorithm . . . . . . . . . . . . . . . . . . . . 86 6.1 Calculating Annealing Parameters [40] . . . . . . . . . . . . . . . . . . . 98 6.2 The Main SA Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.3 The Main HC Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 100 7.1 The HC Routing Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 123 7.2 The Sequential Construction . . . . . . . . . . . . . . . . . . . . . . . . 125 7.3 Parallel Construction: First Route . . . . . . . . . . . . . . . . . . . . . 127 7.4 Parallel Construction: Best Route . . . . . . . . . . . . . . . . . . . . . 128 7.5 Parallel Construction: Best Request . . . . . . . . . . . . . . . . . . . . 129 7.6 Basic Construction Algorithm for the VRP [24] . . . . . . . . . . . . . . 138
- 20. xx List ofAlgorithms 8.1 The Complete Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . 157 8.2 The 2-Stage SA Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 169 9.1 VND(x) Procedure [73] . . . . . . . . . . . . . . . . . . . . . . . . . . 179 9.2 Hybrid GRASP/VND Procedure [73] . . . . . . . . . . . . . . . . . . . 179 9.3 The EPS Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 10.1 Find Initial Solution & Calculate Starting Temperature . . . . . . . . . . 198 10.2 Adaptive VNS-SA (AVNS-SA) Algorithm . . . . . . . . . . . . . . . . . 202 10.3 The VNSSA Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
- 21. xxi Acronyms 1M One LevelExchange Mutation 1-PDP One Commodity Pickup and Delivery Problem ACO Ant Colony Optimization ALNS Adaptive Large neighbourhood Search AVNS-SA Adaptive Variable neighbourhood Search - Simulated Annealing BF Best Fit CDARP Capacitated Dial-A-Ride Problem CEL centre TW - Early TW - Late TW CKKL Evolutionary Approach to the MV-PDPTW in [33] CL Number of Closest neighbours CLE centre TW - Late TW - Early TW CPP Chinese Postman Problem CTSPPD Capacitated Traveling Salesman Problem with Pickup and Delivery CV Capacity Violations CVRP Capacitated Vehicle Routing Problem DARP Dial-A-Ride Problem DLS Descent Local Search DP Dynamic Programming ECL Early TW - centre TW - Late TW ELC Early TW - Late TW - centre TW EPA Exact Permutation Algorithm EPS Evolutionary Perturbation Scheme FCGA Family Competition Genetic Algorithm GA Genetic Algorithm GALIB Genetic Algorithm Library GDP Gross Domestic Product GGA Grouping Genetic Algorithm GIS Geographic Information Systems GLS Guided Local Search
- 22. xxii Acronyms GPS GeographicPositioning Systems GRASP Greedy Randomized Adaptive Search Procedure GRGA Grouping Routing Genetic Algorithm GRS Greedy Random Sequence GUI Graphical User Interface HC Hill Climbing HPT Handicapped Person Transportation IBX Insertion-Based Crossover ILS Iterated Local Search IMSA Iterated Modified Simulated Annealing IRNX Insert-within-Route Crossover LC1 Li & Lim benchmark instances [100]- clustered customers- short TW LC2 Li & Lim benchmark instances [100]- clustered customers- long TW LCE Late TW - centre TW - Early TW LEC Late TW - Early TW -centre TW LNS Large neighbourhood Search LR1 Li & Lim benchmark instances [100]- random customers- short TW LR2 Li & Lim benchmark instances [100]- random customers- long TW LRC1 Li & Lim benchmark instances [100]- random & clustered customers - short TW LRC2 Li & Lim benchmark instances [100] - random & clustered customers - long TW LS Local Search LSM Local Search Mutation MSA Modified Simulated Annealing MTSP Multiple Vehicle Traveling Salesman Problem MV Multiple Vehicle MV-PDPTW Multiple Vehicle Pickup and Delivery Problem with Time Windows MX1 Merge Crossover NN Nearest neighbour N P Non Deterministic Polynomial Time Problems OR Operations Research OR/MS Operations Research/ Management Science P Polynomial Time Problems PBQ Parallel Construction Best Request PBR Parallel Construction Best Route PD Pickup and Delivery
- 23. Acronyms xxiii PDP Pickupand Delivery Problem PDPTW Pickup And Delivery Problem with Time Windows PDPX Pickup And Delivery Crossover PDTSP Pickup And Delivery Traveling Salesman Problem PDVRP Pickup And Delivery Vehicle Routing Problem PF Push Forward PFR Parallel Construction First Route PMX Partially Matched Crossover PS Predecessor-Successor PTS Probabilistic Tabu Search PVRPTW Periodic Vehicle Routing Problem with Time Widows QAP Quadratic Assignment Problem RBX Route Based Crossover RCL Restricted Candidate List SA Simulated Annealing SAT Boolean Satisfiability Problem SBR Swapping Pairs Between Routes SBX Sequence Based Crossover SDARP Single Vehicle Dial-A-Ride Problem SEQ Sequential Construction SPDP Single Vehicle Pickup and Delivery Problem SPI Single Paired Insertion SV Single Vehicle SV-PDPTW Single Vehicle Pickup and Delivery Problem with Time Windows TA Threshold Accepting TS Tabu Search TSP Traveling Salesman Problem TSPB Traveling Salesman Problem with Backhauls TSPCB Traveling Salesman Problem with Clustered Backhauls TSPDDP Traveling Salesman Problem with Divisible Delivery and Pickup TSPMB Traveling Salesman Problem with Mixed Linehauls and Backhauls TSPPD Traveling Salesman Problem with Pickup and Delivery TSPSDP Traveling Salesman Problem with Simultaneous Delivery and Pickup TW Time Window TWV Time Window Violations UOX Uniform Order Based Crossover
- 24. xxiv Acronyms VCX VehicleCopy Crossover VMM Vehicle Merge Mutation VMX Vehicle Merge Crossover VND Variable neighbourhood Descent VNS Variable neighbourhood Search VRP Vehicle Routing Problem VRPB Vehicle Routing Problem with Backhauls VRPCB Vehicle Routing Problem with Clustered Backhauls VRPDDP Vehicle Routing Problem with Divisible Delivery and Pickup VRPMB Vehicle Routing Problem with Mixed Linehauls and Backhauls VRPPD Vehicle Routing Problem with Pickups and Deliveries VRPSDP Vehicle Routing Problem with Simultaneous Delivery and Pickup VRPTW Vehicle Routing Problem with Time Widows WRI Within Route Insertion
- 25. 1 Chapter 1 Introduction Efficient transportationand logistics plays a role in the economic wellbeing of society. Almost everything that we use in our daily lives involves logistic planning, and practically all sections of industry and society, from manufacturers to home shopping customers, require the effective and predictable movement of goods. In today’s dynamic business environment, transportation cost constitutes a significant percentage of the total cost of a product. In fact, it is not unusual for a company to spend more than 20% of the product’s value on transportation and logistics [76]. In addition, the transportation sector itself is a significant industry, and its volume and impact on society as a whole continues to increase every day. In a 2008 report, the UK Department of Transport estimates that freight and logistics sector is worth £74.5 billion to the economy and employs 2.3 million people across 190,000 companies [2]. In the last few decades, advancement in transportation and logistics has greatly improved people’s lives and influenced the performance of almost all economic sectors. Never- theless, it also produced negative impacts. Approximately two thirds of our goods are transported through road transport [2]. However, vehicles moving on our roads contribute to congestion, noise, pollution, and accidents. Reducing these impacts requires coopera- tion between the various stakeholders in manufacturing and distribution supply chains for more efficient planning and resource utilization. Efficient vehicle routing is essential, and automated route planning and transport management, using optimization tools, can help reduce transport costs by cutting mileage and improving driver and vehicle usage. In ad- dition, it can improve customer service, cut carbon emissions, improve strategic decision making and reduce administration costs. Research studying effective planning and optimization in the vehicle routing and sche- duling field has increased tremendously in the last few decades [48] . Advances in tech- nology and computational power has encouraged researchers to consider various problem types and real-life constraints, and to experiment with new algorithmic techniques that can be applied for the automation of vehicle planning activities. A number of these techniques have been implemented in commercial optimization software and successfully used in
- 26. 2 1.1 ResearchProblem and Motivation every day transportation and logistics applications. However, due to the large increase in demand and the growing complexity of this sector, innovations in solution techniques that can be used to optimize vehicle routing and scheduling are still urgently needed. Our research is concerned with developing efficient algorithmic techniques that can be applied in vehicle routing and planning activities. In Section 1.1 we introduce the research problems that we are trying to tackle and emphasize the motivation behind our selection of these problems. Section 1.2 highlights the hypothesis and contribution of the thesis. Section 1.3 gives an overview of the contents of this thesis and indicates the publications related to each part of the research. Finally, some brief remarks about the computational experimentation performed in this research are presented in Section 1.4. 1.1 Research Problem and Motivation The Vehicle Routing Problem (VRP) is at the heart of scientific research involving the distribution and transportation of people and goods. The problem can be generally des- cribed as finding a set of minimum cost routes for a fleet of vehicles, located at a central depot, to serve a number of customers. Each vehicle should depart from and return to the same depot, while each customer should be visited exactly once. Since the introduction of this problem by Dantzig and Fulkerson in 1954 [35] and Dantzig and Ramser in 1959 [35], several extensions and variations have been added to the basic VRP model, in or- der to meet the demand of realistic applications that are complex and dynamic in nature. For example, restricting the capacity of the vehicle and specifying certain service times for visiting clients are among the popular constraints that have been considered while addressing the VRP. One important variant of the VRP is the Pickup and Delivery Problem (PDP). Unlike the classical VRP, where all customers require the same service type, a central assumption in the PDP is that there are two different types of services that can be performed at a customer location, a pickup or a delivery. Based on this assumption, other variants of the PDP have also been introduced depending on, for example, whether the origin/destination of the commodity is the depot or some customer location, whether one or multiple commodities are transferred, whether origins and destinations are paired, and whether people or goods are transported. In addition, some important real-life constraints that have been applied to the basic VRP have also been applied to the PDP, such as the vehicle capacity and service time constraints. Applications of the PDP are frequently seen in transportation and logistics. In addition,
- 27. 1.2 Thesis Hypothesisand Contribution 3 the problem is likely to become even more important in the future, due to the rapid growth in parcel transportation as a result of e-commerce. Some applications of this problem include bus routing, food and beverage distribution, currency collection and delivery bet- ween banks and ATM machines, Internet-based pickup and delivery, collection and distri- bution of charitable donations between homes and different organizations, and the trans- port of medical samples from medical offices to laboratories, just to name a few. Also, an important PDP variant, known as dial-a-ride, can provide an effective means of transport for delivering people from door to door. This model is frequently adopted for the disa- bled, but could provide a greener alternative than the car and taxi to the wider community. Besides road-based transportation, applications of the PDP can also be seen in maritime and airlift planning. As previously mentioned, considerable attention has been paid to the classical VRP and its related variants by the scientific community. Literally thousands of papers have been published in this domain (see for example survey papers [142], [96], [48], [147] and the survey paper in the 50th anniversary of the VRP [97]). Despite this, research on Pickup and Delivery (PD) problems is relatively scarce [137]. A possible reason is the complexity of these problems and the difficulty in handling the underlying problem constraints. In- novations in solution methods that handle different types of PD problems are certainly in great demand. We have selected two important variants of PD problems as the focus of this research. These are: the Pickup and Delivery Problem with Time Windows (PDPTW), and the One-commodity pickup and delivery problem (1-PDP), with more emphasis on the first of these two variants and a thorough investigation of both its single and multiple vehicle cases. The main difference between the two problems is that the PDPTW assumes that origins and destinations are paired, while the 1-PDP assumes that a single commodity may be picked up from any pickup location and delivered to any delivery location. In fact, the problems dealt with in this research are regularly encountered in every day transportation and logistics applications, as will be demonstrated during the course of this thesis. 1.2 Thesis Hypothesis and Contribution Similar to the VRP, the PDP belongs to the class of Combinatorial Optimization (CO) problems, for which there is usually a huge number of possible solution alternatives. For example, the simplest form of the VRP is the Traveling Salesman Problem (TSP), where a salesman must visit n cities starting and ending at the same city, and each city should be visited exactly once. It is required to find a certain route for the salesman such that the
- 28. 4 1.2 ThesisHypothesis and Contribution total traveling cost is minimized. Although this problem is simple for small values of n, the complexity of solving the problem increases quickly as n becomes large. For a TSP problem of size n, the number of possible solutions is n!/2. So, if n = 60 for example, the number of possible solutions that must be searched is approximately 4.2 × 1081 . If one considers the VRP with added problem constraints, the solution process becomes even more complex. Despite the advancement in algorithmic techniques, solving the VRP and similar highly constrained CO problems to optimality may be prohibitive for very large problem sizes. Exact algorithms that may be used to provide optimum problem solutions cannot solve VRP instances with more than 50-100 customers, given the current resources [72]. Approximation algorithms that make use of heuristic and meta-heuristic approaches are often used to solve problems of practical magnitudes. Such approaches provide good solutions to CO problems in a reasonable amount of time, compared to exact methods, with no guarantee to solve the problem to optimality (see Chapter 2 for more details about combinatorial optimization, complexity theory and exact and approximation techniques). Many heuristic and meta-heuristic algorithms have been applied to solving the different variants of PD problems. However, most of these approaches are adaptations of algo- rithms that have been previously developed for the classical VRP. The solution process for such problems is usually divided into two phases: solution construction and solution improvement. In the solution construction phase one or more initial problem solutions is generated, while in the solution improvement phase the initial solution(s) is gradually improved, in a step-by-step fashion, using a heuristic or a meta-heuristic approach. Ap- plying classical VRP techniques to the PDP, though, requires careful consideration, since the PDP is in fact a highly constrained problem that is much harder than other variants of the VRP. We noticed during our literature survey that research tackling PD problems tends to exhi- bit complexity and sophistication in the solution methodology. Probably the main reason behind this is that handling the difficult, and sometimes conflicting, problem constraints is often a hard challenge for researchers. Straightforward heuristic and meta-heuristic algo- rithms cannot normally be applied to PD problems, without augmenting the approach with problem-specific techniques. In many cases, researchers resort to sophisticated methods in their search for good quality solutions. For example, they may hybridize several heu- ristics and meta-heuristics, or utilize heuristics within exact methods. Some researchers use adaptive and probabilistic search algorithms, which undergo behavioural changes dy- namically throughout the search, according to current features of the search space or to recent performance of the algorithm. Dividing the search into several stages, adding a
- 29. 1.2 Thesis Hypothesisand Contribution 5 post-optimization phase, and employing a large number of neighbourhood operators are also popular approaches to solving the PDP. Unfortunately, when complex multi-facet techniques are used, it can be very difficult to assess which algorithm component has contributed most to the success of the overall approach. Indeed, it is possible that some components may not be needed at all. In many papers only the final algorithm is pre- sented, and experimental evidence justifying all its various components and phases is not included. Notwithstanding the run time issues, the more complex the approach, the harder it is for other researchers to replicate. Another important issue to consider with PD problems is solution infeasibility. Due to the many problem constraints that often apply, obtaining a feasible solution ( i.e., a so- lution that does not violate problem constraints) may be a challenge in itself. Since the generation of infeasible solutions cannot be easily avoided during the search, solution techniques from the literature often add a repair method to fix the infeasibility of solu- tions. This will inevitably make the solution algorithm less elegant and slow down the optimization process. In our research, we investigate the potential of using heuristic and meta-heuristic ap- proaches in solving selected important variants of PD problems. Unlike previous research in this field, we aspire to develop solution techniques that can handle this complex pro- blem in a simple and effective way, sometimes even at the expense of a slight sacrifice in the quality of the final solution obtained. The simpler the solution technique, the more it can be integrated with other approaches, and the easier it can be incorporated in real- world optimization tools. In our opinion, to achieve this goal the solution technique must be directed towards handling the underlying constraints efficiently, without complicating the whole solution method. In addition, we aim to provide robust methodologies, to avoid the need for extensive parameter tuning. While we support, and appreciate, the need to compare the performance of our algorithms with the state-of-the-art, we try not to engage in hyper-tuning simply to produce marginally improved results on benchmark instances. The heuristic and meta-heuristic approaches applied in this research focus on two main aspects that we believe can help us solve hard PD problems and achieve good quality solutions, while keeping the overall technique simple and elegant. These aspects are: the solution representation and the neighbourhood moves. The solution representation should reflect the problem components, while being simple to code and interpret. In addition, it should facilitate dealing with the problem constraints by being flexible and easily manageable, when neighbourhood moves are applied to create new solutions during the search. An appropriate solution representation is thus the initial step towards an overall successful solution technique. In addition, ‘intelligent’ neighbourhood moves enable the
- 30. 6 1.2 ThesisHypothesis and Contribution algorithm to progress smoothly exploring promising areas of the search space, and at the same time, avoiding generating and evaluating a large number of infeasible solutions. Specifically, our research mainly concentrates on solution representations and neighbou- rhood moves as portable and robust components that can be used within different heuris- tics and meta-heuristics to solve some hard pickup and delivery problems, and may be adapted for solving other variants of vehicle routing problems as well. Based on this per- ception, we were able to tailor some well-known heuristic and meta-heuristic approaches to solving the selected variants of pickup and delivery problems, i.e., the PDPTW and the 1-PDP. In this research, we employed our key ideas within some famous heuristic and meta-heuristic algorithms, such as Genetic Algorithms (GAs), Simulated Annealing (SA), Hill Climbing (HC) and Variable Neighbourhood Search (VNS), and tried to demonstrate how the proposed constraint handling mechanisms helped guide the different solution methods towards good quality solutions and manage infeasibility throughout the search, while keeping the overall algorithm as simple as possible. The proposed ‘techniques’, though, have the potential of being applicable within other heuristics and meta-heuristics, with no or just minor modifications, based on the particular problem to which they are being applied. The contribution of this thesis is six fold: 1. We have developed a unique and simple solution representation for the PDPTW that enabled us to simplify the solution method and reduce the number of problem constraints that must be dealt with during the search. 2. We have devised intelligent neighbourhood moves and genetic operators that are guided by problem specific information. These operators enabled our solution al- gorithm for the PDPTW to create feasible solutions throughout the search. Thus, the solution approach avoids the need for a repair method to fix the infeasibility of newly generated solutions. 3. We have developed new simple routing heuristics to create individual vehicle routes for the PDPTW. These heuristics can be easily embedded in different solution construction and improvement techniques for this problem. 4. We have developed several new solution construction methods for the PDPTW, that can be easily used within other heuristic and meta-heuristic approaches to create initial problem solutions. 5. We have demonstrated how traditional neighbourhood moves from the VRP litera- ture can be employed in a new way within a meta-heuristic for the 1-PDP, which
- 31. 1.3 Thesis Overview7 helped the algorithm to escape the trap of local optima and achieve high quality solutions. 6. We have applied simple adaptation of neighbourhood moves and search parameters, in both the PDPTW and the 1-PDP, for more efficient searching. Although the techniques adopted in our research were only tried on some pickup and delivery problems, we believe that they can be easily applied to other vehicle routing and scheduling problems with only minor modifications. As will be seen through the course of this thesis, the techniques mentioned above were in most cases quite successful in achieving the objectives that we had in mind when we started our investigation of the problems in hand. Details about these techniques and how we applied them within the heuristic and meta-heuristic framework will be explained in the main part of this thesis, i.e., Chapters 5 to 10. 1.3 Thesis Overview Chapter 1 documents the motivation behind the research carried out in this thesis. The hypothesis and contribution of the thesis are highlighted, and an overview of the structure of the thesis is presented. Chapter 2 provides some background information about complexity theory and algorithm analysis. A quick look at some exact solution methods that can be used to solve combi- natorial optimization problems to optimality is taken, before a more detailed explanation of some popular heuristic and meta-heuristic approaches is provided. The techniques introduced in this chapter include: Hill Climbing (HC), Simulated Annealing (SA), Ge- netic Algorithms (GAs), Ant Colony Optimization (ACO), Tabu Search (TS) and Variable Neighbourhood Search (VNS). Chapter 3 is a survey of vehicle routing problems, with more emphasis on one particular variant that is of interest to our research, the Vehicle Routing Problem with Time Win- dows (VRPTW). For this problem, some solution construction and solution improvement techniques are described. In addition, a quick summary of some published meta-heuristic approaches that have been applied to this problem is provided. Chapter 4 is another literature survey chapter dedicated to one variant of vehicle rou- ting problems that is the focus of this research, namely pickup and delivery problems. A classification of the different problem types is given, and a brief summary of some pu- blished heuristic and meta-heuristic approaches that tackle these problems is provided.
- 32. 8 1.3 ThesisOverview The survey presented in this chapter is intended to be a concise overview. More details about state-of-the-art techniques are presented in later chapters, when related work to each individual problem that we handled is summarized. Chapter 5 is the first of six chapters that detail the research carried out in this thesis. This chapter describes the first problem that we tackled, the Single Vehicle Pickup and Deli- very Problem with Time Windows (SV-PDPTW). A general overview of the problem, and a summary of related work from the literature is given. In addition, a Genetic Algorithm (GA) approach to solving this problem is described in detail. A novel solution representa- tion and intelligent neighbourhood moves that may help the algorithm overcome difficult problem constraints are tried within the GA approach. The experimental findings of this part of the research were published, as a late breaking paper, in GECCO2007 conference [79]. Chapter 6 continues with the research started in Chapter 5 for the SV-PDPTW. Never- theless, a more extensive investigation of the problem is given here. Several heuristic and meta-heuristic techniques are tested for solving the problem, employing the same tools introduced in Chapter 5. Three approaches are applied to solving the problem: genetic algorithms, simulated annealing and hill climbing. A comparison between the different approaches and a thorough analysis of the results is provided. The algorithms and the findings of this part of the research were published in the Journal of Heuristics [84]. Chapter 7 starts the investigation of the more general Multiple-Vehicle case of the Pi- ckup and Delivery Problem with Time Windows (MV-PDPTW). After summarizing some related work in this area, new solution construction methods are developed and compa- red, using the single vehicle case as a sub-problem. The algorithms introduced here act as a first step towards a complete solution methodology to the problem, which will be presented in the next chapter. The different construction heuristics are compared, and conclusions are drawn about the construction algorithm that seems most appropriate for this problem. This part of the research was published in the MIC2009 [81]. Chapter 8 continues the investigation of the MV-PDPTW, by augmenting solution construc- tion, introduced in the previous chapter, with solution improvement. In this chapter, we tried both a GA and an SA for the improvement phase. Several techniques from the first part of our research (explained in Chapters 5 to Chapter 7) are embedded in both the GA and the SA. The GA approach is compared with two related GA techniques from the li- terature, and the results of both the GA and the SA are thoroughly analyzed. Part of the research carried out in this chapter was published in the MIC2009 [80]. Chapter 9 starts the investigation of another important PD problem which is the One- commodity Pickup and Delivery Problem (1-PDP). In this chapter, we explain the pro-
- 33. 1.4 A Noteon Implementation and Computational Experimentation 9 blem, provide a literature review, and highlight some related problems. After that, we try to solve the 1-PDP using an Evolutionary Perturbation Scheme (EPS) that has been used for solving other variants of vehicle routing problems. Experimental results on published benchmark instances are reported and analyzed. Chapter 10 continues the investigation of the 1-PDP, using another meta-heuristic ap- proach, which is a Variable Neighbourhood Search (VNS) hybridized with Simulated Annealing (SA). Also, adaptation of search parameters is applied within this heuristic for more efficient searching. The approach is thoroughly explained and the experimental results on published test cases are reported. Two conference papers that discuss this ap- proach have now been accepted for publication in GECCO2010 (late breaking abstract) [82], and PPSN2010 [85]. In addition, a third journal paper has been submitted to the Journal of Heuristics and is currently under review [83]. Chapter 11 looks beyond the current research phase and discusses how a theoretical scientific research, such as that carried out in this thesis, can be used in real-life com- mercial applications of vehicle routing problems. We give examples of some commercial software tools, and we also highlight important industrial aspects of vehicle routing that the research community should be aware of. An overview of future trends in scientific research tackling this issue is also provided. Chapter 12 summarizes the research undertaken in this thesis and elaborates on the thesis contribution. Some critical analysis of parts of the current research and suggestions for future work are also presented. 1.4 A Note on Implementation and Computational Expe- rimentation All algorithmic implementations presented in this thesis are programmed in C++, using Microsoft Visual Studio 2005. Basic Genetic Algorithms (GAs) were implemented with the help of an MIT GA library GALIB [155]. Most of the computational experimentations were carried out using Intel Pentium (R) CPU, 3.40 GHz and 2 GB RAM, under a Win- dows XP operating system, unless otherwise indicated. The algorithms developed in this thesis were tested on published benchmark data for the selected problems, or on problem instances created in this research. We indicate in the relevant sections of this thesis the exact source of the data, and provide links where applicable.
- 34.
- 35. 11 Chapter 2 Combinatorial Optimization, Heuristics,and Meta-heuristics In this chapter, we introduce an important class of problems that are of interest to resear- chers in Computer Science and Operations Research (OR), which is the class of Combina- torial Optimization (CO) problems. The importance of CO problems stems from the fact that many practical decision making issues concerning, for example, resources, machines and people, can be formulated under the combinatorial optimization framework. As such, popular optimization techniques that fit this framework can be applied to achieve optimal or best possible solutions to these problems, which should minimize cost, increase profit and enable a better usage of resources. Our research handles one class of CO problems, which is concerned with vehicle routing and scheduling. Details about this specific class will be presented in Chapters 3 and 4. In the current chapter we provide a brief description of CO problems, and the related al- gorithm and complexity analysis theory in Section 2.1. We will then proceed to review important techniques that are usually applied to solving CO problems. Section 2.2 briefly highlights some exact algorithms that can be used to find optimal solutions to these pro- blems. On the other hand, heuristic methods that may be applied to find a good solution, which is not necessarily the optimum, are introduced in Section 2.3. We emphasize in this section: Hill Climbing (HC), Simulated Annealing (SA), Genetic Algorithms (GAs), Ant Colony Optimization (ACO), Tabu Search (TS), and Variable Neighbourhood Search (VNS). Finally Section 2.4 concludes this chapter with a brief summary. 2.1 Combinatorial Optimization, Algorithm and Complexity Analysis Combinatorial optimization problems can generally be defined as problems that require searching for the best solution among a large number of finite discrete candidate solutions.
- 36. 12 2.1 CombinatorialOptimization, Algorithm and Complexity Analysis More specifically, it aspires to find the best allocation of limited resources that can be used to achieve an underlying objective. Certain constraints are usually imposed on the basic resources, which limit the number of feasible alternatives that can be considered as problem solutions. Yet, there is still a large number of possible alternatives that must be searched to find the best solution. There are numerous applications of CO problems in areas such as crew scheduling, jobs to machines allocation, vehicle routing and scheduling, circuit design and wiring, solid- waste management, energy resource planning, just to name a few. Optimizing solutions to such applications is vital for the effective operation and perhaps the survival of the institution that manages these resources. To a large extent, finding the best or optimum solution is an integral factor in reducing costs, and at the same time maximizing profit and clients’ satisfaction. Many practical CO problems can be described using well-known mathematical models, for example the Knapsack problem, job-shop scheduling, graph coloring, the TSP, the boolean satisfiability problem (SAT), set covering, maximum clique, timetabling. . . etc. Many algorithms and solution methods exist for solving these problems. Some of them are exact methods that are guaranteed to find optimum solutions given sufficient time, and others are approximation techniques, usually called heuristics or meta-heuristics, which will give a good problem solution in a reasonable amount of time, with no guarantee to achieve optimality. Within this domain, researchers are often faced with the requirement to compare algo- rithms in terms of their efficiency, speed, and resource consumption. The field of algo- rithm analysis helps scientists to perform this task by providing an estimate of the number of operations performed by the algorithm, irrespective of the particular implementation or input used. Algorithm analysis is usually preferred to comparing actual running times of algorithms, since it provides a standard measure of algorithm complexity irrespective of the underlying computational platform and the different types of problem instances sol- ved by the algorithm. Within this context, we usually study the asymptotic efficiency of an algorithm, i.e., how the running time of an algorithm increases as the size of the input approaches infinity. In algorithm analysis, the O notation is often used to provide an asymptotic upper bound of the complexity of an algorithm. We say that an algorithm is of O(n) (Order n), where n is the size of the problem, if the total number of steps carried out by the algorithm is at most a constant times n. More specifically, f(n) = O(g(n)), if there exist positive constants c and n0 such that 0 ≤ f(n) ≤ cg(n), for all n ≥ n0. The O notation is usually used to describe the complexity of the algorithm in a worst-case scenario. Other less
- 37. 2.1 Combinatorial Optimization,Algorithm and Complexity Analysis 13 frequently used notations for algorithm analysis are the Ω notation and the Θ notation. The Ω notation provides an asymptotic lower bound, i.e., f(n) is of Ω(g(n)) if there exist positive constants c and n0 such that 0 ≤ cg(n) ≤ f(n), for all n ≥ n0. On the other hand, the Θ notation provides an asymptotically tight bound for f(n), i.e., f(n) is of Θ(g(n)) if there exist positive constants c1, c2 and n0 such that 0 ≤ c1g(n) ≤ f(n) ≤ c2g(n), for all n ≥ n0 [32]. Figures 2.1(a), 2.1(b) and 2.1(c) demonstrate a graphic example of the O notation, the Ω notation, and the Θ notation respectively.1 (a) The O notation (b) The Ω notation (c) The Θ notation Figure 2.1: Asymptotic Notations. In addition to analyzing the efficiency of an algorithm, we sometimes need to know what types of algorithms exist for solving a particular problem. The field of complexity ana- lysis analyzes problems rather than algorithms. Two important classes of problems are usually identified in this context. The first class is called P (polynomial time problems). 1 In this thesis we use the O notation for the analysis of algorithms when needed, since it is the most commonly used type of asymptotic notations among the three notations introduced in this section.
- 38. 14 2.1 CombinatorialOptimization, Algorithm and Complexity Analysis It contains problems that can be solved using algorithms with running times such as O(n), O(log(n)) and O(nk ) 2 . They are relatively easy problems, sometimes called tractable problems. Another important class is called N P (non-deterministic polynomial time problems). This class includes problems for which there exists an algorithm that can guess a solution and verify whether the guessed solution is correct or not in polynomial time. If we have an unbounded number of processors that each can be used to guess and verify a solution to this problem in parallel, the problem can be solved in polynomial time. One of the big open questions in computer science is whether the class P is equivalent to the class N P. Most scientists believe that they are not equivalent. This, however, has never been proven. Researchers also distinguish a sub class of N P, called the N P-complete class. In a sense, this class includes the hardest problems in computer science, and is characterized by the fact that either all problems that are N P-complete are in P, or not in P. Many N P- complete problems may require finding a subset within a base set (e.g. the SAT problem), or an arrangement (or permutation) of discrete objects (e.g. the TSP), or partitioning an underlying base set (e.g. graph coloring). As mentioned above, these problems belong to the combinatorial optimization class of problems, and sometimes called intractable problems. An optimization problem for which the associated decision problem is N P-complete is called an N P-hard problem. A decision problem is a problem that has only two pos- sible answers: yes or no. For example, if the problem is a cost minimization problem, such that it is required to find a solution with the minimum possible cost, the associated decision problem would be formulated as: “ is there a solution to the problem whose cost is B, where B is a positive real number?”. Figure 2.2 demonstrates how the different complexity classes may relate to each other. For more details about algorithm analysis and complexity theory, the reader is referred to the book by Garey and Johnson [54] and Cormen et al. [32]. Solving CO problems has been a challenge for many researchers in computer science and operations research. Exact methods used to solve regular problems cannot be used to solve CO problems given current resources, since searching among all possible solutions of a certain intractable problem is usually prohibitive for large problem sizes. The natural alternative would be to use approximation methods that give good, rather than optimal, solutions to the problem in a reasonable amount of time. In the next section we briefly consider some exact methods that can be used to solve 2 Running times of O(n) are usually called linear running times. Also, running time of O(nk ) are often referred to as polynomial running times, while running times of O(2n ) are called exponential running times.
- 39. 2.2 Exact Algorithms15 Figure 2.2: Complexity Classes. CO problems. However, since exact algorithms are not practical solution methods for the applications considered in this thesis, our discussion of these techniques will be very brief. On the other hand, a more detailed investigation of approximation algorithms, i.e., heuristics and meta-heuristics, will be presented in Section 2.3. 2.2 Exact Algorithms As previously mentioned, exact algorithms are guaranteed to find an optimum solution for a CO problem if one exists, given a sufficient amount of time. In general, the most successful of these algorithms try to reduce the solution space and the number of different alternatives that need to be examined in order to reach the optimum solution. Some exact algorithms that have been applied to solving CO problems are the following: 1. Dynamic Programming (DP): which refers to the process of simplifying a com- plicated problem by dividing it into smaller subproblems in a recursive manner. Problems that are solved using dynamic programming usually have the potential of being divided into stages with a decision required at each stage. Each stage also has a number of states associated with it. The decision at one stage transforms the current state into a state in the next stage. The decision to move to the next state is only dependent on the current state, not the previous states or decisions. Top- down dynamic programming is based on storing the results of certain calculations, in order to be used later. Bottom-up dynamic programming recursively transforms a
- 40. 16 2.3 HeuristicAlgorithms complex calculation into a series of simpler calculations. Only certain CO problems are amenable to DP. 2. Branch-and-Bound (B&B): is an optimization technique which is based on a sys- tematic search of all possible solutions, while discarding a large number of non- promising candidate solutions, using a depth-first strategy. The decision to discard a certain solution is based on estimating upper and lower bounds of the quantity to be optimized, such that nodes whose objective function are lower/higher than the current best are not explored. The algorithm requires a ‘branching’ tool that can split a given set of candidates into two smaller sets, and a ‘bounding’ tool that computes upper and lower bounds for the function to be optimized within a given subset. The process of discarding fruitless solutions is usually called ‘pruning’. The algorithm stops when all nodes of the search tree are either pruned or solved. 3. Branch-and-Cut (B&C): is a B&B technique with an additional cutting step. The idea is to try to reduce the search space of feasible candidates by adding new constraints (cuts). Adding the cutting step may improve the value returned in the bounding step, and could allow solving subproblems without branching. 2.3 Heuristic Algorithms Given the limitation of exact methods in solving large CO problems, approximation tech- niques are often preferred in many practical situations. Approximation algorithms, like heuristics and meta-heuristics3 , are techniques that solve ‘hard’ CO problems in a rea- sonable amount of computation time, compared to exact algorithms. However, there is no guarantee that the solution obtained is an optimal solution, or that the same solution quality will be obtained every time the algorithm is run. Heuristic algorithms are usually experience-based, with no specific pre-defined rules to apply. Simply, they are a ‘common-sense’ approach to problem solving. On the other hand, an important subclass of heuristics are meta-heuristic algorithms, which are general- purpose frameworks that can be applied to a wide range of CO problems, with just minor changes to the basic algorithm definition. Many meta-heuristic techniques try to mimic biological, physical or natural phenomena drawn from the real-world. 3 In this discussion we use the term approximation/heuristic algorithm to refer to any ‘non-exact’ so- lution method. More accurately, however, the term approximation algorithm is often used to refer to an optimization algorithm which provides a solution that is guaranteed to be within a certain distance from the optimum solution every time it is run, with provable runtime bounds [153]. This may not be necessarily the case for heuristic algorithms, though.
- 41. 2.3 Heuristic Algorithms17 In solving a CO problem using a heuristic or a meta-heuristic algorithm, the search for a good problem solution is usually divided into two phases: solution construction and solution improvement. Solution construction refers to the process of creating one or more initial feasible solutions that will act as a starting point from which the search progresses. In this phase, a construction heuristic usually starts from an empty solution and gradually adds solution components to the partially constructed solution until a feasible solution is generated. On the other hand, solution improvement tries to gradually modify the starting solution(s), based on some predefined metric, until a certain solution quality is obtained or a given amount of time has passed. Within the context of searching for a good quality solution, we usually use the term search space to refer to the state space of all feasible solutions/states that are reachable from the current solution/state. To find a good quality solution, a heuristic or a meta-heuristic al- gorithm moves from one state to another, i.e., from one candidate solution to another, through a process that is often called Local Search (LS). During LS, the new solution is usually generated within the neighbourhood of the previous solution. A neighbou- rhood N(x) of a solution x is a subset of the search space that contains one or more local optima, the best solutions in this neighbourhood. The transition process from one candidate solution to another within its neighbourhood requires a neighbourhood move that changes some attributes of the current solution to transform it to a new solution x′ . A cost function f(x) is used to evaluate each candidate solution x and determine its cost compared to other solutions in its neighbourhood. The best solution within the overall search space is called the globally optimal solution, or simply the optimum solution. Figure 2.3 shows a search space for a minimization function, i.e., our goal is to obtain the global minimum for this function. In this figure three points (solutions) are local optima within their neighbourhoods, A, B and C. The global optimum among the three points is point C. Several modifications to the basic local search algorithm have been suggested to solve CO problems. For example, Baum [10] suggested an Iterated Local Search (ILS) procedure for the TSP, in which a local search is applied to the neighbouring solution x′ to obtain another solution x′′ . An acceptance criterion is then applied to possibly replace x′ by x′′ . Voudouris and Tsang [154] suggest a Guided Local Search (GLS) procedure, in which penalties are added to the objective function based on the search experience. More spe- cifically, the search is driven away from previously visited local optima, by penalizing certain solution features that it considers should not occur in a near-optimal solution. In addition, some local search methods have been formulated under the meta-heuristic fra- mework, in which a set of predefined rules or algorithmic techniques help guide a heuristic
- 42. 18 2.3 HeuristicAlgorithms Figure 2.3: Search space for a minimization problem: local and global optima. search towards reaching high quality solutions to complex optimization problems. Among the most famous meta-heuristic techniques are: Hill Climbing (HC), Simulated Annea- ling (SA), Genetic Algorithms (GAs), Ant Colony Optimization (ACO), Tabu Search and Variable Neighbourhood Search (VNS). For more details about local search techniques and its different variations and applications to CO problems, the reader is referred to [3] and [125]. In what follows we describe some important meta-heuristic techniques that have been widely used for solving CO problems. 2.3.1 Hill Climbing (HC) Hill Climbing is the simplest form of local search, where the new neighbouring solution always replaces the current solution if it is better in quality. The process can thus be visualized as a step-by-step movement towards a locally optimum solution. Figure 2.4 shows how an HC algorithm transitions ‘downhill’ from an initial solution to a local mi- nimum. On the other hand, Algorithm 2.1 shows the steps of an HC procedure applied to a minimization problem.
- 43. 2.3 Heuristic Algorithms19 Figure 2.4: Hill Climbing: downhill transition from an initial solution to a local optimum. Algorithm 2.1: Hill Climbing (HC). 1: Generate an initial solution x 2: repeat 3: Generate a new solution x′ within the neighbourhood of x (x′ ∈ N(x)) 4: ∆ ← f(x′) − f(x) 5: if (∆ < 0) then 6: x ← x′ 7: until (Done){stopping condition is reached} 8: Return solution x HC is frequently applied within other meta-heuristic techniques to improve solution qua- lity at various stages of the search. In our research we tried HC among the selected ap- proaches for solving the Single Vehicle Pickup and Delivery Problem with Time Windows (SV-PDPTW), as will be explained in Chapter 6. HC was also an important part of the solution construction heuristics that we developed for the Multiple Vehicle Pickup and De- livery Problem with Time Windows (MV-PDPTW), as will be explained in Chapter 7. We have also applied it in parts of the algorithms developed for solving the One-Commodity Pickup and Delivery Problem (1-PDP), as will be shown in Chapters 9 and 10.
- 44. 20 2.3 HeuristicAlgorithms 2.3.2 Simulated Annealing (SA) Simulated annealing is a well-known meta-heuristic search method that has been used successfully in solving many combinatorial optimization problems. It is a hill climbing algorithm with the added ability to escape from local optima in the search space. However, although it yields excellent solutions, it is very slow compared to a simple hill climbing procedure. The term simulated annealing is adopted from the annealing of solids, where we try to minimize the energy of the system using slow cooling until the atoms reach a stable state. The slow cooling technique allows atoms of the metal to line themselves up and to form a regular crystalline structure that has high density and low energy. The initial temperature and the rate at which the temperature is reduced is called the annealing schedule. The theoretical foundation of SA was led by Kirkpatrick et al. in 1983 [93], where they applied the Metropolis algorithm [107] from statistical mechanics to CO problems. The Metropolis algorithm in statistical mechanics provides a generalization of iterative im- provement, where controlled uphill moves (moves that do not lower the energy of the system) are probabilistically accepted in the search for obtaining a better organization and escaping local optima. In each step of the Metropolis algorithm, an atom is given a small random displacement. If the displacement results in a decrease in the system energy, the displacement is accepted and used as a starting point for the next step. If on the other hand the energy of the system is not lowered, the new displacement is accepted with a certain probability exp(−E/kbT) where E is the change in energy resulting from the displacement, T is the current temperature, and kb is a constant called a Boltzmann constant . Depending on the value returned by this probability either the new displacement is accepted or the old state is retained. For any given T, a sufficient number of iterations always leads to thermal equilibrium. The SA algorithm has also been shown to possess a formal proof of convergence using the theory of Markov Chains [47]. In solving a CO problem using SA, we start with a certain feasible solution to the pro- blem. We then try to optimize this solution using a method analogous to the annealing of solids. A neighbour of this solution is generated using an appropriate method, and the cost (or the fitness) of the new solution is calculated. If the new solution is better than the current solution in terms of reducing cost (or increasing fitness), the new solution is accepted. If the new solution is not better than the current solution, though, the new solu- tion is accepted with a certain probability. The probability of acceptance is usually set to exp(−∆/T) , where ∆ is the change in cost between the old and the new solution and T is the current temperature. The probability thus decreases exponentially with the badness of
- 45. 2.3 Heuristic Algorithms21 the move. The SA procedure is less likely to get stuck in a local optimum, compared to a simple HC, since bad moves still have a chance of being accepted. The annealing temperature is first chosen to be high so that the probability of acceptance will also be high, and almost all new solutions are accepted. The temperature is then gradually reduced so that the probability of acceptance of low quality solutions will be very small and the algorithm works more or less like hill climbing, i.e., high temperatures allow a better exploration of the search space, while lower temperatures allow a fine tuning of a good solution. The process is repeated until the temperature approaches zero or no further improvement can be achieved. This is analogous to the atoms of the solid reaching a crystallized state. Figure 2.5 shows how occasional uphill moves may allow the SA algorithm to escape local optima in a minimization problem, compared to the HC algorithm shown in Figure 2.4. Figure 2.5: Simulated Annealing: occasional uphill moves and escaping local optima. Applying SA to CO problems also requires choices concerning both general SA para- meters and problem specific decisions. The choice of SA parameters is critical to the performance of the algorithm. These parameters are: the value of the initial temperature T, a temperature function that determines how the temperature will change with time, the
- 46. 22 2.3 HeuristicAlgorithms number of iterations N(t) to be executed at each temperature, and a stopping criterion to terminate the algorithm. As previously mentioned, the initial temperature T(0) is generally chosen high enough so that almost any move is accepted regardless of its fitness. This choice is adopted from the physical analogy and corresponds to heating up a substance until all particles are randomly arranged in a liquid. The temperature update function is usually a proportional temperature function T(t + 1) = αT(t), where t is the current iteration. Typical values of α lie between 0.8 and 0.99. Using such values provides very small decrements of the temperature, which corresponds to slow cooling of the substance until the temperature approaches zero. The number of iterations carried out at each temperature value should be large enough to bring the system to a stable state equating to thermal equilibrium in the physical analogy. Some applications may choose N(t) to be constant for each temperature. The stopping criterion of the algorithm is usually the stagnation of the system when no change in the result can be obtained for a specified number of iterations or temperature changes. Implementing SA also requires a set of problem-specific decisions. These include: iden- tifying the set of feasible solutions to the problem, defining a clear objective function, ge- nerating an initial solution, and defining a neighbourhood operator that generates moves using the current solution. The topology of the neighbourhood structure is also critical to the performance of the SA algorithm. In general, a smooth topology with shallow local optima is favoured over a bumpy topology with many deep local minima. A neighbourhood function is also easy to implement for discrete problems, while implementing a neighbourhood function for conti- nuous problems is more challenging. Constrained problems also raise some difficulties. A choice must be made between restricting the solution space to solutions that conform to the constraints, or allowing solutions that break the constraints at the expense of a suitably defined penalty function. The generic SA algorithm is described in Algorithm 2.2, where the underlying optimization problem is again assumed to be a minimization function. Some modifications to the basic SA algorithm have been suggested. These modifications are intended to provide an improvement of the quality of the solution and/or processing time. One attempt is to store the best solution found so far. Since the SA algorithm accepts solutions probabilistically, it may accept solutions that are worse than the current solution. A good solution found during the run may be discarded because it was not lucky during the acceptance attempt. Storing the best solution found so far prevents the SA algorithm from returning a final solution that is worse than the best solution ever found (e.g. [63]).
- 47. 2.3 Heuristic Algorithms23 Algorithm 2.2: The Simulated Annealing (SA) Algorithm. 1: Generate an initial solution x 2: T ← T0 {initialize current temperature} 3: repeat 4: for (i = 0; i < numIterations; i + +) do 5: Generate a new solution x′ within the neighbourhood of x (x′ ∈ N(x)) 6: ∆ ← f(x′) − f(x) 7: if (∆ < 0) then 8: x ← x′ 9: else 10: p = Random(0, 1) {generate a random number in the interval (0,1)} 11: if (p < exp(−∆/T)) then 12: x ← x′ 13: T ← α × T {reduce current temperature} 14: until (Done){stopping condition is reached} 15: Return solution x Some researches also modify the basic SA algorithm using problem specific information. For example [148] suggest a neighbourhood operator that is able to identify promising areas in the neighbourhood, and give a greater probability to generated moves that fall in the promising areas. SA has also been hybridized with other optimization techniques. This can be done by using another technique to generate a good initial solution that SA can improve, or using SA to generate a good solution that can be used by another search technique. To sum up, SA has several attractive features, especially in difficult optimization problems in which obtaining a good solution, with a reasonable computational effort and proces- sing time, is preferred to an optimal solution with considerably higher cost. The basic advantages of SA are the following: 1. It is very easy to implement, since it just requires a method for generating a move in the neighbourhood of the current solution, and an appropriate annealing schedule. 2. It can be applied to a wide range of problem types. For example, any combinatorial optimization problem can be tackled using SA, if an appropriate neighbourhood structure has been devised. 3. High quality solutions can be obtained using SA, if a good neighbourhood structure and a good annealing schedule have been chosen.
- 48. 24 2.3 HeuristicAlgorithms These benefits, however, may be at the expense of a longer processing time, compared to simple hill climbing. In our research we applied SA to solving the SV-PDPTW, as explained in Chapter 6, and for the MV-PDPTTW, as explained in Chapter 8. We also hybridized SA with a Variable Neighbourhood Search (VNS) approach for solving 1-PDP, as will be shown in detail in Chapter 10. 2.3.3 Genetic Algorithms (GAs) The idea of simulation of biological evolution and the natural selection of organisms dates back to the 1950’s. One of the early pioneers in this area was Alex Fraser with his research published in 1957 [50] [51]. Nevertheless, the theoretical foundation of GAs were esta- blished by John Holland in 1975 [78], after which GAs became popular as an intelligent optimization technique that may be adopted for solving many difficult problems. The theme of a GA is to simulate the processes of biological evolution, natural selection and survival of the fittest in living organisms. In nature, individuals compete for the re- sources of the environment, and they also compete in selecting mates for reproduction. Individuals who are better or fitter in terms of their genetic traits survive to breed and produce offspring. Their offspring carry their parents’ basic genetic material, which leads to their survival and breeding. Over many generations, this favourable genetic material propagates to an increasing number of individuals. The combination of good characteris- tics from different ancestors can sometimes produce ‘super fit’ offspring who out-perform their parents. In this way, species evolve to become better suited to their environment. GAs operate in exactly the same manner. They work on a population of individuals re- presenting possible solutions to a given problem. In traditional GAs, each individual is usually represented by a string of bits analogous to chromosomes and genes, i.e., the parameters of the problem are the genes that are joined together in a solution chromo- some. A fitness value is assigned to each individual in order to judge its ability to survive and breed. The highly fit individuals are given a chance to breed by being selected for reproduction. Thus, the selection process usually favours the more fit individuals. Good individuals may be selected several times in one iteration, while poor ones may not be selected at all. By selecting the ‘most fit’ individuals, favourable characteristics spread throughout the population over several generations, and the most promising areas of the search space are explored. Finally, the population should converge to an optimal or near optimal solution. Convergence means that the population evolves toward increasing uni- formity, and the average fitness of the population will be very close to the highest fitness. During the reproduction phase of a GA, two individuals breed by combining their genes in
- 49. 2.3 Heuristic Algorithms25 an operation called crossover. Not all selected pairs undergo crossover. A random choice is applied, where the likelihood of crossover is some given probability. If crossover is not performed, offspring are produced simply by duplicating their parents. Crossover allows the basic genetic material of parents to pass to their children, who then form the next generation. Another operation that is performed by GAs is mutation. Mutation is applied to each child generated from crossover. With a certain small probability, each gene may be alte- red. Thus, Crossover allows a rapid exploration of the search space by producing large jumps, while mutation allows a small amount of random search. The basic outline of a GA is shown in the following algorithm: Algorithm 2.3: The Basic Genetic Algorithm (GA). 1: Coding: initialize and encode a random population of solutions called chromosomes 2: repeat 3: Fitness Assignment: decode and evaluate the fitness of each chromosome 4: Selection: select some chromosomes from the current population for reproduction, where the selection criterion is based on the fitness of the selected parents 5: Recombination: with some probability apply crossover between the selected parents to produce new children 6: Variation: apply mutation with a small probability to some genes of the newly produced offspring, or to selected members of the population 7: Replacement: integrate the newly generated offspring with the old population to create a new generation 8: until (a certain stopping condition is reached) The traditional GA uses strings of bits to represent chromosomes. In addition, the classical crossover operator is called one-point crossover, where the two mating chromosomes are each cut once at corresponding points and the sections after the cuts are exchanged. However, many different crossover types have been devised, often involving more than one cut point. For example, in a two-point crossover two cut points are chosen randomly in the parent chromosomes. The section between the selected cut points is exchanged between the two children. Another form of crossover is called uniform crossover, in which a random mask of bits is generated. Each gene in the offspring is created by copying the corresponding gene from one of the parents. The parent is selected according to the value of the corresponding bit in the mask. These three crossover operators are illustrated in Figure 2.6. On the other hand, the most popular mutation operators are point mutation, where one
- 50. 26 2.3 HeuristicAlgorithms bit in the chromosome is flipped, and inversion mutation, in which a substring is selected at random and the bits within this substring are inverted. Figure 2.7 demonstrates these two mutation operators. (a) One-point Crossover (b) Two-point Crossover (c) Uniform Crossover Figure 2.6: Crossover. (a) Point Mutation (b) Inversion Mutation Figure 2.7: Mutation.
- 51. 2.3 Heuristic Algorithms27 Chromosome representation as a bit string is not suitable for many problems types, though. For example, a permutation of integers representing nodes or locations has been often used for the TSP and other vehicle routing problems. This kind of representation is called an order-based representation, since the fitness depends on the order in which the genes ap- pear. Different crossover and mutation operators must be used in this case, since classical genetic operators, such as the ones described above, will result in ‘infeasible’ chromo- somes. Goldberg in [64] described several order-based operators, such as the Partially Matched Crossover (PMX). In a PMX, it is not the genes that are crossed, but the order in which they appear, i.e., offspring have genes that inherit order information from each parent. This avoids the problem of generating offspring that violate problem constraints, such as having duplicate cities in a chromosome that represents a solution to the TSP, or having a chromosome with some non-visited cities. Figure 2.8 shows an example of a PMX, where cities 5,6,7 exchange their positions with 2, 3 and 10 respectively. Other crossover operators that can be used for order-based representations are cycle crossover and order crossover (see [64] for details). Figure 2.8: Partially Matched Crossover (PMX). To select two individuals in the population for mating, several selection methods may be applied. For example, in a rank selection, individuals are sorted according to their objective function values, and each individual is assigned a number of offspring that is a function of its rank in the population. Roulette wheel selection picks an individual, with a probability, based on the magnitude of the fitness score relative to the rest of the population. The higher the score, the more likely an individual will be selected, and the probability of the individual being chosen is equal to the fitness of the individual divided by the sum of the fitness values of all individuals in the population. On the other hand, a tournament selection uses the roulette wheel, or another selection method, to select two
- 52. 28 2.3 HeuristicAlgorithms or more individuals, then picks the one with the higher score. Similarly, many replacement schemes can be used to integrate the new offspring and produce the next generation. In a Simple GA, the new generation of offspring comple- tely replace the old population, while a Steady State GA allows generations to overlap, with a certain percentage of replacement. On the other hand, an Incremental GA only allows a small degree of overlap between generations, for example by replacing one or two individuals at each generation. In summary, GAs represent an intelligent search method, since they operate on a popu- lation of solutions and allocate trials to promising areas of the search space. GAs do not depend heavily on information available from the underlying problem. They are easy to connect to existing simulations and models, and can be easily hybridized to generate knowledge-augmented GAs. Using the operations of selection of the fittest, mutation, and crossover, GAs can quickly reach fit individuals (not always the most fit), but who are usually good enough as solutions to problems of a large magnitude. Crossover is consi- dered as the main GA operator. Having to combine two solutions rather than one, makes designing an appropriate crossover operator often more challenging than developing a mutation operator or a simple neighbourhood move. This usually makes GAs implemen- tation more difficult compared to SA or simple HC. In our research we applied GAs to solving the PDPTW, for both the single and multiple vehicle variants, as will be explained in detail in Chapters 5, 6 and 8. We also applied a GA within a perturbation scheme for the 1-PDP, as will be detailed in Chapter 9. 2.3.4 Ant Colony Optimization (ACO) Ant Colony Optimization is a meta-heuristic technique that is inspired by the behaviour of real ants. Its principles were established by Dorigo et al. in 1991 [42]. Real ants coope- rate to find food resources by laying a trail of a chemical substance called ‘pheromone’ along the path from the nest to the food source. Depending on the amount of pheromone available on a path, new ants are encouraged, with a high probability, to follow the same path, resulting in even more pheromone being placed on this path. Shorter routes to food sources have higher amounts of pheromone. Thus, over time, the majority of ants are di- rected to use the shortest path. This type of indirect communication is called ‘stigmergy’ [41], in which the concept of positive feedback is exploited to find the best possible path, based on the experience of previous ants. Figure 2.9 visualizes the foraging behaviour of real ants in their search for food sources.
- 53. 2.3 Heuristic Algorithms29 (a) Two ants start with equal probability of going on either path. The ant on the shorter path arrives more quickly to the food source (b) The density of the pheromone on the shorter path increases more ra- pidly than the pheromone on the longer path. Many ants begin using the path with higher pheromone (c) Over time, the shorter path is almost exclusively used, and the phero- mone on the longer path evaporates Figure 2.9: Foraging behaviour of ants.
- 54. 30 2.3 HeuristicAlgorithms Applying ACO to hard combinatorial optimization problems is attractive, due the flexi- bility and robustness of the algorithm. In fact, the technique can be applied to solving a wide range of problems with only minimal changes to the basic algorithm. In addition, its population based nature allows exploring the search space, while exploiting positive feedback during the search process to find near optimal solutions. In solving a combinatorial optimization problem using an ACO, the problem may be vi- sualized as finding the shortest path in a weighted graph. Artificial ants (software agents) cooperate to find the best path by constructing solutions consisting of graph nodes step- by-step. Each step adds a selected node to the partial solution in a stochastic manner, but biased by the amount of pheromone available on the nodes or the edges connecting them. To this end, problem-specific information is placed in a common memory, which plays the role of the pheromone trail. The information in the memory, i.e., the pheromone trail, is updated by ants at each iteration, allowing ants to cooperate in finding good problem solutions. Pheromone values also diminish over time, similar to the evaporation of real pheromone. Thus, solutions of bad quality are eliminated from further consideration as the search progresses. Algorithm 2.4 shows the main steps of the ACO meta-heuristic. Algorithm 2.4: The Ant Colony Optimization (ACO) Algorithm. 1: Set parameters, initialize pheromone trails 2: while (termination condition not met) do 3: ConstructAntSolutions 4: DaemonActions {optional} 5: UpdatePheromones The steps of Algorithm 2.4 are briefly explained below: 1. ConstructAntSolutions: during this phase, each ant constructs a solution from the set of feasible problem solutions, by adding one solution component at each step of the construction process to the current partial solution. For example, in solving a TSP, the construction process will add one non-visited city at a time to the current solution. The choice of the new solution component depends on a probabilistic rule that takes into account both the pheromone trail of the component at the current iteration and other problem-specific heuristic information. In a TSP, the pheromone trail is associated with the edge connecting the last added city and the potential new city, such that edges previously used by other ants are favoured, due to an increased pheromone value. On the other hand, the heuristic information (sometimes called the attractiveness) will be proportional to the length of the edge connecting the two
- 55. 2.3 Heuristic Algorithms31 cities. Certain parameters are also used in the probabilistic rule to determine the relative importance of the pheromone trail and the heuristic information. During the solution construction, individual ants may update the pheromone trail on the edges they visited in a process called local pheromone update. 2. DaemonActions: in this optional step of the ACO algorithm, some problem-specific action may be required, which cannot be usually performed by a single ant. For example, local search may be applied to optimize the set of generated solutions. The new optimized solutions are then used to decide which pheromone trails to update. 3. UpdatePheromones: this is a global pheromone update process that is performed at the end of each iteration, where updating the pheromone values depends on the quality of the generated solutions in the current iteration. This is usually done by decreasing the pheromone value for all solutions, in a process called evaporation, and increasing the pheromone value of good solutions. Evaporation is a tool that ACO uses to explore new areas of the search space and avoid being trapped in local optima. ACO has become very popular in solving CO problems. For example, its versatility and robustness have been demonstrated in [43] by tailoring the approach to the TSP, the asym- metric TSP, the Quadratic Assignment Problem (QAP), and the job-shop scheduling. It has also been applied to variants of vehicle routing problems (e.g. [39] and [90]). The idea of ‘attractiveness’ and ‘pheromone trails’ were also exploited within other meta-heuristic techniques. For example, in the crossover operator used by [158] for solving the 1-PDP. For further information about the various types of Ant systems and their applications, the reader is referred to the book by Dorigo and Stützle [44]. 2.3.5 Tabu Search (TS) Tabu search is another popular search technique proposed by Glover in 1977 [59]. Since then, it has been widely used for solving CO problems. Its name is derived from the word ‘taboo’ meaning forbidden or restricted. TS, like SA, allows for exploring the search space ‘intelligently’ in an attempt to escape the trap of local optima. Nevertheless, there are three main differences between TS and SA. Firstly, unlike SA, TS only accepts moves within the vicinity of the current solution that improve the objective function. Secondly, TS always searches for the best solution in the current neighbourhood before applying the replacement criterion. Thirdly, the most distinguishing feature of TS is the use of
- 56. 32 2.3 HeuristicAlgorithms a short term memory called a tabu list, in which moves that have been recently visited during the search are recorded. Moves in the tabu list are considered prohibited by the search and cannot be visited again for a certain number of iterations. The idea is to avoid the problem of cycling, meaning that the search may be trapped within the boundaries of a certain neighbourhood region, oscillating among solutions that have been previously visited, as illustrated in Figure 2.10. By prohibiting recently visited moves, the algorithm is forced to explore new areas of the search space in an attempt to escape local optima. Figure 2.10: The problem of cycling. The size of the tabu list is usually fixed, such that some old tabu moves are removed to allow for recording new moves recently visited, and the duration that a move is declared tabu is called its tabu tenure. Hence, the structure of the neighbourhood being searched varies dynamically from one iteration to another. However, always restricting the search to non-tabu moves may prevent some promising search areas from being explored. To avoid this problem, TS often makes use of an aspiration criteria, which allow overriding the tabu status of some moves that look attractive from a search perspective. For example, an aspiration criterion may override the tabu status of a newly generated solution, if its objective value is better than the best solution found so far. To determine the tabu status of certain solutions, it is common in tabu search to identify particular solution features or neighbourhood moves as ‘undesirable’. Accordingly, a newly generated solution holding such features will be considered tabu. Similarly, some
- 57. 2.3 Heuristic Algorithms33 previously identified tabu moves may be prohibited during certain stages of the search. Algorithm 2.5 shows the main steps of TS for a cost minimization problem. Algorithm 2.5: The Tabu Search (TS) Algorithm. 1: Initialize an empty Tabu List T 2: Generate an initial solution x 3: Let x∗ ← x {x∗ is the best so far solution} 4: repeat 5: Generate a subset S of solutions in N(x) {N(x) is the current neighbourhood of x} 6: Select the best neighbourhood move x′ ∈ S, where f(x′) < f(x) 7: if (f(x′) < f(x∗)) then 8: x∗ ← x′ {aspiration condition: if current solution improves best so far, accept it even if it is in the tabu list} 9: x ← x′ {update current solution} 10: T ← T + x′ {update tabu list} 11: else 12: if (x′ ∈ N(x) T) then 13: x ← x′ {update current solution if the new solution is not tabu} 14: if (f(x′) < f(x∗)) then 15: x∗ ← x′ {update the best so far solution if the new solution is better in quality} 16: T ← T + x′ {update tabu list} 17: until (Done){stopping condition is reached} 18: Return solution x∗ It is also sometimes fruitful in TS to make use of an intensification and/or a diversifica- tion mechanism. Intensification tries to enhance the search around good solutions, while diversification tries to force the algorithm to explore new search areas, in order to escape local optima. For example, intensification can be performed by encouraging solutions that have some common features with the current solution. On the other hand, diversification may be enforced by applying a penalty in the objective function, at some stage of the search, to solutions that are close to the present one [156]. Some variations of TS also exist in the literature, for example Probabilistic TS assigns a probability to neighbourhood moves, such that some attractive moves that lower the solution cost are given a higher probability, while moves that result in a repetition of some previous state are given a lower probability [63]. Also, a Reactive TS was proposed by Battiti and Tecchiolli [8] in which the size of the tabu list is adapted dynamically during the search, according to the frequency of repeated moves. TS has been extensively applied to many CO problems. For example, it has been applied
- 58. 34 2.3 HeuristicAlgorithms to scheduling in manufacturing systems in [113], to uniform graph partitioning in [36], to the Quadratic Assignment Problem (QAP) in [140]. In addition, it was applied to many variants of vehicle routing problems, for example in [129], [100], [112], [143] and [99]. For more details about some variations and applications of tabu search, the reader is referred to [62] and [55]. 2.3.6 Variable Neighbourhood Search (VNS) Variable Neighbourhood Search (VNS) is a relatively new meta-heuristic that has been suggested by Hansen and Mladenovi´c in [69] and [70]. The main idea is based on explo- ring the search space by gradually increasing the neighbourhood size, within which a new solution is generated, until a certain stopping condition is reached. In addition, whenever a new solution is generated in the current neighbourhood, a local search is applied to this solution to optimize it before the replacement decision is undertaken. The basic VNS algorithm, as described in [70], is shown in Algorithm 2.6. Algorithm 2.6: The Variable Neighbourhood Search (VNS) Algorithm [70] . 1: Initialization: Select the set of neighbourhood structures Nk, (k = 1, ...kmax), that will be used in the search 2: Generate an initial solution x 3: repeat 4: k ← 1 5: while (k < kmax) do 6: Shaking: Generate a point x′ at random from the kth neighbourhood of x (x′ ∈ Nk(x)) 7: Local Search: Apply some local search method with x′ as initial solution; denote with x′′ the so-obtained local optimum 8: Move or not: if the local optimum x′′ is better than the incumbent, move there (x ← x′′), and continue the search with N1 (k ← 1); otherwise set k ← k + 1 9: until (Done){stopping condition is reached} 10: Return solution x In the above VNS algorithm, the neighbourhood size k increases from 1 to a certain maxi- mum value kmax. The most important step is the Shaking step in which a new point (solution) is generated within the current neighbourhood Nk(x). It is crucial to choose a shaking procedure that will allow enough perturbation of the solution, while preserving, at the same time, the most favourable solution features which should be utilized in obtaining
- 59. 2.3 Heuristic Algorithms35 a near optimum solution as the search progresses. For example, in solving the TSP, sha- king my be performed by displacing or inverting a sequence of k cities in each iteration, i.e., the number of cities to be displaced is the variable neighbourhood size that gradually increases from one iteration to the next. On the other hand, the local search step in the VNS algorithm intensifies the search to obtain a local optimum, which may replace the current solution. For example, in a TSP a 2-Opt improvement heuristic may be applied as a local improvement method (see Section 3.4 for more details about the 2-Opt heuristic). The basic VNS procedure is a descent algorithm with a first acceptance criterion. Ho- wever, the algorithm can be easily changed to a descent/ascent procedure, similar to SA, by accepting bad moves with a certain probability in Step 8 (move or not) of Algorithm 2.6. Also, the first improvement criterion adopted by the basic VNS in the same step, may be changed to a best improvement by selecting the neighbourhood k∗ , which yields the best improvement, among all kmax neighbourhoods. Other variations include starting the neighbourhood size from a certain value kmin rather than 1, and varying the step size, such that increasing the neighbourhood size allows jumps to far away regions of the space [70]. Another important variant of the basic VNS algorithm, called Variable Neighbourhood Descent (VND), was also suggested by Hansen in [70]. The idea is to apply a change of neighbourhood size within the local search as well. The steps of the VND algorithm as described in [70] are shown in Algorithm 2.7. Algorithm 2.7: The Variable Neighbourhood Descent (VND) Algorithm [70] . 1: Initialization: Select the set of neighbourhood structures Nk, (k = 1, ...kmax), that will be used in the local search 2: Generate an initial solution x 3: repeat 4: Set k ← 1 5: while (k < kmax) do 6: Exploration of the neighbourhood: Find the best neighbour x′ of x (x′ ∈ Nk(x)) 7: Move or not: if x′ is better than x, set (x ← x′); otherwise set k ← k + 1. 8: until (Done){no further improvement is obtained} 9: Return solution x The VNS meta-heuristic offers many attractive features that have encouraged researchers to use it for CO problems. First, the algorithm is simple, easy to understand and imple- ment, and, to a large extent, parameter-independent. Second, well-known neighbourhood moves and local search methods can be easily integrated in several ways and applied wi-
- 60. 36 2.4 ChapterSummary thin the VNS framework. Third, it is a robust technique that can be used in a variety of problem types with almost no modification to the basic structure of the algorithm. Fi- nally, it can be easily hybridized with other heuristic and meta-heuristic approaches to solve hard optimization problems. In fact, VNS has been successfully used in solving many well-known CO problems. For example, it has been applied to the TSP and the facility location problem in [70], and to the graph coloring problem in [6]. It has also been extensively applied to vehicle routing problems, for example in [19], [119], [118] and [73]. For more details about advances in VNS and its applications, the reader is re- ferred to the recent work by Hansen et al. [71]. As previously mentioned, our research applied the VNS approach, hybridized with SA, in solving the 1-PDP, as will be explained in detail in Chapter 10. 2.4 Chapter Summary Conventional OR techniques may not be sufficient for solving complex optimization pro- blems, which model a large number of decision making strategies in real-world applica- tions. Heuristic and meta-heuristic techniques, which usually perform well in most prac- tical situations, have become increasingly popular among researchers in the optimization field. In this chapter we introduced combinatorial optimization problems and the theory of al- gorithm and complexity analysis. We then briefly highlighted some exact algorithms that can be used to solve these problems to optimality, for limited problem sizes only. For more practical applications, though, a heuristic or a meta-heuristic approach is usually the preferred option. We reviewed in this chapter some important meta-heuristic techniques like Local Search, Simulated Annealing, Genetic Algorithms, Ant Colony Optimization, Tabu Search, and Variable Neighbourhood Search. In the next chapter, we will introduce an important class of CO problems, which is the class of vehicle routing and scheduling. A particular variant of vehicle routing problems, pickup and delivery problems, is introduced in Chapter 4. Since pickup and delivery problems are the main theme of our research, our investigation of some selected problems from this category continues in Chapters 5 to 10.
- 61. 37 Chapter 3 Vehicle RoutingProblems: A Literature Review Solving the Vehicle Routing Problem (VRP) and its related variants is an important field in the area of Operations Research (OR), which has attracted a growing interest in recent years. Effective decision support tools that can be adopted in logistic planning are cur- rently in great demand, since they can lead to a substantial cost reduction and efficient resource consumption. In addition, they can help reduce the negative environmental im- pacts of transportation. An European Commission White Paper [1] states: A modern transport system must be sustainable from an economic and social as well as an environmental viewpoint. Plans for the future of the transport sector must take account of its economic importance. Total expenditure runs to some EUR 1000 billion, which is more than 10% of gross domestic product. The sector employs more than 10 million people. It involves infrastructure and technologies whose cost to society is such that there must be no errors of judgment. The use of automated route planning and scheduling can lead to huge savings in transpor- tation costs, typically ranging from 5% to 20% [147], which should contribute to boosting the overall economic system. This great potential, together with the advancement in tech- nological and computational powers in the last three decades, has encouraged researchers to experiment with diverse algorithms and address different applications, in order to meet the increase in demand for effective vehicle routing support tools. This indeed resulted in a large increase in the literature dealing with the VRP, especially given that its theme encompasses several intertwined disciplines, such as scientific computing, operations re- search, and business and management [48]. The rest of this chapter is organized as follows: Section 3.1 gives an idea about the dif- ferent types of vehicle routing problems existing in the literature and highlights the diver- sity of published research in this area. Section 3.2 briefly describes one particular VRP va- riant that is of interest to our research: the Vehicle Routing Problem with Time Windows
- 62. 38 3.1 VehicleRouting Problems (VRPs) (VRPTW). Section 3.3 emphasizes heuristic approaches for the solution construction of the VRPTW, while Section 3.4 describes some popular solution improvement heuristics. Section 3.5 then briefly summarizes some meta-heuristic algorithms that were applied to the VRPTW. Finally, section 3.6 concludes this chapter with a short summary. 3.1 Vehicle Routing Problems (VRPs) VRPs generally involve problems that deal with the transportation of goods or people bet- ween depots and customers, where the objective is to design an optimum schedule for one or more vehicles to service the clients with minimum possible operational cost and maxi- mum customer satisfaction. For example, the routing and scheduling plan may try to mi- nimize the number of vehicles used, the total distance traveled, and the labour force used. At the same time, it may try to maximize orders and volumes serviced per unit distance. Moreover, for practical applicability of the problem, a number of constraints are usually added to the basic model. These may include respecting the capacity of the vehicle, ap- plying a certain visiting order, or adhering to preferred service times for clients and to the maximum working hours of drivers. Other practical considerations include service types for customers (delivery and/or collection), number of depots (one/two/multiple), types of operating vehicles (homogeneous/heterogeneous), drivers availability (fixed/variable start times), and whether the information concerning routing requests is known in advance or revealed in real-time. As a generalization of the famous TSP, the VRP is an N P-hard problem [54] [147]. This fact, together with the above mentioned constraints and practical considerations, add to the difficulty of handling the different variants of vehicle routing problems, and contribute to the existence of a wealth of models and algorithms that deal with this problem. Given the large amount of published research tackling the various types of VRPs [48], it becomes increasingly difficult to keep track of all problem types and underlying so- lution methods. However, a general classification of the basic routing and scheduling problems and their interconnections is shown in Figure 3.1. Two main problem catego- ries are distinguished in this figure, based on the observation that routing and scheduling problems are usually represented as graphical networks, in which pickup and/or delivery points are represented as nodes connected with line segments called arcs. The first cate- gory is called node routing problems, in which the service demand is associated with nodes (locations). The second category is called arc routing problems, in which the service demand is associated with the arc connecting two nodes. Arc routing problems involve applications like refuse collection and winter gritting, and are often referred to as
- 63. 3.1 Vehicle RoutingProblems (VRPs) 39 the Chinese Postman Problem (CPP) [66]. The CPP, though, is beyond the scope of our research and will not be addressed further. As Figure 3.1 indicates, the simplest type of node routing problems is the TSP, in which a single vehicle is required to visit a number of nodes, with no restriction on the capacity of the vehicle. It is required in the TSP that the trip starts and ends at the same node, without identifying a particular depot point. The multiple-vehicle variant of the TSP is called the Multiple Traveling Salesman Problem (MTSP). If the demands of locations are added to the problem, the capacity of the vehicle is restricted, and a depot point is identified, it is usually called the Capacitated Vehicle Routing Problem (CVRP). Requesting two different service types (pickup and delivery) in the problem instance, transforms it to the Pickup and Delivery Problem (PDP) category. In addition, a service time window (TW) may be imposed on any of the above variants to allow for more realistic applications of the problem, in which the visiting time of each node is restricted between certain pre-defined bounds. For further details about the main variants of the VRP and both exact and heuristic methods applied to solving it, the reader is referred to the book edited by Toth and Vigo [147]. Figure 3.1: Vehicle routing and scheduling problems.
- 64. 40 3.1 VehicleRouting Problems (VRPs) Recently, Eksioglu et al. in 2009 carried out a taxonomic review of the VRP [48], in which they demonstrated that the growth in the literature published from 1955 to 2005 is almost perfectly exponential, if we eliminate the first two years of data. Despite this large increase in published VRP literature, other OR/MS (Operations Research/Management Sience) disciplines have grown at a much higher rate, which indicates that research in the VRP area is probably more sophisticated and requires high skills in analysis and design. The authors classified the characteristics of different VRPs, which should help understand the extent of the available literature and also identify potential areas where more research seems to be needed. Their taxonomy of the VRP research as it appears in [48] is shown in Figure 3.2. This taxonomy classifies the VRP literature under five major categories: 1. Type of Study: which identifies the nature of the study itself, for example: theore- tical or applied. Some techniques listed under applied literature include exact and heuristic methods. 2. Scenario Characteristic: which indicates the problem characteristics and the ope- rating scenario of the vehicle routing process. For example, under this category, static problems, in which all requests are determined in advance of the solution pro- cess, are distinguished from dynamic problems in which some requests may arrive later (in real-time). Dynamic problems should account for some probable changes in the current progressing routing plan. Other problem characteristics identified here include the type of service (pickup/delivery), and the precedence and coupling requirements 1 . 3. Problem Physical Characteristics: which outlines factors that directly affect the solution, such as number of depots, number of vehicles, capacity and time win- dows constraints. This category also distinguishes node routing problems from arc routing problems. 4. Information Characteristics: which identifies the nature of information presen- ted and accessed by the underlying solution methodology, and is mainly directed to the study of fuzzy and dynamic routing problems. For example, acting under uncertainty, when certain information (e.g. delays/vehicle breakdown) are revealed during an emergency situation, may fall under this category. 5. Data Characteristics: identifies the type of data used to evaluate the solution me- thod, such as real-world or synthesized data. 1 Precedence means that a pickup location must precede its corresponding delivery, while coupling re- quires that both the pickup and its delivery must be served by the same vehicle.
- 65. 3.1 Vehicle RoutingProblems (VRPs) 41 Figure 3.2: Taxonomy of the VRP literature [48].
- 66. 42 3.2 TheVehicle Routing Problem with Time Windows (VRPTW) The above taxonomy gives a general idea about how sophisticated and diverse the VRP literature is. As previously mentioned in the introduction to this thesis (Chapter 1), our research mainly focuses on one particular variant of vehicle routing problems, which are problems that deal with two types of services simultaneously, pickup and delivery (PD). Specifically, we selected two main problems from this category, the Pickup and Delivery Problem with Time Windows (PDPTW) and the One-commodity Pickup and Delivery Problem (1-PDP), with more emphasis on the first of these two variants. In the next chap- ter, we will present a brief summary of some important PD problems, and give examples of published research in this field. On the other hand, we will devote the rest of this chapter to an important and closely related problem in the literature, which is the Vehicle Routing Problem with Time Windows (VRPTW). This problem is significant to our research, be- cause it bears similarities to our selected PD problems, in terms of the constraints involved and the solution approaches used, as well as the objective function. Moreover, it is one of the most well-studied problems in the vehicle routing field. As previously mentioned in Chapter 2, exact algorithms have their limitations in solving large scale combinatorial optimization problems. In the vehicle routing domain, Hasle and Kloster [72] indicate that today’s exact methods cannot consistently solve VRP ins- tances having more than 50-100 customers, which is generally small for most realistic applications. Moreover, given the complexity of the problem, it is very unlikely that an algorithm that solves the problem to optimality in a reasonable time will be developed in the near future. Hence, our research mainly investigates heuristics and meta-heuristics as possible solution methods to the problems of interest. This literature review will, the- refore, focus on the approximation techniques introduced in Chapter 2. In other words, we will highlight selected published research that deals with heuristic and meta-heuristic methods only. Information about research that uses exact methods can be found in several vehicle routing and pickup and delivery problems surveys that we are going to mention throughout this discussion . In this chapter and the next chapter, a general classification and overview of the literature is presented, while more details about previous research par- ticularly related to the individual problems we tackled will be presented in the chapters dedicated to these problems (i.e., Chapters 5 to 10). 3.2 The Vehicle Routing Problem with Time Windows (VRPTW) The VRPTW is an important problem occurring frequently in transportation systems. The problem deals with a number of customer requests that must be dealt with by a fleet of
- 67. 3.2 The VehicleRouting Problem with Time Windows (VRPTW) 43 vehicles. All customers are assumed to require the same type of service (either pickup or delivery but not both) 2 . Each vehicle route must start and end at a central depot, and each customer must be visited exactly once. Two main constraints are strictly enforced in this problem. The capacity constraint requires that the vehicle capacity should not be exceeded at any time, while the time window (TW) constraint requires that each customer must be visited during a pre-specified time window interval, i.e., if the vehicle reaches the customer before the beginning of its TW, it should wait until the allowed service time begins. Similarly, arrival after the deadline of the TW means a violation of problem constraints. The solution objective is usually hierarchical, such that the minimization of the number of vehicles used is a primary objective, followed by minimizing the total travel distance of the operating fleet, or the total schedule duration. Figure 3.3 shows a representation of a small instance of the VRPTW, before solving it (Figure 3.3(a)) and after solution (Figure 3.3(b)). (a) VRPTW - before solution (b) VRPTW - after solution Figure 3.3: The vehicle routing problem with time windows. The VRPTW is one of the most well-studied problems in the vehicle routing research. Both exact and heuristic methods have been widely applied to solving this problem. In our current literature summary, we mostly base our review on the 2-part survey of the VRPTW by Bräysy and Gendreau [20] and [21]. As previously mentioned, our investigation of the problem focuses on heuristic and meta-heuristic solution methods, since exact methods are only limited to small problem sizes. Up-to-date exact algorithms were only capable of solving some of Solomon’s benchmark instances (developed in [141]) of 100 customers 2 In the current discussion, we use the terms requests and customers interchangeably to refer to the set of nodes to be visited. Nevertheless, a distinction between the two terms will be introduced within the context of the PDPTW in Chapters 5 to 8.
- 68. 44 3.3 SolutionConstruction for the VRPTW to optimality. Moreover, most of the solved instances are characterized by having a short schedule horizon (a short TW width). The remaining (unsolved) instances have a long schedule horizon, and are much harder for exact algorithms to solve. For further details about some important exact algorithms in the VRPTW literature, the reader is refereed to Cordeau et al. [28]. Solving the VRPTW generally requires two types of decisions: 1) the assignment or the grouping decision, which means to assign to each vehicle a subset of nodes from the customers set, and 2) the routing and scheduling decision which involves generating a minimum cost route for each vehicle to visit its assigned requests, such that the generated route respects the capacity and TW constraints. It is often observed that for vehicle routing problems of this sort, the assignment decision is usually of more importance than the routing and the scheduling decision, in determining the final solution quality [137]. In addition, the solution process often consists of two phases: solution construction, in which one or more solutions for the problem is generated, and solution improvement, in which the initial solution is improved using a heuristic or a meta-heuristic approach. Both phases are discussed in the following two sections. 3.3 Solution Construction for the VRPTW Solution construction refers to the creation of a set of routes for the vehicles by selecting nodes (customers) and inserting them in one of the partial routes already created, or in a new route. The decision to select a particular node for insertion is usually based on a cost-minimization criterion, and requires that the insertion of the node in a selected route does not cause violations of problem constraints. Two main types of solution construction exist in the VRPTW literature: sequential construction, which builds routes one after the other, and parallel construction which builds several routes at the same time. Sequential construction does not attempt to allocate an additional vehicle unless no more requests can be ‘feasibly’ added to the current vehicle. A parallel construction, on the other hand, initially pre-specifies the number of vehicles that could be used, but more vehicles can be added as needed if the initial estimate of the number of vehicles does not serve all requests without violating the constraints of the problem. Figure 3.4 demonstrates these two variants.
- 69. 3.3 Solution Constructionfor the VRPTW 45 (a) Sequential construction (b) Parallel construction Figure 3.4: Solution Construction. Several famous construction heuristic are described by Solomon in [141]. These are the savings heuristics, the time oriented nearest neighbour heuristic, the insertion heuristics, and the time oriented sweep heuristic. These heuristics are briefly explained below. 3.3.1 Savings Heuristics This method is based on one of the most successful construction heuristics for the VRP, the savings heuristic proposed by Clarke and Wright [26]. The idea is to initially assign each customer individually to one vehicle. Thus, an initial vehicle route will only consist of a trip from the depot to the customer and then back to the depot. If we have two such routes, one serving customers i and another serving customer j, combining the two routes such that the new route will service both i and j will result in a saving of service cost Sij = ci0 + c0j − cij. This idea is illustrated in Figure 3.5. The parallel savings construction algorithm for the VRPTW can be summarized as fol- lows: 1. Step 1: compute saving Sij for every two customers i and j, and sort them from largest to smallest; 2. Step 2: take saving Sij from top of the saving list. Search for a route containing (0, i) and another containing (j, 0). If such routes exist, join them to form route (0, i, j, 0). Solomon in [141] adapted the Clarke and Wright savings heuristic to the VRPTW by taking the route orientation into account, as shown in Figure 3.5 and Step 2 above. In addition the algorithm must check the capacity and time window feasibility at every step
- 70. 46 3.3 SolutionConstruction for the VRPTW Figure 3.5: The savings heuristic. in the heuristic process 3 . A limit on the waiting time is also imposed when routes are joined. A sequential version of the savings heuristic will consider one route at a time, and then implement the best saving that can be achieved if another route is joined with the current route. The sequential savings heuristic can be briefly described as follows: 1. Step 1: compute saving Sij for every two customers i and j, and sort them from largest to smallest; 2. Step 2: consider the current route (0, i, ..., j, 0). Find routes containing (k, 0) and (0, l) that have the best savings Ski and Sjl in the saving list, and select the best one among them for joining with the current route. If no more savings can be implemented for the current route, move to the next route. 3.3.2 Time Oriented Nearest Neighbour Heuristic This heuristic is a sequential construction process that starts by initializing the current route with the depot. Then the customer closest to the depot, in terms of a certain cost measure c, is inserted next in the route. Subsequently, the customer ‘closest’ to the last added customer will be inserted next, if its insertion causes no constraint violation. If 3 Solomon also proposed an important heuristic for speeding the TW feasibility checking called the push forward (PF) heuristic, which is based on calculating the extra time occurring in the schedule due to the insertion of the new node in the route. For more derails see [141].
- 71. 3.3 Solution Constructionfor the VRPTW 47 all remaining un-routed customers cannot be feasibly inserted in the current route, a new route is allocated and the process is repeated. The cost function c used by Solomon in [141] to determine the ‘closeness’ between two customers, is based on both geographical and temporal separation between the two nodes. This measure is described by cij = w1 ×dij +w2 ×Tij +w3 ×vij, where dij is the distance between the two customers4 , Tij is the time difference between the completion of the service at i and the beginning of the service at j, and vij is the urgency of visiting customer j, which is calculated as the time remaining until the deadline of servicing customer j is reached. The weights w1, w2 and w3 satisfy: w1 + w2 + w3 = 1.0, and w1 ≥ 0, w2 ≥ 0 and w3 ≥ 0. 3.3.3 Insertion Heuristics This is another class of sequential construction heuristics described by Solomon in [141], and is based on expanding the current route by inserting one un-routed customer at each iteration. The general idea is demonstrated in Figure 3.6, where the un-routed customer k (left) was inserted between customers i and j (right) in the progressing route. Figure 3.6: The insertion heuristic. More specifically, the insertion heuristic starts by initializing the route under consideration with a seed customer. Other customers are then selected for addition to the current route, 4 One simplifying assumption that is usually made in the VRP is that the speed of the vehicle is constant (often = 1). If this assumption is made, the distance separating i and j (dij) and the travel time between i and j (tij) may be used interchangeably.
- 72. 48 3.3 SolutionConstruction for the VRPTW based on two cost measures to be defined shortly. When no more un-routed customers can be ‘feasibly’ inserted in the current route, the process is repeated for a new route. The seed customer could be selected as the customer farthest from the depot, or as the customer having the earliest allowed starting service time. Three insertion heuristics have been defined by Solomon, based on the criteria used for selecting an un-routed customer to be inserted next in the current route. The first and most successful insertion heuristic (called I1 insertion) proceeds as follows, after initializing the route with a seed customer: 1. Assume the current partial route created is (i0, i1, i2, ...im), where i0 and im re- present the depot; 2. For each un-routed customer u, find its best feasible insertion position in the route, between two adjacent customers i and j. The best insertion position is the one that minimizes the first cost measure, which we will call c1(u). c1(u) is a measure that is calculated based on both the extra travel distance and time delay that happens in the route, due to the insertion of u; 3. Find the best customer u∗ to be inserted in the route, where u∗ is the customer having the maximum value of a second measure, which we will call c2(u). c2(u) is calculated based on both c1(u) (the first measure) and the distance from the depot. The second insertion heuristic (I2) differs from I1 in the definition of the measure c2(u), since it is now a combined measure of the total route distance and time resulting after the insertion of u (rather than the extra distance and time, as in I1). On the other hand, the third sequential insertion heuristic (I3) differs from I1 in the definition of c1(u), since the urgency of servicing the new customer u is now also considered in this measure5 . These insertion heuristics are in fact a generalization of the time oriented nearest neigh- bour heuristic, described in Section 3.3.2, since they allow for inserting a new customer between any two nodes in the route, rather than only at the end. They are sometimes refer- red to in the literature as the cheapest insertion heuristics, with I1 being the most popular among them. A parallel construction heuristic for the VRPTW was proposed by Potvin and Rousseau [121], based on the I1 heuristic described above. In this heuristic several routes are first initialized with seed customers. To determine the number of initial routes, the authors 5 Our presentation of Solomon’s insertion techniques and the cost measures adopted within these heuris- tics is simplified to a large extent, in order to fit the general overview intended here. For further information, the reader is referred to [141] and [20].
- 73. 3.4 Solution Improvementfor the VRPTW 49 first ran the I1 sequential construction algorithm. Further routes are later added as needed if the initial number of routes does not yield a feasible solution. After determining the best (feasible and least cost) insertion position for each un-routed customer, the customer whose insertion in the solution causes the least increase in the overall cost is selected next for insertion. Besides a weighted sum of the extra travel distance and time delay used by Solomon to determine the cost of the insertion, this parallel heuristic adds a regret measure over all routes. The regret measure is a type of ‘lookahead’ estimate of the cost of not inserting the customer immediately in its current best route, i.e., if its insertion was postponed to be carried out later in a different route. Although these construction algorithms have mainly been designed for the VRPTW, re- searchers have also adapted them to different problem variants, such as the PDPTW. More discussion about solution construction heuristics from the perspective of the PDPTW will be presented in Chapter 7. 3.3.4 Time Oriented Sweep Heuristic In this heuristic Solomon [141] suggests a two phase approach to construct a VRPTW solution. The first phase is a clustering phase, in which customers are assigned to vehicles in a way similar to the original sweep heuristic suggested by Gillett and Miller [58]. In this heuristic, a centre of gravity is computed based on the angle between the depot and a randomly selected customer. Then, the remaining customers are assigned to vehicles according to the polar angle separating their locations from the centre of gravity. After the assignment phase, the scheduling phase starts by inserting the selected customers in their respective routes, such that the insertion process follows Solomon’s I1 insertion heuristic, explained in Section 3.3.3. 3.4 Solution Improvement for the VRPTW Solution improvement within the context of the VRPTW refers to the gradual and repea- ted modification of the initial solution until a certain stopping condition is satisfied. The initial solution is usually a feasible solution, obtained using a construction algorithm, such as the ones described in the previous section. One must then define a neighbourhood move that can be applied to the initial/current solution to obtain a new solution within its neigh- bourhood, i.e., the new solution only differs in a few attributes from the current solution. For example, the new solution could be obtained by modifying some edges connecting customers in the current solution. Afterwards, the generated solution is evaluated based
- 74. 50 3.4 SolutionImprovement for the VRPTW on the objective function, and may replace the previous solution if it is smaller in cost. Nevertheless, two main acceptance criteria are usually adopted in this context. The first acceptance criterion selects the first solution found in the neighbourhood of the current solution that improves the objective function. On the other hand, the best acceptance exa- mines all solutions in the neighbourhood of the current solution, and selects the best one among them. The above process is often referred to as local search, and usually results in a local opti- mum that may be very far from the optimal solution. In addition, the quality of the local optimum depends heavily on the quality of the initial solution. Within the vehicle rou- ting literature, edge exchanges are the most popular solution improvement method, first described by Lin [103] for the TSP. Edge exchanges are applied to one route in the current solution, and depend on removing a number k of edges from the current route, and replacing them with another set of k edges. Thus, the process is often called k-exchange. A route that cannot be further improved by a k-exchange is called a k-optimal. Performing all possible k-exchanges on a route requires O(nk ) time. Thus, moves beyond 2 or 3 exchanges are very time consuming. The 2-exchange move is illustrated in Figure 3.7. Figure 3.7: 2-exchange move. Several other improvement methods have been described for the VRPTW, a few of them are modifications to the basic edge-exchange moves. We will briefly mention some of the most important methods, without going into details, since most of them are beyond the
- 75. 3.5 Meta-heuristics forthe VRPTW 51 scope of our research. Potvin and Rousseau [122] introduce a modification to the 2-Opt heuristic of [103], called the 2-Opt* exchange heuristic. 2-Opt* works on two different routes (unlike 2-Opt which exchanges edges belonging to the same route). 2-Opt* tries to combine the two routes without changing the orientation of the tours, by appending the last customers of the second route after the first customers of the first route. Also, Or [114] modified the 3-Opt operator of [103] and introduced the Or-Opt operator for the TSP. In this operator, three edges in the original tour are replaced by three new ones without changing the orientation of the tour. Inter-route operators for the VRP were introduced by Savelsbergh [136] and used by Pos- ser and Shaw [123] for the VRPTW. These are the re-locate, exchange and cross opera- tors. The relocate operator moves a customer from one route to another. The exchange swaps two visits in different routes and the cross is similar to the 2-Opt* proposed in [122]. Other neighbourhood operators are the λ-exchange of Osman [115], the CROSS- exchange of Taillard et al. [143], the GENI-exchange of Gendreau et al. [56], ejection chains of Glover [60] [61] and cyclic k-transfers of Thompson and Psaraftis [146]. In addition, other improvement heuristics include the Large Neighbourhood Search (LNS) of Shaw [139]. This algorithm removes a large number of customers from their current routes, and re-schedules them at optimal cost. The removed customers could be selected at random, or based on some relatedness measure that takes into account customers demands or starting times. Schrimpf [138] used a similar approach and calls it ruin and recreate. Finally, Bräysy [18] introduces a modified ejection chain approach that also considers re-ordering of the routes. For the interested reader, more details and illustrations of the different types of solution improvement heuristics are presented in [20]. Similar to solution construction approaches, solution improvement methods have also been adapted and applied to PD problems. Some examples of their application to these problems will be presented in the upcoming chapters, when related work to our selected problems is investigated. 3.5 Meta-heuristics for the VRPTW Since our research is particularly focused on meta-heuristic techniques, we highlight in this section some papers that use meta-heuristic approaches for the VRPTW, and explain briefly some important common features among them. Table 3.1 summarizes research
- 76. 52 3.5 Meta-heuristicsfor the VRPTW that uses Tabu Search (TS), Simulated Annealing (SA), Ant Colony Optimization (ACO), Variable Neighbourhood Search (VNS), and Guided Local Search (GLS). As Table 3.1 indicates, most meta-heuristic techniques apply some cheapest insertion method for the construction of the initial solution, with Solomon’s insertion being the most popular among them. On the other hand, solution improvement is usually per- formed using well-known neighbourhood moves like 2-Opt, 2-Opt*, 3-Opt, Or-Opt, λ- interchange. . . etc. To accelerate the optimization, the search is sometimes limited to neighbourhoods having particular features. For example, in [53] only moves involving arcs close in distance are allowed, while the approach in [22] only considers selected routes and selected customers for merging during the solution construction process. Some approaches allow infeasibility of solutions (e.g. [99]) and penalize the violation in the objective function. In addition, many algorithms apply a specific strategy for reducing the number of vehicles. For example, the algorithm in [53] removes customers from routes having a small number of nodes and inserts them into other routes, using Or-Opt exchanges. Also, in [11] a 2-phase approach is applied, where the first phase is an SA procedure intended to minimize the number of routes, while the second phase focuses on minimizing total distance. On the other hand, the ACO of [52] uses two colonies, one for the minimization of the number of vehicles and the other to minimize total travel distance. Different diversification mechanisms have also been applied within the various meta- heuristics to improve solution quality, so that the algorithm is directed towards exploring new areas of the search space. For example, in [129] the search is diversified using an adaptive memory, where routes taken from best solutions found during the search are sto- red. Some of these routes are combined to form solutions that will act as new starting solutions for the TS. Also, in [143] diversification is applied by penalizing frequently per- formed exchanges. On the other hand, the approach in [22] adopts a Guided Local Search (GLS) diversification mechanism that penalizes certain solution features, if they are going to bring the search close to previously visited solutions. Intensification of the search around best solutions discovered during the search have also often been considered. For instance, in [101] the SA algorithm is forced to start from the current best solution several times, while the ACO in [52] uses global pheromone updating, to further explore search areas around the best solutions obtained. Also, in [143] intensifying the search is done by selecting the best routes and re-ordering customers within these routes, using Solomon’s I1 insertion. A post-optimization phase is sometimes added to the basic meta-heuristic to further en- hance best solutions. For example, the approach in [129] performs a post-optimization
- 77. 3.5 Meta-heuristics forthe VRPTW 53 using an exact set-partitioning algorithm on routes obtained from the adaptive memory. Also, in [143] the post-optimization technique of [56] is applied to each route of the final solution obtained. In many cases, hybridization of more than one meta-heuristic technique is adopted. For example, in [101] a tabu-embedded SA is used. Also, in [25] the annealing process is enhanced using a varying size tabu-list, and the authors in [11] use both SA and LNS. To complete our review of meta-heuristic techniques for the VRPTW, Table 3.2 summa- rizes some Genetic Algorithm (GA) approaches that have been applied to this problem. Almost all of these techniques hybridize GAs with some other heuristic, local search, or another meta-heuristic approach for solution construction and/or solution improvement. Solution representation is usually based on integer representation, rather than the classical bit string encoding. Thus, genetic operators are directly applied to problem solutions. The initial population is often created randomly, or by adapting some well known construction methods. For example, in [144], customers are grouped randomly, and then the cheapest insertion method of [65] is used for routing each group of customers. The fitness function is usually based on solution cost, i.e., number of routes, total dis- tance and total duration. However, since some approaches allow violations of problem constraints, the fitness function should take this into account. For example, in [16], the fitness considers the number of un-serviced customers in an infeasible solution. Also, in [15] a penalty is added in the objective function to violated constraints. On the other hand, the selection scheme often applied is roulette-wheel selection. However, other selection methods have also been used. For instance, a ranking scheme is used in [120]. Regarding recombination, most of the GA techniques apply specialized crossover opera- tors, since traditional genetic operators do not usually fit the VRPTW. For example, the authors in [120] propose a Sequence-Based Crossover (SBX) and a Route-Based Cros- sover (RBX), while the GA in [16] applies special crossover operators that depend on a global precedence relationship among genes, such as the distance or time windows orde- ring. On the other hand, mutation is often based on traditional local search operators like 2-Opt, 2-Opt*, Or-Opt and LNS. In addition, mutation is sometimes applied to reduce the number of routes in the solution, as in [120] and [15]. To sum up, meta-heuristic approaches to the VRPTW often embed solution construction and improvement methods as described previously. However, they usually obtain bet- ter quality solutions than algorithms that apply straightforward solution construction and improvement techniques on their own. This often comes at the expense of greater com- plexity in implementation and increased computation time, though. Bräysy and Gendreau
- 78. 543.5Meta-heuristicsfortheVRPTW Table 3.1: Meta-heuristicsfor the VRPTW. Paper Algorithm Solution Construction Solution Improvement Remarks Garcia et al. [53] TS Solomon’s insertion 2-Opt*, Or-Opt Parallel implementation - Only allows moves involving arcs close in distance Rochat and Taillard [129] TS Solomon’s insertion & 2-Opt 2-Opt, relocate Uses adaptive memory containing routes obtained from best solutions visited during the search, with the purpose of providing new starting solutions Taillard et al. [143] TS Solomon’s insertion CROSS Decomposes solutions into subsets of routes, based on a polar angle associated with the centre of gravity of each route - TS applied to each subset separately - Diversification by penalizing frequently performed exchanges - Intensification by reordering customers within best routes using I1 insertion Lau et al. [99] TS Relocate from a holding list where all customers are initially stored Exchange - relocate Allows violation of constraints for a penalty in the objective function - Penalty parameters adjusted dynamically Chiang and Russell [25] SA The parallel construction of Russell [133] λ-interchange and k-node interchange Enhancing of the annealing process via a varying size tabu-list Li et al. [101] Tabu- embedded SA Insertion and exten- ded sweep heuristics of Solomon Shifting and exchange of customer segments bet- ween and within routes SA restarts from current best solution several times - Reduce routes by reordering customers and inserting them into other routes - Diversification by random shifts and exchange of customer segments Bent and Van Hentenryck [11] SA and LNS Not specified 2-Opt, Or-Opt, relocate, exchange and 2-Opt* 2-phase approach: SA to minimize number of routes and LNS to minimize total distance Gambardella et al. [52] ACO Nearest-neighbour heuris- tic with probabilistic rules CROSS 2 colonies: first one minimizes number of vehicles, and the second minimizes total distance - Both cooperate in updating best solution Bräysy [19] VNS Solomon’s Insertion and the parallel construction of Russell [133] Or-Opt exchanges and CROSS-exchanges Reducing number of routes using an ejection-chain phase - Second VNS phase for improving total distance Bräysy et al. [22] Threshold accepting (TA) & Guided Lo- cal Search (GLS) Savings heuristic SPLIT (of routes) - limited CROSS - limited Or-Opt Construction algorithm only considers selected routes and selected customers for merging - Initial phase to minimize number of routes, based on simple customer re-insertion - Further improvement using TA & GLS - TA allows local search moves that worsen the objective function, provided it is within a certain threshold limit - GLS penalizes certain solution based on some solution features (e.g. long edges), not considered part of a near-optimal solution
- 79. 3.5Meta-heuristicsfortheVRPTW55 Table 3.2: GeneticAlgorithms for the VRPTW. Paper Initial Population Crossover Mutation Remarks Thangiah [144] A set of K seed angles are planted, where K is the number of vehicle. Sec- tor rays are drawn originating from the depot to each seed angle - Routing done using cheapest insertion of Golden and Stewart [65] Exchange a randomly selected por- tion of the bit string between chro- mosomes Randomly change bit value Cluster first route second approach cal- led GIDEON - Each chromosome is a set of possible clustering scheme and the GA used to improve the clustering - Routes improved by λ-interchange Potvin and Bengio [120] Chromosome is a problem solution crea- ted using Solomon’s cheapest insertion Sequence-Based Crossover (SBX): two sub-tours from two parents are linked together - Route-Based Crossover (RBX) replaces a route of one parent by another route from the second parent - Repair operator applied for infeasible offspring Reduce number of routes by in- serting customers from a ran- domly selected short route into another - Local search muta- tion using Or-Opt exchanges The approach is named GENEROUS Blanton and Wainwright [16] Solution coded as a permutation of customers - Insertion heuristic used to construct solution Specialized crossover that uses glo- bal precedence relationship among customers (TW/distance) Randomly exchange 2-edges The GA searches for a good ordering of customers. while construction of fea- sible solutions is handled by a greedy heuristic Berger et al. [15] Random insertion heuristic - Chromo- somes are problem solutions - Two po- pulations: POP1 used to minimize dis- tance and POP2 minimizes constraint violations Insertion-based crossover (IBX): combining k routes from parent1 (one at a time) with subsets of customers from parent2 - Insert- within-route crossover (IRNX): si- multaneously combines k routes from parent1 with subsets of custo- mers from parent2 Large Neighbourhood Search (LNS) mutation - Each custo- mer considered for reinsertion in an alternate route. eliminate shortest route - Reorder custo- mers within a route Simultaneous evolution of two popula- tions (minimize distance & constraint violations) - POP2 has smaller number of routes than POP1 - When a new better solution is obtained from POP2, POP1 is replaced by POP2 Repoussis et al. [128] Discrete arc-based representation of in- dividuals combined with a binary vec- tor of strategy parameters - Individuals created using greedy insertion from a Restricted Candidate List (RCL), with a probabilistic element Multiple-parent recombination ap- plied only to strategy parameters not to problem solutions Mutation rate determined by strategy parameters evolved during the search - Mutation done using “ruin and recreate” (λ + µ) Evolution Strategy (ES). Star- ting from µ individuals, at each iteration a new intermediate population of λ indi- viduals is produced by mutation - Evo- lutionary search is based only on muta- tion - Each offspring is further optimi- zed using route elimination and a local search (based on TS) to improve total distance
- 80. 56 3.6 ChapterSummary in [21] identified several techniques that seem to give good results when applied within the different meta-heuristic approaches. Some of these techniques are: 1) saving best solutions found during the search, 2) varying the neighbourhood structure, with the size of the neighbourhood being typically small, 3) using a memory structure to facilitate the search, 4) applying a specific strategy for reducing the number of routes in the solution and 5) hybridization of different heuristics and meta-heuristics. 3.6 Chapter Summary Vehicle routing problems have gained considerable attention in the last few decades. This is mainly due to the increase in complexity in transportation and logistics demands, and the urgent need for optimizing client services, reducing operational costs, and limiting the negative environmental impacts that may result from the non-optimal planning of vehicles and their routes. This chapter provided a general classification of vehicle routing problems existing in the literature and highlighted the diversity of problem types, solution methods, and the huge amount of published research that deals with this problem. We emphasized in this chapter one particular vehicle routing problem that is of interest to our study, the Vehicle Routing Problem with Time Windows (VRPTW). This problem is closely related to pickup and delivery problems that we investigated in our research, in terms of: problem definition, solution methods, objective function and constraint handling approaches. We presented in this chapter a summary of some important heuristic and meta-heuristic algorithms that have been applied to solving the VRPTW. The next chapter is devoted to an overview of pickup and delivery problems, which are the main focus of our research. We will present a general classification and highlight some published research in the field, as an introduction to a more in-depth analysis of our selected pickup and delivery problems, which will follow during the course of this thesis.
- 81. 57 Chapter 4 Pickup andDelivery Problems: A Literature Review We dedicate this chapter to a literature summary of Pickup and Delivery (PD) problems, since they are the main theme of our research. In addition, there are various types of pi- ckup and delivery problems and different classifications of these problems that distinguish them from other variants of vehicle routing problems. PD problems are widely applicable in areas such as: the transportation of raw materials from suppliers to factories, Internet-based pickup from sellers and delivery to buyers, food and beverage collection and delivery, post and parcel delivery, newspaper distribution, and airline and bus scheduling. In addition, an important related variant is the dial-a- ride problem and the handicapped persons’ transportation, where people instead of goods are transported, giving rise to customer inconvenience issues that should be taken into consideration while constructing a solution. Similar to other vehicle routing problems, pickup and delivery problems have attracted the attention of researchers in the last few decades. Nevertheless, published research tackling these problems is still relatively limited, compared to other variants of vehicle routing problems [137]. Several important surveys of pickup and delivery problems have appeared in the literature. The oldest ones are by Solomon and Desrosiers [142], and Savelsbergh and Sol [137]. Two other surveys appeared recently in [14] and [117]. A survey dedicated to the dial-a-ride problem is in [29]. Another survey on the one-to-one pickup an delivery problems (see Section 4.2 for a description of this problem category) is provided in [31]. In our literature summary, we mostly rely on the information and classification provided by Parragh et al. in 2008 [117], unless otherwise indicated. Parragh et al. [117] classify the different types of pickup and delivery problems under two classes: 1) problems that deal with the transportation of goods from pickup customers (also called backhaul customers) to the depot, and from the depot to delivery customers (also called linehaul customers). This class is called the Vehicle Routing Problem with
- 82. 58 4.1 TheVehicle Routing Problem with Backhauls (VRPB) Backhauls (VRPB), and 2) problems that deal with the transportation of goods/people between two pickup and delivery points. This class is called the Vehicle Routing Pro- blem with Pickup and Delivery (VRPPD). Sections 4.1 and 4.2 describe in detail these two categories. Section 4.3 briefly highlights some meta-heuristic techniques that have been applied to pickup and delivery problems. Finally, Section 4.4 concludes this chapter and introduces the rest of the thesis. 4.1 The Vehicle Routing Problem with Backhauls (VRPB) As mentioned above, this class deals with PD problems that involve the transportation of goods from the depot to customers and vice versa. These problems are further subdivided into four categories, generally based on the required visiting order of pickups and delive- ries, and whether a customer can demand both types of services. In all cases the objective is to minimize the total cost (e.g. total travel distance) of the routing plan, while adhering to some pre-specified problem constraints. For example, respecting the capacity of the vehicle at all times, and ensuring that the vehicle starts, if needed, with a load that is equal to the total load to be delivered. 1. The vehicle Routing Problem with Clustered Backhauls (VRPCB): where all delivery customers must be visited before all pickup customers. The single vehicle special case of the same problem is denoted by (TSPCB). 2. The Vehicle Routing Problem with Mixed Linehauls and Backhauls (VRPMB): where mixed visiting of pickup and delivery customers are allowed. The single vehicle case of the same problem is denoted by (TSPMB)1 . If the capacity of the vehicle is greater than or equal to the total sum of demands of pickup and delivery customers, the problem coincides with a simple TSP. 3. The Vehicle Routing Problem with Divisible Delivery and Pickup (VRPDDP): where each customer is both a pickup and a delivery customer. In addition, two visits to the same customer is allowed. The single vehicle variant of the same pro- blem is denoted by (TSPDDP). A possible scenario in this variant is that the vehicle could initially visit a few delivery customers to get rid of some of the load it is car- rying. Afterwards, the vehicle could visit another set of customers to perform both 1 More details about this problem will be presented in Chapter 9, since this problem is closely related to the One-commodity Pickup and Delivery Problem (1-PDP) that we handled in our research. Nevertheless, we refer to it in our research by the Traveling Salesman Problem with Pickup and Delivery (TSPPD), the name most commonly used for this problem in the literature (e.g. [110] and [75]).
- 83. 4.1 The VehicleRouting Problem with Backhauls (VRPB) 59 delivery and collection service at each, and finally a collection service would be performed at customers that were initially visited for delivery. This type of solution is called a ‘lasso’ solution. It is also observed in [117] that a solution method of the VRPMB (the second category) can be applied to the VRPDDP if each custo- mer demanding both delivery and pickup was modeled by two vertices, one for the delivery and another for the pickup. Very little research has been applied to this problem. 4. The Vehicle Routing Problem with Simultaneous Delivery and Pickup (VRPSDP): where each customer demands both a delivery and a pickup, but, unlike the VRPDDP (the third category), only one visit is allowed for each customer. The single vehicle variant is denoted by (TSPSDP). We summarize in Table 4.1 the most important features of the different variants of the VRPB described in this section. In this table, we indicate the problem name, whether the customer is allowed to demand both pickup and delivery, whether delivery customers must all be visited before pickup customers, or otherwise that mixed sequences of visits is allowed. The table also shows whether more than one visit is permitted, and if split- ting the demand of one customer is allowed. Finally, the last column of the table gives examples of selected published papers that dealt with the underlying problem, together with the solution method adopted. We used the notations: (LNS) for Large Neighbou- rhood Search, (TS) for Tabu Search, and (VNS) for Variable Neighbourhood Search. If the paper handled the single vehicle case, we indicate this by (SV) and if it handled the multiple vehicle case, we indicated this by (MV). The authors in [117] concluded from their survey that the LNS algorithm of [132] seems to be the most flexible and accurate meta-heuristic to date, since it can be applied to seve- ral problems in the VRPB class, and it provided new best solutions for several benchmark instances. On the other hand, they selected the heuristic in [111] among the fastest algo- rithms applied to the VRPB.
- 84. 604.1TheVehicleRoutingProblemwithBackhauls(VRPB) Table 4.1: TheVehicle Routing Problem with Backhauls (VRPB). Problem Customer both P&D Deliveries Visited be- fore Pickups Mixed Visi- ting One Custo- mer Visit Demand Splitting Example Papers and Solution Methods VRPCB TSPCB no yes no yes no Gendreau et al. [57]- heuristic (SV) Thangiah et al. [145]- heuristic (MV) Mladenovi´c and Hansen [108]- VNS (SV) Ropke and Pisinger [132]- LNS (MV) VRPMB TSPMB no no yes yes no Mosheiov [110]- heuristic (SV) Nagy and Salhi [111]- heuristic (MV) Ropke and Pisinger [132]- LNS (MV) VRPDDP TSPDDP yes no yes no (at most two) yes Hoff and Løkketangen [77]- TS (SV) Salhi and Nagy [134] - heuristic (MV) VRPSDP TSPSDP yes no yes yes no Alshamrani et al. [4]- heuristic (SV) Nagy and Salhi [111]- heuristic (MV) Ropke and Pisinger [132]- LNS (MV)
- 85. 4.2 The VehicleRouting Problem with Pickups and Deliveries (VRPPD) 61 4.2 The Vehicle Routing Problem with Pickups and Deli- veries (VRPPD) This class refers to problems that deal with the transportation of goods/people between a pickup and a delivery point. Two subclasses can be distinguished in this category: 1. The Pickup and Delivery Vehicle Routing Problem (PDVRP): where pickup and delivery points are unpaired, and one type of commodity is transferred. Thus, pi- ckup customers will supply the demands needed by the delivery customers. The single vehicle variant of this problem is denoted by (PDTSP). The objective func- tion in this problem category is to minimize the total cost of the route(s), without violating the capacity constraints. It should be noted that the PDTSP is the same as the 1-PDP investigated in our research, as explained in detail in Chapters 9 and 10. 2. The Classical Pickup and Delivery Problem (PDP) and the Dial-a-Ride Pro- blem (DARP): where a transportation request is associated with both a pickup (ori- gin) and a delivery (destination), i.e., customers are paired and the demand of the pickup is the same as the demand of its associated delivery. Unlike the PDP, the DARP additionally takes customer inconvenience into account. The single vehicle cases of these problems are denoted by SPDP and SDARP respectively. The ob- jective function for these problems usually tries to minimize the number of vehicles used (in the multiple-vehicle case) as a primary objective, followed by reducing the overall travel distance or service duration. The final solution should also adhere to all underlying problem constraints, such as the vehicle capacity and the visiting time restrictions. Our research handles the PDP problem with the addition of the time window constraint (denoted by PDPTW) in Chapters 5 and 6 for its single vehicle variant (denoted in our research by SV-PDPTW), and Chapters 7 and 8 for its multiple vehicle variant (denoted in our research by MV-PDPTW).
- 86. 62 4.2 TheVehicle Routing Problem with Pickups and Deliveries (VRPPD) We provide in Table 4.2 a summary of the most important features pertaining to the VRPPD. We indicate in this table the subcategory name, whether the customers are paired or not, whether a homogeneous commodity or multiple commodities are considered, and whether the problem is primarily associated with the transportation of people or goods. In addition, we highlight the most important constraints that are usually taken into consi- deration while handling the problem. These include: 1) the precedence constraint, in which a pickup request is required to precede its corresponding delivery, 2) the coupling constraint, which requires that the pickup and delivery pair must be visited by the same vehicle (in case of the multiple-vehicle variant), 3) the capacity constraint, which ensures that the load carried by the vehicle at any given time does not exceed its capacity, 4) the time window constraint, which requires that a customer has to be served in a pre-specified time interval, and 5) the maximum ride time constraint, which ensures that when people are transported, the time they spend in the vehicle does not far exceed the direct travel time between their origin and destination. Finally, the table gives examples of some published papers that deal with the problem. Again, we use (SV) for the single vehicle case and (MV) for the multiple-vehicle variant. (LNS) refers to a Large Neighbourhood Search algorithm, and (ALNS) refers to an Adaptive version of LNS, (HC) is a Hill Climbing approach, (GA) is a Genetic Algorithm, (GGA) is a Grouping Genetic Algorithm, (SA) is a Simulated Annealing algorithm, and (TS) is a Tabu Search algorithm. In our opinion, the GA in [158], the LNS in [12], the ALNS in [131], and the GGA in [126] seem to be among the state-of-the-art algorithms applied to the VRPPD, in terms of their solution quality and/or robustness or flexibility. More details about these approaches and other important research in the field will be described in the following chapters, where we address specific pickup and delivery problems in our research. Before we conclude our review of pickup and delivery problems, though, we present in the next section a brief summary of some meta-heuristic techniques that have been applied to this problem class.
- 87. 4.2TheVehicleRoutingProblemwithPickupsandDeliveries(VRPPD)63 Table 4.2: TheVehicle Routing Problem with Pickups and Deliveries (VRPPD). Problem Paired Custo- mers One Com- modity Type of Commo- dity Precedence Coupling Capacity TW Max Ride Time Example Papers and Solution Methods PDVRP PDTSP no yes goods no no yes no no Hernández-Pérez and Salazar-González [75]- heuristic (SV) Zhao et al. [158] - GA (SV) Dror et al. [45]- heuristic (MV) PDP SPDP yes no goods yes yes yes yes no Jih and Hsu [89]- GA (SV) Hosny and Mumford [84] - HC & SA & GA (SV) Li and Lim [100]- TS (MV) Bent and Van Hentenryck [12] - LNS (MV) Ropke and Pisinger [131] - ALNS (MV) Hosny and Mumford [80]- GA (MV) DARP SDARP yes yes people yes yes yes yes yes Psaraftis [124]- heuristic (SV) Jaw et al. [87]- heuristic (MV) Baugh et al. [9]- SA (MV) Rekiek et al. [126]- GGA (MV) Jørgensen et al. [91]- GA (MV)
- 88. 64 4.3 Meta-heuristicAlgorithms for Pickup and Delivery Problems 4.3 Meta-heuristic Algorithms for Pickup and Delivery Problems Similar to all routing and scheduling problems, pickup and delivery problems have at- tracted the interest of researchers who deal with meta-heuristic techniques, since exact algorithms can only be used for small problem sizes and may not be useful for large prac- tical applications. For example, the approach in [46], for solving the multiple-vehicle PDPTW, was only capable of solving problem sizes of up to 55 customers using an exact algorithm. Also, the exact algorithm in [74], solved PDTSP instances of up to 60 cus- tomers. On the other hand, heuristic and meta-heuristic approaches are able to solve problems having several hundreds of nodes, which is common in every day routing and scheduling demands for some industrial applications. For example, Paragon software sys- tems recently reported, in their October 2009 newsletter2 , that their optimization software helped Sainsbury’s to manage its transport requirement, involving daily deliveries from 19 distribution centers to 527 supermarkets and 276 smaller stores. Many heuristic and meta-heuristic techniques that have been successfully applied to the VRP have been adapted to fit the PDP variant. Table 4.3 summarizes some meta-heuristic techniques for pickup and delivery problems. In addition to the notations used for the meta-heuristic techniques in Table 4.2, we use (GRASP) for a Greedy Randomized Adap- tive Search Procedure, and (VND) for a Variable Neighbourhood Descent algorithm. On the other hand, Table 4.4 summarizes some GA research on pickup and delivery problems. It is important to emphasize again that the review presented here is just intended as a quick introduction to the literature that deals with pickup and delivery problems. As previously mentioned, further investigation of specific work that is closely connected to our research will be presented in later chapters, when we address certain variants of pickup and deli- very problems. 2 http://www.paragonrouting.com/cms/assets/pdf/directionsissue22.pdf
- 89. 4.3Meta-heuristicAlgorithmsforPickupandDeliveryProblems65 Table 4.3: Meta-heuristicsfor Pickup and Delivery Problems. Paper Problem Algorithm Solution Construction Solution Improvement Remarks Hernández- Pérez and Salazar- González [73] PDTSP (SV) Hybrid GRASP & VND Selecting the next ele- ment for insertion from a Restricted Candidate List (RCL), with a probabilis- tic element in the choice of the next node 2-Opt and 3-Opt in the VND phase - Move for- ward and move backward in a post optimization phase Combining two optimization heuristics: 1) GRASP (Greedy Ran- domized Adaptive Search Procedure), which is based on a repe- tition of a construction phase and a local search phase, 2) VND (Variable Neighbourhood Descent), which is a variant of the VNS, where the local optimum found acts as the new starting point for the local search Landrieu et al. [95] PDPTW (SV) TS Simple insertion heuristic A swap move and an insertion move Two tabu search methods compared: a regular deterministic Tabu Search (TS), and a Probabilistic Tabu Search (PTS) Nanry and Barns [112] PDPTW (MV) Reactive TS Cheapest insertion Single Paired Insertion (SPI): move all predeces- sor nodes to better feasible locations, then in- sert successor nodes in the best possible positions - Swapping Pairs Between Routes (SBR): ex- change predecessor nodes between vehicles then successor nodes - Within Route Insertion (WRI): re-order nodes within the same route Reactive tabu search - Allows tuning of search parameters, such as a short-term memory length, based on an assessment of visited solutions during the search Li and Lim [100] PDPTW (MV) Tabu- embedded SA Modified Solomon’s inser- tion heuristic - Initializes each route with a P&D pair that satisfies a set of criteria, based on combi- ned TW intervals and dis- tance from the depot PD-Shift: moves a P&D pair from route1 to route2, and moves another pair from route2 to route1 - PD-Swap: removes a P&D pair from each route, and then reinserts each pair in the other route - PD-Rearrange: removes and then reinserts a P&D pair in the same route Main meta-heuristic is a tabu-embedded SA with K-restarts, i.e., the algorithm stops when the number of iterations without impro- vement reaches a pre-defined value K - To prevent cycling, the SA records the accepted solutions in a tabu list Urban [149] PDPTW (MV) Guided SA Assigning only one pickup and delivery pair to each route. A guided SA is then applied to improve this ini- tial solution by trying to bundle requests into the available routes Removing a selected request from its current route and inserting it into another - Exchange of requests served on different routes - Repositio- ning of a pickup and delivery pair in its same route The cost function takes into consideration both wage and non-wage related traveling and operating costs of the vehicles - The selection of neighbourhood move depends on a certain probabilistic criterion - Calculated measurements control the selection of customers and routes during the generation of the new solution Cordeau and Laporte [30] DARP (MV) TS Assigning requests to ve- hicles randomly and inser- ting source location follo- wed by destination at the end of the route Moving a request from one route to another Re-insertion of the removed request in its original route is forbid- den by for a number of iterations - Diversification penalizes fre- quently occurring moves in the objective function - Intensification performed by removing all requests from their respective routes, and re-inserting them in the best possible positions
- 90. 664.3Meta-heuristicAlgorithmsforPickupandDeliveryProblems Table 4.4: GeneticAlgorithms for Pickup and Delivery Problems. Paper Problem Initial Population Crossover Mutation Remarks Zhao et al. [158] PDTSP (SV) Nearest-neighbour construction heuristic A new pheromone-based cros- sover - Edges between last node and potential new node that have proved successful in the past are favoured 3-exchange procedure Initial population is optimized using a 2-Opt move - Phe- romone trails are updated each generation - Offspring is optimized using a 2-Opt local search. Jih and Hsu [89] PDPTW (SV) A solution is encoded as a permuta- tion of locations Two traditional order-based crossover operators - Two merge crossover operators, MX1 and MX2, that use a global precedence vector to guide the inheritance process, as in [16] Swap two genes selected at random - Select two random sites in the chromosome and reverses the sub-route between them - Shuffle genes that pre- cede a location for which a vio- lation of constraints is obser- ved A Family Competition Genetic Algorithm (FCGA) - Each individual of the population plays the role of a family fa- ther in turn - Another randomly selected individual plays the role of mate for the family father Jung and Haghani [92] PDPTW (MV) Chromosome representation is ba- sed on assigning a 4 digit code to each request - First digit represents the number of the vehicle - Remai- ning 3 digits are used to sort re- quests according to the visiting or- der of the vehicle - A pickup request always has a sorting code that is less than its corresponding delivery Two crossover points genera- ted, and the resulting segments swapped between the two pa- rents Change the first digit of a pi- ckup and delivery pair to ano- ther digit, i.e., assign the re- quest to another vehicle - Ve- hicle merging also used to re- duce the number of vehicles Customers inconvenience is taken into consideration by assigning a penalty in the objective function to the arrival of the vehicle before or after the specified TW Pankratz [116] PDPTW (MV) Each gene represents a group of re- quests that are assigned to one ve- hicle - Routes are represented by an independent data structure associa- ted with each gene Vehicles with their assigned re- quests removed from one pa- rent and inserted in the other parent Removes a random cluster from a chromosome, and re- assigns its requests to other clusters A Grouping Genetic Algorithm (GGA), where the GA is only responsible for the grouping (assignment) problem - Routing is handled using an adaptation of Solomon’s cheapest insertion - Chromosome ‘clean up’ needed after the genetic operators to remove duplicate vehicles and re- peated assignment of requests, in addition to re-assigning requests that are no longer assigned Rekiek et al. [126] DARP (MV) Chromosome representation is group oriented, where each group represents a vehicle to which a set of requests is assigned traditional GGA crossover as in [49] Create a new group - Remove an existing group - Move items between groups - An inversion operator is also used to change the location of groups in the chromosome A Grouping Genetic Algorithm (GGA) applied to the Handicapped Person Transportation (HPT) problem, which focuses on minimizing clients’ inconvenience - The insertion heuristic used to insert requests in routes is the best-fit (BF) heuristic - An unserved client is inserted in the vehicle that will grant him the best service without vio- lating problem constraints
- 91. 4.4 Chapter Summary67 4.4 Chapter Summary In this chapter, we reviewed an important category in the VRP literature, which deals with pickup and delivery problems. These problems are gaining more attention every day, due the continuous need for optimizing routing and scheduling costs in real-life applications that require two different service types for clients. We briefly analyzed basic problem features and general existing classifications. We also highlighted some published research in the field, mainly focusing on heuristic and meta-heuristic approaches. In the following chapters, we will start discussing the specific problems that we addressed in our research. The Pickup and Delivery Problem with Time Windows (PDPTW) will be analyzed in detail in Chapters 5 and 6 for its single vehicle variant (SV-PDPTW), and in Chapters 7 and 8 for its multiple vehicle variant (MV-PDPTW). On the other hand, the One-commodity Pickup and Delivery problem (1-PDP) will be thoroughly discussed in Chapters 9 and 10.
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- 93. 69 Chapter 5 The SingleVehicle Pickup and Delivery Problem with Time Windows: Introduction and a Genetic Algorithm Approach The Single Vehicle Pickup and Delivery Problem with Time Windows (SV-PDPTW) is a frequently encountered problem in public and goods transport systems. However, only a few researchers seem to have tackled it, possibly due to the difficulty in managing the different underlying problem constraints. As previously mentioned in the introduction to this thesis (Chapter 1), our research concentrates on representational issues and neigh- bourhood moves within a simple meta-heuristic framework. For this problem, we adopt a solution representation that depends on a duplicate code for both the pickup location and its corresponding delivery. This simple representation will guarantee the satisfaction of the precedence constraint, among the pickup and delivery pair, throughout the search. We also present intelligent neighbourhood moves, that are guided by the time window, ai- ming to overcome the difficult timing constraint efficiently and produce feasible and good quality solutions in a reasonable amount of time. This chapter covers two aspects: 1) a detailed analysis of the problem and research chal- lenges and motivations, as well as a summary of some related work on the SV-PDPTW and 2) the initial development of our key ideas using a Genetic Algorithm (GA) as a can- didate solution methodology. The next chapter presents a comparison of three different approaches to the SV-PDPTW: a genetic algorithm approach, a simulated annealing ap- proach, and a simple hill climbing heuristic, which all employ the same representation and neighbourhood move that our initial GA approach adopts. The work related to the current chapter, i.e., the initial GA technique, was presented in the GECCO’07 conference, as a late breaking paper [79]. Also, the research tackling other heuristics and meta-heuristics applied to the SV-PDPTW, as described in the next chapter, was published in the Journal
- 94. 70 5.1 Problemand Motivation of heuristics [84]. The rest of this chapter is organized as follows: Section 5.1 explains the problem tackled in this part of the research and the motivation behind our selection of the problem and the underlying solution approaches. Then, Section 5.2 formally defines the SV-PDPTW, while Section 5.3 provides a brief summary of some related work from the literature. Section 5.4 highlights the contribution of our research, and focuses on the solution repre- sentation and neighbourhood move suggested to facilitate dealing with the SV-PDPTW. Sections 5.5 presents the details of the proposed GA approach to solving this problem, together with the experimental findings of this part of our research. Finally, Section 5.6 concludes the chapter with our intended future work. 5.1 Problem and Motivation The Pickup and Delivery Problem with Time Windows (PDPTW) is an important variant of vehicle routing problems that is likely to assume even greater prominence in the fu- ture. Current concerns over global warming, resource depletion and the social impact of traffic congestion and pollution, are driving companies, government organizations and re- searchers to improve the efficiency of logistics and distribution operations. In addition, the rapid growth in parcel transportation as a result of e-commerce is likely to have an increasing impact. As previously mentioned in Chapter 4, an important related variant of the PDPTW is the dial-a-ride problem, which is concerned with the transportation of people, especially the elderly and the disabled, from their origins to their destinations, while minimizing customer inconvenience. The SV-PDPTW is a special case of the PDPTW dealing with a number of customer requests that must be satisfied by one vehicle with a known capacity. The route of the vehicle usually starts and ends at a central depot. A request must be collected from a pickup location before being dropped off at a corresponding delivery location, and every pickup and delivery location is associated with a specific time window during which it must be served. If the vehicle arrives earlier than the beginning of the designated time window interval, it must wait until the requested service time begins. All requests must be served in a way that minimizes the total travel cost of the vehicle, without violating precedence, capacity and time windows constraints [137]. In addition to being a sub-problem in the PDPTW, the SV-PDPTW has also practical ap- plications, where small scale companies or individuals may operate one vehicle to serve a set of clients, as for example in a dial-a-ride service. In other applications, the underlying
- 95. 5.1 Problem andMotivation 71 vehicle could be a helicopter or a small ship. Also, in some multiple vehicle applications, the assignment of requests to vehicles may be restricted by some commodity, vehicle, driver or client conditions. In such cases, certain requests must be assigned to a speci- fic vehicle, making the optimization of a single vehicle route necessary for reducing the overall cost of the logistic operation. Moreover, the analysis of the SV-PDPTW is essen- tial in developing an insight into the more general multiple-vehicle case, and establishing comprehensive solution algorithms. As a constrained version of the TSP, the SV-PDPTW is known to be N P-hard [95], with the presence of time windows making the problem particularly complicated. Since exact algorithms are too slow for large problem sizes, heuristic and meta-heuristic approaches seem to be reasonable alternatives. Several heuristics and meta-heuristics are potentially suitable for this problem. Genetic Algorithms (GAs) are known for their robustness, pa- rallelism, and their ability to perform reasonably well on a wide variety of problems, including ordering and grouping problems, as well as highly constrained problems [64] [105]. Thus, exploring GAs to solve the SV-PDPTW would seem to be a justified option. Another alternative meta-heuristic approach that could be suitable for this problem is Si- mulated Annealing (SA), which is analogous to the annealing of solids. As previously discussed in Chapter 2, this approach has been widely applied to many optimization pro- blems, successfully transforming ‘random’ low quality solutions to stable high quality optimized solutions. One appealing feature of simulated annealing is that it is very easy to implement, since it only requires a method for generating a move in the neighbourhood of the current solution, and an appropriate annealing schedule. A third solution alterna- tive is simple Hill Climbing (HC) that first creates a candidate solution and then iteratively tries to perturb this solution to improve it. In our research, we mainly focus on developing a meta-heuristic framework for the SV- PDPTW rather than on a specific algorithmic paradigm. Within the context of the meta- heuristic, we employ a problem-specific solution representation and guided neighbou- rhood moves, aiming to overcome and deal efficiently with the hard problem constraints. Our key ideas were adopted in all three approaches highlighted above, the GA, SA and HC. In all cases, our view was to design an intelligent and robust solution approach that can handle all problem constraints efficiently, while keeping the overall algorithm as simple as possible, a feature often overlooked in most up to date solution methodolo- gies. Ultimately, we aim to obtain feasible and high quality solutions in an acceptable amount of time.
- 96. 72 5.2 TheSV-PDPTW 5.2 The SV-PDPTW Let G = (N, A) be a digraph. The node set is N = {ni ∈ N|i = 0, 1, 2, ..., m}, such that m is an even index. The node n0 denotes the depot, and each ni, i = 1, 2, ...m denotes a customer location. It is assumed that a customer’s request rk, k = {1, 2, ...m/2} consists of a pickup and delivery (P&D) pair. We can consider, without loss of generality, that the set N+ = {ni ∈ N|i = 1, 2, ..., m/2} represents pickup locations, and the set N− = {ni ∈ N|i = (m/2) + 1, ..., m} represents delivery locations, such that the pickup location ni has the corresponding delivery location ni+(m/2). Thus, N = N+ ∪ N− and |N+ | = |N− | = m/2. Each location ni is associated with: • A customer demand qi, such that qi > 0 for a pickup location, qi < 0 for a delivery location and qi + qj = 0 for the same customer’s pickup and delivery locations (q0 = 0); • A service time si (s0 = 0), which is the time needed to load or unload a customer demand; • A Time Window (TW) [ei, li] during which the location must be served, and li ≥ ei. For each pair of nodes < ni, nj > a travel time tij and/or a travel distance dij are specified. We assume here a symmetric case, i.e., tij = tji and dij = dji. In addition, only edges satisfying the TW are allowed. Thus the arc set is A = {< ni, nj > |ni, nj ∈ N, ni = nj, t0i + si + tij < lj}. The vehicle has a limited capacity C. The capacity constraint ensures that the total load carried by the vehicle at any given time does not exceed its capacity. The vehicle’s journey should start and end at the depot, while each location is to be visited exactly once. The time window constraint requires that a location must be serviced within the specified TW, i.e., if the vehicle reaches the location before the earliest service time ei, it must wait until ei. The precedence constraint requires that each pickup location must precede the corresponding delivery location in the visiting order. The objective function varies depending on the application. In general, one or more of the following objectives are minimized as far as possible: the total traveling distance, the total route duration, or the drivers’ total waiting time. Figure 5.1 shows a simplified representation of a small instance of this problem before and after solving it.
- 97. 5.3 Related Work73 (a) SV-PDPTW - before solution (b) SV-PDPTW - after solution Figure 5.1: The single vehicle pickup and delivery problem with time windows. 5.3 Related Work As indicated above, there are various approaches to handle pickup and delivery problems. Some approaches are exact and guarantee to solve the problem to optimality, while others are approximations and attempt to find an acceptable solution in a reasonable amount of time. For the SV-PDPTW, an exact algorithm is the dynamic programming approach of [124]. However, this technique has a time complexity of O(n2 3n ) (where n is the number of locations) and for this reason is normally limited to solving small problems of up to about 10 requests (20 locations). Also, the authors in [38] are able to provide exact solutions to the single vehicle dial-a-ride problem, with precedence, capacity and time windows constraints, using dynamic programming. Their algorithm uses a set of states (S, i), such that S is a subset of nodes from the node set, and i is a selected node. State (S, i) is defined only if there is a feasible route that visits all nodes in S and terminates at i. To reduce the state space, their dynamic programming algorithm uses sophisticated state elimination criteria, based on the state (S, i). Computational experimentation indicated that the elimination criteria work best when the time windows are tight and the vehicle capacity is small, such that narrowing the constraints in their approach helped to reduce the usual exponential running time of dynamic programming to a linear running time. This algorithm can solve instances of up to 40 requests (80 locations). Approximation algorithms, in which a heuristic or a meta-heuristic is developed to deal
- 98. 74 5.3 RelatedWork with the problem, make it possible to cope with much larger instances. Among the heu- ristics for solving the SV-PDPTW is the one suggested in [23]. This heuristic involves a 2-phase algorithm. In the initial phase a feasible solution is constructed, and in the second phase this solution is improved. In both phases a variable depth arc exchange procedure is performed, as a neighbourhood move, in which the number of arcs to be exchanged is not determined in advance, but calculated dynamically during the search. In the route construction phase, the time constraint may be violated as long as the pre- cedence and capacity constraints are satisfied. In the improvement phase, however, only feasible solutions are permitted and route duration is used as an objective function. Du- ring the improvement phase, the feasibility of a tour (in terms of precedence and capacity) is verified using a set of global variables and an algorithm of time complexity O(n2 ). To check the feasibility of the time constraint and determine the promising arc exchanges, a forward time slack is computed at each node to identify the permissible shift in departure time that can be introduced without causing violations of time windows for other nodes in the route. For problem sizes up to 38 customers, a near optimal solution can be reached by this algorithm in less than 150 seconds. Unfortunately, success is not guaranteed, and low quality or even infeasible solutions can result, if the 2-phase approach gets trapped in a poor local optimum. To handle this possible situation, the authors present an alterna- tive approach, which uses SA. This algorithm accepts time window violations in the early stages, but penalizes them more severely as the search progresses. The SA approach is able to obtain good quality solutions, albeit with a relatively high processing time. The work reported in [88] also deals with the SV-PDPTW. This time, however, a hybrid strategy is proposed which combines an exact method with a genetic algorithm and both static and dynamic cases are considered. The static case assumes that all requests are determined in advance of the route construction process, while the dynamic case allows some requests to arrive during the construction of the route. The approach in this work consists of three consecutive stages: a pre-planned module, a dynamic programming mo- dule, and a GA. The pre-planned module arranges requests and prepares information for the dynamic programming module. The role of the dynamic programming module is to create a set of sub-routes, which it will eventually pass on to a temporary result pool, where the GA module will pick them up, installing these unfinished sub-routes to esta- blish its initial population. In the GA module, a solution is encoded as a permutation of locations, and four crossover operators are compared: two traditional order-based cros- sover operators, and two merge crossover operators, MX1 and MX2, that use a global precedence vector to guide the inheritance process, as explained in [16] and later in this chapter. The mutation operator is applied only when the offspring is identical to one of its parents. Three mutation operators are used: one swaps two genes selected at random, the
- 99. 5.3 Related Work75 second selects two random sites in the chromosome and reverses the sub-route between the selected sites, finally the third mutation shuffles the genes that precede a location for which a violation of constraints is observed. If an infeasible solution, violating the prece- dence constraint, is generated as a result of any genetic operator, this solution is repaired by swapping the corresponding pickup and delivery pair. The experimental results on data sets ranging from 10 to 50 requests indicate the merge crossover operator MX1 ge- nerally performed better than the other crossover operators tested. The first two mutation operators also achieved better results than the third mutation operator. In a more recent work [89], the same authors use similar genetic operators to the ones used in [88], but in the context of a Family Competition Genetic Algorithm (FCGA). Again, an order-based representation is adopted. The idea is to allow each individual of the po- pulation to play the role of a family father in turn. Another randomly selected individual plays the role of mate for the family father. The two individuals are combined to produce an offspring in a regular GA fashion. The selection of the mate and the reproduction is repeated for a chosen number of iterations to produce a family of offspring. Only the best offspring in the family survives and is added to a temporary population of champions. The new generation is chosen from among the best individuals in both the original population and the champions of the families. Comparing the performance of the FCGA and a tradi- tional GA on data sets created by the authors (ranging from 10 to 100 requests), the results indicate that the merge crossover operators MX1 and MX2 generally worked better in the context of a traditional GA than a FCGA, possibly due to premature convergence in the latter case. On the other hand, a traditional uniform order-based crossover (UOX) worked better within the framework of a FCGA than a traditional GA, possibly due to its uniform, non greedy, nature of exploring the search space. The authors in [95] present a heuristic based on tabu search to solve the SV-PDPTW. The algorithm first creates a route respecting precedence and capacity constraints, using a simple insertion heuristic. The generated route may be infeasible in terms of the time window constraint, though. Then, two tabu search methods are compared in transforming the initial route to a feasible route, with minimum total distance. The two tabu search methods investigated are a regular deterministic Tabu Search (TS), and a Probabilistic Tabu Search (PTS), which is based on the same principles as the deterministic one with the addition of a buffered memory of potential moves and introducing some probabilistic criteria for the selection of the next move. The creation of a neighbouring solution in both tabu searches is based on classical neighbourhood moves, a swap operation and an insertion operation, which respect the precedence and the capacity constraints. Test results on problem instances created by the authors (from 10 to 40 customers) demonstrate the superiority of TS compared to the PTS in both run time and solution quality. Possible
- 100. 76 5.4 SV-PDPTW:Research Contribution reasons behind the superiority of TS were not indicated by the authors, though. For the problem sizes tested in their approach, a feasible solution could be reached by both tabu search methods in a reasonable amount of time. However, for larger problem sizes (more than 60 locations), the authors predict that reaching a feasible solution could possibly take more than one hour. According to the authors, an improvement in processing time could be achieved by adapting the parameters of the method, and a better analysis of the neighbourhood structure. 5.4 SV-PDPTW: Research Contribution Our research tries to overcome some gaps currently existing in the literature that tackles the SV-PDPTW. Existing approaches usually face many difficulties, among them are so- lution representations that need frequent repair during the search process to correct infea- sibility. In addition, handling the difficult constraints puts a huge burden on the search algorithm, usually making it quite complicated, especially if the researcher chooses to limit the search to only feasible solutions in the neighbourhood. This would indeed re- quire time consuming computations to ensure the satisfaction of all constraints in every step during the search process, which will inevitably add to the overhead of the algorithm, making it slow or inefficient for most practical situations. The main challenge that we are faced with in this research is the development of an appro- priate solution representation, reflecting the problem and its constraints in a simple way to avoid complicating our algorithms. In addition, the representation should be coupled with intelligent neighbourhood operators that are capable of directing the search towards high quality and feasible solutions. A major objective is to avoid generating and evaluating a large number of infeasible solutions (that violate one or more of the hard constraints), in order to maximize the efficiency of the search process. Our proposed approach for handling these challenges is presented in the following two subsections. 5.4.1 The Solution Representation A suitable solution representation for this kind of problem is not as obvious as it seems. The PDPTW is an ordering problem in which a solution could be represented as a permu- tation of locations, representing an order in which these locations will be visited. In the PDPTW, however, the issue of precedence should be addressed in the representation of the solution, because for this particular problem, no delivery location is allowed to precede
- 101. 5.4 SV-PDPTW: ResearchContribution 77 its corresponding pickup location. Nevertheless, the precedence order may not be main- tained following the application of a neighbourhood move to a solution. For example, a simple swap of locations could disturb the precedence order and result in an infeasible so- lution. Consequently, a repair method would be needed to restore the solution feasibility, as done, for example, in [88] and [89]. Inevitably, such an approach would increase the processing time and complicate the algorithm. We have developed a solution representation which avoids the precedence issue: we sim- ply assign the same code to both the pickup and its associated delivery location, and rely on a simple decoder to identify its first occurrence as the pickup and the second as the deli- very. This straightforward representation eliminates the problem of backtracking to repair an infeasible solution, and solves the precedence constraint issue in an effective way. As a result, more effort can be directed towards harder constraints such as the vehicle capacity and time windows. Figure 5.2 shows an example of a solution with 3 requests1 following this representation. Figure 5.2: Solution Representation. 5.4.2 The Neighbourhood Move As previously mentioned, while creating a neighbouring solution during the search pro- cess, we are faced with a major challenge: the possible violation of one or more of the problem constraints that may follow such moves. A neighbourhood move should be in- telligent enough to direct the search towards high quality and feasible solutions, and thus avoid valuable time being wasted evaluating a large number of infeasible solutions. When designing such moves, all problem constraints should be considered. Nevertheless, in our research we found the time window constraint the most difficult to deal with. Recall that our duplicate encoding scheme renders the precedence constraint trivial. Furthermore, the capacity constraint tends to be easy to satisfy in most problem instances, because half of 1 Recall from Section 5.2 that a request refers to a pair of locations (pickup and delivery).
- 102. 78 5.5 AGenetic Algorithm Approach for Solving the SV-PDPTW the locations visited in any route are delivery locations whose demands are removed from the vehicle. In our research we adopt neighbourhood search operators that apply ‘randomness’, yet at the same time take account of the time windows, and bring the more ‘urgent’ locations earlier in the visiting order, where this is possible. This neighbourhood idea is adopted in all our search algorithms, with slight variations depending on the context in which it is applied. The rest of this chapter and the next chapter are dedicated to our suggested solution me- thodologies for handling the SV-PDPTW. The following section addresses our problem- specific genetic algorithm that we adopted for solving this problem. On the other hand, our investigation of the SV-PDPTW continues in Chapter 6, which presents the proposed operators within the framework of three different heuristic and meta-heuristic techniques to deal with this problem. 5.5 A Genetic Algorithm Approach for Solving the SV- PDPTW The focus of this part of our research is to investigate the potential of using genetic al- gorithms to solve the SV-PDPTW. In particular, herein we experiment with a duplicate gene encoding that guarantees the satisfaction of the precedence constraint, between the pickup and the delivery, throughout the search. Thus, our chromosome encoding simply follows the representation depicted in Figure 5.2. In addition, several problem-specific genetic operators are tested and compared on a number of data sets with various sizes. The rest of this section presents the details of our GA approach. Section 5.5.1 describes the fitness function that the GA relies on during the evolutionary process. Section 5.5.2 introduces our suggested genetic operators. Section 5.5.3 explains the procedure adopted to create test data for the problem. Finally, Section 5.5.4 details the experimental findings of this part of the research, 5.5.1 The Fitness Function Following [89], the fitness function treats the constraints as soft constraints, meaning that an infeasible solution that violates either the capacity and/or the time window constraint
- 103. 5.5 A GeneticAlgorithm Approach for Solving the SV-PDPTW 79 will be penalized by an added term in the fitness function. The fitness function of a route r is: F(r) = w1 × D(r) + w2 × TWV (r) + w3 × CV (r), (5.1) where D(r) is the total route duration including the waiting time and the service time at each location, TWV (r) is the total number of time window violations in the route, and CV (r) is the total number of capacity violations. w1, w2, and w3 are weights in the range [0, 1] assigned to each term in the fitness function, and w1 + w2 + w3 = 1.0. The fitness function will thus try to minimize infeasibility as well as the total route du- ration. Also, the choice of appropriate weights depends on the importance of each term in the fitness function. Specifically, the highest penalty must be imposed on the number of time window violations, since it is the most difficult constraint to satisfy in the SV- PDPTW. We used the following values for the weights of the fitness function: w1 = 0.001, w2 = 0.7, and w3 = 0.299. Our experimentation with different values in the range [0, 1] indicated that if a smaller weight value was assigned to the TWV term, the best individual resulting at the end of the evolutionary process is often infeasible, in terms of violating the time window constraint. 5.5.2 The Operators Two crossover operators have been used in our GA, the first crossover operator is a Merge Crossover (MX1), first suggested in [16]. The second crossover operator is a new pro- blem specific crossover that we called the Pickup and Delivery Crossover (PDPX). Also, two mutation operators were used in our research. A regular gene swap mutation and another constraint oriented directed mutation. These four operators are explained below. Crossover MX1 Crossover: Several crossover operators were considered potentially suitable for this kind of problem. The first crossover operator we tried follows the merge crosso- ver operators suggested in [16] for the VRPTW2 , and used in [88] and [89] for the SV- PDPTW. Unlike traditional crossover operators for ordering problems, which depend only on the order of genes in a chromosome, the merge crossover operators depend on a glo- bal precedence among genes, such as the time window or distance ordering. Traditional 2 Recall from Chapter 3 that in the VRPTW, the same service type applies to all locations, either pickup or delivery.
- 104. 80 5.5 AGenetic Algorithm Approach for Solving the SV-PDPTW order-based crossover operators are not very effective for highly constrained problems like the SV-PDPTW, since they frequently produce infeasible solutions. Merge crossover operators, however, were shown to be superior to the traditional ones for these types of problems [16] [88] [89]. In the current research we have slightly modified the MX1 operator used by [88] and [89]. Instead of creating just one child, giving priority to a parent’s gene having an earlier time window lower bound, we have created two children: the first child favouring genes that have an earlier lower bound, while the second child favours genes that have an earlier upper bound. The idea is that visiting a location just before its deadline could be more beneficial than visiting it as early as possible in its allowed interval. This may help reduce the waiting time that would result if the vehicle arrives too early at a location, and may consequently decrease the total route duration. Creating two children instead of one was suggested in [16], and may help improve the quality of the new generation and accelerate the optimization process. To illustrate how the MX1 operator works, assume that the vector shown in Figure 5.3 defines the precedence order, in terms of the lower bound of the time window, among all pickup and delivery locations, where a (+) following a request number indicates that this is its pickup location and a (-) indicates the delivery location for the same request. Figure 5.3: Time window precedence vector. Figure 5.4 illustrates the steps of the MX1 operator. Consider the two parent solutions, P1 and P2, shown in Figure 5.4(a). The MX1 crossover starts by decoding the parent solutions to differentiate the pickup locations from the delivery locations, as illustrated in Figure 5.4(b). Then, the MX1 proceeds as follows: • Step 1: since 2+ has a higher precedence than 3+ (as indicated by the precedence vector depicted in Figure 5.3), the child will inherit 2+ as the first gene, as shown in Figure 5.4(c); • Step 2: to maintain feasibility, 2+ in P2 will be swapped to the first location. Now the second gene in both parents is identical, so it is copied to the child, as shown in
- 105. 5.5 A GeneticAlgorithm Approach for Solving the SV-PDPTW 81 Figure 5.4(d), and we move on to the next gene in order; • Step 3: now since the next gene in P2 (1-) has a higher precedence than 3+ of P1, 1- is copied to the third gene in the child, and 1- is swapped with 3+ in P1, as shown in Figure 5.4(e); • Step 4: continuing in the same manner, we obtain the child C1 shown in Figure 5.4(f), together with its final format, using the duplicate encoding. Note that after the MX1 crossover, the last 2 genes of C1 were out of order (4- followed by 4+ ), but this is of no concern since the first one is automatically considered as the pickup. (a) MX1 - two parents before decoding (b) MX1 - two parents after decoding (c) MX1 - Step 1 (d) MX1 - Step 2 (e) MX1 - Step 3 (f) MX1 - Step 4 Figure 5.4: Merge Crossover (MX1).
- 106. 82 5.5 AGenetic Algorithm Approach for Solving the SV-PDPTW The second child is created in a similar manner but with the precedence vector defined by the upper bound of time window intervals instead. PDPX Crossover: In addition to the MX1 operator, we designed a new problem-specific crossover operator for the SV-PDPTW. The idea is that, since any crossover operator essentially tries to create a child that inherits half of its genes from the first parent and half of its genes from the second parent, we will try to transmit pickup locations to the child in an order close to the order of their appearance in the first parent, and try to transmit delivery locations to the child in an order close to the order of their appearance in the second parent. A second child is created by simply reversing the role of parents. This kind of crossover might prove useful in satisfying the capacity constraint in particular, since the satisfaction of this constraint is mainly dependent on the ordering of pickups and deliveries. Figure 5.5 illustrate how the PDPX crossover works. (a) PDPX - Step 1 (b) PDPX - Step 2 (c) PDPX - second child Figure 5.5: Pickup and Delivery Crossover (PDPX). Starting from the two parent solutions, P1 and P2, shown in Figure 5.5(a), the PDPX crossover proceeds as follows: • Step 1: first all pickups (first occurrences of genes) will be copied to the child in the exact order and locations as the first parent, as shown in Figure 5.5(a); • Step 2: now, we want to take the order of delivery locations from the second parent. Since a delivery is always the second occurrence, we will start processing P2 from
- 107. 5.5 A GeneticAlgorithm Approach for Solving the SV-PDPTW 83 the last gene rather than the first. If we encounter a gene that appeared in the child only once, i.e., it has already been picked up, we will copy it in the first available location in the child (also processed starting from the last gene). If, on the other hand, we encounter a gene that has already appeared twice in the child, this gene will be ignored because pickup and delivery has already occurred for this gene, and we move on to the next gene in order. Following this, we obtain the child shown in Figure 5.5(b). The second child is obtained by taking the pickups order from the second parent and the deliveries order from the first parent, as shown in Figure 5.5(c). Mutation As mentioned above, we experimented with two mutation operators in our GA. The first is a random gene swap mutation, which selects two genes at random and swaps them. Using the duplicate gene encoding eliminates the possibility of infeasible solutions, in terms of precedence order, that may result following such a swap. We also implemented, a new problem oriented mutation operator, named Directed Mu- tation. As mentioned above, it appears that the TW constraint is the most challenging to satisfy among all other problem constraints. To deal with the TW constraint, a mutation operator can try to bring a location that may be more urgent, to an earlier point in the vi- siting order. This may result in a better ordering of locations, which should be beneficial in satisfying the timing requirement. Instead of a traditional random swap of two genes, this new mutation operator only swaps genes if they are out of order in terms of their late TW bounds, i.e., if the later one has a deadline that precedes the earlier one. The directed mutation operator is shown in Figure 5.6. Figure 5.6: Directed Mutation.
- 108. 84 5.5 AGenetic Algorithm Approach for Solving the SV-PDPTW 5.5.3 Test Data Since there are no standard benchmark test cases available for the SV-PDPTW, our GA was tested on a data set obtained from the authors of [89]. This data set has a number of customer requests ranging from 10 to 100 (20 to 200 locations). In addition, for a more extensive and thorough testing of the operators, we have created a new data set with larger numbers of customer requests ranging from 100 to 200 (200 to 400 locations). To create test data for the SV-PDPTW, it is essential to ensure the existence of at least one solution that satisfies all problem constraints. Similar to [89], our algorithm first creates a route that respects precedence and capacity constraints, then a time window interval is generated for each location based on the arrival time realized in the created route. Unlike [89], however, we rely on our GA to create the initial feasible solution, while they create their starting feasible solution randomly. More specifically, the following steps were followed to create the test data: 1. Generate a random vehicle capacity within a certain predetermined range; 2. Generate random x and y coordinates for the depot; 3. For all pickup and delivery locations: (a) Generate random x and y coordinates; (b) Generate a random demand (load) within a certain allowable range, such that the demand of a delivery location is the same as the demand of its correspon- ding pickup location, but with a negative sign; (c) Assume a very large time window interval that could not possibly be violated; 4. Run the GA to obtain a feasible solution. Note that, the precedence constraint is always satisfied in any generated solution, thanks to our duplicate gene enco- ding. Moreover, due to the nature of the problem, the satisfaction of the capa- city constraint can be easily accomplished by our genetic operators. The timing constraint can also be easily satisfied because the TW intervals are very large at this point; 5. Calculate the arrival time at each location in the feasible route obtained; 6. Create a random time window interval for each location such that the arrival time falls within the created time window. The width of the permitted time window interval should be determined in advance.
- 109. 5.5 A GeneticAlgorithm Approach for Solving the SV-PDPTW 85 Algorithm 5.1 describes the initial steps carried out for the creation of test data for the problem without the final time window intervals, i.e., Steps 1 to 3 above. In this algorithm, the allowable ranges for the random values were determined empirically. Note that all ranges were scaled according to the number of requests currently generated, i.e., the larger the number of requests, the larger the allowable range for the vehicle capacity and the demand of the location. Algorithm 5.2 describes the procedure for determining the TW intervals, i.e., Step 6 above. Algorithm 5.1: Create Test Data. 1: Choose values for constants MinCapacity, MaxCapacity, MinCoord and MaxCoord {We used MinCapacity = 1, MaxCapacity = 10, MinCoord = 0 and MaxCoord = 200} 2: Choose value for constant MinTW {A very small value (we used 0)} 3: Choose value for constant MaxTW {A very large value (we used 100000)} 4: Let nrequests = n/2 be the number of requests in the set, where n is the total number of nodes (locations) in the data set, excluding the depot 5: V ehicleCap = Random(MinCapacity×nrequests, MaxCapacity×nrequests){Func- tion Random generates a random value between its two parameters} 6: AverageDemand = V ehicleCap/nrequests {The average demand of a request} {Generate coordinates, demand, and TW for the depot and the pickup locations} 7: for (pickloc = 0; pickloc <= nrequests; pickloc + +) do 8: x[pickloc] = Random(MinCoord, MaxCoord){The x-coordinate} 9: y[pickloc] = Random(MinCoord, MaxCoord){The y-coordinate} 10: if (pickloc = 0) then 11: demand[pickloc] = 0 {The demand of the depot is 0} 12: else 13: demand[pickloc] = Random(AverageDemand×m1, AverageDemand×m2) {The choice of m1 and m2 depends on how big we want the demand of the location} 14: ETW[pickloc] = MinTW{The lower bound of the TW} 15: LTW[pickloc] = MaxTW{The upper bound of the TW} {Generate coordinates, demand, TW for delivery locations} 16: for (delloc = nrequests + 1; delloc <= n; delloc + +) do 17: x[delloc] = Random(MinCoord, MaxCoord) 18: y[delloc] = Random(MinCoord, MaxCoord) 19: demand[delloc] = −demand[delloc − nrequests]{Same as its pickup demand but -ve} 20: ETW[delloc] = MinTW{The lower bound of the TW} 21: LTW[delloc] = MaxTW{The upper bound of the TW}
- 110. 86 5.5 AGenetic Algorithm Approach for Solving the SV-PDPTW Algorithm 5.2: Generate Time Windows. 1: Choose value for constant CurrentWidth {We used 10} 2: Calculate the arrival time of the vehicle at each location in the feasible route obtained by the GA 3: for (loc = 1; loc <= n; loc + +) do 4: ETW[loc] = Arrival[loc] − Random(CurrentWidth × m1, CurrentWidth × m2) {The choice of m1 and m2 depends on whether we want the time window interval to be wide or narrow} 5: if (ETW[loc] < 0) then 6: ETW[loc] = Arrival[loc] − Random(1, Arrival[loc]) 7: LTW[loc] = Arrival[loc] + Random(CurrentWidth × m1, CurrentWidth × m2) 5.5.4 Experimental Results We implemented a steady state GA with a replacement percentage of 100% and a popu- lation size of 1000. Algorithm 5.3 outlines the GA used in our implementation. Algorithm 5.3: The SV-PDPTW Genetic Algorithm. 1: Initialize a random population POP of candidate solutions to the SV-PDPTW, using the duplicate encoding of pickup and delivery locations 2: while (stopping condition is not reached) do 3: for (i=0; i<NumCrossovers;i++) do 4: Select parents P1 and P2 from POP, using roulette wheel selection 5: Apply crossover to parents P1 and P2 to produce two children C1 and C2 6: With some probability, apply mutation to C1 and C2 7: Update POP by integrating the new generation and eliminating some worst individuals {i.e., steady state GA with overlapping populations} To test our GA, different combinations of crossover and mutation operators were compa- red on two data set samples. The first sample, which we will call SET 1, is the sample obtained from [89] and includes 30, 80, 90 and 100 customer requests3 . It should be no- ted that while solving the problem for this data set, the authors of [89] assumed that the vehicle’s journey is open path. Thus, although the vehicle should start its journey from the depot, it could end at any of the delivery locations. In order to compare our results with their results we also make this assumption in our GA approach. An alternative, and more ‘standard’, assumption is for the vehicle to start and end at the depot. 3 Source: http://wrjih.wordpress.com/2006/12/09/pdptw-test-data/
- 111. 5.5 A GeneticAlgorithm Approach for Solving the SV-PDPTW 87 The second data set used to test our algorithm, which we will call SET 2, is obtained from the instances created by us, and includes 130, 170, and 200 customer requests 4 . Note that the number of locations is always double the number of requests. The algorithm was run 10 times on each test case, with a crossover rate of 1.0 and a mutation rate of 0.45 . The following combinations of crossover and mutation operators were compared: 1. MX1 crossover and random swap mutation. 2. MX1 crossover and directed mutation. 3. PDPX crossover and random swap mutation. 4. PDPX crossover and directed mutation. 5. Directed mutation without any crossover6 . 6. MX1 crossover without any mutation. The genetic algorithm was run to convergence or for a maximum of 3000 generations if no convergence can be reached. Since robustness is one of our main goals in this research, the GA was run with the same parameters on all problem instances, i.e., it was not specifically tuned or optimized for each instance separately. The results for the 10 runs are recorded in Table 5.1 as follows, for each test case under each GA version: the best score found (in terms of total route duration only), and the percentage of feasible solutions obtained during the 10 runs. The last column of the table shows the previous best result. For SET 1, the previous best results are the ones reported in [89], during their computational experimentation with these problem instance7 , while for SET 2, the previous best results are the best results achieved in the current experiment. A score followed by a (*) indicates that this solution is infeasible in terms of time window constraint violation. When the obtained result is the same as or better than the previous best result, it is highlighted in boldface. In terms of the objective function values, Table 5.2 shows the average, best, and worst objective function values, as calculated by Equation 5.1, for all versions of our GA. the 4 Source: http://users.cs.cf.ac.uk/M.I.Hosny/PDP.zip 5 A higher than usual mutation rate was found necessary to avoid being trapped in a local optimum. One reason could be the duplicate encoding, a side effect of which is that mutation may swap identical genes producing the same offspring. 6 Mutation rate was set to 1 and crossover rate to 0 7 The results shown are truncated as they appear in [89].
- 112. 885.5AGeneticAlgorithmApproachforSolvingtheSV-PDPTW Table 5.1: GAResults Summary (Total Route Duration). Data Task MX1&Swap MX1&Directed PDPX&Swap PDPX&Directed Directed Mut MX1 Prev Best Best %Feas Best %Feas Best %Feas Best %Feas Best %Feas Best %Feas SET 1 30 3696.51 100 3738.73 100 3863.27* 0 3734.54 100 3747.47 100 3741.10 100 3713 80 7838.21 100 7858.72 100 9163.92* 0 7838.21 100 7879.35 90 7867.68 80 7849 90 8619.01 100 8619.01 100 10057.0* 0 8623.23 100 8662.20 100 8619.01 100 8618 100 10600.10 100 10600.10 100 11886.10* 0 10608.20 80 10608.20 90 10600.10 90 10600 SET 2 130 13856.20 100 15913.10 20 19185.40* 0 14791.70 60 14679.20 100 21626.0* 0 13856.20 170 19861.30 100 20674.50 60 25817.10* 0 20359.30 90 20470.20 90 30575.60* 0 19861.30 200 24512.80 80 26058.10 60 32009.70* 0 24911.20 40 25091.70 70 37148.50* 0 24512.80
- 113. 5.5AGeneticAlgorithmApproachforSolvingtheSV-PDPTW89 Table 5.2: GAResults Summary (Objective Function Values). Task MX1&Swap MX1&Directed PDPX&Swap Avg Best Worst SD Avg Best Worst SD Avg Best Worst SD 30 3.72 3.70 3.74 0.02 3.74 3.74 3.74 0 15.82 13.66 17.34 1.19 80 7.84 7.84 7.85 0 7.88 7.86 7.89 0.01 81.52 72.86 87.10 4.36 90 8.62 8.62 8.62 0 8.62 8.62 8.62 0 102.78 94.76 114.5 6.68 100 10.60 10.60 10.60 0 10.60 10.60 10.60 0 114.43 101.49 127.16 8.90 130 14.06 13.86 14.23 0.14 56.33 15.91 108.21 35.03 124.0 118.59 128.68 3.88 170 20.17 19.86 20.6 0.24 21.98 20.67 23.95 1.04 183.66 174.22 192.86 5.93 200 24.72 24.51 25.4 0.34 29.27 26.06 47.70 6.61 225.55 214.71 245.10 8.32 Avg 12.82 12.71 13.01 0.11 19.78 13.35 30.1 6.10 121.11 112.9 130.39 5.61 Task PDPX&Directed Directed Mutation MX1 Avg Best Worst SD Avg Best Worst SD Avg Best Worst SD 30 3.77 3.73 3.80 0.02 3.80 3.75 3.86 0.03 3.74 3.74 3.75 0 80 7.91 7.84 7.97 0.04 8.0 7.88 8.63 0.22 8.03 7.87 8.59 0.29 90 8.69 8.62 8.78 0.06 8.70 8.66 8.76 0.03 8.62 8.62 8.62 0 100 10.80 10.61 11.38 0.3 10.73 10.61 11.37 0.23 10.67 10.60 11.30 0.22 130 15.42 14.79 16.59 0.66 15.14 14.68 15.49 0.27 124.53 111.23 132.67 6.85 170 20.84 20.36 22.07 0.49 20.95 20.47 23.03 0.74 178.85 169.48 189.61 6.51 200 25.91 24.91 28.06 1.01 25.46 25.09 26.35 0.39 229.47 214.25 240.92 6.96 Avg 13.33 12.98 14.09 0.37 13.26 13.02 13.93 0.27 80.56 75.11 85.07 2.98
- 114. 90 5.5 AGenetic Algorithm Approach for Solving the SV-PDPTW Standard Deviation (SD) value of the objective function for the 10 runs is also shown in this table. As can be observed in Table 5.1, the best results are obtained from the MX1 crossover and random swap mutation. Together they achieved a 100% feasibility rate in all test cases except for the largest task (200 requests) in which the feasibility rate was 80%. In two cases the results obtained were even better than the previous best results reported in [89]. These are the results for test cases 30 and 80. Table 5.2 also shows clearly that the overall average objective function results of the MX1 crossover and swap mutation were better than those obtained by all other GA versions, for all categories demonstrated in the table. Table 5.1 shows that the worst results are obtained by the PDPX crossover and swap mu- tation. These two operators together failed to achieve any feasible results, in terms of the time window constraint, even for the small sized tasks. Also, the overall average objective function results for these two operators were considerably worse than the results obtained by all other GA versions, as Table 5.2 indicates. These inferior results may be explained by noticing that both PDPX crossover and swap mutation lacked any guidance towards the time window ordering. Although they were able to satisfy the capacity constraint in all test cases, they failed to satisfy the time window constraint. Clearly, they worked blindly with respect to the timing order, and consequently were not able to find an ordering that services all requests in their desired intervals. When directed mutation replaced the random swap mutation, however, the results drama- tically improved for the PDPX. Table 5.1 shows that PDPX together with directed mu- tation were able to obtain feasible solutions in all test cases. The feasibility ratio ranged from 40% for the largest task to 100% for the smallest task. The overall results, though, were not able to beat the results obtained from the MX1 crossover and swap mutation, which can be seen in Table 5.2. Figure 5.7 sustains the above observation, by showing how directed mutation was able to guide the search and helped the PDPX crossover to find much better solutions for the largest task tested (200 requests), when used in place of the random swap mutation. The situation is different, however, when directed mutation replaced random swap mu- tation with the MX1 crossover. As shown in Tables 5.1 and 5.2, MX1 crossover and directed mutation produced lower quality solutions than those obtained by MX1 cros- sover with the random swap mutation. Possibly, the introduction of directed mutation caused the algorithm to rapidly converge to a local optimum. Figure 5.8 bears this out, and shows the rapid convergence of the MX1 crossover with directed mutation compared to the gradual convergence of the MX1 crossover with the random swap mutation.
- 115. 5.5 A GeneticAlgorithm Approach for Solving the SV-PDPTW 91 0 500 1000 1500 2000 2500 3000 0 50 100 150 200 250 300 350 400 Generation BestFitness PDPX&Swap Mutation PDPX&Directed Mutation Figure 5.7: PDPX & Swap against PDPX & Directed - Task 200. 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 Generation BestFitness MX1&Swap Mutaion MX1&Directed Mutation Figure 5.8: MX1 & Swap against MX1 & Directed- Task 200.
- 116. 92 5.5 AGenetic Algorithm Approach for Solving the SV-PDPTW The MX1 crossover thus seems capable of guiding the GA towards feasible solutions without the help of an ‘intelligent’ mutation operator. Its success is possibly due to the fact that it creates not only one but two children that have a better visiting order than their parents. These children are biased towards the more urgent requests in terms of the allowed service time. All that this crossover needs is a simple random swap to introduce a bit of a diversity in the population. Any more ‘intelligent’ interference does not seem to help and actually appears to drive the search away from better solutions. The importance of the random swap mutation to help introduce diversity can also be noticed in Tables 5.1 and 5.2, by observing the results when MX1 worked alone without any mutation. The results in Table 5.1 show that MX1 failed to find some previously achieved best results in the small tasks (SET 1), and even failed to achieve any feasible results in all large tasks (SET 2). In addition, the overall average objective function results produced by MX1 alone were considerably worse than the results produced when MX1 worked together with the random swap mutation, which can be clearly realized from Table 5.2. Finally, when directed mutation was tested without any crossover, the results obtained were, to some extent, surprising. As shown in Table 5.1, this mutation alone was able to obtain feasible solutions for all test cases. The feasibility rate was 90% or 100% for all test cases except for the largest one, in which the feasibility rate was 70%. Table 5.2 shows that in terms of the objective function results, the GA with directed mutation obtained the second best results (after the MX1 and swap mutation results). Both tables indicate that this mutation is an intelligent operator that has a great potential, since it was able to obtain good quality and feasible solutions without the help of any crossover operator. It should be noted, though, that when directed mutation was tested with MX1 crossover, only one swap movement was needed to inject the necessary level of diversity and to reduce the chance of being trapped in a local optimum. On the other hand, when directed mutation was tested alone or with the PDPX crossover, it was found that a larger number of swaps was needed to achieve good results. For these cases, the number of swaps was taken to be a random number between 1 and the number of requests in the data set. Table 5.3 shows the average number of generations of the 10 runs needed to reach the best individual in the population, for all tasks and all GA versions. It is clear from this table that the GA with MX1 crossover (alone) had the smallest average number of generations, in all test cases. This again indicates the lack of diversity in the population in this GA version, due to the absence of mutation, which led to the rapid convergence to a low quality solution. On the other hand, the GA with PDPX crossover and random swap mutation had the largest average number of generations in most test cases. This could
- 117. 5.6 Summary andFuture Work 93 be the result of some kind of ‘chaotic’ search that these two operators followed, since they lacked any guidance towards the promising areas of the search space, as previously discussed. Also, the GA with directed mutation (alone) had the largest average number of generations in two test cases (130 and 200 requests). This is most probably due to the absence of crossover, which is usually beneficial in exploring wide areas of the search space and finding good solutions within a fewer number of generations. Similarly, the GA with PDPX and directed mutation needed a relatively large number of generations in all test cases. This sustains our previous observation that the crossover operator in this GA version seems to be ineffective, and the search could be relying mostly on directed mutation to obtain good quality solutions. Table 5.3: Average Number of Generations to Reach the Best Individual. Task Mx1 & Swap Mx1 & Directed PDPX & Swap PDPX & Directed Directed Mutation MX1 30 31 15 704 39 35 13 80 116 15 943 414 267 14 90 23 18 1811 421 341 17 100 17 14 1696 617 562 14 130 444 230 1000 893 1432 53 170 446 382 2000 1769 1572 54 200 851 365 2000 2268 2320 49 5.6 Summary and Future Work In this part of our research we introduced the first variant of pickup and delivery problems that we tried to handle, namely the single vehicle with time windows. After describing the problem and the motivation behind the research in addition to a summary of some related work, we explained our view regarding the solution representation and the neighbourhood moves that we believe can help conquer the difficult problem constraints. We then exami- ned the suggested ‘ideas’ in the context of a problem-specific genetic algorithm. Specifically, the encoding used in our GA is a duplicate gene encoding that guarantees the satisfaction of the precedence constraint throughout the search, alleviating the need for backtracking to solve the infeasibility that may result following any genetic manipu- lation. Four genetic operators were tested: a 2-child merge crossover (MX1) guided by
- 118. 94 5.6 Summaryand Future Work precedence of both time window bounds, a new pickup and delivery crossover (PDPX) that depends on the order in which the pickup and delivery locations appear in parent solutions, a regular random swap mutation, and a directed mutation that swaps genes according to the urgency of service time. The experimental results on two data set samples indicate that the MX1 crossover and the directed mutation are each effective as genetic operators (although they do not work very well together). We believe their success is due to the guidance they take from the time window information. On the other hand, the PDPX crossover was not able to find fea- sible solutions to any of the test instances, when tested in combination with the random swap mutation. This was probably due to the absence of guidance towards the desired service time. However, the results for PDPX dramatically improved, when used in combi- nation with directed mutation. Indeed, the directed mutation operator seems to show great promise, and would appear to be useful in guiding the search towards feasible solutions, in cases where a crossover operator is disruptive, ineffective or absent. Our upcoming research will attempt to investigate the potential of this operator for the SV-PDPTW in the context of other heuristics and meta-heuristics. The details of this investigation is presented in the next chapter.
- 119. 95 Chapter 6 Several Heuristicsand Meta-heuristics Applied to the SV-PDPTW Taking into consideration the challenges of developing an appropriate solution represen- tation and effective neighbourhood moves to facilitate dealing with the constraints of the SV-PDPTW, in this part of our research we investigate and compare three approaches to the problem. The first approach is a GA approach, similar to that introduced in Chapter 5. As previously mentioned, this approach adopts a solution representation having the same code for both the pickup and its associated delivery, and applies some problem-oriented genetic operators. The second solution methodology we investigated is Simulated An- nealing (SA), which adopts the same solution representation as the one used in the GA. Also inspired by the GA mutations, two SA neighbourhood strategies are tested: a ran- dom blind move, and an intelligent move that is directed by the time window. The third approach is a simple Hill Climbing (HC) heuristic that creates a starting solution and then tries to improve it, replacing the current solution with better solutions generated during the search. The same solution representation and the directed neighbourhood move used in both the GA and the SA are also adopted in the HC. As previously indicated in Chap- ter 5, the details of the three approaches and the results of this part of our research were published in the Journal of Heuristics [84]. Before we start the discussion of our solution methodologies, we present, in the following section, the objective function used in this part of the research. The three sections that introduce our selected approaches to the problem are Sections 6.2 (GA), 6.3 (SA) and 6.4 (HC). Section 6.5 then details our findings when the three approaches were analyzed and compared, and Section 6.6 elaborates on some further analysis of the SA algorithm, which is the most promising approach among those tested. Finally, Section 6.7 concludes this chapter with some suggested future work.
- 120. 96 6.3 The3-Stage Simulated Annealing (SA) Approach 6.1 The Objective Function In this part of the research, we slightly modified the objective function used by [89] and also by our GA in Chapter 5 (i.e., Equation 5.1). The current objective function additio- nally penalizes the amount of time delay, together with the number of time window and capacity violations. The idea is that adding a penalty for the extent of total tardiness in the route could possibly direct the search towards better quality and feasible solutions, especially if the problem size increases. Thus, the objective function of a route r is now described by the following equation: F(r) = w1 × D(r) + w2 × TWV (r) + w3 × CV (r) + w4 × TD(r), (6.1) where D(r) is the total route duration including the waiting time, if the vehicles arrives at a location before the start of its service time. TWV (r) is the total number of time window violations in the route. CV (r) is the total number of capacity violations, and TD(r) is the total amount of delay if the vehicle arrives at a location later than the specified deadline. w1, w2, w3 and w4 are weights in the range [0, 1], and w1 + w2 + w3 + w4 = 1.0. While experimenting with this objective function, we found that in order to eliminate infeasibility, a higher penalty should be imposed on the time window violations and the total delay than the capacity violations or the total route duration. We used the following values for the penalty weights in our objective function: w1 = 0.001, w2 = 0.6, w3 = 0.099, and w4 = 0.3. 6.2 The Genetic Algorithm (GA) Approach The genetic algorithm used in the current research is identical to the GA used in the first part of the research, as presented in Chapter 5. Nevertheless, here we only applied the two most promising genetic versions. Namely, the GA with MX1 crossover and random swap mutation, and the GA with directed mutation only. Also, the GA adopted here uses the new objective function described in Equation 6.1, instead of the previous objective function (Equation 5.1). 6.3 The 3-Stage Simulated Annealing (SA) Approach As previously explained in Section 2.3.2, the theoretical foundation of simulated annea- ling was established in [93]. Since then, the algorithm has been successfully used in
- 121. 6.3 The 3-StageSimulated Annealing (SA) Approach 97 solving many combinatorial optimization problems, including vehicle routing and sche- duling problems. To solve the SV-PDPTW using SA, we first need to construct an initial solution, and then try to optimize this solution using a method analogous to the annealing of solids. Some researchers choose to apply different heuristic techniques to create an initial solution with minimum constraint violations, for example [23] and [95]. In our research, we construct an initial solution s by simply generating a large number of random routes and selec- ting the route with minimum cost (according to Equation 6.1) to be our starting solution1 . This technique is straightforward to implement and also computationally inexpensive. Al- though the quality of the initial solution may be poor, the main focus of the algorithm is on the route improvement phase, accomplished during the SA search. A good neighbou- rhood move should be capable of directing the SA algorithm to reach a final high quality solution, irrespective of the quality of the starting solution. The solution representation adopted in the SA approach is the same as the solution representation of the GA, i.e., assigning the same code to the pickup and its delivery as shown in Figure 5.2. The choice of an appropriate annealing schedule is critical to the performance of SA. Ideally it is desirable to devise a scheme that is adaptable for all test cases and problem sizes, eliminating the need for ‘arbitrary’ parameter tuning, which can be very time consu- ming. Following [40], we created the annealing parameters for each test case individually as shown in Algorithm 6.1. To generate a neighbouring solution s′ in this algorithm, a simple random swap between two different locations in s is performed. The main SA algorithm is shown in Algorithm 6.2. This algorithm starts with the same solution s used to calculate the annealing parameters in Algorithm 6.1. This time, ho- wever, to get a new state s′ from the current state s, an intelligent neighbourhood move, that depends on the upper bound of the time window, is used. First two locations in the current solution are selected at random. Then, these two locations are swapped, only if their deadlines are out of order, i.e., if the latter has a more urgent deadline than the earlier one. This is exactly the same idea adopted in the directed mutation used in the GA (see Section 5.5.2 and Figure 5.6), and its purpose is to arrange requests in a way that will best satisfy the timing constraint. 1 We generated 10000 random solutions and selected the best one among them.
- 122. 98 6.3 The3-Stage Simulated Annealing (SA) Approach Algorithm 6.1: Calculating Annealing Parameters [40]. 1: Given an initial solution s {Initialize Pstart the starting acceptance probability and Pend the final acceptance probabi- lity} 2: Let Pstart = a large value {We used 0.999} 3: Let Pend = a small value {We used 0.001} 4: ∆avg ← 0 {Generate n neighbouring solutions of s} 5: for (i = 0; i < n; i + +) do 6: Select two random locations in s 7: Swap the current two locations in s to get a new solution s′ 8: ∆ ← |cost(s′) − cost(s)| 9: ∆avg ← ∆avg + ∆ 10: ∆avg ← ∆avg/n 11: T0 ← −∆avg/ log(Pstart) {T0 is the initial temperature} 12: TN ← −∆avg/ log(Pend) {TN is the final temperature} 13: α ← exp(log(TN )−log(T0))/N {α is the temperature reduction factor, and N is the number of iterations desired} 14: Return T0, TN and α Algorithm 6.2: The Main SA Algorithm. 1: Given an initial solution s, a starting temperature T0, and a temperature reduction factor α 2: T ← T0 {Initialize the current temperature} 3: repeat 4: Select two random locations in s 5: if (The latter location has a smaller value of the upper bound of the time window) then 6: Swap the current two locations in s to get a new solution s′ 7: ∆ ← cost(s′) − cost(s) 8: if (∆ < 0) then 9: s ← s′ 10: T ← α × T {Reduce the current temperature} 11: else 12: p = Random(0, 1) {generate a random number in the interval (0,1)} 13: if (p < exp(−∆/T)) then 14: s ← s′ 15: until (Frozen){Stop when no improvement is achieved for a pre-specified number of iterations (we used 5000 iterations)} 16: Return s
- 123. 6.3 The 3-StageSimulated Annealing (SA) Approach 99 During any iteration of the main SA algorithm, if the currently selected locations are not out of sequence in terms of their deadlines, however, no swap of locations will take place. As a result, the current solution will remain intact, and no change in cost is evaluated. The temperature value T also remains unchanged and is carried forward to the next itera- tion. In fact, as will be explained shortly, the present temperature value is only reduced following the replacement of the current solution s with a new improved solution s′ . Nevertheless, since the final solution obtained after the main SA procedure may still be a low quality or an infeasible solution, we extended our algorithm with two further SA stages, in order to more fully exploit all the guidance that the time windows can give us. The second stage adopts a neighbourhood move depending on the lower bound of the time window instead of the upper bound, while the third stage uses a neighbourhood move based on the centre of the time window interval. Specifically, the second and third SA stages will only differ from the first SA stage (which is shown in Algorithm 6.2) in Step 5 of the algorithm, since the lower bound and the centre of the time window will be used respectively in these two stages. The idea is that, each of these different neighbourhood moves may help introduce some improvement to the fitness of the current solution, for example by reducing the total delay or the total waiting time, which could ultimately lead to obtaining high quality solutions. Each new SA stage starts from the final solution reached by the end of the previous stage, and its starting temperature is the final temperature reached by the previous stage. We chose to start with a route improvement move that is guided by the upper bound of time window, in order to reduce the total number of time window violations in the route by visiting the more urgent requests first. However, a different improvement order could also be attempted (See Section 6.6 for a further investigation of different orders of time window moves). In the first two stages of the SA procedure (in which the neighbourhood move depends on the upper and then the lower bound of the time window), the temperature reduction sche- dule was slow. As shown in Step 10 of Algorithm 6.2, we chose to reduce the temperature only when a better solution was found, i.e., several neighbouring solutions are explored for the same temperature value. The idea is to allow the algorithm a thorough exploration of the neighbourhood by accepting a large number of worse moves during the early stages of the search. However, during the final stage of the SA (the one that adopts the centre of the time window to perturb the solution), the solution was approaching stability, and there was a danger of losing the best solution obtained if the temperature was slowly reduced. Thus, during this final stage, the temperature reduction was rapid. This was achieved by reducing the temperature at every iteration of the search, which will make the probability
- 124. 100 6.4 TheHill Climbing (HC) Approach of accepting worse solutions very small. To evaluate the performance of the 3-stage SA algorithm described above, it was compa- red with another simple SA algorithm in which the same annealing schedule and a slow cooling is used, but the neighbourhood move was a simple random swap of locations that does not take the timing order into consideration. The comparison results are reported in Section 6.5. 6.4 The Hill Climbing (HC) Approach The final approach we used to solve the SV-PDPTW problem is a simple hill climbing heuristic. The algorithm basically has two phases: a route construction phase, and a route improvement phase. As with our SA algorithm, the initial solution is simply created by generating a large number of random solutions and selecting the best generated solution (i.e., minimum cost solution according to Equation 6.1). Again this procedure will avoid unnecessary complications and increased processing time that may result if we try to ge- nerate a high quality solution with minimum infeasibility. A guided route improvement phase, which repeatedly replaces the current solution with better solutions generated du- ring the search, should be able to transform the starting low quality and possibly infeasible solution to a high quality and feasible solution. Our main HC algorithm is described in Algorithm 6.3. Algorithm 6.3: The Main HC Algorithm. 1: Given an initial solution s 2: repeat 3: for (Each possible pair of locations in s) do 4: if (The latter location has a smaller value of the upper bound of the time window) then 5: Swap the current two locations in s to get a new solution s′ 6: ∆ ← cost(s′) − cost(s) 7: if (∆ < 0) then 8: s ← s′ 9: until (Done){Stop if no improvement has been achieved in the previous pass} 10: Return s Again, in our HC we adopt the same solution representation and neighbourhood moves that were used in both the GA and the SA. Similar to the SA algorithm, the HC algorithm was divided into three stages. The first stage generates a neighbouring solution by swap- ping the two locations currently under consideration, only if their deadlines (i.e., the upper
- 125. 6.5 Experimental Results101 bounds of the time window) are out of order. On the other hand, if the deadlines are in sequence, the current solution remains intact, and two new locations are considered in the next iteration. When no further improvement is possible using this neighbourhood move, the second stage starts from the final solution achieved in the first stage and repeats the same HC procedure, but with a neighbourhood move that adopts the lower bound of the time window to decide the swapping. Finally, the third stage starts with the final solution obtained in the second stage, but with a neighbourhood move that swaps locations if they are out of order in terms of the centre of the time window interval. 6.5 Experimental Results To test our algorithms, we used the same data set samples on which we tested our GA, as previously explained in Section 5.5.4. Each of the following algorithms was run 10 times on each test case: 1. The GA with MX1 crossover and random swap mutation (GA1). 2. The GA with directed mutation only (GA2). 3. The 3-stage SA (SA1). 4. The simple random-move SA (SA2). 5. The Hill Climbing (HC) algorithm . All algorithms were run on all problem instances using the same set of parameters (e.g. GA parameters, simulated annealing schedule, and stopping conditions), i.e., no particular tuning of parameters was performed for each test case separately. The results for the 10 runs are recorded in Table 6.1 as follows: for each test case under each algorithm, the best result found (in terms of total route duration only), and the percentage of feasible solutions obtained during the 10 runs. The last column of the table shows the previous best results, attributed to [89] and our GA approach reported in Chapter 5. Whenever the result achieved in the current experiment is better than the previous best result, it is highlighted in boldface.
- 126. 1026.5ExperimentalResults Table 6.1: ResultsSummary for all Algorithms (Total Route Duration). Data Task GA1 GA2 SA1 SA2 HC Prev Best Best %Feas Best %Feas Best %Feas Best %Feas Best %Feas SET 1 30 3738.73 100 3751.21 100 3299.40 100 3684.21 100 3772.82 70 3696.51 80 7838.21 100 7867.68 100 7267.0 100 7852.25 80 7847.67 90 7838.21 90 8619.01 100 8619.01 100 8292.67 100 8619.01 100 8671.54 100 8618.0 100 10600.10 100 10600.10 90 10544.60 90 10611.10 70 10600.10 90 10600.10 SET 2 130 14041.0 100 14343.0 100 13826.0 100 14081.20 90 14031.10 100 13856.20 170 20618.10 100 20333.80 100 19690.90 100 19964.80 100 20003.60 100 19861.30 200 25799.50 100 24730.0 100 23841.10 100 24461.60 100 24304.30 100 24512.80
- 127. 6.5 Experimental Results103 As shown in Table 6.1, the best route duration results were achieved by the 3-stage SA (SA1). In all test cases the best results achieved by this version of SA were better than the previous results achieved in [89] and by our GA explained in Chapter 5. The results were also superior to the results achieved by all the other algorithms tested in the current research. A very high feasibility rate and good quality solutions were obtained in all tasks tested, with an improvement of approximately 11%, over the previous best2 , for the smallest task and approximately 3% for the largest task. In terms of the overall objective function values, Table 6.2 shows the average, best, and worst objective function values, as calculated by Equation 6.1, for the GA1 and GA2 algorithms. the Standard Deviation (SD) value of the objective function for the 10 runs is also shown in this table. Table 6.3 shows the same objective function information for the remaining three algorithms (SA1, SA2, and HC). Table 6.2: Results Summary for GA1 and GA2 (Objective Function Values). Task GA1 GA2 Avg Best Worst SD Avg Best Worst SD 30 3.74 3.74 3.74 0 3.81 3.75 3.89 0.05 80 7.86 7.84 7.87 0.01 7.92 7.87 7.97 0.04 90 8.62 8.62 8.62 0 8.69 8.62 8.77 0.05 100 10.60 10.60 10.60 0 11.08 10.60 14.69 1.27 130 14.51 14.04 15.28 0.35 14.85 14.34 15.16 0.25 170 20.94 20.62 21.25 0.21 20.67 20.33 20.88 0.16 200 26.20 25.80 26.45 0.21 25.04 24.73 25.33 0.21 Avg 13.21 13.04 13.4 0.11 13.15 12.89 13.81 0.29 Tables 6.2 and 6.3 also support the conclusions drawn from Table 6.1. Observing the overall average results in the last row of Table 6.2 and 6.3, we can clearly see that the SA1 algorithm outperformed all other heuristics in the average, best and worst objective function values. On the other hand, its overall average SD value was only slightly larger than the overall average SD of the GA1 algorithm. The GA1 algorithm obtained a smaller SD, due to the fact that the population converged to a sub-optimal solution in 3 out of the 7 test cases (30, 90 and 100 requests), as shown in Table 6.2. 2 The improvement percentage is calculated using the formula: ((R∗ − R)/R∗ ) × 100, where R is the current result and R∗ is the previous best result.
- 128. 1046.5ExperimentalResults Table 6.3: ResultsSummary for SA1, SA2 and HC (Objective Function Values). Task SA1 SA2 HC Avg Best Worst SD Avg Best Worst SD Avg Best Worst SD 30 3.53 3.30 3.66 0.1 3.71 3.68 3.74 0.02 3.83 3.77 3.96 0.06 80 7.63 7.27 7.85 0.19 9.25 7.85 18.78 3.43 9.55 7.85 24.45 5.24 90 8.46 8.29 8.60 0.11 8.69 8.62 8.75 0.04 8.69 8.67 8.72 0.02 100 10.63 10.54 11.0 0.13 21.49 10.61 75.31 20.70 10.90 10.60 13.40 0.88 130 13.95 13.83 14.12 0.11 18.68 14.08 59.05 14.19 14.51 14.03 15.08 0.37 170 20.06 19.69 20.43 0.22 20.16 19.96 20.46 0.16 20.31 20.0 20.89 0.25 200 24.37 23.84 24.68 0.25 24.71 24.46 24.94 0.15 24.63 24.30 24.99 0.22 Avg 12.66 12.39 12.91 0.16 15.24 12.75 30.15 5.53 13.2 12.75 15.93 1.01
- 129. 6.5 Experimental Results105 It appears from these results that the SA1 algorithm was able to escape the trap of local optima and to gradually find better solutions by progressing from one stage to another. Each stage of the 3-stage SA (SA1) seems to contribute to the improvement process. To illustrate this, Table 6.4 shows the progress of the 3-stage SA in the best run for our largest task of 200 requests. The table shows the total route duration, the number of capacity violations (CV) and the number of time window violations (TWV) at the beginning of the run, and after the termination of each of the three stages. It is clear from this table how the solution progressively improves after each stage. Table 6.4: 3-stage SA Progress for Task 200. Stage Route Duration CV TWV Initial Solution 52969.3 64 387 After Stage 1 37768.2 0 344 After Stage 2 23841.1 0 3 After Stage 3 23841.1 0 0 Regarding the other four algorithms (GA1, GA2, SA2 and HC), it appears from Tables 6.1, 6.2 and 6.3 that the best results obtained by both versions of the GA were slightly worse than the results obtained by the random-move SA (SA2) and the HC in most test cases. The GAs, nevertheless, achieved a better feasibility ratio, and their overall average objective values were better than the average obtained by the SA2 algorithm. The tables also show that the performance of the HC algorithm was in general very good. Its overall average objective function result (shown in the last row of Table 6.3) was comparable to those obtained by the GAs, but better than the overall average objective of the SA2 algo- rithm. The SA2 algorithm has the worst overall average objective value, obviously due to the higher percentage of infeasible solutions obtained by this algorithm, as indicated in Table 6.1. Similar to the 3-stage SA, the three stages applied in the HC algorithm seem to contribute to guiding the search towards better solutions. To see this, consider again the largest task of 200 requests. Table 6.5 shows the progress of the best run for this task from one stage to another, highlighting the total route duration and the number of capacity and time window violations. It is again clear from this table how each stage played its role in improving the solution it received from the previous stage.
- 130. 106 6.5 ExperimentalResults Table 6.5: HC Progress for Task 200. Stage Route Duration CV TWV Initial Solution 55013.0 117 387 After Stage 1 26462.9 1 1 After Stage 2 24315.9 0 0 After Stage 3 24304.3 0 0 Figure 6.1 also bears out the above observations. This graph shows the average objective value (as calculated by Equation 6.1) achieved during the 10 runs, by each of the algo- rithms tested. For all test cases, the graph demonstrates the superiority of the 3-stage SA, since its average objective value is the lowest compared to the other algorithms tested. 30 80 90 100 130 170 200 0 5 10 15 20 25 30 Task Size AverageObjectiveValue GA1 GA2 SA1 SA2 HC Figure 6.1: Average objective value for all algorithms. It should be noted that infeasible solutions are seldom produced by our algorithms. Ho- wever, since an infeasible solution usually has a very large objective value, the presence of one or more such solutions often results in a large increase in the average objective va- lue (as an example, notice the average objective value produced by the random-move SA algorithm (SA2) for tasks 100 and 130 shown in Figure 6.1). Specifically, if the infeasible solution has a very large number of TW violations, its overall objective value will be very
- 131. 6.5 Experimental Results107 high, given the large penalty imposed on the TW violations in Equation 6.1, and also the penalty on the amount of delay. On the other hand, a feasible solution will only have the route duration (multiplied by its small weight) as its total objective value. Figure 6.2 shows the best run of the HC algorithm against the best run of 3-stage SA (SA1) algorithm for the largest task of 200 requests. The graph demonstrates the current objective value, as calculated by Equation 6.1, in each iteration of the run. It seems in this graph that the 3-stage SA was able to explore a wider area of the search space, albeit with a larger number of iterations needed to reach convergence. Figure 6.3 shows the best run of the random-move SA algorithm (SA2) against the best run of the 3-stage SA (SA1), again showing the current objective value in each iteration of the run for the 200-requests task. In this graph, it is clear that the 3-stage SA was immediately directed towards lower cost solutions during the early phases of the run. The random- move SA, however, spent quite a long time during these early phases investigating low quality solutions, and only started discovering the promising areas of the search almost halfway through the run3 . 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 0.5 1 1.5 2 2.5 x 10 6 Iteration ObjectiveValue HC SA1 Figure 6.2: HC against 3-Stage SA (SA1) for task 200. 3 It should be noted that the objective values only appear to go to zero in Figures 6.2 and 6.3, due to the scale of the y-axis. However, the final objective values are in fact very small. This is because the solution reached by the end of each run in both figures is a feasible solution whose cost is only measured by the duration of the route (multiplied by its small weight), according to Equation 6.1.
- 132. 108 6.5 ExperimentalResults 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10 6 Iteration ObjectiveValue SA2 SA1 Figure 6.3: Random-move SA (SA2) against 3-Stage SA (SA1) for task 200. Table 6.6 shows the average processing time in seconds for all the algorithms tested. The table shows that the HC algorithm has the fastest average processing time among all algorithms, in all test cases. The table also indicates that the overall average processing time of the 3-stage SA was slightly longer than the overall average processing time of the random-move SA. In addition, the table clearly indicates that both GA versions were very slow compared to all other algorithms, which is expected due to the need to maintain a large population of individuals throughout the search. The SA and HC algorithms, on the other hand, have lower computational costs due to focusing on a single solution. Finally, comparing the results obtained by the 3-stage SA with the results obtained by pre- vious researchers for the same problem, our algorithm seems impressive, in terms of run time as well as solution quality. We appreciate the difficulty in making such comparisons when different platforms have been used. Nevertheless, our run times are to a large extent acceptable. For example, as shown in Table 6.6, the average run time of the 3-stage SA for the 100-requests task was 44.5 seconds, while the best algorithm tried in [89] appears to have an average CPU time of approximately 300 seconds for the same task.
- 133. 6.6 The 3-StageSA: Further Investigation 109 Table 6.6: Average Processing Time in Seconds. Task GA1 GA2 SA1 SA2 HC 30 92.6 42.65 5 4.9 1 80 114.5 73.15 23.8 11.7 4.3 90 121 105.65 39.1 14 5.4 100 43.55 111.9 44.5 14.4 6.2 130 422 581 26.3 30.1 9.2 170 479.7 699.4 38.8 42.1 13.8 200 1018.8 760.95 46.4 53.9 19.4 Average 327.44 339.24 31.99 24.44 8.47 6.6 The 3-Stage SA: Further Investigation As mentioned above, the 3-stage SA seems to be quite a promising approach. Thus, it would be justified if we perform a thorough investigation of this algorithm by trying to examine all its potential. The choice of the sequence of the three neighbourhood moves was only based on our judgment. Starting with a move that is guided by the upper bound of the time window may give the algorithm better guidance, since it would potentially reduce the infeasibility incurred as a result of over delay. This, however, may not be the best choice. A different order could prove more beneficial in the optimization process, either in the quality or the speed of the final solution obtained. Thus, this part of our research tries to investigate all possible sequences of neighbourhood moves, by testing them on selected tasks from the data sets used in the previous part of the research. Our aim is to reach a conclusion as to whether there is any significance to the order of the underlying neighbourhood moves. The following discussion details our experimental findings when the different sequences of stages were investigated. 3-Stage SA Experimental Results: To evaluate the performance of the 3-stage SA when different orders of neighbourhood moves are considered, we selected 4 tasks from our two data set samples. These are tasks 30, 90, 130, and 200. For each task, we ran the 3-stage SA algorithm under each possible sequence 10 times. In the upcoming analysis of results, we use a combination of the first letters of the three time window (TW) moves to denote the sequence under consideration. For example (LEC) denotes the sequence: Late time window followed by Early time
- 134. 110 6.6 The3-Stage SA: Further Investigation window then Centre time window. Table 6.7 shows the average objective value4 , of the 10 runs achieved by the different TW orders for all tasks. Table 6.8 shows the best and worst average objective values achieved in each task, together with the sequence of moves that obtained these averages. This table also shows the best objective value achieved in each task and the sequence which obtained this best result. The last column of the table shows the previous best objective value obtained using the preliminary experimentation order (LEC), as explained in Section 6.5. Table 6.7: Average Objective Value for all Sequences. Task LEC LCE ELC ECL CEL CLE 30 3.53 3.75 3.49 3.70 3.57 3.78 90 8.46 8.62 8.55 8.52 8.48 8.75 130 13.95 14.03 13.97 13.96 14.22 14.19 200 24.37 24.40 24.30 24.27 24.32 24.49 Table 6.8: Best and Worst Results. Task Best Avg Obj Worst Avg Obj Best Obj Prev Best Obj 30 3.49 - ELC 3.78 - CLE 2.97 - ELC 3.30 - LEC 90 8.46 - LEC 8.75 - CLE 8.23 - CEL 8.29 - LEC 130 13.95 - LEC 14.22 - CEL 12.93 - ECL 13.83 - LEC 200 24.27 - ECL 24.49 - CLE 23.43 - ELC 23.84 - LEC As shown in Table 6.8, the sequence LEC (used in Section 6.5) achieved the best average in 2 tasks, tasks 90 and 130. On the other hand, the sequence CLE achieved the worst average objective in 3 tasks, 30, 90 and 200, while the sequence CEL achieved the worst average for task 130. Note that both sequences that obtained the worst averages start with the centre of the time window, while all sequences that obtained the best averages started with either ends of the TW interval. Possibly, adopting the centre of time window as the starting phase may not be the best choice for our algorithm. This deduction, however, still needs further evidence, specially when we notice in Table 6.8 that the sequence CEL was able to achieve the best solution among all sequences for task 90. For the remaining three 4 As calculated by Equation 6.1.
- 135. 6.6 The 3-StageSA: Further Investigation 111 tasks, 30, 130 and 200, the sequences that obtained the best results all started with a move that depends on the early TW bound. Apart from the above observations, there seems to be no significant difference between the averages obtained by the different sequences. This can also be seen in Figure 6.4, which shows graphically the same information as Table 6.7. The graph shows that all sequences obtained more or less comparable averages in all test cases. Task 30 Task 90 Task 130 Task 200 0 5 10 15 20 25 Task Size AverageObjectiveValue LEC LCE ELC ECL CEL CLE Figure 6.4: Average objective value for all sequences . Also, observing the resulting feasibility ratio of all sequences, we found that the feasibility ratio obtained by all sequences was 100% for all tasks, with only one exception, the sequence CLE. This sequence had an 80% feasibility for task 30, and a 90% feasibility for task 90. Again this may provide an additional evidence that starting with the centre of the time window may, in some cases, drive the algorithm away from the promising solutions. Possibly, the middle of the time window is not a very good indicator of how urgent the request is. Thus, starting with it may lead to a solution that does not have the best visiting order of requests and may have a large number of time window violations. This order may be difficult to change later in the run. It can also be noticed in Table 6.8 that the best results obtained in the current experiment were better than the previous best results in all test cases. A possible indication of this is that the 3-stage SA is a stable and robust algorithm which consistently produces high quality solutions even if different orders of neighbourhood moves were employed. Moreover, it should be noted that in most test cases the three stages seem to contribute positively to the optimization process. Each stage often added further improvement to the
- 136. 112 6.6 The3-Stage SA: Further Investigation result of the previous stage, although the extent of improvement obviously varies depen- ding on several factors, such as the quality of the starting solution, the current annealing schedule, and the neighbourhood structure. Table 6.9 shows the progress of the SA optimization process during the three stages, for the best run of each task. The table demonstrates the objective value5 , the number of capacity violations (CV), and the number of time window violations (TWV) before and after each stage of the algorithm. The sequence of TW stages that obtained these best solutions is also shown next to the size of each task. This table shows that, in some cases, for example task 90, the best solution reached by the end of one stage may be lost during the following stage. This is normal in any SA algorithm, and especially when slow cooling is employed, as we did in our first two stages6 . Nevertheless, the final result obtained by the end of the last stage was better than the lost solution. Obviously, when the algorithm starts a fresh stage, using a new move, after the stagnation of the previous stage, it may escape a local optimum, only by passing through worse solutions. In terms of processing speed, nevertheless, we might be able to reach a slightly better distinction. Table 6.10 shows the average processing time in seconds for the different sequences, together with the best average speed and the sequence that achieved this best average. The table shows that the sequence LCE achieved the best average speed in 3 cases, 30, 130, and 200. Table 6.10: Average Processing Time in Seconds for all Sequences. Task LEC LCE ELC ECL CEL CLE Best Average Sequence of Best 30 5.0 1.5 29.2 5.0 5.6 1.8 1.5 LCE 90 39.1 21.6 18.9 18.1 19.7 15.9 15.9 CLE 130 26.3 18.0 45.8 36.9 29.5 33.4 18.0 LCE 200 46.4 33.7 85.1 73.1 63.4 37.7 33.7 LCE 5 The objective value shown in Table 6.9 is calculated by Equation 6.1. Notice that this value is very large for the initial solution, due to the presence of a very large number of capacity and TW violations, which are penalized in the objective function. When the violations are removed by the end of the run, though, the remaining objective value will only be the route duration multiplied by the weight assigned to it. 6 See the discussion in Section 6.3.
- 137. 6.6The3-StageSA:FurtherInvestigation113 Table 6.9: SolutionProgress through Stages. Task Initial Solution After Stage 1 After Stage 2 After Stage 3 Obj CV TWV Obj CV TWV OBJ CV TWV OBJ CV TWV 30 - ELC 18689.70 4 48 2.97 0 0 2.97 0 0 2.97 0 0 90 - CEL 258255.0 83 171 8.84 0 0 9.68 0 1 8.23 0 0 130 - ECL 792709.0 23 232 112.24 0 2 12.93 0 0 12.93 0 0 200 - ELC 2345240.0 130 382 696.82 0 7 23.44 0 0 23.43 0 0
- 138. 114 6.7 Summaryand Future Work 6.7 Summary and Future Work In this part of our research, we investigated three different approaches to the SV-PDPTW. First, a GA approach equipped with problem-specific genetic operators was implemented. One GA version had a TW directed merge crossover working together with a random swap mutation, while the other depended only on an intelligent directed mutation that is guided by the TW. Second, two versions of SA were tested: the first operated in three stages and adopted neighbourhood moves that are guided by the time window, and a second adopted a random unguided neighbourhood move. The third approach is an HC heuristic which also operated in a manner similar to the 3-stage SA, but only accepted better solutions encountered during the search. The experimental results indicated that both GA versions had more or less comparable results in terms of both quality and processing speed. This could only indicate that the directed mutation operator is an intelligent operator that can guide the search towards better solutions even without the help of any crossover, which is usually the main GA operator. However, the performance of the GA in general was slightly inferior to the other algorithms tested in this research, in terms of the quality of the best solution in most test cases, but more so in terms of processing speed. On the other hand, the 3-stage SA seems to be superior to all other algorithms tested in this research in the quality of the solutions obtained. Moreover, its results were better than the best results from previous research on the problem, both in terms of quality and speed, in all test cases. However, this SA version was slightly slower than the random-move SA version and the HC heuristic tested. The success of the 3-stage SA is possibly due to its dependence on intelligent neighbourhood moves that contributed to guiding the search towards better solutions. These neighbourhood moves were also successful in the context of hill climbing but with a less dramatic effect than their effect in the context of simulated annealing, possibly due to the trap of local optima. However, using hill climbing can still give us good quality solutions in a very short processing time, which may be preferable in real world applications, and if the algorithm is used repeatedly, as usually done when the problem is generalized to the multiple vehicle case. We also further investigated the 3-stage SA algorithm by examining all possible time win- dow sequences and analyzing their outcome on selected instances. The results in general indicated that the 3-stage SA is a stable algorithms that usually produces high quality solutions under various orders of stages. Nevertheless, all sequences seem to produces comparable average results in most test cases. It was not possible to reach a definite conclusion as to whether one or more sequences are preferred during the application of
- 139. 6.7 Summary andFuture Work 115 the SA algorithm. Some results, though, indicated that the initial phase should probably start with either bounds of the TW interval, rather than its centre. In some cases, this will probably provide the algorithm a push ahead towards more promising solutions. In terms of processing speed, however, the sequence LCE seems to have the fastest average speed in most test cases. The ideas introduced so far for handling the SV-PDPTW problem constraints can be incor- porated in any heuristic or meta-heuristic approach that tackle this problem, and also its multiple vehicle variant, as will be shown later in this thesis. The robustness and portabi- lity of these simple tools have been demonstrated on several occasions within the current research. Our next plan is to broaden our scope and investigate the more general Multiple Vehicle Pickup and Delivery Problem with Time Windows (MV-PDPTW). The work done so far for the single vehicle case will be integrated within our upcoming study. The details of this investigation are presented in the next two chapters.
- 140.
- 141. 117 Chapter 7 New SolutionConstruction Heuristics for the Multiple Vehicle Pickup and Delivery Problem with Time Windows The Multiple Vehicle Pickup and Delivery Problem with Time Windows (MV-PDPTW) is a generalization of the SV-PDPTW introduced in Chapters 5 and 6. Thus, the problem definition in Section 5.2 applies to the MV-PDPTW except that we assume here the avai- lability a homogenous fleet of vehicles, with identical capacity, to serve the requests. In addition to the precedence, capacity and time window constraints, applied to the single vehicle case, the multiple vehicle case has an added coupling constraint. This constraint ensures that the pickup location and its corresponding delivery are served by the same vehicle. Also, the objective function of the MV-PDPTW usually tries to minimize the number of vehicles used in the solution, as a primary objective, followed by minimizing the total traveling distance and/or the total schedule duration. Figure 7.1 shows a simpli- fied representation of a small instance of this problem before and after solving it. Figure 7.1: MV-PDPTW before solution (left) and after solution (right).
- 142. 118 7.1 SolutionConstruction for the MV-PDPTW The MV-PDPTW is known to be N P-hard [131], and the presence of many constraints makes the problem particularly complicated. Indeed, generating feasible and good quality solutions to the problem in a reasonable amount of time is often a hard challenge for re- searchers. The MV-PDPTW is both a grouping problem (assigning requests to vehicles), and a routing problem (finding the best route for each vehicle). Thus, an intelligent solu- tion methodology should be able to handle these two aspects efficiently. Similar to other vehicle routing problems, researchers usually try to solve the problem in two stages: the first stage is solution construction, while the second is solution improvement. This chapter is dedicated to the solution construction phase of the MV-PDPTW, while our investigation of the solution improvement phase is detailed in the next chapter. The findings of this part of our research were published in the MIC2009 conference [81]. The rest of this chapter is organized as follows: Section 7.1 gives a general idea about solution construction for the MV-PDPTW and introduces our upcoming research in this area. Section 7.2 summarizes some related work. Section 7.3 emphasizes the motivation and the objectives of this part of our research. Section 7.4 explains the routing algorithm embedded within the different construction heuristics suggested. Section 7.5 details the construction heuristics implemented in this research. Section 7.6 reports the experimental results of the algorithms tested. Section 7.7 sheds light on some implementation issues and complexity analysis of the suggested algorithms. Finally, Section 7.8 concludes this chapter with our future plans. 7.1 Solution Construction for the MV-PDPTW Like the construction heuristics for the VRPTW, a solution construction algorithm for the MV-PDPTW usually selects at each iteration an un-assigned customer whose insertion is predicted to cause the least increase in the overall cost of the solution. The selected cus- tomer is then inserted in its best (least cost) feasible insertion position found among all available routes. This kind of insertion may require multiple calculations to estimate the effect of the insertion, in terms of the increase in travel distance and time delay, on all cus- tomers already existing in that particular route who could be affected by the insertion (see Section 3.3 for more details about construction heuristics). Additional decisions during the construction of the solution include whether to build routes sequentially or in parallel, and possibly the selection of seed customers to initialize the routes. Some construction algorithms order customers before the insertion, and the initial order is also an important factor that may affect the quality of the generated solution. Common approaches include sorting customers according to the distance from the depot, or according to the time win-
- 143. 7.1 Solution Constructionfor the MV-PDPTW 119 dow. Finally, the selection of a cost function to assess the quality of the whole solution1 during the construction is sometimes needed, so that insertions which greatly affect the solution cost can be identified and appropriately handled. While these considerations also apply to the general VRPTW, where all requests are of the same type (either pickups or deliveries), the MV-PDPTW in itself entails additional considerations. This is due to the presence of a pair of related locations for each indivi- dual request and the precedence and coupling issues resulting thereof. For example, the decision regarding the best insertion position for a certain request should ideally take both the pickup and the delivery into account. The sorting criteria for requests may likewise be based on either the pickup or the delivery location, or perhaps combine both. It is also frequently the case with the MV-PDPTW that the initial solution is drastically changed during the improvement phase. For example, both the algorithms in [12] and [131] produ- ced very good results using an approach that is based on a Large Neighbourhood Search (LNS). The algorithm removes and then relocates a large number of requests (30% - 40%) in each iteration. This could possibly indicate that sophisticated construction algorithms, that are usually time consuming, parameter dependent, and hard to implement, may not actually merit their cost, compared to more straightforward and faster algorithms. In an attempt to overcome the difficulties inherent in the construction of a feasible so- lution, which are mainly due to the hard problem constraints and the complex problem- specific decisions, we propose, in this part of our research, four different construction heuristics that aim to build initial feasible solutions to the MV-PDPTW. To the best of our knowledge, our research is the first attempt in the literature to analyze and compare different initial solution construction methods for the MV-PDPTW. The aim of the re- search is to decide which construction algorithm has more potential as a preliminary step towards a complete solution methodology to the problem. A promising construction al- gorithm should demonstrate a suitable balance between quality of the generated solution, processing speed and simplicity of implementation. All suggested algorithms make use of a simple and efficient routing algorithm to generate feasible individual vehicle routes. This routing heuristic was based on our investigation of the SV-PDPTW, as previously detailed in Chapters 5 and 6. These algorithms, never- theless, differ in whether the construction of vehicle routes is performed sequentially or in parallel. They also differ in the criteria according to which the next un-routed request is selected for insertion in a particular route. 1 For example, the cost function may combine the number of routes and the total cost of each route.
- 144. 120 7.2 RelatedWork 7.2 Related Work As previously mentioned in Section 3.3, solution construction, for vehicle routing pro- blems in general, can be done either sequentially or in parallel. A sequential construction builds routes one after another, while a parallel construction builds a number of routes simultaneously. Figure 7.2(a) shows a typical solution, consisting of 8 locations, that is constructed sequentially, while Figure 7.2(b) shows a solution with the same number of locations that is constructed in parallel. (a) Sequential construction (b) Parallel Construction Figure 7.2: Solution Construction. To construct initial solutions for the MV-PDPTW sequentially, researchers usually adap- ted Solomon’s sequential insertion heuristics of the VRPTW [141]. A weighted sum of the extra travel distance and total time delay resulting from the insertion is often used to estimate the cost of the insertion (see Section 3.3.3 for more details about Solomon’s insertion heuristics). This type of construction was used by [100] for the MV-PDPTW, and was followed by a solution improvement phase called a tabu-embedded simulated annealing. As previously mentioned in Section 3.3.3, a parallel construction heuristic was first intro- duced in [121] for the VRPTW. In a parallel construction, several routes are initialized with seed customers and requests are subsequently inserted into any of the initialized routes. Accordingly, the algorithm needs an initial estimate of the number of vehicles to be used. Routes are later added as needed if the initial estimate does not yield a feasible solution. The authors also introduced an additional complex measure in the cost function, which is called “a generalized regret value”, in order to compare the difference between the cost of an immediate insertion versus a postponed insertion. Requests with a large regret value are given priority of insertion. The regret value is a kind of a ‘look-ahead’ measure to estimate the difference between inserting the request in its best route and in- serting it in its second best route. The measure may be generalized by comparing the
- 145. 7.3 Research Motivationand Objectives 121 insertion costs of the best k routes of a particular request, rather than only the best two routes. In a sense, we choose the insertion (of a request) that we will ‘regret’ most if it was not performed now. This regret measure was also used by [131] for the MV-PDPTW, and was embedded within an Adaptive Large Neighbourhood Search (ALNS) technique to improve the solution quality. The work in [102] presents a sequential construction algorithm for the MV-PDPTW. The algorithm repeats a cycle of three components. The first component is a constructor, which uses a sequential greedy algorithm to add pairs of customers in the order they appear in a priority sequence that is initially random. The analyzer afterwards analyzes the solution and assigns a certain ‘blame’ value for each customer based on its contribution to the total solution cost. Finally, the prioritizer reorders the customers, such that customers with a high blame value are moved forward in the priority sequence. A parallel construction heuristic that solves the MV-PDPTW is presented in [104]. The algorithm starts by finding the largest set of customers, where it is impossible to serve any two customers with the same vehicle due to constraint violation. These customers are then used as seed customers in the initial set of routes. To insert the remaining custo- mers afterwards, the algorithm takes into consideration the effect of insertion on both the classical increase in distance measure, and also the remaining time window slack in the route, i.e., priority is given to insertions that do not use much of the available time slack, allowing for more feasible latter insertions. The authors also use a non-standard measure of the visual attractiveness of the route to select the most desired insertions, by trying to minimize the number of crossings (intersection points) between the generated routes. 7.3 Research Motivation and Objectives We noticed during our literature survey of the MV-PDPTW that researchers who adopt a 2-phase approach (i.e., construction of an initial solution, followed by an improvement phase) to solving the problem often pay more attention to the solution improvement phase, such that the results of the initial solution construction phase are seldom reported, if at all. This makes it difficult to assess the contribution of the construction method to the success or the failure of the overall algorithm. It is also important to note that the role of the construction algorithm is not only limited to the initialization phase. The construction algorithm is often applied at various stages during the improvement phase to create or modify new or partial solutions, as done for example in [12] and [116]. Therefore, a good choice of the construction algorithm is vitally important.
- 146. 122 7.4 TheRouting Algorithm The research reported in this part of the thesis will help identify the construction heuris- tic(s) that seems to be most appropriate for this problem, and decide whether sophisticated and computationally expensive methods actually warrant their cost, as opposed to other simpler and less expensive algorithms. In the following section we explain our simple routing algorithm, which is the core of the different construction algorithms proposed in this research. Section 7.5 then discusses in detail these construction algorithms. 7.4 The Routing Algorithm A crucial part of the MV-PDPTW is the routing algorithm that will generate a feasible route for each individual vehicle. A major concern is how to handle all problem constraints efficiently. The routing algorithm we used here was selected based on our research on the SV-PDPTW, detailed in Chapters 5 and 6. This algorithm relies on an iterative impro- vement of individual routes, and is embedded in the overall constructive algorithm that could either be sequential or parallel. The main difference between our routing algorithm and other routing (insertion) heuristics in the literature is that our algorithm does not try to find the best insertion position for each request in the route, but accepts any feasible insertion. As a result, many complex calculations and problem-specific decisions, that are related to the association between the pickup and the delivery, can be avoided. For example, our algorithm eliminates the bias towards either the pickup or the delivery loca- tion, which is one of the major drawbacks of ‘classical’ insertion methods. Clearly, when the best insertion position for one location (pickup or delivery) is chosen first, the choices available for its partner will be restricted accordingly. Based on the representation attempted in our SV-PDPTW heuristics, rather than repre- senting the visiting order of locations in each route by a one-dimensional permutation of all the different locations, we assign the same code to both the pickup location and its delivery, and we refer to the pair as a request. We then rely on a simple decoder to al- ways identify the first occurrence of the code as the location of pickup and the second as the location of its delivery. This representation will handle both the precedence and the coupling constraints of the MV-PDPTW, since it will be no longer necessary to ensure that both the pickup location and its delivery are assigned to the same vehicle, and that the pickup comes before the delivery in the visiting order. On the other hand, the capacity and time windows constraints may be violated in this representation, but they are penalized in the objective function, as will be explained shortly (see Equation 7.1 below). Recall from Chapter 5, that the different routing algorithms for an individual vehicle tried to improve the current route by rearranging nodes based on their time window intervals.
- 147. 7.4 The RoutingAlgorithm 123 For the purpose of the MV-PDPTW, we selected as a route improvement heuristic, the simple Hill Climbing (HC) algorithm which tries to gradually modify the current route until no further improvement is possible. In fact, we chose the HC algorithm to improve individual routes, because it was very fast compared to the other heuristics investigated for the SV-PDPTW, and it also gave good quality results. As such, it seemed more suitable for repeated application within the various construction algorithm, as will be explained in Section 7.5. Also, only one stage of time window improvement was applied here to accelerate the generation of multiple routes. The neighbourhood move used by the HC algorithm in this part of our research depends only on the upper bound of the time window interval to decide the swapping of locations. Algorithm 7.1 describes this simple heuristic. Algorithm 7.1: The HC Routing Algorithm. 1: Given a route r 2: repeat 3: for (Each possible pair of locations in r) do 4: if (The latter location is more urgent in its upper time window bound) then 5: Swap the current two locations in r to get a new route r′ 6: ∆ ← cost(r′) − cost(r) 7: if (∆ < 0) then 8: r ← r′ 9: until (Done){Stop if no improvement has been achieved in the previous pass} The cost function to evaluate the quality of a route r, in Step 6 of the HC algorithm, is described by the following equation: F(r) = w1 × D(r) + w2 × TWV (r) + w3 × CV (r) , (7.1) where D(r) is the total route duration, including the waiting time and the service time at each location. TWV (r) is the total number of time window violations in the route, and CV (r) is the total number of capacity violations. The constants w1, w2, and w3 are weights in the range [0,1], and w1 + w2 + w3 = 1.0. As previously mentioned, the largest penalty should be imposed on the time window violations, in order to direct the search towards more feasible routes. We used the following weights for the route cost function: w1 = 0.201, w2 = 0.7 and w3 = 0.099 2 . 2 Comparing the current weights to those assigned in Equation 5.1, the penalty on the capacity violations was reduced, while the weight assigned to the total route duration was increased. This is due to the fact that routes in the multiple vehicle case are usually shorter than routes in the single vehicle case, and the satisfaction of the capacity constraint tends to be easier as a result.
- 148. 124 7.5 SolutionConstruction Heuristics 7.5 Solution Construction Heuristics In all our construction heuristics we first start by sorting customers according to the dis- tance from the depot (farthest first). However, since in our approach we deal with custo- mers in pairs, where each pair consists of a pickup location and its associated delivery, the distance measure, in relation to the depot, could either be the distance between the depot and the pickup location, or the distance between the depot and the delivery location. We arbitrarily chose the distance separating the depot and the delivery location for the initial order of requests. 7.5.1 The Sequential Construction Algorithm The sequential construction heuristic tries to build routes one after another. Requests are taken one by one in order, and each request (pickup and delivery pair) is initially inserted at the end of the current route. Our HC routing heuristic (Algorithm 7.1) is then called to try to improve the current route. If the HC algorithm returns an improved route that can ‘feasibly’ accommodate the newly inserted pair, this insertion is accepted and we move on to the next request. However, if the improved route is still infeasible, the newly inserted pair is removed from the current route to wait for another insertion attempt in a new route3 . Thus, unlike the ‘traditional’ insertion methods, our algorithm relies on the HC heuristic to improve the quality of the current route, without actually having to calculate the cost of each and every possible insertion position in order to select the best one among them. Algorithm 7.2 describes the sequential construction procedure. It is important to note in Step 7 of this algorithm that, besides overcoming the precedence and the coupling issues, inserting a request (a pickup and delivery pair) at the end of the route has the added advantage of accelerating the insertion process, since two locations instead of one are simultaneously inserted. 3 Although the HC algorithm is usually very successful in ‘deciding’ whether a new pair can be feasibly inserted in the route, there is still a small chance that the wrong decision is made. In other words, a pair may be removed from the route although it could have been feasibly inserted there, had the cost of all possible insertion positions been calculated and the best position identified, as done in ‘traditional’ construction methods.
- 149. 7.5 Solution ConstructionHeuristics 125 Algorithm 7.2: The Sequential Construction. 1: Let M ← 0 {M is the number of vehicles used} 2: repeat 3: Initialize an empty route r 4: M = M + 1 5: for (All unassigned requests) do 6: Get the next unassigned request i 7: Insert request i at the end of the current route r 8: Call the HC routing heuristic (Algorithm 7.1) to improve r 9: if (r is a feasible route) then 10: Mark i as inserted 11: else 12: Remove i from r 13: until (All requests have been inserted) 7.5.2 The Parallel Construction Algorithms As mentioned previously, for a parallel construction, several routes are considered simul- taneously for inserting a new request, and an initial estimate of the number of vehicles is required. Potvin and Rousseau in their parallel construction algorithm for solving the VRPTW [121], estimate the initial number of vehicles by first running Solomon’s sequen- tial construction [141]. The number of vehicles in the resulting solution is then used as an estimate of the initial number of vehicles for the parallel heuristic. In our research, we adapted the parallel construction heuristic of the VRPTW in [121] to the MV-PDPTW. However, to avoid extra processing time, we estimated the initial number of vehicles using a simple formula that divides the total demand of the pickup requests in the problem instance by the capacity of the vehicle, as shown in Equation 7.2. M = ⌊( i∈N+ qi)/C⌋ , (7.2) where M is the estimated initial number of vehicles, N+ is the set of pickup customers, qi is the demand (load) of a pickup request, and C is the capacity of the vehicle. However, this estimate seems to be more suitable for instances with a critical schedule horizon, i.e., those having a short time window width. Instances with more flexible (long) time window intervals, on the other hand, may require fewer vehicles to start with. More specifically, when the width of the time window interval is short, there will be a limited number of alternative feasible locations for scheduling the request. Accordingly, instances with such
- 150. 126 7.5 SolutionConstruction Heuristics a property usually need a large number of vehicles to be able to serve all requests without any constraint violation, as opposed to instances having a long time window width. Thus, to be able to handle the different types of test cases, we introduced a small modification to this formula for some problem instances, as will be explained in Section 7.6. Similar to the parallel approach for the VRPTW in [121], which initializes each route with a seed customer, our parallel algorithms initialize each route with a seed request (pickup and delivery pair) from the sorted list of requests. We then take the remaining requests in order and attempt to insert the next request in one of the partial routes created. If the next request cannot be feasibly inserted in any of the already created routes, a new route is added to accommodate this request. This process is repeated until all requests have been inserted. As previously explained, traditional parallel construction algorithms for both the VRPTW and the PDPTW, usually select the customer who has the current minimum insertion cost among all remaining un-routed customers to be inserted next. This cost is often a measure of the extra travel time and distance, which would result from inserting the customer in the best possible (feasible and minimum cost) insertion position found in all available routes. Our parallel algorithms, on the other hand, differ among each other in how they select the next request to be inserted, and also in selecting the route in which this request will be inserted. Following is an explanation of the different parallel construction algorithms proposed in our research. Parallel Construction - First Route: In our first parallel construction algorithm, the next request in order is inserted in the first route in which a feasible insertion of this request is found, i.e., no attempt is made to find the best route for the current request. Thus, our first parallel construction uses a fast first acceptance criterion for insertion. Algorithm 7.3 describes this procedure.
- 151. 7.5 Solution ConstructionHeuristics 127 Algorithm 7.3: Parallel Construction: First Route. 1: Calculate M (the initial estimate of the number of vehicles) 2: Initialize M routes with seed customer pairs from the sorted list of customers 3: for (All remaining unassigned requests) do 4: Get the next unassigned request i 5: r = 0 {start with the first route} 6: while ((r < M) and (i not yet inserted) ) do 7: Insert the request i at the end of the current route r 8: Call the HC routing heuristic (Algorithm 7.1) to improve r 9: if (r is a feasible route) then 10: Mark i as inserted 11: else 12: Remove i from r 13: r = r + 1 14: if (i was not inserted) then 15: Initialize a new route r′ 16: M = M + 1 {increase the number of vehicles} 17: Insert the request i in the new route r′ 18: Mark i as inserted Parallel Construction - Best Route: In our second parallel construction algorithm, the next request in order is inserted in the best route in which a feasible insertion of this request is found. The best route for each request is the route that causes the least increase in the overall cost of the solution (the routing schedule) due to the insertion process. To calculate the overall cost of the solution, we used an objective function that is suggested by Bent and Hentenryck in [12]. The objective function consists of three components: the first component tries to minimize the number of vehicles used in the solution, the second component tries to minimize the total distance traveled, while the third component is a measure that tries to maximize the square of the number of nodes visited by each vehicle. This last component is intended to favour routes that are rather full and those that are rather empty, as opposed to an even distribution of nodes among routes. The idea is to try to get rid of some vehicles that are under-utilized during subsequent route improvement phases. The objective function of a solution S is described by Equation 7.3. O(S) = α × M + β × r∈S Dist(r) − γ × r∈S |r|2 , (7.3)
- 152. 128 7.5 SolutionConstruction Heuristics where M is the number of vehicles used in the current solution, Dist(r) is the total dis- tance traveled by each vehicle, and |r| is the number of nodes visited by each vehicle. The constants α, β, and γ are weights in the range [0, 1] assigned to each term in the objective function, and α + β + γ = 1.0 . In our research we try to minimizing the number of ve- hicles as our primary objective followed by the total distance, thus we chose α > β > γ. We used the following weight values for Equation 7.3: α = 0.7, β = 0.29 and γ = 0.01. Algorithm 7.4 describes the second parallel construction algorithm. It is important to note, in Step 10 of Algorithm 7.4, that since the insertion process only affects one route, the calculation of the new solution cost does not require evaluating all routes in the current solution. The calculation is simply done by removing the old cost of the current route (before the insertion), and adding the new cost resulting from the insertion. Algorithm 7.4: Parallel Construction: Best Route. 1: Calculate M (the initial estimate of the number of vehicles) 2: Initialize M routes with seed customer pairs from the sorted list of customers 3: for (All remaining unassigned requests) do 4: Initialize LocalMin to an arbitrary large value 5: for (r = 0; r < M; r + +) do 6: Get the next unassigned request i 7: Insert the request i at the end of the current route r 8: Call the HC routing heuristic (Algorithm 7.1) to improve r 9: if (r is a feasible route) then 10: calculate ∆cost {∆cost is the change in solution cost due to the insertion (where cost is estimated using Equation 7.3) } 11: if (∆cost < LocalMin) then 12: LocalMin = ∆cost 13: r∗ = r { r∗ is the current best vehicle for request i} 14: Remove i from r {temporarily remove i until the insertion cost of the current request in all routes is calculated} 15: if (r∗ is found) then 16: Insert i in r∗ 17: Mark i as inserted 18: else 19: Initialize a new route r′ {Since no feasible insertion is found for i in any of the available routes, allocate a new route} 20: M = M + 1 {increase the number of vehicles} 21: Insert the request i in the new route r′ 22: Mark i as inserted
- 153. 7.5 Solution ConstructionHeuristics 129 Parallel Construction - Best Request: Our next parallel construction heuristic does not only try to find the best route for each request, but also selects the best un-routed request to be inserted next. The best un- routed request is the one whose insertion (in its best route) causes the least increase in the overall cost of the solution, where the solution cost is again evaluated using Equation 7.3. Algorithm 7.5 describes this procedure. Algorithm 7.5: Parallel Construction: Best Request. 1: Calculate M (the initial estimate of the number of vehicles) 2: Initialize M routes with seed customer pairs from the sorted list of customers 3: repeat 4: Initialize GlobalMin to an arbitrary large value 5: for (All remaining unassigned requests) do 6: Initialize LocalMin to an arbitrary large value 7: for (r = 0; r < M; r + +) do 8: Get the next unassigned request i 9: Insert the request i at the end of the current route r 10: Call the HC routing heuristic (Algorithm 7.1) to improve r 11: if (r is a feasible route) then 12: calculate ∆cost 13: if (∆cost < LocalMin) then 14: LocalMin = ∆cost 15: r∗ = r { r∗ is the current best route for request i} 16: Remove i from r 17: if (r∗ is found) then 18: if (LocalMin < GlobalMin) then 19: GlobalMin = LocalMin 20: i∗ = i {i∗ is the current best request} 21: v∗ = r∗ {v∗ is the best vehicle (route) for i∗} 22: else 23: Initialize a new route r′ 24: M = M + 1 25: Insert i in the new route r′ 26: Mark i as inserted 27: if (i∗ is found) then 28: Insert i∗ in v∗ 29: Mark i∗ as inserted 30: until (All requests have been inserted)
- 154. 130 7.6 ComputationalExperimentation 7.6 Computational Experimentation 7.6.1 Characteristics of the Data Set To test our algorithms, we used several instances from the benchmark data of the MV- PDPTW created by Li and Lim in [100]. The authors of [100] created this data set based on Solomon’s test cases of the VRPTW in [141]. There are 6 different categories of problem instances in this data set: LR1, LR2, LC1, LC2, LRC1, and LRC2. Problems in the LR category have randomly distributed customers, problems in the LC category have clustered customers, and problems in the LRC category have partially random and partially clustered customers. On the other hand, problems identified with the number ‘1’ have a short scheduling horizon (tight time window width), while problems iden- tified with the number ‘2’ have a long scheduling horizon (large time window width). Each category has 6 different problem sizes: 100, 200, 400, 600, 800, and 1000 cus- tomers4 . There are between 56-60 files from each problem size. The total number of files in the data set is 354. The data and the best known results can be downloaded from http://www.sintef.no/Projectweb/TOP/Problems/PDPTW/Li--Lim-benchmark/. For the purpose of testing our algorithms we selected the first 6 files from each category for each problem size. The total number of files used to test our algorithms is 216. The files used for testing our algorithms are summarized in Table 7.1. As mentioned in Section 7.5.2, we used a simple formula (Equation 7.2) to estimate the initial number of vehicles needed for the parallel construction heuristics. However, during our preliminary experimentation, we found that this estimate does not suit the different types of problem instances. Apparently, problems that have a long schedule horizon al- low for a more flexible visiting schedule, and generally require fewer vehicles. We also found during our experimentation that an underestimate of the initial number of vehicles is usually preferred to an overestimate, since reducing the total number of vehicles used is our primary concern. As a result, to estimate the initial number of vehicles for problems with a long schedule horizon (problems of category ‘2’) we reduced our initial estimate by 50%. Thus, Equation 7.4 was used instead of Equation 7.2. M = ⌊(1/2)( i∈N+ qi)/C⌋ . (7.4) However, this estimate is to some extent arbitrary and remains under consideration for future reassessment. 4 The original data set in [100] contained only 56 100-customers problems. Larger problem sizes were later added to the original data set.
- 155. 7.6 Computational Experimentation131 Table 7.1: Test Files. Category 100 customers 200 customers 400 customers LC1 LC101 to LC106 LC1-2-1 to LC1-2-6 LC1-4-1 to LC1-4-6 LC2 LC201 to LC206 LC2-2-1 to LC2-2-6 LC2-4-1 to LC2-4-6 LR1 LR101 to LR106 LR1-2-1 to LR1-2-6 LR1-4-1 to LR1-4-6 LR2 LR201 to LR206 LR2-2-1 to LR2-2-6 LR2-4-1 to LR2-4-6 LRC1 LRC101 to LRC106 LRC1-2-1 to LRC1-2-6 LRC1-4-1 to LRC1-4-6 LRC2 LRC201 to LRC206 LRC2-2-1 to LRC2-2-6 LRC2-4-1 to LRC2-4-6 Category 600 customers 800 customers 1000 customers LC1 LC1-6-1 to LC1-6-6 LC1-8-1to LC1-8-6 LC1-10-1 to LC1-10-6 LC2 LC2-6-1 to LC2-6-6 LC2-8-1to LC2-8-6 LC2-10-1 to LC2-10-6 LR1 LR1-6-1 to LR1-6-6 LR1-8-1 to LR1-8-6 LR1-10-1 to LR1-10-6 LR2 LR2-6-1 to LR2-6-6 LR2-8-1 to LR2-8-6 LR2-10-1 to LR2-10-6 LRC1 LRC1-6-1 to LRC1-6-6 LRC1-8-1 to LRC1-8-6 LRC1-10-1 to LRC1-10-6 LRC2 LRC2-6-1 to LRC2-6-6 LRC2-8-1 to LRC2-8-6 LRC2-10-1 to LRC2-10-6 7.6.2 Comparing the Construction Heuristics Throughout this discussion, we use the following notations to refer to each algorithm 1. Sequential Construction: SEQ 2. Parallel Construction - First Route: PFR 3. Parallel Construction - Best Route: PBR 4. Parallel Construction - Best Request: PBQ Since the construction algorithms are all deterministic, each algorithm was run only once on each test file. Table 7.2 shows the percentage of time each algorithm produced the smallest number of vehicles (Min-Vehic), and the smallest total distance (Min-Dist), that are found in the current experiment, over all 216 problem instances5 . Table 7.3 shows the average number of vehicles, the average total distance, and the average processing time (in seconds), produced by each algorithm for each problem size separately. 5 Some ties are produced and counted in the results.
- 156. 132 7.6 ComputationalExperimentation Table 7.2: Frequency of Generated Best Solutions . Algorithm Min-Vehic Min-Dist SEQ 48% 31% PFR 24% 6% PBR 19% 8% PBQ 49% 56% The following observations can be realized from Tables 7.2 and 7.3: • Regarding the number of vehicles generated, SEQ and PBQ produced the best re- sults, with SEQ producing better results than PBQ in large size problems, while both PFR and PBR were slightly inferior in this respect. • Regarding the total distance traveled, PBQ was able to beat all other algorithms, followed by PBR and SEQ. • PFR produced the worst average distance in all test cases, but it was slightly better than PBR in the average number of vehicles used. • SEQ and PFR have comparable average processing time in all test cases. Their processing time on average is faster than the other two algorithms, with PBQ being the slowest among all. In summary, the results in Tables 7.2 and 7.3 suggest that PFR and PBR are inferior to SEQ and PBQ, both in terms of the number of vehicles used and the total distance traveled6 . As a result, PFR and PBR can be eliminated from further consideration, and we can focus our attention on SEQ and PBQ. As can be noticed from the average results in the last row of Table 7.3, SEQ produced better results than PBQ in the number of vehicles used. The PBQ algorithm, however, was able to beat the SEQ algorithm in minimizing the total distance traveled. This was obviously due to the fact that the SEQ algorithm was more concerned with fitting the largest possible number of requests in each vehicle before allocating a new one, while the PBQ algorithm relied on a cost function that has the total travel distance among its components. The PBQ algorithm was, nevertheless, much slower than the SEQ algorithm. 6 However, PBR was able to slightly improve upon PFR with respect to the total distance traveled, while PFR was slightly better than PBR in the number of vehicles used.
- 157. 7.6ComputationalExperimentation133 Table 7.3: AverageResults for all Algorithms. Problem Size SEQ PFR PBR PBQ Vehic Dist Time Vehic Dist Time Vehic Dist Time Vehic Dist Time 100-customers 11.78 2662.92 0.02 11.83 2767.19 0.02 11.83 2711.89 0.03 11.69 2564.09 0.34 200-customers 17.33 8887.08 0.08 17.69 8954.06 0.08 18.17 8816.33 0.1 17.14 8132.84 3.62 400-customers 33.56 22215.14 0.32 34.64 23010.53 0.3 34.69 21898.96 0.38 33.72 19758.38 26.93 600-customers 48.22 44949.4 0.72 49.89 46644.49 0.69 50.69 45234.96 0.86 49.53 41791.82 147.57 800-customers 63.53 74650.07 1.24 65.94 77895.32 1.23 66.44 74056.45 1.65 64.89 68713.31 438.97 1000-customers 77.25 108513.19 1.88 81.97 115106.93 1.93 81.75 108662.01 2.54 81.58 103751.31 952.34 Average 41.95 43646.30 0.71 43.66 45729.75 0.71 43.93 43563.43 0.92 43.09 40785.29 261.62
- 158. 134 7.6 ComputationalExperimentation The average processing time of the SEQ algorithm ranged from 0.02 seconds for 100- customers problems to 1.88 seconds for 1000-customers problems. The PBQ algorithm, on the other hand, had a processing time ranging from 0.34 seconds to 952.34 seconds for the same problem types, which indicates beyond doubt the huge difference in the computational effort needed for both algorithms. It should also be noted that the SEQ algorithm neither requires an initial estimate of the number of vehicles, nor does it need a solution evaluation mechanism during the construc- tion process (the SEQ is unlike the PBR and the PBQ algorithms for instance, in which the cost of the whole solution must be calculated using Equation 7.3 at each step of the construction process). The only advantage that the PBQ algorithm offers, which is a slight reduction in the total travel distance, does not seem to justify its added cost in terms of the complexity of the algorithm and the increase in processing time. Another advantage of the SEQ algorithm is that it can be easily adapted to population-based heuristics or meta-heuristics by randomizing the initial order of requests to generate different diverse solutions. The PBQ algorithm, on the other hand, is expected to produce a limited diver- sity, even if the initial order of requests is randomized, because of the selection criteria and the cost function it relies on during the insertion process. Most likely, requests that are hard to insert, and thus cause a large increase in the solution cost, will always remain the same, despite the change in the insertion order. 7.6.3 Comparing with Previous Best Known Although our algorithms are primarily intended for constructing initial solutions to the MV-PDPTW, it would still be useful to compare our results with the best known solutions. This would give us a general idea about the expected effort in the solution improvement phase. Table 7.4 shows the relative gap (in percentage) between the average results produced by both the SEQ algorithm and the PBQ algorithm and the average best known results. We used the following formula to calculate the relative gap gap = ((Result − BestKnown)/BestKnown) × 100 (7.5) The relative gap is measured with respect to both the number of vehicles and the distance7 . The table shows that the SEQ algorithm produced, on average, a slightly smaller gap with respect to the number of vehicles, and a slightly larger gap with respect to the total 7 For example, a gap of 50% in the average number of vehicles means that the result of the construction heuristic produced 50% more vehicles than the best known result.
- 159. 7.6 Computational Experimentation135 distance. Together with the fact that the SEQ algorithm is quite simple and fast compared to the PBQ algorithm, the results in Table 7.4 would again seem to justify its preference as a solution construction method over the PBQ algorithm. Table 7.4: Average Relative Distance to Best Known. Problem Size Vehic-Gap Dist-Gap SEQ PBQ SEQ PBQ 100-customers 58% 57% 146% 137% 200-customers 63% 61% 187% 163% 400-customers 66% 67% 207% 173% 600-customers 64% 69% 206% 185% 800-customers 66% 70% 205% 181% 1000-customers 65% 74% 191% 178% Average 64% 66% 191% 170% It may also be beneficial to analyze the results produced by the construction heuristics for each benchmark category separately. This may give an insight into what problem types would require more effort in the solution improvement phase. Figure 7.3 shows the ave- rage gap produced by the SEQ algorithm for all tasks, organized by problem categories. Figure 7.4 shows the average gap produced by the same algorithm with respect to the distance traveled. Both figures show that the SEQ construction heuristic seems to be more ‘successful’ in instances with a short schedule horizon, i.e., instances identified with ‘1’ in the data set, since these instances always have a smaller gap than instances of type ‘2’. Regarding the primary objective, which is the number of vehicles used, the algorithm seems to do a better job for instances that have clustered customers, as opposed to instances that have random or partially random customers. It is clear that instances in the LC category always have the smallest gap compared to other problem types. Problems with random customers and a long time window interval appear to be the most challenging for the SEQ algorithm, and possibly all solution algorithms. The reason could be that the solution space for these problems seems to be larger, due to the randomness of locations and the large width of time windows involved in this case. It also appears from both graphs that the gap in the number of vehicles is inversely proportional to the gap in the total travel distance, in most test cases. It is also worth mentioning that the results in Table 7.4 and Figures 7.3 and 7.4 indicate
- 160. 136 7.6 ComputationalExperimentation 100 200 400 600 800 1000 0 50 100 150 200 250 Problem Size AverageVehicleGap% LR1 LC1 LRC1 LR2 LC2 LRC2 Figure 7.3: SEQ algorithm - average vehicle gap for all problem categories. 100 200 400 600 800 1000 0 50 100 150 200 250 300 350 400 450 500 Problem Size AverageDistanceGap% LR1 LC1 LRC1 LR2 LC2 LRC2 Figure 7.4: SEQ algorithm - average distance gap for all problem categories. that a lot of work still needs to be done in the improvement phase, in order to reach the anticipated standard for the final problem solutions. This is evident by the relatively large gap between the current initial solutions and the final best known results. Designing an ‘intelligent’ improvement phase seems to be inevitable, in order to cope with the difficult problem constraints and the various types of problem instances. Finally, Table 7.5 shows the average processing time of the SEQ algorithm, for each
- 161. 7.7 The SEQAlgorithm: Complexity Analysis and Implementation Issues 137 problem size in each benchmark category. It appears in this table that problems involving random customers with a long schedule horizon, LR2 and LRC2, generally require a longer processing time than the other problem categories, which sustains our previous observation regarding the large solution space for these problems. Table 7.5: Average Processing Time (seconds) of the SEQ Algorithm for all Tasks. Category Problem Size 100 200 400 600 800 1000 LR1 0.02 0.06 0.22 0.53 0.95 1.54 LC1 0.02 0.05 0.22 0.48 0.92 1.43 LRC1 0.02 0.06 0.21 0.50 0.93 1.44 LR2 0.03 0.13 0.43 1.04 1.71 2.72 LC2 0.03 0.06 0.39 0.61 1.06 1.57 LRC2 0.02 0.12 0.47 1.12 1.84 2.53 7.7 The SEQ Algorithm: Complexity Analysis and Im- plementation Issues Before we conclude this part of our research, we present in this section some remarks concerning the complexity and feasibility checking of the SEQ algorithm in relation to the common construction methods. Analyzing our SEQ algorithm we find that: the routing heuristic in Algorithm 7.1 needs O(n2 ) time for accessing each pair of locations in the route, where n is the number of requests in the problem instance. Also, the cost function of Equation 7.1, which checks the feasibility of the whole route as well, needs O(n) time. The SEQ algorithm (Algorithm 7.2) needs O(n) since its major iteration processes all requests in order. This will make the run-time complexity of the whole algorithm O(n4 ). For the sake of comparison, Algorithm 7.6 describes a basic construction method for vehicle routing problems in general. The algorithm appears in [24].
- 162. 138 7.7 TheSEQ Algorithm: Complexity Analysis and Implementation Issues Algorithm 7.6: Basic Construction Algorithm for the VRP [24]. 1: N = set of unassigned customers 2: R = set of routes {initially contains one route} 3: while N = ∅ do 4: p∗ = −∞ 5: for j ∈ N do 6: for r ∈ R do 7: for (i − 1, i) ∈ r do 8: if (Feasible(i, j) and Profit(i, j) > p∗) then 9: r∗ = r 10: i∗ = i 11: j∗ = j 12: p∗ = Profit(i, j) 13: Insert(i∗, j∗) {insert j∗ between (i∗ − 1) and i∗} 14: N = N j∗ 15: Update(r∗) According to [24], Algorithm 7.6 is of O(n3 ), provided that the feasibility and profitability can be performed in constant time. Feasibility makes sure that the current position adheres to all problem constraints, while Profitability is usually measured as a weighted combination of extra travel distance and time delay resulting from the insertion. When it is more profitable to insert a customer in a new route, a new route will be allocated. As explained in [24], a special algorithm can be applied to reduce the TW feasibility test of the VRPTW from a linear time to a constant time. The same algorithm can also be applied to the PDPTW. Nevertheless, this algorithm requires that extra information is kept for each customer already existing in the route. In general, two quantities have to be maintained: the earliest and the latest possible times the service can take place, relative to the customer’s current location in the route. This information is not fixed and subject to change after each insertion, which accounts for the existence of an update function to maintain the desired quantities (Step 15 of Algorithm 7.6). Unlike the VRPTW, however, checking the capacity feasibility for the PDPTW can only be done in linear time [86], due to the presence of two different types of customer services in the route. As a result, the basic construction algorithm when applied to the PDPTW also results in O(n4 ) complexity. The algorithm can also include a selection of seed customers for route initialization, which usually does not change the complexity of the algorithm. As mentioned above, it is possible to reduce the TW feasibility check for the PDPTW from a linear time to a constant time, by maintaining and frequently updating extra route
- 163. 7.8 Summary andFuture Work 139 information, as done for example in [37]. However, since our route cost function (Equa- tion 7.1) tests the feasibility of both the TW and the capacity concurrently, it would be redundant to calculate and store additional service timing information to accelerate the TW feasibility test, since an O(n) testing would still be needed for the capacity feasibi- lity. In addition, since our SEQ algorithm accepts any feasible insertion, it does not have to check the feasibility nor estimate the profitability of each and every possible insertion position, as done in Algorithm 7.6. In our algorithm, the cost of the route as a whole will be calculated, if at all, only if the route has been changed. This is due to the restriction imposed by the TW condition in Step 4 of Algorithm 7.1. Finally, during the insertion process, i.e., Step 7 of the SEQ algorithm (Algorithm 7.2), two locations (a pickup and delivery pair) are simultaneously inserted, then Algorithm 7.1 handles the feasibility checking and the improvement of the underlying route altogether. Besides overcoming the precedence and the coupling issues, this insertion has the added advantage of accelerating the solution construction process, since only half the number of locations is processed in the main iteration of Algorithm 7.2. On the other hand, the parallel construction algorithms implemented in this research seem to be one order of magnitude higher than the SEQ algorithm, due to the presence of an extra loop that passes through all available vehicles, although the number of vehicles is always less than n (the number of nodes in the data set). 7.8 Summary and Future Work In this research we investigated several initial solution construction heuristics for the MV- PDPTW, aiming to identify the best heuristic that can be used as part of a comprehensive solution methodology. In our opinion, existing approaches in the literature often overlook and perhaps underestimate this vital component of the overall solution algorithm. The experimental results on a large number of benchmark instances indicate that the se- quential construction heuristic (SEQ) seems to be the most favourable solution construc- tion method, which can be by easily embedded in a heuristic or a meta-heuristic technique to reach final good quality solutions. With just a few simple lines of code, and without a pre-determined number of vehicles or a solution evaluation mechanism, this algorithm produced good quality results, that are sometimes even better than the results obtained by the most sophisticated parallel algorithm tested in our research (the PBQ algorithm). The SEQ algorithm also had an impressive speed, with a processing time that is at most 6% of
- 164. 140 7.8 Summaryand Future Work the time needed by the PBQ algorithm, making it even more suitable for population-based solution algorithms. The experimental results, nevertheless, show that probably a costly improvement phase is still needed to achieve final good quality solutions, as evident by the relatively large gap to best known results produced by the SEQ construction algorithm. This, however, further supports the need for a fast solution construction method to achieve an overall reasonable computation time for the complete solution algorithm. The construction algorithms developed in this part of our research are distinguished by their simplicity and ease in coding and replication, compared to many construction me- thods that are adopted from the VRPTW literature. All of our algorithms are general por- table frameworks that can be used within other heuristics and meta-heuristics that solve the PDPTW and its related variants. In the next part of our research, we will start investigating the solution improvement phase for the MV-PDPTW, using the SEQ algorithm for the solution construction phase. Details of our investigation are presented in the next chapter.
- 165. 141 Chapter 8 Two Approachesfor Solving the Multiple Vehicle Pickup and Delivery Problem with Time Windows: A Genetic Algorithm and a Simulated Annealing In this chapter we continue our investigation of the MV-PDPTW, introduced in Chapter 7. This part of our research will augment the solution construction heuristic developed in the previous chapter with an improvement method, using both a Genetic Algorithm (GA) approach and a Simulated Annealing (SA) heuristic. This chapter is organized as follows: Section 8.1 highlights the motivation behind the re- search carried out in this part of the thesis. Section 8.2 provides a brief summary of some related work from the literature. Section 8.3 discusses genetic algorithms as a solution approach form the perspective of the MV-PDPTW, and introduces our specific GA sug- gested for the improvement phase of this problem. Within this section, we describe the solution representation and the objective function used in Section 8.3.1, how the initial population is created in Section 8.3.2, the genetic operators in Section 8.3.3, the overall GA in Section 8.3.4, some attempts to improve the results of the GA in Section 8.3.5, and finally Section 8.3.6 reports the experimental results of the GA when tested on published benchmark data and compared with previous GA attempts from the literature. After that, Section 8.4 explains the SA approach adopted for solving the problem, together with its experimental results compared to the results of the GA approach. Finally, Section 8.5 gives some concluding remarks and possible future work. Our findings, related to the GA approach, in this part of our research were published in the MIC2009 conference [80].
- 166. 142 8.2 RelatedWork 8.1 Research Motivation As previously mentioned in Chapter 2, Genetic Algorithms (GAs) are intelligent search methods that have been successfully used for solving many hard combinatorial optimiza- tion problems. In the first part of our research, though, the 3-stage SA approach performed better than the GA for the SV-PDPTW, especially in terms of processing time. Despite this, using a GA for the improvement phase of the MV-PDPTW still appeared to be pro- mising for several reasons: Firstly, GAs have been used for solving the related VRPTW, producing good results in many cases, for example [144], [120], [16] and [15]. Secondly, the parallel nature of GAs and its population based mechanism can probably make it more appropriate than other meta-heuristic approaches (that focus on improving only one solution), for solving very hard optimization problems. The multiple-vehicle case of the PDPTW is in fact considerably harder than the single vehicle case [137], which makes applying GAs an attractive option for solving this problem. In addition, our simple sequential construction algorithm (SEQ), developed in Chapter 7, seems to be most appropriate for population- based meta-heuristics, due to its simplicity, speed and its potential for creating a diverse population by randomizing the initial order of requests. Finally, as previously mentioned in the introduction to this thesis (Chapter 1), the main focus of our research is on developing appropriate representations and neighbourhood moves that can help guide the search towards good quality solutions and manage infea- sibility throughout the search. If the appropriate techniques are designed, they should be widely applicable, and may be used within other heuristics or meta-heuristics, for solving the underlying problem. Accordingly, we tried to follow the same approach used in our research for solving the SV-PDPTW, by first developing the representation and neigh- bourhood moves (represented as genetic operators here) that we believe can handle the difficult problem constraints. These operators are first tried within a GA approach for sol- ving the MV-PDPTW, but they were also adapted and employed within an SA approach, as previously done for the single vehicle case. 8.2 Related Work When dealing with the MV-PDPTW, some researchers use exact methods to solve the problem to optimality, but these are limited to small size problems. For example, the work presented in [46] is an optimization techniques which formulates the problem as a set partitioning problem, and then employs a column generation method to solve a linear
- 167. 8.2 Related Work143 relaxation of this problem to optimality. Problem sizes of up to 55 customers and 22 ve- hicles were solved using this approach. Also, the work in [130] presents a new mixed integer programming formulation for both the MV-PDPTW and the dial-a-ride problem, and the formulations are solved using two branch-and-cut algorithms. New problem ins- tances have been created, and the results of the algorithms were compared with upper bound solutions obtained by applying the heuristic in [131] to the same problem instances. Problem sizes up to 194 nodes were solved to optimality. Other approaches adopt approximation techniques to deal with large size problems. As ex- plained in the previous chapters, the solution process, in this case, often starts by construc- ting one or more initial solutions to the problem, usually using techniques adopted from the VRPTW construction algorithms, and then these solutions are improved using heuris- tics or meta-heuristics. In general, Simulated Annealing (SA) and Tabu Search (TS) have been the most popular approaches for solving the MV-PDPTW. The work in [112] is one of the first attempts to solve the MV-PDPTW. The technique is based on a reactive tabu search which allows the tuning of the search parameters, such as a short-term memory length, based on an assessment of visited solutions during the search. The algorithm also tries to detect and escape possible local optima. The solution repre- sentation is a vector that includes customer nodes separated by vehicle nodes to which customers are assigned. The objective function consists of three components: the total schedule duration, the number of capacity violations, and the number of time window violations. Thus, during the search process, overloads and tardiness in the solutions are treated as soft constraints, while precedence and coupling constraints are strictly enfor- ced. To construct an initial feasible solution, a predecessor-successor pair (PS) is inserted in the best feasible insertion position in the current vehicle’s route. If no feasible insertion can be found, a new vehicle route is added to the solution. Three neighbourhood moves are used in this algorithm. The first move is the “Single Paired Insertion” (SPI), which tries to move all predecessor nodes to better feasible locations in other routes. Successor nodes are inserted after their predecessors in the best possible positions, although vehicle capacity or time windows may still be violated. The second move is the “Swapping Pairs Between Routes” (SBR), which first tries to exchange a pair of predecessor nodes bet- ween two different vehicles, and then exchanges their successor nodes. The last move is the “Within Route Insertion” (WRI), which tries to reorder nodes in the same route to re- duce infeasibility in the solution and improve its quality. The sequence of neighbourhood moves is selected dynamically during the search, based on a calculated relationship bet- ween the average time window length (the tightness of the time window) and the average route duration.
- 168. 144 8.2 RelatedWork The authors in [100] present a tabu-embedded simulated annealing approach to solve the MV-PDPTW. To construct an initial feasible solution, they modified Solomon’s Insertion heuristic [141] by initializing each route with a pickup and delivery (PD) pair that satisfies a set of criteria, based on combined time window intervals and distance from the depot. Each un-routed PD pair is then inserted into the partial route in the best possible feasible and minimal cost insertion position. The objective function tries to minimize, in priority order, the number of vehicles used, the total travel distance, the total service duration, and the drivers’ total waiting time. To create a neighbouring solution, three local search methods are presented. The first is a “PD-Shift” operator. In this operator, a PD pair is first removed from route 1 and then it is inserted in route 2. After that, another PD pair is removed from route 2 to be inserted in route 1. The second operator is a “PD-Swap” operator, which simultaneously removes a PD pair from each route, and then reinserts each pair in the other route. The third is a “PD-Rearrange” operator which first removes and then reinserts a PD pair in the same route. In all three moves, only feasible insertions are allowed. For a more thorough neighbourhood search, they extend their local search method to a descent local search (DLS), which tries to improve the current solution for a number of iterations. When no further improvement is possible, the DLS algorithm returns the best current solution. The main meta-heuristic algorithm is a tabu-embedded simulated annealing procedure with K-restarts, i.e., the algorithm stops when the number of iterations without improvement reaches a pre-defined value K. To prevent cycling, their simulated annealing procedure records the accepted solutions in a tabu list. The authors generated test data for the problem based on Solomon’s test cases for the VRPTW [141]. The authors consider their approach as the first efficient attempt to solve practical size MV-PDPTW instances. Their generated data set also became the standard benchmark test data for the MV-PDPTW. The work in [98] presents a two-phase method to solve the MV-PDPTW. In the first phase the algorithm creates an initial feasible solution using a strategy that combines the benefits of Solomon’s classical insertion heuristic [141], and a sweep heuristic, first suggested in [58], which was particularly adapted to fit the MV-PDPTW. The authors call their insertion heuristic “Partitioned Insertion Heuristic”. The objective functions is based on the number of vehicles used and the total travel distance. Their local search is based on three moves, similar to the moves described in [112], as explained above. The second phase of the algorithm is a tabu search that tries to improve the current initial solution using a set of composite neighbourhood moves. In a relatively recent work [12], the authors present a two-stage algorithm to deal with the MV-PDPTW. The first stage uses a simple SA approach, whose focus is to minimize the number of vehicles used in the solution. The second stage uses a Large Neighbourhood
- 169. 8.2 Related Work145 Search (LNS) strategy, first suggested in [139], to minimize the total travel cost of the entire solution. The primary objective function (used in the LNS stage) minimizes the number of vehicles first, then the total travel cost. However, according to the authors, using this objective function is more effective in reducing the total travel cost than mini- mizing the number of routes in the solution. Therefore, they added an SA phase to boost the performance of the overall algorithm by reducing the number of routes in the solution first. To this end, they used a different objective function in the SA stage than the one used by the LNS. The SA objective function included three components: the first com- ponent tries to minimize the number of routes. The second component tries to maximize the number of routes that are rather full and those that are rather empty, with a view to eli- minating routes with few customers during the search. Finally, the third component of the SA evaluation function reduces the total travel cost1 . The neighbourhood move adopted in their SA is a simple pickup and delivery pair relocation operator, which was also used by previous researchers (e.g. [98], [100] and [112]). The LNS of their algorithm is ac- tually a sequence of local searches, where a group of customers are selected for relocation based on a certain relatedness measure between pickup customers. The neighbourhood generated after the relocation of customers is then exhaustively explored, using a branch and bound algorithm, to find its best solution. The best solution found replaces the current solution, if it has a better objective value. The algorithm was tested on benchmark data obtained from [100], where the results show the effectiveness of the approach since it was able to produce many new best solutions for instances with 100, 200, and 600 customers. An interesting solution methodology is presented in [131], where a variant of the MV- PDPTW is considered. It is assumed here that the starting and ending depots of a vehicle’s journey need not be the same, and the depots could be different for different vehicles. It is also assumed that the number of vehicles is limited, so it may not be possible to serve all requests, in which case the unserved requests are placed in a request bank. Thus, mi- nimizing the number of unserved requests in the request bank is one of the objectives of the algorithm, added to minimizing the number of vehicles used and the total travel dis- tance. It is also assumed that there could be some special requests that cannot be served by some vehicles, for example if transferring the request requires a special characteris- tic that is not available in some vehicles (e.g. transferring frozen food or some medical samples). The basic solution methodology in this work is a Large Neighbourhood Search (LNS) technique, used in both [139] and [12], where a number of requests are first remo- ved then re-inserted into the solution. Unlike previous researchers, however, the authors apply a number of different heuristics for both the removal and the insertion methods. 1 Recall that this 3-component evaluation function is the same as the cost function used by our PBR and PBQ construction heuristics (Equation 7.3), as explained in Section 7.5.2
- 170. 146 8.2 RelatedWork They present three different removal heuristics. First, the “Shaw Removal” removes re- quests according to a certain relatedness measure, which is based on differences between distances, arrival times, loads and the numbers of vehicles that can serve these requests. Second, the “Random Removal” heuristic removes randomly selected requests. Finally, the “Worst Removal” heuristic removes requests with high costs, i.e., requests that appear to be placed in unfavourable positions in the solution. They also present two different insertion techniques to insert the removed requests in the partial routes. The basic greedy heuristic inserts the request with minimum insertion cost in the best possible insertion positions. On the other hand, the regret insertion heuristic gives priority for insertion to requests whose insertion would seem to be more costly if it was delayed2 . To select which heuristic to use, the authors assign weights to the different heuristics and use a roulette wheel selection method. These weights are adaptively adjusted during the search based on the performance of the different heuristics employed. Accordingly, the authors call their algorithm an Adaptive Large Neighbourhood Search (ALNS), as opposed to the basic LNS search method which only applies one removal and one insertion heuristic. During the search, a newly generated solution replaces the current solution using an SA acceptance criterion. Also, similar to [12], the authors first apply a preliminary stage to minimize the number of vehicles used in the solution before applying the basic ALNS. During this stage, they create an initial solution using a sequential insertion heuristic. They then try to remove one or more routs from this solution, using the LNS algorithm, until the minimum possible number of routes is reached. The algorithm was tested on the benchmark data of [100]. In addition, new problem instances were created, which takes into account their modified variant of the MV-PDPTW. When compared with the LNS, which uses only one removal and one insertion heuristic, the ALNS seems to be superior. In addition, their ALNS was able to outperform previous heuristics applied to the MV-PDPTW. To the best of our knowledge, the LNS technique of [12] and the ALNS of [131] are the current state-of-the-art, since they have both produced best known results for many benchmark problem instances. Besides SA and Tabu Search, GAs have been applied by some researchers for solving the MV-PDPTW and the related dial-a-ride problem. In the early GA approach by [92], the chromosome representation is based on assigning a four digit code to each location. The first digit of this code represents the number of the vehicle to which the location is assigned. Thus, both the pickup location and its associated delivery will be assigned the same first digit. The other three digits are used to sort locations according to the visiting order of the vehicle, and a pickup location always has a sorting code that is less than its corresponding delivery. The crossover operator works by generating two crossover points, 2 See Section 7.2 for more details about the regret measure of the parallel construction heuristics.
- 171. 8.2 Related Work147 Figure 8.1: A chromosome in a GA that handles both grouping and routing for the MV-PDPTW. and swapping the resulting segments between the two parents. The mutation operator, simply changes the first digit of a pickup and delivery pair to another digit, i.e., assigns the request to another vehicle. Another operator called vehicle merging is also used to reduce the number of vehicles. The algorithm was tested on 24 randomly generated problems with numbers of requests ranging from 5 to 30. The authors in [33] introduce an attempt to handle the MV-PDPTW using an evolutionary algorithm, for both the grouping and the routing aspects of the problem. The solution representation is a list of vehicles routes, where each route consists of a sequence of pickup and delivery locations, and all problem constraints are enforced in the solutions throughout the search. Figure 8.1 shows how a typical chromosome may look in their suggested evolutionary approach. The objective function tries to minimize the number of vehicles in the solution, the total travel distance and the total travel time. Two crossover operators are tried, one exchanges fragments of routes between parents, while the other exchanges complete routes. Infeasible solutions that may be created following crossover are repaired by removing repeated requests or adding requests that are not served. If this is not possible, the offspring is discarded. Two mutation operators are presented, the first tries to reduce the number of vehicles in a solution by selecting a route and moving all its requests to other routes in the same solution. The second mutation tries to improve a route by rearranging its requests. The work in [116] presents what seems to be a first attempt to apply a Grouping Genetic Algorithm (GGA) to the MV-PDPTW. In a GGA, the genetic representation is based on a set of genes, where each gene represents a group of objects rather than a single object. For the MV-PDPTW, it is assumed that each gene represents a group of requests that are assigned to one vehicle. Thus an individual solution only covers the grouping aspect of the problem. The routing aspect, on the other hand, is handled by an independent data
- 172. 148 8.2 RelatedWork structure associated with each gene. Figure 8.2 is a visual representation of how a chro- mosome in their GGA approach may look. The objective function is to minimize the total travel distance, irrespective of the number of vehicles used. The crossover is an adapta- tion of the original crossover mechanism for the GGA, presented in [49], where clusters (vehicles with their assigned requests) are removed from one parent and inserted into the other parent. This is then followed by a chromosome clean up, to remove duplicate ve- hicles and the repeated assignment of requests, in addition to re-assigning requests that are no longer assigned. The mutation operator removes a cluster from a chromosome, and re- assigns its requests to other clusters, creating additional clusters if necessary to maintain feasibility. The embedded insertion heuristic applied in several stages of this algorithm is based on inserting a request in the best feasible and minimum cost position, among all possible insertion positions in the chromosome 3 . In a recent work [126], another GGA was applied to the Handicapped Person Transportation problem (HPT), which focuses on minimizing clients inconvenience. Figure 8.2: A chromosome in a GA that handles only grouping for the MV-PDPTW. The authors in [91] deal with the dial-a-ride problem using a cluster-first route-second approach. The clustering is handled using a GA, and the routing is handled using a space- time nearest neighbour heuristic. In the clustering phase, the chromosome representation is a two-dimensional array where rows represent routes and columns represent customers. A cell values of 1 or 0 indicates whether or not a customer is assigned to a particular route. Crossover is performed by selecting two random routes from parent solutions and creating a new child route using a traditional uniform crossover. The remaining routes in the child are inherited from the first parent without change. The newly created child may be infeasible, though. Accordingly, a repair method is followed to assign unassigned customers or remove duplicate assignments of customers. The mutation operator just 3 More details about [33] and [116] will be given in Section 8.3.6 when our algorithm is compared with their approach.
- 173. 8.2 Related Work149 moves a random customer to another cluster (route). On the other hand, the work in [34] tries to use a GA for the whole dial-a-ride problem, i.e., for both the grouping and the routing phases. In their representation a gene consists of a vehicle-passenger pair. They used a one-point crossover and a Partial Match Crossover (PMX). For mutation they use a bit-level mutation and a 2-Opt operator. A recent paper [37] presents an indirect search method for the MV-PDPTW. The idea is to separate the meta-heuristic search strategy from the feasibility checking routines, to simplify and accelerate the optimization process. This technique facilitates the appli- cation of simple problem-independent moves during the meta-heuristic search, while a problem-specific greedy decoder handles the construction of the corresponding feasible solution using the information (encoding) given from the meta-heuristic search engine. For example, in the PDPTW, the encoding is a permutation of transportation requests, which determines the sequence in which these requests will be scheduled. Given a certain permutation, the greedy decoder creates a feasible solution (schedule) using a sequential construction algorithm and a cheapest insertion rule. To create a neighbouring solution during the meta-heuristic search, a simple 2-exchange move is applied to the current per- mutation and the new permutation is supplied again to the greedy decoder to create the corresponding new schedule. A comparison of the indirect search technique with two special-purpose algorithms for the MV-PDPTW shows that the technique is competitive in both solution quality and speed. This may indicate the potential of applying the tech- nique to other rich combinatorial optimization problems. The above literature summary shows that the MV-PDPTW is in fact a hard problem, for which most solution methods tend to be rather complex and hard to explain and repli- cate. Developing an appropriate solution technique is indeed a challenge for researchers. An intelligent solution methodology should be able to handle both the grouping and the routing aspects of the problem efficiently. In addition, in both the construction and the im- provement phases of the problem, the researcher is faced with many decisions that should be made. These decisions include, among others, the components of the objective func- tion, the permissibility of infeasible solutions, how to adhere with the constraints of the problem, how to generate new solutions, and what acceptance criteria should be applied to replace the current solution during the improvement phase. It is thus often difficult to discover good quality solutions that do not violate any problem constraint, in a reasonable processing time.
- 174. 150 8.3 AGenetic Algorithm (GA) for the MV-PDPTW 8.3 A Genetic Algorithm (GA) for the MV-PDPTW Although Genetic Algorithms (GAs) have been successfully used for solving many rou- ting and scheduling problems, research using GAs for solving the MV-PDPTW is gene- rally scarce. In addition, the results reported by most GA techniques attempted are often disappointing in some respects. As previously explained, the MV-PDPTW consists of two related problems: the grouping or the clustering problem tries to find the best allocation of requests to vehicles, while the routing problem is concerned with finding the best feasible route for each vehicle, given the requests assigned to it. When trying to solve this problem using a GA, it is often hard to tackle these two aspects simultaneously. Moreover, a major issue is finding a suitable genetic encoding and designing intelligent genetic operators that are capable of handling all the difficult problem constraints and may encourage the forma- tion of meaningful genetic building blocks, which may help in generating individuals of better fitness [116]4 . In the MV-PDPTW, the genetic operators should be smart enough to transfer the favourable genetic traits from parent solutions to their offspring, while trying to avoid the frequent generation and evaluation of infeasible problem solutions. As pre- viously mentioned, infeasible solutions are handled in many solution algorithms using a repair method to fix infeasibility during the search, which will inevitably increase the processing time and complicate the algorithm. Most previous GA research, for example [116] for the MV-PDPTW and [91] for the dial- a-ride, tried to tackle the difficulties encountered in the GA encoding and operators by allowing the GA to handle only the grouping aspect. The routing aspect, on the other hand, was handled by an independent routing algorithm that is hidden from the GA and is called when a chromosome is decoded. The genetic operators in this case are usually general-purpose and do not apply any problem-specific knowledge. Attempts to use a GA for both the grouping and the routing aspects, for example [33] for the MV-PDPTW and [34] for the dial-a-ride problem, generally produced discouraging results. We present in this part of our research our experimentation with a new GA for solving the MV-PDPTW. Our GA tries to face the challenge of handling both the routing and the grouping aspects of the problem simultaneously. Unlike the most popular approaches in the literature, in which the GA is only aware of how requests are clustered, but is not aware of how they are routed, our chromosome representation more naturally accommo- dates each group of requests together with suggested routes. By explicitly monitoring and manipulating all the solution information, we aim to preserve the distinctive cha- racteristic of GAs in identifying the desirable genetic material and transferring it from 4 For details about the “GA schema theorem” and “the building block hypothesis”, the reader is referred to [64].
- 175. 8.3 A GeneticAlgorithm (GA) for the MV-PDPTW 151 generation to generation during the evolutionary process. Our GA, thus, does not rely on a separate decoder for interpreting the chromosome contents and creating the subordinate routing information for each group of requests. Instead, the algorithm has a simple em- bedded construction heuristic that allows individual routes to dynamically change, within the chromosome itself, during the search . Also, with our solution representation in mind, we developed new simple genetic operators. Using problem-specific knowledge, such as the quality of the generated routes, these operators try to create good quality feasible so- lutions throughout the search. In addition, since no parallel repair method is needed to fix the infeasibility of solutions, the overall algorithm is simple and elegant, a feature often missing from most up-to-date solution algorithms. Besides comparing our GA with the SA approach that will be introduced in the next section, we also compare here in detail our GA approach with what seems to be the only other two GA attempts in the recent literature for solving the MV-PDPTW. We highlight, based on the experimental findings, the promising aspects of our approach, and also point out to where further improvement could be achieved. 8.3.1 The Solution Representation and the Objective Function As mentioned in the introduction to this thesis (Chapter 1), one of our goals in this re- search is to develop a solution representation that facilitates handling the difficult problem constraints. Following our approach for solving the SV-PDPTW, as explained in Chapter 5, we adopt a simple representation for each individual route. A route is simply a list of visited locations in order. However, when we assign requests to each route, both the pickup and its delivery location are given the same code. For more details, the reader is referred to Section 5.4.1. In our GA, the chromosome represents a problem solution. It is simply a collection of indi- vidual routes. Thus, each gene is actually a complete vehicle route. Both the route (gene) length and the chromosome length are variable depending on the number of requests to be visited and the number of vehicles in the solution. Thus our representation is just a problem solution upon which the genetic operators are directly applied. Our chromosome is very much like the chromosome structure depicted in Figure 8.1, but only differs in assigning the same code to both the pickup and the delivery. Thus, the code of each re- quest will appear twice in the chromosome, once showing where its pickup location will be visited and another for its delivery location. Figure 8.3 shows a typical chromosome in our GA.
- 176. 152 8.3 AGenetic Algorithm (GA) for the MV-PDPTW Figure 8.3: Our chromosome representation for the MV-PDPTW. Most solution methods from the literature, for example [100], try to minimize the number of vehicles used in the solution as a primary objective, followed by either or both the total distance traveled and the total service duration. We used the following objective function of a solution S to achieve this goal: O(S) = N(S)2 × TotDist(S) × TotDur(S) , (8.1) where N(S) is the number of vehicles used in the solution, TotDist(S) is the total dis- tance traveled by all vehicles, and TotDur(S) is the total schedule duration, which in- cludes the total travel time, the waiting time of the vehicles, and the service time at each location. The number of vehicles is squared in this objective function, so that solutions that use more vehicles will have a considerably higher cost than solutions that use less vehicles. Initial experimentation with this objective function indicated that it adequately serves the purpose of giving priority to minimizing the number of vehicles. In addition, no fine tuning or adjustment of different weights is needed in this objective function. 8.3.2 The Initial Population To create a solution for our GA, requests are first placed in a relocation pool in a random order, before being inserted into the solution. Based on our investigation of several solu- tion construction methods, as explained in Chapter 7, we chose the sequential construction algorithm (SEQ), to create each solution in the initial GA population. The same algorithm was also used to create or modify solutions during the evolutionary process, as will be ex- plained later when we address the genetic operators. The underlying Hill Climbing (HC) routing heuristic and the overall sequential construction algorithm are explained in detail in Sections 7.4 and 7.5.1, respectively, and outlined in Algorithms 7.1 and 7.2. The cost function, in Step 6 of the HC algorithm (Algorithm 7.1), is used to evaluate the quality of each route, and is different from the objective function (8.1) used to eva-
- 177. 8.3 A GeneticAlgorithm (GA) for the MV-PDPTW 153 luate the overall solution (chromosome). This cost function tries to minimize the total route duration as well as the degree of infeasibility in capacity and TW constraints. The cost function of a route r is described by the following equation (same as Equation 7.1, repeated for convenience): F(r) = w1 × D(r) + w2 × TWV (r) + w3 × CV (r) , (8.2) where D(r) is the total route duration, TWV (r) is the total number of time window violations in the route, and CV (r) is the total number of capacity violations, while w1, w2 and w3 are assigned weights in the range [0, 1], and w1 + w2 + w3 = 1.0. For this route cost function, we used the same weights used in Equation 7.1, i.e., w1 = 0.201, w2 = 0.7 and w3 = 0.099. Our routing algorithm allows routes to dynamically change during the search, i.e., pre- vious routing decisions (locations) for some requests already existing in the route may be altered as new requests are added to the route. The new routing information is copied back to the chromosome whenever a change in the route occurs. 8.3.3 The Genetic Operators As previously mentioned in the introduction to this thesis, we try to focus on neighbou- rhood moves (represented by genetic operators here) as a tool for satisfying the hard pro- blem constraints, and maintaining the feasibility of solutions throughout the search. Two crossover operators and one mutation operator have been designed to achieve this purpose. In what follows we explain in detail our proposed genetic operators. Mutation The mutation operator, which we will call the Vehicle Merge Mutation (VMM), simply tries to merge requests from two randomly selected vehicles. The idea is to try to reduce the number of vehicles by distributing the requests among already existing vehicles, or possibly combining two vehicles into one. Figure 8.4 demonstrates the steps of the VMM. Figure 8.4(a) shows that vehicles v2 and v3 were randomly selected for merging from the current solution. The requests belonging to them will then be placed in a relocation pool in a random order. The remaining requests in the solution, i.e., vehicles v1 and v4 will be copied to the new solution to form a partial solution. Figure 8.4(b) shows that the requests in the relocation pool are then re-inserted into the partial solution using the SEQ construction, i.e., Algorithm 7.2, and the final
- 178. 154 8.3 AGenetic Algorithm (GA) for the MV-PDPTW mutated solution is constructed. In our mutation operator, the new solution replaces the old one only if it is better in quality, i.e., if it has a lower objective function value. (a) VMM - two vehicles selected for merging, and their request placed in the relocation pool. Remaining routes copied to the new solution without change (b) VMM - requests in the relocation pool re-inserted into the partial solution using the SEQ algorithm Figure 8.4: Vehicle Merge Mutation (VMM).
- 179. 8.3 A GeneticAlgorithm (GA) for the MV-PDPTW 155 Crossover Two crossover operators have been designed in our research. The first crossover operator, which we will call the Vehicle Merge Crossover (VMX), is similar to the mutation ope- rator described above. However, instead of merging two vehicles from the same solution, the VMX tries to merge two vehicles selected at random, one from each parent solution. The second crossover operator, which will call the Vehicle Copy Crossover (VCX), tries to copy complete routes from the parent to the child. The number of routes to be copied is a random number between 1/4 to 1/2 the number of routes in the first parent. To select routes for inheritance, the VCX tries to select the ‘good’ routes. It is generally desirable to copy routes that serve a large number of requests, since our main objective is to reduce the number of vehicles. Accordingly, the VCX first ranks routes based to the number of nodes served in each route. The larger the number of nodes served the higher the rank of the route. Routes with the same number of nodes are ranked according to the total distance traveled, in which case routes with a shorter distance are more favourable than the longer ones. Figure 8.5 demonstrate the steps of the VCX. Figure 8.5(a) shows that vehicles v1 and v4 were selected from Parent1, depending on the ranking criterion described above. These two vehicles are then copied to the first child. Figure 8.5(b) shows that the remaining requests that have not been included in Child1 (those highlighted in Parent2), will be copied in the same order of their appearance in Parent2 and placed in a relocation pool. The requests in the relocation pool are then sent to the SEQ construction algorithm and used to form a set of new routes. These new routes will afterwards be appended to the routes already existing in Child1, which were inherited from Parent1. Child2 is created similarly by reversing the roles of parents. Our early experimentation indicated that the presence of both crossover operators was necessary for improving the results and satisfying the objective function.
- 180. 156 8.3 AGenetic Algorithm (GA) for the MV-PDPTW (a) VCX - selected routes copied from parent1 to the child (b) VCX - remaining requests copied from Parent2 to the relocation pool, and used to create new child routes Figure 8.5: Vehicle Copy Crossover (VCX).
- 181. 8.3 A GeneticAlgorithm (GA) for the MV-PDPTW 157 8.3.4 The Complete GA Algorithm 8.1 shows the outline of the complete evolutionary algorithm that we used to solve the MV-PDPTW. Algorithm 8.1: The Complete Genetic Algorithm. 1: Initialize a population POP of candidate solutions to the MV-PDPTW, using the SEQ construction algorithm (Algorithm 7.2) 2: for (a pre-specified number of generations) do 3: for (i=0; i<NumCrossovers;i++) do 4: Select parents P1 and P2 from POP, using roulette wheel selection 5: Randomly select crossover type (VCX or VMX) 6: Apply the selected crossover to parents (P1, P2) to produce child C1 7: Apply the selected crossover to parents (P2, P1) to produce child C2 8: With some probability, apply mutation (VMM) to C1 and C2 9: Update POP by integrating the new generation and eliminating some worst individuals {i.e., steady state GA with overlapping populations} 8.3.5 Attempts to Improve the Results Before we report the experimental findings of this part of our research, we describe in this section some other operators and modifications to the solution approach that we ex- perimented with, in an attempt to reach the best possible results. These attempts are briefly explained below, in addition to some concluding remarks about their performance. However, since the benefits realized from adopting these approaches did not meet our an- ticipations, they were abandoned in favour of the simpler approach described in Sections 8.3.1 to 8.3.4. Vehicle Removal Mutation This mutation operator is based on the GGA of [116]. One vehicle is removed and its requests are placed in the relocation pool in a random order, where the SEQ algorithm tries to re-insert them into the solution. This may result in reducing the number of vehicles, if all the requests from the pool were accommodated into other vehicles. The approach in [116] selected a random vehicle for removal, while in our research we selected the vehicle with the minimum number of nodes.
- 182. 158 8.3 AGenetic Algorithm (GA) for the MV-PDPTW Large Neighbourhood Search (LNS) Mutation This mutation operator is based on the Large Neighbourhood Search (LNS) move used in the work by [131], which is an adaptation of the same operator used by Shaw in [139] for the VRPTW. The idea is to remove a number of requests from the solution and then try to re-insert these requests back. Usually a large number of nodes (30% to 40%) are removed. However, since in our research this operator is used as a mutation operator, it would be more appropriate to reduce the number of requests removed. In our experimentation we found that 5% to 10% of the total number of nodes produce reasonable results. Two variations of this operator were tried in our research, as advised by [131]. One ver- sion of this mutation selects the nodes to be removed randomly (random removal), while the other version selects the nodes that seem to be located in the wrong positions in the solution (worst removal). To find these requests, we calculate the cost of the solution (the objective function), with and without each request in turn, i.e., the request is temporarily removed from the solution to calculate its effect on the overall cost. The requests are then sorted in decreasing order of the change in cost. The requests to be removed are taken, in order, starting from the top of the list and then placed in a relocation pool in a random order. The removed requests are then re-inserted in the solution using the SEQ algorithm. The ‘worst removal’ version of the LNS algorithm gave slightly better results than the ‘random removal’ version. A Memetic Algorithm We also tried to improve the performance of the GA by introducing local search at various stages of the evolutionary process. This idea is based on the memetic algorithm described in [94]. We tried both the vehicle removal mutation and the LNS mutation (explained above), as neighborhood moves to create a new solution within the local search. We also used an SA acceptance for solution replacement. More specifically, in this memetic algorithm, local search is performed on some indivi- duals in the population (according to a certain probability), for example at the beginning of each generation. The local search repeatedly applies a selected ‘mutation’ operator on the individual for a pre-specified number of trials. Within the local search, a new solu- tion replaces the current solution using an SA acceptance criterion, i.e., a better solution always replaces the current solution, while a worse solution may replace the current solu- tion with some small probability. The best solution found during the local search is finally the one that replaces the initial solution (the individual that we started with).
- 183. 8.3 A GeneticAlgorithm (GA) for the MV-PDPTW 159 Our experimentation indicated that the more suitable operator to be used within the local search seems to be the LNS operator. It should also be noted that a small population size (e.g. 30 individuals) would be most appropriate for this memetic algorithm, due to the large increase in processing time that result from the repeated application of the local search. Allowing Infeasible Solutions In our attempt to improve the results, we also tried to add some infeasible solutions to the set of allowable solutions. The idea is that sometimes good solutions lie in the vicinity of infeasible solutions, so by allowing infeasibility we may be able to climb out of local optima. In this version of our algorithm, whenever a solution construction is needed during the initialization phase of the population, or during the course of the evolutionary operators, there was a 50% chance that the construction process will allow infeasibility. In order to restrict the amount of infeasibility and reduce the search space, we only allo- wed violations in the time window but not in the capacity constraint. To do that, several modifications to the algorithm were needed. Firstly, we had to restrict the number of nodes allowed in each individual vehicle. Other- wise, we could end up with a solution in which all requests are served by one vehicle. This is due to the fact that even if violating the capacity is not permitted, the algorithm would still be able to find a solution in which all requests are served, without any capacity violations, using only one vehicle. For example, a solution in which each pickup is fol- lowed by its delivery will always satisfy the capacity constraint. Specifically, the allowed number of nodes, for each vehicle individually, was set to a random number ranging from 1 to m, where m was arbitrarily chosen to be 20% of the total number of nodes in the problem instance. Thus, nodes are added to the current vehicle until the predetermined limit on the number of nodes for this vehicle is reached, at which stage a new vehicle has to be allocated. Secondly, the objective function has to be changed in order to penalize infeasibility in the solution. The following equation was used O(S) = w1 × N(S)2 + w2 × TotDist(S) + w3 × TWV (S). (8.3) Where N(S) is the number of vehicles in the solution, TotDist(S) is the total distance traveled by all vehicles in the solution, and TWV (S) is the number of time window violations in the solution. w1, w2 and w3 are weights in the range [0,1] and w1 +w2 +w3 =
- 184. 160 8.3 AGenetic Algorithm (GA) for the MV-PDPTW 1.0. As with all other similar objective functions in our research, the penalty on the time window violations has to be large in order to get rid of infeasible solutions as the search progresses. Thirdly, the genetic operators had to be modified. We modified the VCX crossover opera- tor by ranking vehicles according to the percentage of infeasibility in each vehicle, rather than the number of nodes visited and the total distance, as done previously when only feasible solutions were allowed. The VMX crossover was used without modification. We also used the LNS mutation, after modifying it to remove the requests that have a time window violation and try to re-insert them again in the solution, i.e., the worst requests are now the requests that violate the time window. The vehicle removal mutation was also used without modification. Results of the Improvement Attempts Based on our experimentation, we can rank the effectiveness of the above improvement attempts (from best to worst) as follows: the memetic algorithm, the LNS operator, the vehicle removal mutation and finally allowing infeasible solutions. The improvement introduced by these variants, or any combination of them, though, did not warrant, in general, the increase in the complication of the overall algorithm. These variants seem to cause an improvement in some test cases, but may cause a degradation in the results of other test cases. Accordingly, to keep the overall algorithm as simple as possible, which is the main philosophy in our research, we decided not to include any of these variants in the final testing version of the proposed GA. Thus, only the operators described in Section 8.3.3 and the final GA described in Section 8.3.4, were adopted in the final version. The experimental results of this final version are detailed in the following subsection.
- 185. 8.3 A GeneticAlgorithm (GA) for the MV-PDPTW 161 8.3.6 GA Experimental Results To test our algorithm, we implemented a steady state GA with a 95% replacement. The following parameters were used: population size= 500, crossover probability= 0.6, mu- tation probability= 0.05, and the number of generations= 300. In cases where crossover is performed either VCX or VMX is selected at random. We used the standard 56 (100- customers) benchmark instances, created by Li & Lim in [100]. For the different types of problem instances in the data set, the reader is referred to Section 7.6.1. The algorithm was run 10 times on each problem instance. We compared our algorithm, which we will call the Grouping-Routing GA (GRGA), with the GA in [33], denoted by CKKL5 , and the grouping GA in [116], denoted by GGA. To the best of our knowledge, they are the only GA approaches that have been attempted in the literature for the MV-PDPTW and applied to the published benchmark data of [100]. The results in [116] are also close to the best known results, so we found that a comparison with their results will be sufficient for the purpose of this part of our research6 . Before we report our experimental findings, though, we present in Table 8.1 a comparison of the most distinctive features of the three algorithms under consideration. Figures 8.6 and 8.7 show the best results achieved by the three algorithms in terms of the number of vehicles and the total distance traveled. The two figures show that our GA clearly achieves better results than the CKKL algorithm in almost all test cases. There are only 5 cases in which our algorithm produced one more vehicle than the number of vehicles produced by the CKKL. Moreover, all our total distance results were better than the results of the CKKL. On average, the improvement of our results compared to the results of the CKKL in the number of vehicles is approximately 16%, while the average improvement in the total travel distance is approximately 36%. Also, our results are also close to the results of the GGA in the number of vehicles produced, with only few exceptions. Nevertheless, the resulting total distance is larger than the resulting total distance of the GGA in most test cases. This is even more noticeable in instances with a long time window width, i.e., instances of category ‘2’. 5 We thank the authors of [33] for providing us with the data files containing their detailed results. 6 The detailed results of the GRGA approach and a comparison with best known results will be presented in Section 8.4
- 186. 162 8.3 AGenetic Algorithm (GA) for the MV-PDPTW Table 8.1: Comparison between the GRGA, GGA and CKKL Algorithms GA GRGA GGA CKKL General Approach - A GA handles both the grouping and the routing aspects of the problem. - All problem informa- tion is explicity monito- red and manipulated by the GA. - A GA only handles the grouping aspect of the problem. - The routing informa- tion is hidden from the GA and created when the chromosome is de- coded. - A GA handles both the grouping and the routing aspects of the problem. - All problem informa- tion is explicity monito- red and manipulated by the GA. Encoding - A Chromosome has a variable number of genes. - Each gene is a vehicle route (a sequence of visited nodes). - Same code for pickup and delivery (P&D), and a parser to traverse the route and identify each. - A Chromosome has a variable number of genes. - Each gene is a cluster of requests assigned to one vehicle. - A sperate data struc- ture and an insertion heuristic are used to create individual routes. - A Chromosome has a variable number of genes. - Each gene is a vehicle route (a sequence of vi- sited nodes). Routing or Insertion Heuristic - Insert P&D pair at end of route, and improve the route using an HC algorithm. - The routing decisions of requests in the route may be changed by the genetic operators. - New routes are copied back to the chromo- somes, during recombi- nation and mutation. - Examine all feasible insertions for the P&D pair in all routes, and select the insertion that causes minimal additio- nal cost. - The routing decisions of nodes already exis- ting in the route are sta- tic and are not chan- ged by the genetic ope- rators. - The P&D pair is inser- ted in a feasible route position. - Position of insertion could be modified later using a local search mu- tation. Continued on Next Page. . .
- 187. 8.3 A GeneticAlgorithm (GA) for the MV-PDPTW 163 Table 8.1 . . . Continued from Previous Page GA GRGA GGA CKKL Crossover - Vehicle Copy Cros- sover (VCX) & Ve- hicle Merge Crossover (VMX). - Crossover operators are aware of, and make use of, the routing in- formation of each gene. - Offspring is always feasible, and no repair method needed. - Adaptation of the ge- neral GGA crossover, where crossover is not aware of the routing information of each gene. - Consecutive set of clusters are selected from the first parent and inserted in the second parent. - Chromosome cleanup is needed to correct in- feasibility of offspring. - Sequence Based Cros- sover (SBX): fragments of two routes are se- lected from each parent and joined together to form a new route. - Route Based Crosso- ver (RBX): two selec- ted routes are exchan- ged between the two parents. - If possible, infeasibi- lity of offspring is re- paired. Otherwise, off- spring is discarded. Mutation - Vehicle Merge Muta- tion (VMM). - Mutation is performed on the offspring created by crossover with a cer- tain probability. - Remove one vehicle and reassign its re- quests. - Mutation is performed on the offspring created by crossover with a certain probability. - One-Level exchange Mutation (1M): re- moves one vehicle and reassigns its requests. - Local Search Muta- tion (LSM): tries to find better locations for requests in a randomly selected route. - Mutation is performed on a randomly chosen individual. Objective Function - Minimize the number of vehicles, followed by total distance and total duration. - Minimize total travel distance, irrespective of the number of vehicles. - Minimize a weigh- ted sum of the number of vehicles, total dis- tance and total duration (equal weights are assi- gned).
- 188. 164 8.3 AGenetic Algorithm (GA) for the MV-PDPTW LC101 LC201 LR101 LR201 LRC101 LRC201 0 5 10 15 20 25 Problem NumberofVehicles GRGA CKKL GGA Figure 8.6: Total number of vehicles produced by all algorithms. LC101 LC201 LR101 LR201 LRC101 LRC201 500 1000 1500 2000 2500 3000 3500 4000 4500 Problem TotalDistance GRGA CKKL GGA Figure 8.7: Total travel distance produced by all algorithms.
- 189. 8.3 A GeneticAlgorithm (GA) for the MV-PDPTW 165 When we try to analyze the computational results in the light of the differences between the three algorithms summarized in Table 8.1, we will realize the following. First, since both the GRGA and the CKKL algorithms have the same chromosome structure and the same components of the objective function, then the obvious success of the GRGA com- pared to the CKKL algorithm must be due to the routing algorithm and/or the genetic operators used in the former. The success of the GRGA to produce solutions with less vehicles in most test cases could be attributed to the presence of two genetic operators the are specifically designed for this purpose and heavily applied during the search, namely the VMM mutation and the VMX crossover. On the other hand, the 1M mutation, used for reducing the number of vehicles in the CKKL, is only performed occasionally during the search. Also, the noticeable success of the GRGA in terms of reducing the total travel distance, could be attributed to the routing algorithm that is called whenever a route is created or modified to try to improve the quality of the route by reducing its overall cost. This again is in contrast to the LSM mutation of the CKKL that tries to improve the route, but is only called occasionally during the search. The VCX crossover, used in the GRGA, also seems to do a better job than the RBX crossover used in the CKKL. The RBX merely exchanges one route between parents, while the VCX tries to be selective when transfer- ring routes from the parent to the child, by choosing routes that serve a large number of nodes with the smallest possible distance. It also seems that the SBX crossover, used in the CKKL, may not be suitable for the genetic representation used. Since the gene is ac- tually a complete route, it would seem more appropriate to transfer a collection of routes rather than route fragments between parents. We will now try to analyze the reasons behind the sub-optimal results achieved by the GRGA compared to the GGA, specially in terms of the total travel distance7 . First of all, one of the major differences between the two algorithms is the objective function. The objective function of the GRGA includes the number of vehicles, the total distance, and the total duration, while the objective function of the GGA only includes the total dis- tance. Thus, the best solution as far as the GRGA is concerned must balance all three components, i.e., it takes into consideration other parameters involved in the routing sche- dule like the total waiting time of the vehicle at each location and the total service time. The second main difference is the routing heuristic used in both algorithms. The routing (insertion) algorithm of the GGA tries to find the best insertion position for each newly inserted request in all available routes, while we use a simple and fast algorithm that ac- cepts any feasible insertion. The attempt to improve the route, using the HC algorithm, is only local to each route and does not involve comparing the insertion cost with other 7 It should also be noted that our algorithm was run only 10 times on each test case, while the GGA of [116] was run 30 times and the best result was selected.
- 190. 166 8.3 AGenetic Algorithm (GA) for the MV-PDPTW routes. It seems that our routing algorithm could probably use the help of an additional local search operator (e.g. 2-Opt), to try to improve the routes, which should in turn help reduce the number of vehicles. In our opinion, these two issues are the main reasons for the inability of our approach to reach the solution quality of the GGA. On the other hand, the genetic operators in both seem to be comparable, since they either try to reduce the number of vehicles, or copy complete routes from parents to children. However, we think that our VCX crossover may be more suitable than the crossover of the GGA for this problem type, because transferring a sequence of routes is not really meaningful, since the order of routes in the chromosome is irrelevant. Also, as mentioned previously, our algorithm seems not able to cope with instances of type ‘2’, as evident by the large gap between our results and the results of the GGA in the total travel distance. This could be explained if we recall that our routing algorithm had a neighbourhood move, which was guided by the TW. It seems that the routing algorithm was thus capable of dealing better with instances in which the TW constraint is hard to satisfy, i.e., those with a tight TW width. This neighbourhood move may not be sufficient to improve the route in problems with a large TW width, because of the availability of many different feasible orders of nodes. An alternative neighbourhood move may be needed in that case. For example, the neighbourhood move could take into account not only the bound(s) of the time windows of the swapped locations, but also whether the swapping can improve other route characteristics, such as the total waiting time, the total travel time, or the extra travel distance. The results obtained for instances of type ‘2’ also sustain our previous observation in Section 7.6.3, regarding the large solution space for this specific type of problem instances. Finally, the average processing time needed by our algorithm, over all 56 problem ins- tances, was 176.9 seconds, which is comparable to the processing time of the grouping GA in [116] having an average of 167.1 seconds. Nevertheless, the separation of the data structure used for individual routes in [116] from the actual chromosome, seems to be favourable than our representation which includes all vehicle routes in the chromosome, as far as processing time is concerned, since the transfer of complete routes during the recombination and mutation operators is definitely time consuming. The authors in [33], on the other hand, do not report their processing time.
- 191. 8.4 Simulated Annealing(SA) for the MV-PDPTW 167 8.4 Simulated Annealing (SA) for the MV-PDPTW As previously mentioned in Section 8.1, we will try in this part of our research to adopt some of the neighbourhood operators used in the GA approach for the MV-PDPTW within an SA approach. As expected, the solution representation used in the SA is similar to the chromosome representation used in the GA (explained in Section 8.3.1). The same objective function (Equation 8.1) is also used here. The initial solution from which the SA will progress is created by generating a number of random solutions8 , in a manner similar to the creation of an initial genetic population (as explained in Section 8.3.2), and the best solution among them, in terms of the objective function value, is selected. To allow for an adaptive calculation of the SA parameters for each problem instance in- dividually, we again used the approach proposed by [40], as previously done for the SV- PDPTW (see Section 6.3 and Algorithm 6.1). Thus, the annealing parameters are calcu- lated based on the average value of ∆cost, where ∆cost is the difference in the objective function value between some randomly generated solutions for the current problem ins- tance. In our approach, the same set of random solutions created for the selection of the starting solution, were also used for the purpose of calculating the annealing parameters. A crucial part of any SA algorithm is the neighbourhood move that will be used to generate a new solution. In this part of the research, we experimented with different neighbourhood moves from the ones tried in our GA approach. Two neighbourhood moves were found to be the most appropriate for the SA approach: 1. The Large Neighbourhood Search (LNS) Move: the LNS move is similar to the LNS mutation operator explained in Section 8.3.5, which is inspired from the Adaptive Large Neighbourhood Search (ALNS) approach of [131]. As previously explained, the idea is to remove and then re-insert a large number of requests in each application of the move. The authors in [131] recommend that 30% to 40% of the total number of nodes is removed in each iteration. In our LNS move, we slightly increased the allowed range for the number of removed requests, in order to explore a wider area of the search space. The number of removed requests in our LNS move ranges between 20% to 50% of the total number of requests. In addition, we adopted here the “Worst Removal” variant from the three types of removal heuristics applied in [131] (see Section 8.2 for more details about the different removal heuristics of [131]). In our approach, the requests removed are those that are estimated to cause 8 100 random solutions were created in this experiment.
- 192. 168 8.4 SimulatedAnnealing (SA) for the MV-PDPTW a large increase in the cost of their respective routes, i.e., they may be inserted in unfavourable positions in the solution. Thus, to determine the cost of each request, the total travel distance of the route to which the request belongs is calculated, with and without the request under consideration, and the difference in the route distance is used as a measure of the request cost. Requests having the highest costs are selected for removal from the current solution. After this, the removed requests are placed in a relocation pool in a random order, before they are inserted back in the solution using the SEQ construction algorithm, hoping to find better insertion positions for them in the new solution. 2. The Vehicle Merge (VM) Move: this move is identical to the Vehicle Merge Mu- tation (VMM), explained in Section 8.3.3. The vehicle merge move selects two vehicles at random from the current solution. Then, the requests belonging to them will be temporarily placed in a relocation pool in a random order, before they are re-inserted in the solution using the SEQ algorithm as shown in Figure 8.4. Similar to the idea of the 3-stage SA for the SV-PDPTW (explained in Section 6.3), our SA approach to the MV-PDPTW operates in two stages. In the first stage, the LNS move is used to generate a new solution, while in the second stage the VM move is used instead. Changing the SA move in this manner allows the search process to slightly perturb the current solution, before trying to re-optimize it, which may help in escaping local optima. The second SA stage starts from the final solution obtained in the previous stage and from the final temperature reached by the end of the previous stage. During each stage, the best so far solution is saved and each SA stage terminates when no improvement is realized in the best solution for a consecutive number of iterations9 . Also, during each stage, the current temperature value is reduced in each iteration of the SA algorithm. For a more extensive searching, the two SA stages are repeated several times. Again, the repetition only stops when the best obtained solution reaches a stage of stagnation and does not improve for a number of consecutive attempts of applying the two stages10 . We also found during our computational experimentation that there seems to be no signifi- cance to the order of application of the two moves. Thus, as long as the two stages are repeated, the SA may be started from either move. The overall 2-stage SA approach is shown in Algorithm 8.2. 9 100 consecutive attempts without improvement was used to terminate each SA stage. 10 10 consecutive attempts without improvement was used to stop the repetition of the 2-stage SA ap- proach
- 193. 8.4 Simulated Annealing(SA) for the MV-PDPTW 169 Algorithm 8.2: The 2-Stage SA Algorithm. 1: Find an initial solution (InitSol) and calculate the annealing parameters 2: BestSol ← InitSol {Initialize the best so far solution} 3: Initialize MaxAttempts to a small number {We used 10} 4: NoImprovement ← 0 5: repeat 6: OldCost ← Objective(BestSol) {Calculate the cost of the solution using Equation 8.1} 7: BestSol ← SALNS(BestSol) {Perform SA on the current best solution, using the LNS move, and return the best found solution} 8: BestSol ← SAV M (BestSol) {Perform SA on the current best solution, using the VM move, and return the best found solution} 9: NewCost ← Objective(BestSol) {Calculate the cost of the new solution using Equation 8.1} 10: if (NewCost is not better than OldCost) then 11: NoImprovement + + 12: else 13: NoImprovement ← 0 14: until (NoImprovement reaches MaxAttempts) SA Experimental Results Our SA algorithm was tested on the same 100-customers test cases used for testing our GA, as explained in Section 8.3.6. Similar to the GA, the SA algorithm was run 10 times on each test case. Table 8.2 shows the best result, in the 10 runs, achieved by both the GA and the SA for each test case. The best result is the one having the minimum number of vehicles, and for the same number of vehicles, the one having the minimum total travel distance. The better obtained result between the two algorithms is highlighted in boldface. In addition, the processing time in seconds is also shown in the table. The last two columns of the table show the current best known results for these problem instances, which are published in: http://www.sintef.no/Projectweb/TOP/Problems/PDPTW/Li--Lim-benchmark/ The best known results are attributed to [100], [12], and [135]. The last two rows of the table show, respectively, the overall average results of the corres- ponding column, and the percent difference (gap) between our average and the average of the best known results, in terms of both the number of vehicles and the total distance.
- 194. 170 8.4 SimulatedAnnealing (SA) for the MV-PDPTW Table 8.2: GA & SA Best Results (100-Customers) GA Results SA Results Best Known Name Vehic Dist Time Vehic Dist Time Vehic Dist LC101 10 828.94 34.8 10 828.94 45.72 10 828.94 LC102 11 978.48 109.97 11 945.88 57.99 10 828.94 LC103 10 1310.3 131.3 10 1238.57 226.67 9 1035.35 LC104 10 1203.1 145.28 9 1328.29 201.8 9 860.01 LC105 10 828.94 85.03 10 828.94 40.41 10 828.94 LC106 10 844.58 87.69 10 828.94 41.52 10 828.94 LC107 10 828.94 85.66 10 828.94 46.7 10 828.94 LC108 10 949.96 85.52 10 828.04 64.66 10 826.44 LC109 10 827.82 83.7 10 849.08 52.63 9 1000.6 LC201 3 591.56 261.17 3 591.56 3.47 3 591.56 LC202 5 1261.81 304.77 4 1186.64 58.34 3 591.56 LC203 5 1957.86 420.83 5 1903.04 75.36 3 585.56 LC204 5 1770.26 627.95 4 2194.2 119.03 3 590.6 LC205 3 591.56 283.27 3 591.56 8.02 3 588.88 LC206 4 681.35 216.52 4 626.89 11.61 3 588.49 LC207 3 766.62 293.88 3 701.72 24.77 3 588.29 LC208 3 604.51 320.33 3 604.7 15.19 3 588.32 LR101 19 1667.68 80.67 19 1667.68 10.22 19 1650.8 LR102 17 1627.73 90.61 17 1627.91 48.41 17 1487.57 LR103 14 1619.68 103.23 14 1525.99 86.06 13 1292.68 LR104 11 1262.3 231.22 11 1335.91 146.14 9 1013.39 LR105 15 1433.79 76.48 15 1450.98 49.74 14 1377.11 LR106 14 1564.06 95.33 13 1458.71 114.17 12 1252.62 LR107 12 1356.72 99.78 12 1353.44 195.7 10 1111.31 LR108 13 1380.93 108.16 12 1353.05 235.58 9 968.97 LR109 13 1448.14 85.14 13 1449.38 71.34 11 1208.96 LR110 13 1362.74 94.81 12 1323.12 151.78 10 1159.35 LR111 13 1431.07 103.59 12 1299.28 278.8 10 1108.9 LR112 12 1339.66 95.56 11 1237.92 100.78 9 1003.77 LR201 4 1783.1 194 4 1841.72 65.25 4 1253.23 LR202 5 2035.85 335.24 4 2083.77 177.84 3 1197.67 LR203 5 2135.47 500.69 4 2312.46 409.73 3 949.4 Continued on Next Page. . .
- 195. 8.4 Simulated Annealing(SA) for the MV-PDPTW 171 Table 8.2 . . . Continued from Previous Page GA Results SA Results Best Known Name Vehic Dist Time Vehic Dist Time Vehic Dist LR204 4 2017.64 541.06 4 2059.11 181.25 2 849.05 LR205 4 1939.61 289.36 4 2123.62 78.56 3 1054.02 LR206 4 2128.15 554.16 4 2134.15 206.45 3 931.63 LR207 4 2191.28 615.16 4 2360.2 185.77 2 903.06 LR208 4 2064.23 723.53 4 2059.02 276.97 2 734.85 LR209 5 1787.82 275.5 4 2058.16 106.94 3 930.59 LR210 5 1928.42 313.3 4 2175.39 164.81 3 964.22 LR211 4 1782.99 401.3 4 1886.82 196.34 2 911.52 LRC101 16 1806.27 81.13 16 1806.27 26.11 14 1708.8 LRC102 15 1840.05 94.08 14 1776.15 80.47 12 1558.07 LRC103 13 1599.7 118.67 13 1558.38 122.19 11 1258.74 LRC104 13 1535.82 110.97 12 1464.11 279.77 10 1128.4 LRC105 16 2009.51 91.36 16 1862.36 88.17 13 1637.62 LRC106 13 1688.86 87.11 14 1672.87 53.06 11 1424.73 LRC107 14 1637.27 93.05 12 1452.42 153.31 11 1230.15 LRC108 13 1535.48 101.88 12 1379.59 148.47 10 1147.43 LRC201 5 2230.74 144.81 5 2327.42 89.98 4 1406.94 LRC202 5 2442.78 286.58 5 2525.27 171.39 3 1374.27 LRC203 5 2335.04 395.59 5 2413.63 400.44 3 1089.07 LRC204 5 2049.8 628.95 4 2323.05 289.55 3 818.66 LRC205 6 2162.85 162.3 5 2605.27 148.02 4 1302.2 LRC206 5 2181.3 191.58 5 2400.12 40.58 3 1159.03 LRC207 5 2346.32 245.63 5 2319.27 172.53 3 1062.05 LRC208 5 1983.58 247.73 5 2195.22 68.34 3 852.76 AVG 8.75 1562.52 226.2 8.43 1592.23 124.37 7.18 1036.68 GAP% 22% 51% - 17% 54% - - - Table 8.2 shows that SA achieved better results than the GA in 31 out of the 56 problem instances. In terms of the overall average results, SA achieved a slightly better average in the number of vehicles and a slightly worse average in the total distance traveled, which can also be seen from the relative difference to best known results, indicated in the last row of the table. In terms of the average processing time, though, the result was in favour
- 196. 172 8.4 SimulatedAnnealing (SA) for the MV-PDPTW of SA, with approximately 45% less average processing time than the GA. Figure 8.8 shows the average objective function value (as calculated by Equation 8.1) of the 10 runs, for each problem instance, obtained by both the GA and the SA. LC101 LC201 LR101 LR201 LRC101 LRC201 0 0.5 1 1.5 2 2.5 x 10 9 Problem AverageObjectiveValue GA SA Figure 8.8: Average objective value for the GA and the SA algorithms. In general the results in Table 8.2 and Figure 8.8 indicate that both the SA and GA obtai- ned comparable results when tested on the 100-customers instances for the MV-PDPTW. This could be explained by realizing that both algorithms adopted similar neighbourhood moves. The VM move used in SA is exactly the same as the VMM mutation used in the GA. In addition, the LNS move used in SA is also comparable to the crossover operators used in the GA. In the GA, the crossover operators generally try to remove requests be- longing to some selected routes from parent solutions and insert them back into the child solution. This process is similar to what the LNS move performs in SA, when requests are removed and then re-inserted into the solution. Moreover, selecting ‘good’ routes to be transferred to the child in the VCX crossover is also analogous to removing ‘bad’ requests in the LNS move adopted in SA. The results of both algorithms, though, were inferior to the best known results, especially in terms of the total travel distance. This indicates that both algorithm could probably make use of an additional local search method, such as 2-Opt edge exchange, to improve the quality of the routes and reduce the travel distance at some stage during the run.
- 197. 8.5 Summary andFuture Work 173 8.5 Summary and Future Work In this chapter we investigated both a GA and an SA for solving the MV-PDPTW. Our GA approach tried to face the challenge of allowing the GA and its operators to be aware of and manipulate both the grouping and the routing aspects of the problem. A challenge that most previous GA research on this problem has achieved little success with, or has been avoided completely by allowing the GA to tackle only one problem aspect. We first tried a simple representation and an intelligent neighbourhood move to handle the routing part of the problem. For the grouping part, on the other hand, we designed new genetic operators that try to exploit problem-specific information and create new solutions from existing ones, while maintaining the feasibility of solutions throughout the search. Our operators overcome the difficult problem constraints, and avoid at the same time the need for a repair method to fix infeasible solutions, a technique that previous GAs and most other heuristic and meta-heuristic techniques have been relying on to maintain feasibility. Overall, though, our algorithm is a simple and straightforward GA technique, and the genetic operators developed here are applicable to other related routing problems. We compared our results with two previous GA attempts to solve the problem, the CKKL algorithm of [33] and the GGA algorithm of [116]. The experimental results show that our algorithm was able to greatly improve upon the results of the CKKL algorithm, using just a few simple modifications in the routing algorithm and the genetic operators, and applying problem-specific information. The improvement was evident in both main ob- jectives, the number of vehicles and the total travel distance. However, our results are still behind the results of the GGA in most test cases. The three algorithms have been tho- roughly analyzed and possible reasons behind the differences in performance have been identified. The GA was also compared with a 2-stage SA approach that uses two different neigh- bourhood moves repeatedly. The neighbourhood moves used within this SA approach are analogous to the genetic operators used previously in the GA, since they rely on removing and then re-inserting requests or merging selected vehicles. Both the GA and the SA obtained comparable results when tested on published benchmark instances, which is mostly due to their reliance on similar neighbourhood operators. Ho- wever, neither algorithm was capable of competing favourably with best known results. In general, it appears that our representation and neighbourhood operators in both algorithms are doing a fair job in guiding the search towards better solutions. However, to cope with the difficulty of the problem and the different types of problem instances, both approaches still need further improvement. For this purpose, different neighbourhood moves could be
- 198. 174 8.5 Summaryand Future Work attempted in the route improvement heuristic, for example by taking some route charac- teristics into consideration when swapping locations, such as the resulting total waiting time or the total route duration. Moreover, a local search method, such as 2-Opt or 3-Opt, could be added to improve the quality of the routes and reduce the total travel distance. In terms of processing time, though, the GA was clearly much slower than the SA, which is expected, due to the overhead of maintaining a large population of solutions. Having concluded our investigation of the PDPTW both its single and multiple vehicles variants, in the following chapter we move on to another interesting and important variant of pickup and delivery problems, the One-Commodity Pickup and Delivery Problem (1-PDP).
- 199. 175 Chapter 9 The One-CommodityPickup and Delivery Problem: Introduction and an Evolutionary Perturbation Scheme The One-Commodity Pickup and Delivery Problem (1-PDP) is another important problem in transportation and logistics systems. The 1-PDP deals with supplying and collecting one type of commodity from a number of customers, some of them are designated as pickup customers and the others as delivery customers. Each pickup customer provides a certain amount of the commodity, while each delivery customer consumes a certain amount of the same commodity, i.e., goods collected from pickup customers can be de- livered to any delivery customer. All customers are served by one vehicle with a limited capacity, and the journey of the vehicle should start and end at a central depot. The de- pot can supply or consume any additional amount of the commodity that is not supplied or consumed by the customers. Our goal is to find a feasible and minimum cost route for the vehicle, such that all customers are served without violating the vehicle capacity constraint. This problem has attracted our interest for several reasons. First, there are many applica- tions for this problem in practice. For example, the commodity could be milk that should be collected from farms and delivered to factories with no restriction on the origin and the destination of the product, or it could be money that should be distributed between the branches of a bank [74]. It can also model any logistic situation in which some wa- rehouses have an extra supply of some commodity, while others are in short of the same commodity. A typical situation is when some hospitals need to transfer a certain medicine to other hospitals, who are in short of this medicine. For example, an H1N1 vaccination or treatment could be transferred in urgent epidemic circumstances [106]. Second, this problem has not been adequately explored in the literature. Since the in- troduction of the 1-PDP in [74], only very few papers seem to have handled it. Finally, there are other important problems in the literature that are closely related to the (1-PDP).
- 200. 176 9.1 The1-PDP For example, the Traveling Salesman Problem with Pickup and Delivery (TSPPD) (see Section 9.3 for more details about this and other related problems). Our initial attempt to handle the 1-PDP was based on a simple Evolutionary Perturba- tion Scheme (EPS) that has proved successful for other routing problems, such as the TSP and the Capacitated Vehicle Routing Problem (CVRP). We made some preliminary experimentation with this technique for the 1-PDP and tested it on published benchmark data. The details of the approach and its experimental results are reported in this chapter. The outcome of the approach, though, did not meet the anticipated standard. Some ana- lysis of possible shortcomings of the algorithm and future directions are also discussed here. Nevertheless, A more successful heuristic for solving the 1-PDP, will be detailed in the next chapter. The rest of this chapter is organized as follows. Section 9.1 formally defines the 1-PDP, while Section 9.2 is a brief summary of related research in this area. Some problems that are related to the 1-PDP are presented in Section 9.3. The EPS technique attempted for solving this problem is explained in Section 9.4, together with its experimental findings. Finally, Section 9.5 concludes with a summary and some thoughts for future work. 9.1 The 1-PDP Based on the definition provided in [73], the 1-PDP is characterized by having a set of customers i(i = 1, 2, ..., n), where customer 1 is the depot. V = 1, 2...n is the vertex set, and E is the edge set. For each pair of locations (i, j) the travel cost cij is known in advance. For each customer there is an associated demand qi, such that qi > 0 for a pickup customer and qi < 0 for a delivery customer. The depot is considered as a customer that supplies or consumes any amount of the product that is not supplied or consumed by the set of customers, and the demand of the depot will be calculated as q1 = − n i=2 qi. All customers are to be served by one vehicle with a limited capacity Q > 0, and the capacity of the vehicle should at least be equal to the maximum customer demand (whether a pickup or a delivery), i.e., Q ≥ maxi∈V {|qi|}. A feasible path for the vehicle is a path that travels from the first customer to the last customer, and visits each customer exactly once without exceeding its capacity. The au- thors in [73] more formally describe the feasibility of a path P through the sequence of customers i1, ..., ik, with k ≤ n. If we assume that lj(P) is the load of the vehicle after visiting the jth customer, and l0(P) = 0, then P is feasible if and only if max j=0,...k {lj(P)} − min j=0,...k {lj(P)} − Q ≤ 0. (9.1)
- 201. 9.2 Related Work177 In [158], more explanation of Equation 9.1 is given by indicating that minj=0,...k{lj(P)} could be either 0 or negative, given the existence of some negative customers’ demands. Accordingly, there are two possible conditions for a feasible route: (i) if minj=0,...k{lj(P)} = 0, Equation 9.1 will simply become maxj=0,...k{lj(P)} ≤ Q. (ii) if minj=0,...k{lj(P)} < 0 (i.e., there is a shortage in the required delivery demands), we can assume that | minj=0,...k{lj(P)}| will be supplied by the depot when the vehicle starts its journey. Thus, | minj=0,...k{lj(P)}| will be added to the demands of all nodes, making minj=0,...k{lj(P)} = 0 again, and we return to the first case, i.e., maxj=0,...k{lj(P)} ≤ Q. Although the 1-PDP is N P-hard (indeed it coincides with the TSP when the vehicle ca- pacity is large enough), checking the feasibility of a path can be done in a linear time. It is also important to note that the feasibility of the path is independent of its orientation, such that a path that is feasible/infeasible if traversed forward, will be also feasible/infeasible if traversed backward [75]. This feature allows a solution construction algorithm to start a tour from any customer node, visit each node, including the depot, exactly once and return to the first node. This closed circular tour will correspond to starting and ending at the depot (see Section 9.4.3 for an example of a construction algorithm). 9.2 Related Work Since this problem is N P-hard, exact algorithms are only suitable for small problem sizes. For example, [74] presented a branch-and-cut exact algorithm to solve instances of up to 60 customers. To deal with large size problems, the same authors tried heuristic approaches in [75]. Two heuristic approaches have been presented in their work. The first approach starts with a construction heuristic that is based on an adaptation of a TSP Nearest-Neighbour (NN) insertion heuristic. However, travel distances were modified using a special formula that intends to penalize the use of edges connecting customers of the same type, i.e., edges connecting pickup customers together or delivery customers together. It was thought that this approach would be more likely to lead the construction algorithm to find feasible solutions. An improvement phase then follows, using 2-Opt and 3-Opt edge exchanges, to try to improve the feasibility and/or the total travel distance. The process of optimization is repeated several times, each time a new initial solution is constructed using a different starting node from the node list. The second approach is an incomplete optimization procedure, based on the branch-and-cut approach presented in [74] to find the best solution in a restricted feasible region. In [73] a heuristic approach, named hybrid GRASP/VND, is proposed. The approach is based on combining two optimization heuristics that have been successfully applied
- 202. 178 9.2 RelatedWork to combinatorial optimization problems. The first is called GRASP (Greedy Randomi- zed Adaptive Search Procedure), and is based on a repetition of a construction phase and a local search phase. The construction process works by selecting the next element for insertion from a Restricted Candidate List (RCL), which has been previously created according to the benefit of inserting each node, and depends on the current state of the solution. There is also a probabilistic element in the choice of the next node from the RCL. On the other hand, the Variable Neighbourhood Search (VNS) approach is based on systematically changing the neighbourhood move, during the optimization process, each time a local optimum is reached. The new neighbourhood move is usually of higher order than the previous move, and it is applied to the same starting solution. Variable Neigh- bourhood Descent (VND) is a variant of VNS, where the local optimum found acts as the new starting point for the local search (see Section 2.3.6 for more details about the VNS and the VND approaches). For the 1-PDP, the hybrid GRASP/VND approach in [73] is basically a GRASP, where the local search is performed using a VND procedure. In this heuristic, two VND al- gorithms have been applied with different neighbourhood moves in each algorithm. The first VND (called VND1) is applied in the improvement phase after the construction of the initial solution. The neighbourhood move used in VND1 is a classical 2-Opt, which is then followed by a 3-Opt move whenever a local optimum is reached. After the basic GRASP/VND, a further post-optimization phase is performed, using a second VND (cal- led VND2). This time, however, the neighbourhood moves applied are a move forward and a move backward operators respectively. In the move forward operator, a customer is moved from its current position to a further position in the route. So for example, if a customer in position i is moved to position j with j > i, this requires that all customers in positions i + 1, . . . , j must be shifted backwards one position. The locations of other customers in the route do not change. The move backward operator works similarly, but j is now a position that precedes i in the tour, and intermediate customers have to be shifted forward one position. The VND algorithm used in this heuristic is shown in Algorithm 9.1 and the hybrid GRASP/VND algorithm is shown in Algorithm 9.2, both adopted from [73].
- 203. 9.2 Related Work179 Algorithm 9.1: VND(x) Procedure [73]. 1: for (k = 1; k < max; k + +) do 2: x′ ← LocalSearch(x, Nk(x)) 3: if (x′ is better than x) then 4: x ← x′ 5: Return x Algorithm 9.2: Hybrid GRASP/VND Procedure [73]. 1: while stopping condition is not satisfied do 2: x ← GreedyRandomizedInitSol() {construction phase} {improvement phase} 3: x ← V ND1(x) {edge-exchange neighbourhoods} 4: if x is feasible and improves the best solution x′ then 5: x′ ← x {post-optimization} 6: x′ ← V ND2(x′) {vertex-exchange neighbourhood} 7: Return x′ A GA approach was introduced in [158] to solve the problem. The algorithm first starts by creating a population of feasible solutions using a new nearest-neighbour construction heuristic. The initial population is then optimized using a 2-Opt neighbourhood move. The most distinguishing feature of the algorithm is a new pheromone-based crossover operator, inspired from the Ant Colony Optimization (ACO) technique, where pheromone trails are updated each generation. During crossover, the selection of the next node to be inserted in the child is based on a probabilistic rule that takes into account the pheromone trail of the edge connecting the last inserted node and the potential new node, such that edges that have proved successful in the past are favoured. The crossover operator also considers the distance between the two nodes, as well as the demand of the new node and the current maximum and minimum loads carried by the vehicle. The offspring is further optimized using a 2-Opt local search, in which only feasible solutions are accepted, and only the closest neighbours to the current node are considered for edge exchanges. The mutation operator is based on a 3-exchange procedure. Since there are 5 legal ways in which 3 nodes could be exchanged, the move that yields the best result is the one performed in the mutation operator. The algorithm was tested on the benchmark data that are created by [75], producing the best so far results in most test cases. The work done in [106] presents a simulation environment to solve the problem. The algo- rithm starts with a Modified Simulated Annealing (MSA) algorithm, followed by a manual
- 204. 180 9.3 RelatedProblems improvement phase. The SA approach is called Iterative Modified Simulated Annealing (IMSA). The idea is to create multiple neighbourhood feasible solutions Y from the cur- rent feasible solution X by performing a number of changes on X. The changes are done iteratively on X, starting from 1 change to a maximum of nChangesmax. For each change value, a fixed number of Y solutions is created, and this is done at the same SA tempera- ture. This whole process is then repeated in an outer loop for a number nCountsmax of iterations. The initial solution X is created based on a Greedy Random Sequence (GRS) algorithm, which is a nearest-neighbour heuristic with some randomness in selecting the nearest customer or the second nearest customer. A final manual optimization is then performed to improve the solution. Using a Graphical User Interface (GUI), the user se- lects a subgraph (of less than 15 nodes) from the optimal result obtained, upon which an improvement algorithm is applied. This is achieved by performing an Exact Permutation Algorithm (EPA) for small size problems, or improving a selected sub-route for large size problems. The algorithm was tested on some instances created in [7], and was found to be useful for practical applications. 9.3 Related Problems Hernández-Pérez and Salazar-González in [75] presented an extensive survey of routing problems that are closely connected to the 1-PDP. Based on this survey, some important problems that are related to the 1-PDP are: • The Capacitated Traveling Salesman Problem with Pickup and Delivery (CTSPPD): which is a special case of the 1-PDP, in which the demand of each pickup or delivery customer is restricted to one unit (e.g. [5]). • The Capacitated Dial-a-Ride Problem (CDARP): in which the vehicle should move one-unit of commodity (for example a person) between pairs of customers (e.g. [124]). • The Pickup and Delivery Traveling Salesman Problem (PDTSP): which is the same as the CDARP, but there is no restriction on the vehicle capacity (e.g. [127]). • The Traveling Salesman Problem with Backhauls (TSPB): in which all delivery customers must be visited before all pickup customers (e.g. [57]). • The Traveling Salesman Problem with Pickup and Delivery (TSPPD): due to the importance of this problem and its close connection to the 1-PDP, the following discussion provides more details about this problem.
- 205. 9.4 Solving the1-PDP Using an Evolutionary Perturbation Scheme (EPS) 181 The Traveling Salesman Problem with Pickup and Delivery (TSPPD)1 This problem was introduced in [110], and it is assumed here the commodity collected from pickup customers is different from the commodity delivered to delivery customers. The depot supplies all the demand of the delivery customers and collects all the supplies from the pickup customers, and both products must be accommodated in the vehicle wi- thout exceeding its capacity. An example of an application of this problem is when empty soft drink bottles are to be collected from homes or shops and delivered to the depot, and at the same time full bottles are supplied by the depot to be delivered to some customers [75]. For a TSPPD to be feasible, the vehicle capacity should at least be the maximum among the sum of the pickup demands and the delivery demands. This condition is not required for the 1-PDP, though, since the vehicle capacity could even be as small as the largest customer demand [75]. In mathematical terms, if we let K = max{ i∈V :qi>0 qi, − i∈V :qi<0 qi}, then for a TSPPD to be feasible Q must be ≥ K. However, we can always assume, as mentio- ned in [110], that the TSPPD is in ‘standard form’, meaning that K = Q. Accordingly, we can also assume that in a feasible TSPPD route, the vehicle starts at the depot with a full load, delivers all the demand needed by the delivery customers, and at the same time collects the goods provided by the pickup customers. Finally, the vehicle will arrive at the depot fully loaded again, having collected all the goods from the pickup customers. This condition does not occur in the 1-PDP, since the product can be collected from the pickup customers and delivered to the delivery customers, with the depot only supplying or absorbing any additional amount [75]. In [74] it is observed that a TSPPD can be solved using the same algorithm as the 1-PDP if the TSPPD instance is transformed by duplicating the depot into two dummy 1-PDP customers, one collecting all the quantity supplied by the pickup customers, and the other providing all the quantity needed by the delivery customers. 9.4 Solving the 1-PDP Using an Evolutionary Perturba- tion Scheme (EPS) In this part of our research we are trying to handle the 1-PDP using a technique based on ‘perturbation’ of the problem instance, first introduced in Codenotti et al. [27] for solving large instances of the TSP. The idea is to introduce a small perturbation to the 1 This problem was briefly mentioned in our literature survey of pickup and delivery problems in Section 4.1 under the name: The Traveling Salesman Problem with Mixed Linehauls and Backhauls (TSPMB).
- 206. 182 9.4 Solvingthe 1-PDP Using an Evolutionary Perturbation Scheme (EPS) original problem instance P to transform it to a new instance P′ , for example by making a small change in the coordinates of the cities to be visited in the TSP. The new coordinates, which in turn will result in new separating distances between the nodes, will then be used to construct a solution for the new instance P′ . The new solution is then evaluated, possibly after applying some local search method to improve it, relative to the original problem instance P to potentially replace the initial solution2 . Figure 9.1(a) shows a typical product of a nearest-neighbour construction algorithm to an original small TSP instance, while Figure 9.1(b) shows an improved solution obtained after perturbing the coordinates of the original instance. This idea was extended in [151], [152], [17] and [150], by applying a search algorithm to find a perturbation that will give a better solution. For example, a GA can be applied to optimize the set of perturbed coordinates. A simple greedy algorithm could then be used to create a solution for the TSP, using the perturbed coordinates rather than the original coordinates. For the 1-PDP, we tried to apply the same idea of optimizing the set of perturbed coor- dinates, as done in [150] for the TSP, and in the PhD thesis by Matthew Morgan3 for the Capacitated Vehicle Routing Problem (CVRP) [109]. We were motivated by the success of this simple algorithm on these routing problems, as evident by the experimental fin- dings reported. For both problems, it was demonstrated that the technique was able to produce high quality solutions compared to simple solution construction heuristics wi- thout perturbation. The results obtained also compared favourably to published results from the literature in terms of both solution quality and processing time. Its applicability for the 1-PDP seemed to be viable, since a simple construction heuristic, such as the one described in [158], can be easily embedded within the evolutionary perturbation scheme, in which nodes’ coordinates could be perturbed and optimized using a simple GA tech- nique. More specifically, in our perturbation scheme, we use a straightforward GA technique with a chromosome representing a list of customer locations, with each gene consisting of the x and y coordinates of the location. Traditional crossover and mutation operators are then applied to optimize the set of perturbed coordinates. The perturbation of each coordinate is done by applying a small change to the coordinate within a pre-specified radius, as suggested in [109]. As mentioned above, the underlying construction heuristic, 2 A local search heuristic, applied to a solution for the new instance P′ could be ‘fooled’ into finding better solutions for the original instance, because a local optimum for the original instance is not necessarily a local optimum for the new transformed instance. 3 Matthew Morgan is a member of our research group (Scientific Computing and Optimization) in Cardiff School of Computer Science & Informatics, UK.
- 207. 9.4 Solving the1-PDP Using an Evolutionary Perturbation Scheme (EPS) 183 5 10 15 20 25 5 10 15 20 25 X Y (a) TSP solution - original instance 5 10 15 20 25 5 10 15 20 25 X Y Original Insstance Perturbed Instance (b) TSP solution after perturbation Figure 9.1: TSP solution before and after perturbation. used to create a solution using the perturbed coordinates, is a simple nearest-neighbour insertion heuristic, similar to the heuristic introduced in [158]. The created solution is then evaluated relative to the original coordinate set, and the cost of the solution is used as an objective function for the perturbed coordinates that were used to create the solution. Algorithm 9.3 describes the basic steps of our Evolutionary Perturbation Scheme (EPS).
- 208. 184 9.4 Solvingthe 1-PDP Using an Evolutionary Perturbation Scheme (EPS) Algorithm 9.3: The EPS Algorithm. 1: Initialize a population POP of perturbed coordinates from the original coordinates of pro- blem instance P 2: while (stopping condition is not satisfied) do 3: for (i=0; i<NumCrossovers;i++) do 4: Select parents P1 and P2 from POP, using roulette wheel selection 5: Apply crossover to parents (P1, P2) to produce child C1 6: Apply crossover to parents (P2, P1) to produce child C2 7: with some probability, apply mutation to C1 and C2 8: Generate a solution s1 from child C1 and a solution s2 from child C2, using a simple problem-specific construction heuristic 9: decode solutions s1 and s2 to produce two real solutions r1 and r2 for the original pro- blem instance P 10: Possibly apply an improvement heuristic to r1 and r2 {improvement heuristic is only applied to small problem sizes, due to time limitations} 11: Evaluate r1 and r2 and assign fitness values to C1 and C2 accordingly 12: Update POP by integrating the new generation and eliminating some worst individuals {i.e., steady state GA with overlapping populations} Although the technique in general is simple, there are some parts that need careful consi- deration during the implementation of this algorithm. A critical issue is how to deal with infeasible solutions created during the search, and how to assign a suitable objective value for these solutions, since the total travel distance will no longer reflect the true cost of an infeasible solution. In the following subsections, we explain the basic components of the EPS. 9.4.1 The Encoding The chromosome in our GA is an ordered list of customer locations starting from customer 1 to n. Each gene in the chromosome consists of the x and y coordinates of the customer location. Figure 9.2 shows the chromosome representation for both the original and the perturbed coordinates, depicted in Figure 9.1.
- 209. 9.4 Solving the1-PDP Using an Evolutionary Perturbation Scheme (EPS) 185 Figure 9.2: EPS chromosome representation. 9.4.2 The Initial Population A chromosome is created by applying a small perturbation on the coordinates of each customer location, and storing the perturbed coordinates in the corresponding gene. The process is repeated for all individuals in the population. The perturbation scheme follows the recommendation of [109], such that each coordinate is shifted a small distance within a pre-specified radius. This radius is calculated as a function of the distance between the current node and its nearest neighbour. More specifically the new coordinates are given by the following equation: newx = oldx + (D × S) × r, newy = oldy + (D × S) × r, (9.2) where r is a random number between −1 and 1, D is the distance between the current node and its nearest neighbour, and S is a scaling factor used to control the permitted shift. 9.4.3 The Nearest Neighbour Construction (NN-Construction) The construction algorithm is called whenever the objective function of the chromosome has to be calculated. The chromosome is first used to create a distance matrix reflecting the current perturbed coordinates that are stored in the chromosome. This distance matrix is then passed to a simple greedy nearest-neighbour construction (NN-Construction) al- gorithm to create a corresponding solution. The construction algorithm we adopted in our research is the one described in [158] to create their initial GA population, with a slight modification. This algorithm works as follows.
- 210. 186 9.4 Solvingthe 1-PDP Using an Evolutionary Perturbation Scheme (EPS) 1. Given CL 4 , a user defined parameter, generate the list of CL closest neighbours to each node in the customer list, based on the current distance matrix; 2. Let m = 1, insert the depot node as the first customer T(m) in the tour, and initialize both MaxLoad and MinLoad of the vehicle to be equal the demand of the depot; 3. If m = n, where n is the total number of customers including the depot, then stop; otherwise go to Step 4; 4. Search the CL closet neighbours of T(m) for feasible candidates that have not been inserted before and can be added to the end of the tour without violating the vehicle capacity. If there are such customers, select the customer i having the largest demand (in absolute value) among them, and insert it as T(m + 1), and update MaxLoad and the MinLoad of the vehicle, go to Step 5; otherwise go to Step 6; 5. m = m + 1, go to Step 3; 6. Search all customers that have not appeared in T for feasible candidates for inser- tion. If there are such customers, select the closest customer to the last inserted customer and insert it as T(m + 1) and update MaxLoad and the MinLoad of the vehicle, go to Step 5; otherwise stop and declare an infeasible solution; 7. Return m ≤ n, the number of nodes that have been successfully inserted in the solution before infeasibility (if any) is encountered. The difference between our algorithm and the construction algorithm described in [158], and used to create a genetic population of random solutions to the 1-PDP, is that they start their tour from a randomly selected customer, while we always start the tour from the depot. We chose to fix the starting node to try to make the quality of the constructed solution only a function of the perturbed coordinates used to create the solution. Any randomness in the creation of the solution will interfere in the quality of the solution and will not reflect the true fitness of the perturbed coordinates. Thus, the learning process of the GA from generation to another may be obstructed. Similarly, some degree of randomness in Step 6 of the algorithm that was adopted in [158] has been eliminated in our algorithm. This is in fact essential in our scheme, because the solution constructed from the perturbed coordinates is not saved during the evolutionary process. Making the construction algorithm deterministic, by eliminating all randomness in the selection of nodes, will enable our algorithm to use the best perturbed coordinates (individual) to re-construct the best solution, when the evolutionary process terminates. In addition to 4 CL stands for a constant that determines a selected number of closet neighbours to a certain node.
- 211. 9.4 Solving the1-PDP Using an Evolutionary Perturbation Scheme (EPS) 187 removing randomness, our algorithm stops once the solution becomes infeasible, while in [158], they repeat the construction algorithm starting from a new node, if the previous attempt resulted in an infeasible solution. The selection of the customer with the largest demand in Step 4 of the construction algo- rithm is justified in [158] by giving priority in the insertion order to nodes that may be difficult to insert, because of their large demand. Also, as mentioned in Steps 4 and 6, the minimum load and the maximum load carried by the vehicle are updated each time a new node is inserted. This step is needed for the feasibility check of each candidate node. More specifically, according to Equation 9.1, to check the feasibility of insertion, the difference between MaxLoad and MinLoad after serving the candidate node, should not exceed its capacity, otherwise the potential candidate cannot be feasibly inserted at the end of the tour. 9.4.4 The Objective Function To calculate the fitness of each chromosome in the population, the following steps are per- formed. First, a new distance matrix is calculated based on the set of perturbed coordinates stored in the chromosome. Then, this new distance matrix is passed to the construction algorithm, described in Section 9.4.3, to create a solution S that takes into account the new distances between the nodes. Finally, the objective function of a chromosome (C) is calculated using the following equation F(C) = ((n + 1) − m) × Dist(S) (9.3) where n is the total number of nodes, m is the number of nodes that have been success- fully inserted in the solution before infeasibility (if any) is encountered, and Dist(S) is the ‘true’ total distance traveled when the solution S is decoded relative to the original coordinate set. This objective function combines both the total travel distance and the degree of infeasi- bility in the generated solution. If the created solution is feasible, all nodes would have been inserted successfully, i.e. m = n. Accordingly, the fitness of this solution will be Dist(S). On the other hand, if the solution is infeasible, Dist(S) will be multiplied by a factor that increases with the degree of infeasibility. The smaller the number of nodes that have been successfully inserted, the larger the multiplication factor. Note also that in case of an infeasible solution, the distance value will be smaller than the distance of a feasible solution, because not all nodes have been successfully inserted.
- 212. 188 9.4 Solvingthe 1-PDP Using an Evolutionary Perturbation Scheme (EPS) 9.4.5 The Operators Mutation The mutation operator first selects the number of nodes (genes) to be mutated. This num- ber was chosen to be a random number between 1 and 20% of the total number of nodes in the chromosome. Then, the mutation operator applies Equation 9.2 to each gene selected at random. Crossover Different traditional crossover operators have been tried in our research. The most ef- fective crossover operator seems to be the EvenOdd crossover operator, which works by selecting alternative genes from each parent and inserting them in the corresponding lo- cations in the child. The roles of the parents are reversed for the second child. 9.4.6 Computational Experimentation We implemented a steady state GA, with an 80% replacement, and the following parame- ters: population size= 200, number of generations=1000 for large problems (more than 60 customers) and 200 for small problems (from 20 to 60 customers), probability of muta- tion= 0.3, probability of crossover=0.8, and the perturbation scaling factor S in Equation 9.2 was set to 0.6, after some initial experimentation with different values. The neighbou- rhood size CL was chosen to be 2, i.e., only the closest two neighbours are checked for feasibility of insertion before all other nodes are checked. We found, experimentally, that this ‘tight’ neighbourhood size helped to reduce the runtime, since the neighbourhood list, which has to be created for all nodes whenever the construction algorithm is called, will be very small in size. Also, it appeared from trying several other values that this small value gave better results even for large size problems. The Data Set: The algorithm was tested on instances created by [75]. There are 2 types of problem instances. Small instances have a number of customers n in {20, 30, 40, 50, 60}. For these instances, the optimum is known and was obtained using the exact method proposed in [74]. There are also large instances with n in {100, 200, 300, 400, 500}. For each combination of n and a different vehicle capacity Q in {10, 15, 20, 25, 30, 35, 40, 45, 1000}, 10 problem instances have been created and given the letters {‘A’ to ‘J’}. So for example, a problem instance named N100q20A, means the first instance (A) in
- 213. 9.4 Solving the1-PDP Using an Evolutionary Perturbation Scheme (EPS) 189 the 100-customers category (N100) with vehicle capacity 20 (q20). The data set and the results obtained in [75] and [73] can be downloaded from the Pickup and Delivery Site of Hernández-Pérez 5 : http://webpages.ull.es/users/hhperez/PDsite/index.html Experimental Results: We briefly summarize here the experimental results of the EPS on test cases with the tightest vehicle capacity. These are the problems with Q = 10, and considered to be the hardest problems in the data set. We summarize the results on all problem sizes from 20 to 100. Our perturbation heuristic was run 20 times on each of these instances and the best result was recorded. Larger problem size, however, were found to be very time consuming and their testing was dealt with differently as will be explained later. It should also be noted that there is a slight difference between the final testing version of the heuristic between small instances and large instances. This was based on preliminary experimentations and was intended to give the best possible results. For small instances (from 20 to 60 customers), the solution obtained from the NN-construction heuristic, ex- plained in Section 9.4.3, was optimized using a Hill Climbing (HC) approach that uses a simple node swap neighbourhood move. The objective function (Equation 9.3) then uses the quality of the optimized solution as a fitness of the perturbed coordinates from which the solution was created. This is the same approach applied in the perturbation scheme of the CVRP in [109]. For larger problems, however, this technique could not be used due to the great increase in processing time. Therefore, the objective function was only based on the quality of the constructed solution without optimization. An optimization phase, though, was performed only on the best individual after the termination of the whole GA process. Thus, the set of perturbed coordinates that are represented in the best individual in the final generation, were used to construct the best solution obtained, using the NN- construction algorithm. This solution is then further optimized using an HC approach that uses a simple move forward operator, which was suggested in [73] (see Section 9.2 for more details about the move forward operator) . The results summarized here are the final results obtained after this optimization phase. Results on Problem Sizes 20-100 Customers: Experimental results on problem sizes ranging from 20-100 customers indicated that the perturbation heuristic was able to achieve the optimal result in only 16 out of the 50 test instances, where the optimum is known (i.e., for problem sizes 20-60). Also, the optimum was achieved only for the smallest size pro- 5 New best results were obtained by the GA in [158], but they do not appear in the pickup and delivery site yet.
- 214. 190 9.4 Solvingthe 1-PDP Using an Evolutionary Perturbation Scheme (EPS) blems of 20 and 30 customers. The heuristic, on the other hand, was unable to compete with previous heuristics from the literature for all larger size problems. The average re- lative difference between our results and the optimal or best known results ranged from 2% for 40 customers to 12% for 100 customers. The heuristic was also slower than the heuristic in [158]. For example the average run time for 100-customers problems was 43.8 seconds compared to 21.12 seconds reported by [158] for the same problem types. Thus, the experimental results clearly indicate that the EPS achieved a limited success in dealing with the 1-PDP problem instances, especially as the problem size increases. The- refore, testing the algorithm on even larger size problems did not seem worthwhile. The perturbation scheme in all problem instances, though, always produced feasible solutions. This indicated to us that the heuristic may be used in cases where a quick and feasible solution is needed irrespective of its quality. Results on Problem Sizes 200-500 Customers: For these problem sizes, we shifted our attention to testing the potential of the EPS in transforming infeasibility to feasibi- lity, rather than competing with best known results. Hence, we only ran the algorithm for one generation and compared the result obtained, in terms of the quality of the solu- tion constructed from the best individual in the population, with the result obtained when the NN-construction heuristic was run only once to produce an initial solution (using the original problem coordinates without perturbation). Remember that the NN-construction heuristic is deterministic in our scheme, since we removed all the randomness involved in the construction of such solution. Accordingly, there is only one initial solution that can be constructed using the original coordinates. If the NN-construction managed to insert all nodes in the solution whilst maintaining feasibility, the created solution is feasible. Otherwise, the quality of the created solution is determined by the number of nodes that have been successfully inserted before infeasibility is encountered. In the current expe- riment, the algorithm was again run 20 times on each problem instance, and we counted the number of times the EPS heuristic was able to generate a feasible solution after one generation. The results of this experiment showed that the NN-construction heuristic failed to produce feasible solutions in all cases except one, which is the instance N400q10F. On the other hand, the EPS heuristic was able to find feasible solutions for all problem instances after one generation. In most test cases, the EPS was able to find a feasible solution in all 20 runs. In 9 out of the 40 problem instances the EPS heuristic did not produce a feasible solution in all runs. The minimum number of feasible solutions obtained were 3 out of the 20 runs.
- 215. 9.5 Summary andFuture Work 191 These results, in general, seem to indicate that the EPS heuristic was in fact successful in transforming the frequent infeasibility problem of the NN-construction heuristic to feasi- bility. The processing time needed was also reasonable with an average of 1.13 seconds for 200-customers problems to 7.18 seconds for 500-customers problems. 9.5 Summary and Future Work In this part of the research we investigated a new heuristic for the 1-PDP. The heuristic is a perturbation scheme that transforms the problem instance into a new one by performing a small shift in the coordinates of the nodes in the problem instance. A simple construction heuristic is used to construct a solution to the problem from the perturbed coordinates rather then the original coordinates, hoping to produce a better quality solution. The perturbed coordinates were also optimized using a simple GA technique. The experimental results on a large number of test cases of different sizes indicated that the heuristic was in most cases able to improve the quality of the initial constructed solution to a large extent, removing infeasibility in all test cases. However, the final results obtained were in general of lesser quality than the best known or optimal results. The algorithm was also relatively slow, which is mostly due to the nature of the GA involved. It should be noted that the EPS algorithm is highly sensitive to several underlying factors. The most important element seems to be the embedded construction heuristic. Previous attempts in the literature that used a similar technique, for example [109], made a consi- derable effort in testing several construction methods for the CVRP, to identify the best construction heuristic that can be used within the EPS. Nevertheless, the literature of the CVRP is relatively well-established and very rich with such heuristics. The literature on the 1-PDP, on the other hand, is scarce and only a small number of construction heuris- tics have been previously attempted. We chose from them the construction algorithm that seems to be relatively simple and fast, and also gave reasonable results within the context of other meta-heuristic techniques. This construction heuristic, however, apparently pro- duce frequent infeasible solutions, as illustrated by the results obtained on large problem sizes. Having a large number of infeasible solutions will lower the quality of the overall population of perturbed coordinates, and make identifying the set of perturbed coordinates that lead to feasible and good quality solutions a difficult task for the EPS. In addition, the EPS is also highly sensitive to the perturbation parameters and the selected perturbation neighbourhood shape and size around the coordinates. Fine tuning of these parameters was also a considerable part of the research done in [109]. In our research, we
- 216. 192 9.5 Summaryand Future Work simply relied on the final recommendations of [109] for their CVRP. However, obtaining the best perturbation parameters for the 1-PDP could allow the heuristic to achieve better results, but will, nevertheless, be very time consuming. Another possibility for improvement is to modify the EPS, such that only feasible solution are allowed in the evolutionary process. Alternatively, the construction algorithm could be slightly adjusted, such that it will continue inserting nodes in the solution even after infeasibility is encountered. The objective function, then, should take into account the amount of infeasibility in the solution. This approach could allow the perturbation scheme to more easily transform the infeasibility to feasibility and improve the overall quality of the generated solutions. The viability of this ‘modified’ construction technique was demonstrated within the context of our VNS heuristic that was applied to the 1-PDP, as will be detailed in the next chapter.
- 217. 193 Chapter 10 Solving theOne-Commodity Pickup and Delivery Problem Using an Adaptive Hybrid VNS/SA Heuristic In our attempt to investigate possible heuristics to solve the 1-PDP, and perhaps achieve better results than those achieved by the EPS heuristic, Variable Neighbourhood Search (VNS) seemed a reasonable choice that may have some potential. First, the idea is simple and can be easily adapted to the problem in hand. Second, it has been successfully ap- plied to many routing and scheduling problems, for example to the TSP in [70] and the VRPTW in [19] and [119], and recently to the Periodic Vehicle Routing Problem with Time Windows (PVRPTW) in [118]. In this chapter, we explain our investigation of VNS for solving the 1-PDP. The VNS algorithm introduced here is hybridized with Simula- ted Annealing (SA) to escape local optima. We also employ adaptation of some search parameters for more efficient searching. Section 10.1 briefly describes the VNS meta-heuristic in general, and highlights previous 1-PDP research that uses this approach. Section 10.2 explains in detail our proposed heuristic, which we will call an Adaptive Hybrid VNS/SA (AVNS-SA) technique for solving the 1-PDP. An outline of the complete algorithm is presented in Section 10.3. Experimental results of the algorithm are presented in Section 10.4, and a summary and some future directions will be presented in Section 10.5. 10.1 Variable Neighbourhood Search (VNS) and its Ap- plication to the 1-PDP As previously mentioned in Section 2.3.6, VNS is a relatively new meta-heuristic that has been introduced by Hansen and Mladenovi´c in [69] and [70]. The idea is to generate
- 218. 194 10.1 VariableNeighbourhood Search (VNS) and its Application to the 1-PDP new solutions that are distant from the incumbent solution, by systematically increasing the neighbourhood size within which the search is performed. The new solution replaces the current solution if it is better in quality. This way, many favourable characteristics of the incumbent solution will be preserved in the new generated solution. In addition, a local search is performed on the new solution to reach a local optimum within the current neighbourhood. For convenience, we repeat here the basic steps of the VNS algorithm as described in [70]: • Initialization: Select the set of neighbourhood structures Nk, (k = 1, ...kmax), that will be used in the search; find an initial solution x; choose a stoping condition; • Repeat the following until the stopping condition is met: 1. Set k ← 1; 2. Repeat the following steps until k = kmax: (a) Shaking: Generate a point x′ at random from the kth neighbourhood of x (x′ ∈ Nk(x)); (b) Local Search: Apply some local search method with x′ as initial solution; denote with x′′ the so obtained local optimum; (c) Move or not: if the local optimum x′′ is better than the incumbent, move there (x ← x′′ ), and continue the search with N1 (k ← 1); otherwise set k ← k + 1. A variant of VNS, called Variable Neighbourhood Descent (VND)1 , has been tried for the 1-PDP as part of the heuristic proposed by Hernández-Pérez and Salazar-González in [73]. In their approach, the VND did not include a shaking phase, but only a local search that changes the neighbourhood move from 2-Opt to 3-Opt, whenever a local optimum is reached. The VND was embedded within another heuristic called GRASP (Greedy Ran- domized Adaptive Search Procedure) that repeatedly generates an initial solution upon which the VND is applied. Their approach is described in Algorithms 9.1 and 9.2 in the previous chapter. The heuristic in [73] achieved promising results that were better than the results obtained by the same authors using their previous heuristics suggested in [75]. However, their algorithm apparently was not fully capable of escaping the trap of local optima. This is evident by the fact that they had to use a post-optimization phase to im- prove the final result. According to the authors, this post-optimization often made the difference between beating the results obtained by their previous heuristic in [75] or not. 1 See section 2.3.6 for more details about the VND algorithm.
- 219. 10.2 The AVNS-SAHeuristic 195 The hybrid GRASP/VND heuristic was also outperformed by the GA of [158], in most test cases. A possible shortcoming of the hybrid GRASP/VND heuristic is the absence of a shaking phase, which should help the diversification of the incumbent solution and allow the suc- ceeding local search method to escape local optima. In our proposed approach, we try to apply the basic VNS, with both the shaking and the local search, hoping to overcome the limitation of this previous VND attempt on the 1-PDP. 10.2 The AVNS-SA Heuristic To apply VNS to the 1-PDP, several choices should be made. These are outlined and explained below: 1. The construction heuristic used to create the initial solution: The construction algorithm we used is the same as the algorithm proposed by [158], which is the nearest-neighbour construction (NN-construction) heuristic, explained in Section 9.4.3. 2. The set of neighbourhood structures used for shaking: This is in fact the core of the VNS, and is the tool for diversification of the search. It is essential that the shaking procedure provides a balance between enough perturbation of the current solution, and also maintaining the most favourable characteristics of this solution. The shaking procedure should also allow the systematic change of the neighbou- rhood size. In our algorithm, we chose a simple move that displaces a sequence of nodes with or without inversion2 , where the number of nodes to be displaced could act as the variable neighbourhood size parameter that changes during the search. 3. The local search procedure used to optimize the current solution: This proce- dure is the tool that the VNS uses for intensification of the search. In our algorithm, we chose as a local search a simple 2-Opt edge exchange algorithm, based on the famous procedure proposed by Lin in [103]. We adopted the implementation des- cribed in [157] for the 2-Opt edge exchanges, having a computational complexity of O(n2 ), where n is the number of nodes in the problem instance. In addition, our local search is based on exhaustively testing all possible edge exchanges, and using best improvement as a replacement strategy. 2 Inversion means to reverse the sequence of nodes in a manner similar to the inversion mutation operator shown in Figure 2.7(b)
- 220. 196 10.2 TheAVNS-SA Heuristic 4. The acceptance criterion used by the VNS to accept a new solution as a replace- ment of the incumbent solution: A descent only criterion will only accept a better solution, while a descent-ascent criterion will also accept a worse solution with some probability. Both techniques can be used within the VNS procedure as explai- ned in [70]. In our approach we chose the second criterion, such that the acceptance is based on an SA acceptance probability, given a certain current temperature value. In what follows we describe in detail how our proposed AVNS-SA approach was planned and developed. 10.2.1 The Initial Solution To choose the initial solution that will be subject to the VNS, a number of solutions were created using the NN-construction heuristic (adopted from the algorithm in [158] and explained in Section 9.4.3), and the best solution was selected. However, in order to create different solutions, we re-introduced the randomness that has been removed in the EPS heuristic, for reasons previously explained. Specifically, the starting node was chosen at random, and also there was a small probability of choosing a random customer, rather than the nearest customer, in Step 6 of this algorithm (refer to the steps of the NN-construction algorithm in Section 9.4.3). Also, rather than stopping the construction algorithm once infeasibility is encountered, we changed the algorithm so that if all remaining nodes are infeasible for insertion, the closest one to the last inserted node is selected, even if it is infeasible, hoping to fix the infeasibility later during the search. Thus, all nodes will be inserted in the solution and the degree of infeasibility is taken into consideration when calculating the cost of the solution. The value of CL, which is the neighbourhood size used to create the matrix of nearest neighbours to each node, was chosen to be 15 in the construction algorithm of [158]. In our approach, though, we tried all values of CL from 2 to 15 on selected test cases from both the small and the large size instances3 . The value of CL that worked best for all problem instances tried was 7. Also, unlike the EPS heuristic, where a new neighbourhood matrix was created with each set of perturbed coordinates, the current approach requires only one static matrix to be created at the beginning of the run. Accordingly, a very ‘tight’ neighbourhood size was no longer needed to improve the run time4 . For convenience, we show in detail below the steps of our construction algorithm: 3 We selected test cases: N20q10A, N60q10A, N100q10A, and N500q10A. 4 Recall that the value chosen for CL in the construction algorithm of our EPS heuristic, explained in Chapter 9, was 2.
- 221. 10.2 The AVNS-SAHeuristic 197 1. Given CL, a user defined parameter, generate the list of CL closest neighbours to each node in the customer list; 2. Let m = 1, insert a random node as the first customer T(m) in the tour, and initialize both MaxLoad and MinLoad of the vehicle to be equal to the demand of this customer; 3. If m = n, where n is the total number of customers including the depot, then stop; otherwise go to Step 4; 4. Search the CL closet neighbours of T(m) for feasible candidates that have not been inserted before and can be added to the end of the tour without violating the vehicle capacity. If there are such customers, select the customer having the largest demand (in absolute value) among them, insert it as T(m + 1), and update MaxLoad and the MinLoad of the vehicle, then go to Step 5; otherwise go to Step 6; 5. m = m + 1, go to Step 3; 6. Search all customers that have not appeared in T for feasible candidates for inser- tion. If there are such customers, select the closest feasible customer to the last inserted customer with probability 0.7, or select a random feasible customer with probability 0.3; if there are no remaining feasible customers, select the closest in- feasible customer to the last inserted node; insert the selected node as T(m+1) and update MaxLoad and the MinLoad of the vehicle; go to Step 5. 10.2.2 The Objective Function The objective function used to estimate the solution quality was set to F(S) = (NCV (S) + 1) × Dist(S), (10.1) where NCV (S) is the number of capacity violations along the route, i.e., the number of nodes at which the feasibility check, described by Equation 9.1, is not satisfied. At these locations, the vehicle could be ‘carrying’ an excess of its allowed capacity, or could be in shortage of the necessary quantity to be delivered. The larger the number of capacity violations, the larger the cost of the solution. Dist(S) is the total distance of the solution, given the current visiting order of nodes. If there are no capacity violations in the route, i.e., the route is feasible, the total distance will be the sole measure of the solution quality. Thus, this multiplicative objective function will penalize the number of capacity violations in the route, by significantly increasing the cost of solutions that have a large number of
- 222. 198 10.2 TheAVNS-SA Heuristic capacity violations, compared to feasible solutions, whose cost will only be measured by the total distance traveled. In addition, no arbitrary fine tuning of penalty weights is needed for this objective function. 10.2.3 The Initial SA Temperature As previously mentioned, we used an SA acceptance criterion within the VNS approach. A critical part of any SA algorithm is the SA schedule, especially the starting temperature and the temperature reduction factor. These parameters significantly affect the perfor- mance of the SA algorithm, and their adjustment to fit a wide range of problem instances and sizes is a daunting task and very time consuming. To calculate the SA starting tem- perature for each problem instance individually, we again adopted the approach proposed by [40]. Recall that this approach was used in both our SA algorithms that handled the SV-PDPTW and the MV-PDPTW, as explained in Chapters 5 and 8. The procedure for creating the initial solution and calculating the SA temperature is shown in Algorithm 10.1. Algorithm 10.1: Find Initial Solution & Calculate Starting Temperature. 1: Create a starting solution s using the NN-construction heuristic 2: s∗ ← s {s∗ is the best so far solution} {Initialize Pstart, the starting acceptance probability} 3: Let Pstart = a large value {We used 0.9} 4: ∆avg ← 0 {Generate n solutions using the NN-construction (We used n = 1000)} 5: for (i = 0; i < n; i + +) do 6: generate a new solution s′ using the NN-construction heuristic 7: if (Objective(s′) < Objective(s∗)) then 8: s∗ ← s′ 9: ∆ ← |Objective(s′) − Objective(s)| 10: ∆avg ← ∆avg + ∆ 11: ∆avg ← ∆avg/n 12: T0 ← −∆avg/ log(Pstart) 13: Return s∗, the initial solution from which the AVNS-SA algorithm will progress, and the initial SA temperature T0
- 223. 10.2 The AVNS-SAHeuristic 199 10.2.4 The Shaking Procedure As mentioned previously, we chose as a shaking procedure a displacement of a sequence of nodes with some probability of inverting this sequence (a 50% chance of inversion was adopted in our algorithm). This move is popular in the VRP literature, and especially for solving the TSP, but to the best of our knowledge it has not been tried before for the 1-PDP. Our VNS algorithm passes the current neighbourhood size (NhSize) as a parameter to the shaking procedure, which will in turn use this parameter as the maximum possible number of nodes that will be displaced. Specifically, the number of nodes to be displaced is a random number between 1 and NhSize. So even for large values of NhSize, small sequences of nodes could still be displaced, and in fact there is a ‘bias’ toward such small moves, since they have a chance of being executed in all neighbourhood sizes. This is intended to prevent a large disruption of the current solution, and is recommended by some VNS implementations, as in [119] for the VRPTW. Both the starting position of the selected sequence and its new position within the route are chosen at random. 10.2.5 The Maximum Neighbourhood Size Our VNS algorithm repeats the shaking followed by the local search for the current solu- tion for all NhSize = 1, 2, 3...NhSizemax, where NhSizemax is the maximum sequence of nodes that could be displaced. NhSizemax should be chosen in a way that allows enough perturbation of the solution without disturbing its favourable characteristics. Na- turally, the value for NhSizemax in our shaking procedure must be smaller than n, where n is the total number of nodes in the current problem instance. Also, to make NhSizemax adaptable for any problem instance, it must be calculated relative to n, and not fixed for all instances. However, large values of NhSizemax will increase the computational cost and slow down the optimization process. So, in order to reduce the computational cost, we tried different fractions of n (n/2, n/3, n/4), on selected problem instances from both the small and large problem instances5 . These values of NhSizemax, though, still resul- ted in a very long processing time for large problem instances. We then tried the value 2 × √ n, which was recommended by [106], for the number of changes (nChangesmax) in their Iterated Modified Simulated Annealing (IMSA) approach to the 1-PDP, as pre- viously explained in Section 9.2. NhSizemax = 2 × √ n gave the best results among all values tested, in terms of the balance between processing time and solution quality, for all instances tried in this experiment. 5 Again we used test cases: N20q10A, N60q10A, N100q10A, and N500q10A.
- 224. 200 10.2 TheAVNS-SA Heuristic 10.2.6 A Sequence of VNS Runs During our early experimentation with the VNS algorithm, we found that to reach good solutions for the 1-PDP, several runs of the VNS procedure should be performed. Each run starts from the final solution obtained in the previous run. The VNS could be repeated for a fixed number of iterations, or until no improvement is realized in the current solution for a number of attempted iterations. We chose the second approach, and stopped the repetition of the VNS when no improvement happens in 5 consecutive iterations. However, we realized that during the first VNS run, improvement happens quickly for most neighbourhood sizes, even for the large ones among them. Subsequent VNS runs, though, usually respond only to smaller changes in the solution. In other words, smaller neighbourhood sizes seem to be more beneficial in subsequent VNS runs, since larger changes may cause a disturbance of the current solution and reduce its quality. Accordin- gly, after each VNS run, NhSizemax was reduced by a fraction of its value. To choose the reduction scheme, we tried the fractions 1/2, 1/3, 1/4 and 1/5 of NhSizemax, on the same selected test instances previously indicated in Sections 10.2.1 and 10.2.5. We finally chose the reduction factor 1/4, based on the quality of the results obtained. As previously mentioned, the VNS procedure is repeated several times, and only stops when no improvement happens in the current solution for a number of consecutive ite- rations. Thus, reducing NhSizemax cannot continue indefinitely, because otherwise la- ter runs may not perform any shaking at all. In order to maintain a reasonable num- ber of nodes to be displaced in the shaking procedure during later runs, the reduction of NhSizemax is repeated until it reaches a certain minimum value, at which stage no fur- ther reduction is performed, and the VNS procedure uses the current NhSizemax for all remaining runs. The minimum value of NhSizemax was chosen to be the same as the re- duction factor, i.e., NhSizemax/4, in order to reduce the number of parameters that need to be adjusted in the algorithm. 10.2.7 Stopping and Replacement Criteria for Individual VNS Runs The VNS procedure repeats the shaking and the local search for all values of NhSize = 1, 2, 3...NhSizemax. However, in some cases, the current solution may not respond to changes in the neighbourhood size and reach a stage of stagnation. Therefore, rather than indiscriminately increasing NhSize up to the pre-specified maximum, we chose to also end each VNS run when the solution has not changed for a certain number of consecutive attempts of increasing NhSize. For more flexibility and robustness, the
- 225. 10.2 The AVNS-SAHeuristic 201 number of attempts should be chosen relative to the current maximum neighbourhood size (NhSizemax), and not fixed for all problem instances. Similar to how the reduction scheme was chosen (as explained in Section 10.2.6), the number of attempts was again set to NhSizemax/4. Thus, the VNS will be adaptive in the sense that it will stop the shaking and the local search cycle, when no benefit seems to be realized from increasing NhSize. It is also adaptive from another perspective, since NhSizemax passed to the VNS is not fixed and depends on the current stage of the run, as previously discussed in Section 10.2.6 Also, as indicated before, we chose an SA acceptance criterion to replace the current solution within each VNS run. Initial experimentation showed that accepting worse so- lutions with some probability improved the final result obtained, although it is more time consuming. Thus, the SA component of the algorithm works by allowing the new solution resulting after the shaking and the local search to replace the current solution, even if it is worse in quality. The acceptance criterion is the same as the usual SA acceptance and depends on the difference between the objective value of the new solution and the current solution, and the present temperature value. In our algorithm, the VNS procedure repeats the shaking and the local search for the same neighbourhood size (NhSize) for a number of trials. The number of trials is incremented only when a solution worse than the incumbent appears in the current NhSize. When the number of trials reaches a certain pre-defined limit, the shaking and the local search cycle stops for the current NhSize, and the VNS moves on to the next NhSize. Also for each NhSize, the current temperature is decremented in the current iteration each time the new solution is worse than the current solution. 10.2.8 Updating the SA Starting Temperature Normally, by the end of each complete VNS run, the SA temperature would have reached a small value that should not permit the acceptance of any worse solutions. If we started the new VNS run with such a small value, there would be no benefit to the SA acceptance, since all worse solutions would be rejected. On the other hand, starting a new VNS run with the initial temperature too high is also not beneficial, since many worse solutions would be accepted, possibly causing the destruction of the current solution. To achieve a balance between these two situations, the final temperature value reached in the current VNS run was doubled before the beginning of the next VNS run.
- 226. 202 10.3 TheComplete AVNS-SA Algorithm 10.3 The Complete AVNS-SA Algorithm To put it all together, Algorithm 10.2 shows the main Adaptive VNS-SA (AVNS-SA) heuristic, which will invoke the VNSSA procedure, described in Algorithm 10.3. Algorithm 10.2: Adaptive VNS-SA (AVNS-SA) Algorithm. 1: Find an initial solution (InitSol) and the starting temperature (StartTemp) using Algorithm 10.1. 2: NhSizemax ← 2 × √ n, where n is the number of nodes 3: Decrement ← NhSizemax/m1 {We used m1 = 4} 4: MaxStagnation ← NhSizemax/m2{We used m2 = 4} 5: Initialize MaxAttempts to a small number {We used 5} 6: NoImprovement ← 0 7: repeat 8: NewSol = V NSSA(InitSol, NhSizemax, StartTemp, MaxStagnation) 9: if (NhSizemax > Decrement) then 10: NhSizemax ← NhSizemax − Decrement 11: else 12: NhSizemax ← Decrement 13: if (NewSol is not better than InitSol) then 14: NoImprovement + + 15: else 16: NoImprovement ← 0 17: InitSol ← NewSol 18: StartTemp ← StartTemp × 2 19: until (NoImprovement reaches MaxAttempts)
- 227. 10.3 The CompleteAVNS-SA Algorithm 203 Algorithm 10.3: The VNSSA Algorithm. 1: Input: InitSol, NhSizemax, StartTemp, MaxStagnation 2: Output: a new, possibly improved, solution X 3: k ← 0 { Initialize the current neighbourhood size k } 4: Stagnation ← 0 5: NumTrials ← LIMIT {LIMIT is the maximum allowed number of trials for the current neighbourhood size (we used 30 trials)} 6: X ← InitSol 7: repeat 8: k + + { Increment the current neighbourhood size} 9: Trials ← 0 10: while (Trials < NumTrials) do 11: Shaking(X, XI, k) {displacing a sequence of nodes in X up to a maximum of k, with or without inversion. The result is stored in XI} 12: LocalSearch(XI, XII) {local search is done on XI using 2-Opt. The result is stored in XII} 13: if (Objective(XII) < Objective(X)) then 14: X ← XII 15: else 16: Accept XII using SA acceptance probability 17: StartTemp ← StartTemp × α {Decrement current temperature (we used α = 0.99)} 18: Trials + + 19: end while 20: if (X did not change in the last iteration (i.e., for the current neighbourhood size k)) then 21: Stagnation + + 22: else 23: Stagnation = 0 24: until (Stagnation = MaxStagnation) or (k = NhSizemax) 25: Return X
- 228. 204 10.4 ExperimentalResults 10.4 Experimental Results To test the performance of our AVNS-SA algorithm, we used the same test cases described in Section 9.4.6. We ran the algorithm 5 times on each test case from 20-300 customers. On the other hand, only one run was performed on test cases of 400 and 500 customers, due to time limitation. In this experiment a number of computers with different specifica- tions were used to run the algorithm. For this reason, the run times we quote in this section will vary according to the platform. Nevertheless, our timings give useful estimates of the time requirements of the algorithm. Table 10.1 shows the results achieved by the AVNS-SA algorithm for small size problems of 20-60 customers. As previously explained in Section 9.4.6, a problem instance named N20q10A, for example, means the first instance (A) in the 20-customers category (N20), with vehicle capacity 10 (q10). Table 10.1 shows the best result obtained in the 5 runs, the number of times the best result appeared in the 5 runs (Num Seen), and the average result of the 5 runs. Finally the table also shows the optimum result found by the exact algorithm in [74]. Table 10.1: AVNS-SA Results (20-60 Customers) Name Best Num Seen Average Optimum N20q10A 4963 4 4974.6 4963 N20q10B 4976 5 4976 4976 N20q10C 6333 3 6390.4 6333 N20q10D 6280 4 6341 6280 N20q10E 6415 5 6415 6415 N20q10F 4805 3 4808.6 4805 N20q10G 5119 5 5119 5119 N20q10H 5594 5 5594 5594 N20q10I 5157 2 5195.2 5130 N20q10J 4410 5 4410 4410 N30q10A 6403 1 6455 6403 N30q10B 6603 5 6603 6603 N30q10C 6486 1 6576.2 6486 N30q10D 6652 2 6746 6652 N30q10E 6070 5 6070 6070 N30q10F 5737 4 5817.4 5737 N30q10G 9371 1 9388.2 9371 Continued on Next Page. . .
- 229. 10.4 Experimental Results205 Table 10.1 . . . Continued from Previous Page Name Best Num Seen Average Optimum N30q10H 6431 1 6451.4 6431 N30q10I 5821 3 5909 5821 N30q10J 6271 3 6344.2 6187 N40q10A 7173 2 7246.2 7173 N40q10B 6557 1 6621.2 6557 N40q10C 7528 3 7548.2 7528 N40q10D 8073 1 8206.2 8059 N40q10E 6928 4 6941.4 6928 N40q10F 7506 1 7577.4 7506 N40q10G 7669 2 7707.4 7624 N40q10H 6791 2 6870.6 6791 N40q10I 7215 1 7267.8 7215 N40q10J 6512 4 6516.4 6512 N50q10A 6987 3 7031.2 6987 N50q10B 9488 1 9603.6 9488 N50q10C 9110 1 9178.4 9110 N50q10D 10464 1 10678 10260 N50q10E 9492 1 9644 9492 N50q10F 8684 2 8747.2 8684 N50q10G 7126 1 7240.8 7126 N50q10H 8885 1 8982.6 8885 N50q10I 8404 1 8486.8 8329 N50q10J 8456 3 8638.6 8456 N60q10A 8602 2 8646.8 8602 N60q10B 8514 2 8571.8 8514 N60q10C 9483 1 9553.8 9453 N60q10D 11061 1 11324.2 11059 N60q10E 9487 1 9638 9487 N60q10F 9063 1 9250 9063 N60q10G 8998 2 9101.8 8912 N60q10H 8424 1 8473.4 8424 N60q10I 9524 1 9577.2 9394 N60q10J 8844 1 8982.2 8750
- 230. 206 10.4 ExperimentalResults Table 10.1 shows that the algorithm was able to achieve the optimum results at least once in the 5 runs for 39 out of the 50 test cases. These are shown in boldface in the table. The maximum relative difference to the optimum was less than 2% among the 11 cases where the optimum was not found, which was for test case N50q10D. The processing time was reasonable with an average ranging from 0.66 seconds for 20 customers problems to 47.79 seconds for 60 customers problems. Table 10.2 shows the results of the AVNS-SA algorithm on large size problems, from 100 to 500 customers. The table shows the best result achieved and the average result of the 5 runs. Note that this average is replaced by the best result for problems of size 400 and 500, since the algorithm was run only once on these problems. Finally, the previous best known results are also shown in the last column. Only 6 out of the previous best known results are attributed to [73], which are the results for test cases (N300Q10C, N400Q10A, N500Q10A, N500Q10D, N500Q10E, N500Q10H). All remaining previous best results were found by the GA in [158]. The best result achieved by the AVNS-SA is shown in boldface if it was better than the previous best known result. Table 10.2: AVNS-SA Results (100-500 customers) Name Best Average Previous Best N100q10A 11741 12173.8 11828 N100q10B 13066 13410.6 13114 N100q10C 13893 14073.8 13977 N100q10D 14328 14567.2 14253 N100q10E 11430 11823.6 11411 N100q10F 11813 11947 11644 N100q10G 12025 12118 12038 N100q10H 12821 12844 12818 N100q10I 14025 14278.6 14032 N100q10J 13476 13642.8 13297 N200q10A 17690 17849.2 17686 N200q10B 17618 17887.8 17798 N200q10C 16535 16626.6 16466 N200q10D 21228 21594.2 21306 N200q10E 19220 19485.2 19299 N200q10F 21627 21677.4 21910 N200q10G 17361 17634 17712 N200q10H 20953 21191.4 21276 Continued on Next Page. . .
- 231. 10.4 Experimental Results207 Table 10.2 . . . Continued from Previous Page Name Best Average Previous Best N200q10I 18020 18328.2 18380 N200q10J 19016 19240.4 18970 N300q10A 22940 23163 23242 N300q10B 22473 22920.4 22934 N300q10C 21183 21454 21800 N300q10D 25220 25500.6 25883 N300q10E 26636 26934 27367 N300q10F 24042 24290.6 24826 N300q10G 23683 23945 23868 N300q10H 21555 21824.6 21625 N300q10I 23871 24110.2 24513 N300q10J 22503 22688.8 22810 N400q10A 30657 30657 31486 N400q10B 24248 24248 24262 N400q10C 27853 27853 28741 N400q10D 23750 23750 24508 N400q10E 24798 24798 25071 N400q10F 26625 26625 26681 N400q10G 23925 23925 23891 N400q10H 25628 25628 25348 N400q10I 28262 28262 28714 N400q10J 24847 24847 26010 N500q10A 27904 2790 28742 N500q10B 26612 26612 26648 N500q10C 30247 30247 30701 N500q10D 29875 29875 30794 N500q10E 29978 29978 30674 N500q10F 28527 28527 28882 N500q10G 26171 26171 27107 N500q10H 35805 35805 36857 N500q10I 30247 30247 30796 N500q10J 30428 30428 31255
- 232. 208 10.4 ExperimentalResults Table 10.2 shows that the AVNS-SA algorithm was able to improve the previous best known results for 50% of the 100 test cases, 70% of the 200 test cases, 100% of the 300 test cases, 80% of the 400 test cases, and finally 100% of the 500 test cases. The overall average of the results in the 5 runs for all test cases of size 100 is 13087.94, which is only 1% worse than the average result of the GA in [158], having a value of 12954.16. Moreover, our overall average for the 200 test cases is 19151.44, while the overall average of the heuristic in [158] for the same test cases in 10 runs is 19339.48, i.e., our results account for an improvement of approximately 1%. On the other hand, the overall average of our results for the 300 test cases was 23683.12, with an improvement of more than 2% compared the overall average of their results for the same test cases, which is 24224.28. Also, our average for the results of the 10 test cases of size 400 was 26059.3, which accounts for an approximately 2% improvement over the average of the best results of [158], having the value 26490.4. In addition, our average result of the 10 test cases of size 500 was 29579.4. This is an improvement of approximately 3% over the average of the best results of [158], having the value 30377.1 for the same instances. These results also indicate that our algorithm performs even better on larger size problems. The average processing time in this experiment ranged from approximately 542.22 seconds for 100 customers instances to 151103.04 seconds for 500 customers instances. To further test the robustness of the AVNS-SA algorithm, we performed an additional experiment by running the algorithm on 100-customers problems with a vehicle capacity of 20 and 40. The algorithm was run 10 times on each test case. The results of this experiment is shown in table 10.3. The table shows the best result achieved among the 10 runs, the average result of the 10 runs, and the best known results published in [73]. Results of the algorithm that are the same or better than the best known results are shown in boldface. The results in the table show that the algorithm was able to achieve better than the previous best known results in 4 out of 10 test cases, for vehicle capacity Q = 20. It was also able to achieve the same result as the best known result for 6 out of 10 cases, for vehicle capacity Q = 40. The later results might as well be the optimum results, since they have also appeared as best results in the heuristic of [75], and each one appeared more than once among our 10 runs experiment. The average processing time for Q = 20 instances was 155.76 seconds, and for Q = 40 instances was 146.17 seconds.
- 233. 10.4 Experimental Results209 Table 10.3: AVNS-SA Results (100 Customers with Q=20 and Q=40). Name Best Average Best Known N100q20A 8650 8768.6 8616 N100q20B 9533 9628.8 9536 N100q20C 9954 10093 9993 N100q20D 10015 10351 10064 N100q20E 8864 9710.35 8838 N100q20F 9004 9127.9 9029 N100q20G 8986 9138.7 8865 N100q20H 9561 9692.3 9495 N100q20I 10017 10411 10005 N100q20J 9769 9873.5 9742 N100q40A 7938 8005.8 7938 N100q40B 8124 8234.1 8124 N100q40C 8441 8508.9 8441 N100q40D 8336 8402.9 8264 N100q40E 7960 8037 7960 N100q40F 8074 8131.9 8074 N100q40G 8181 8237.9 8168 N100q40H 7992 8010.5 7992 N100q40I 8478 8551.3 8440 N100q40J 8261 8311.2 8255 Finally, to test the effect of the SA acceptance on the performance of the algorithm, we ran the heuristic twice on test case N100q10A. In the first run the SA acceptance was used, and the result of each VNS iteration was recorded. The final result produced was 11751. In the second run, we removed the SA acceptance, and used an HC criterion, where only solutions of lower cost are accepted in each VNS iteration. The final result produced was 12313. Figure 10.1 shows the performance of both variants. It is clear from the figure that the SA version explored a wider area of the search space, before finally reaching a better result than the one achieved by the HC variant, which converged to a suboptimal solution. The processing time, though, was in favour of the HC variant, with 218.33 seconds compared to 552.11 seconds for the SA variant.
- 234. 210 10.5 Summaryand Future Work 0 10 20 30 40 50 60 70 80 90 100 110 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 x 10 4 Iteration Score SA HC Figure 10.1: SA against HC for N100q10A. Contrary to the exceptional results achieved by our AVNS-SA algorithm, its processing time in general was to a large extent disappointing. For example, the average processing time for 100 customers problems was 542.22 seconds, while the processing time reported by [158] was 21.12 seconds for the same problem category. 10.5 Summary and Future Work In this part of the research, we investigated a VNS approach to the 1-PDP problem. The algorithm is distinguished by performing the VNS repeatedly, each time starting from the final solution obtained in the previous run. Also, the algorithm is adaptive, in the sense that the maximum neighbourhood size allowed in each VNS run is not fixed and depends on the current stage of the run. Early runs are allowed to perform wider jumps in the solution space from the current solution, using large neighbourhood sizes. Later runs, on the other hand, are only allowed smaller explorations of the search space, in the vicinity of the current solution, to maintain the solution quality. The stopping criterion for each VNS run is also adaptive and depends on the improve- ment realized in the current solution. The VNS is terminated when a further increase the neighbourhood size seems unhelpful. During each VNS run, an SA acceptance criterion is used to allow the algorithm to escape local optima, and explore a wider area of the search space. In addition, we used a new neighbourhood move in the shaking part of
- 235. 10.5 Summary andFuture Work 211 the algorithm, which has not been tried before for the 1-PDP. This move, which is based on a displacement of a sequence of nodes with or without inversion, seems to work per- fectly well within the context of the VNS for this problem, as evident by the final results obtained for the overall heuristic. Experimental results on a large number of problem instances indicate that our algorithm outperforms previous heuristics in most hard test cases, where the vehicle capacity is smallest. This is especially noticeable for large problem sizes. The algorithm was able to achieve the optimum results for all but few test cases in the small size problems, and was able to improve the previous best known results for 90% of the large test cases. The algorithm is also robust enough, since it performs equally well on a wide range of problem instances, e.g. instances with a different vehicle capacity, without the need for any parameter adjustment. These distinguished results, though, come at the expense of computation time. Although we cannot provide an accurate analysis at this stage, because of the use of different pro- cessors to run the experiments, we recognize that the running time of the algorithm is rather too long. Of course, this is more noticeable for very large problem sizes, i.e., more than 200 customers. In the future, we will continue investigating possible techniques to reduce the run time. Some attempts include reducing the number of VNS runs, and changing the stopping cri- teria for each individual run. For example, our computational experimentation indicated that some problem instances needed fewer than 5 consecutive attempts (without impro- vement) to reach the best results. However, for other instances, reducing the maximum number of attempts to less than 5 may cause the algorithm to prematurely stop and pro- duce lower quality result. More investigation of the best termination criterion is therefore needed to reduce the overall processing time. Other possible improvement attempts, with respect to the run time, should be oriented towards the local search procedure, since it is the most time consuming part of the algorithm. For example, we can try to reduce the number of calls to this algorithm, or make it optimize only part of the solution rather than whole solution. This could possibly be done by changing the exhaustive search com- ponent of the 2-Opt algorithm to some random or selective search that does not process all nodes in the solution. A better analysis of the neighbourhood structure could also be beneficial in enhancing the execution time, for example by eliminating solutions that may not seem promising from further processing. Having completed our investigation of several heuristics and meta-heuristics on selected pickup and delivery problems, we present in the next chapter a brief discussion of how this research relates to commercial transportation software and industrial applications.
- 236.
- 237. 213 Chapter 11 The Researchand Real-Life Applications Efficient transportation has become vital for today’s dynamic society. In the European Union, the transportation sector constitutes more than 10% of Gross Domestic Product (GDP) and employs more than 10 million people [1]. Transportation volume continues to increase rapidly every day, as a result of economic growth and globalization, compared to a limited expansion in roads and networks capacities. Transportation demand is not always geographically balanced. In addition, the lack of coordination between manufacturers, shippers and carriers in supply chains can lead to inefficient usage of natural and human resources. In many cases, transportation plan- ning is done manually. However, the advent of today’s technology- including high speed computers, digital cellular phones, Geographic Information Systems (GIS), Geographic Positioning Systems (GPS), navigation and tracking technologies, wireless data commu- nication, digital mapping and web-based services- has increased the demand for more efficient commercial software for route planning. If applied on a large scale, commercial software can lead to enormous savings, both economically and environmentally. Vendors of software tools claim that the reduction in cost may range from 5% - 30% [76]. Given the huge volume of today’s transportation, such cost reduction is in fact very significant. Besides cost reduction, efficient routing can greatly reduce the environmental impact of transportation. For example, in a recent survey of UK brewery, Paragon Software Sys- tems, Inc.1 identified savings of more than 2.5 million miles- corresponding to 3,700 tons of CO2- a year, as a result of more efficient routing [68]. To meet this demand, research in vehicle routing and scheduling has grown substantially in the last few decades [48]. As previously discussed in Chapter 3, a huge number of problem variants, different problem constraints, and operating scenarios have been inves- tigated. In fact, research in this field is central to the development of efficient decision 1 http://www.paragonrouting.com/
- 238. 214 11.1 IndustrialAspects of Vehicle Routing support tools that can be adopted in the transportation industry. Despite this, research in vehicle routing is often accused of being too idealistic. The majority of published research tackles simplified problems, based on, for example, Euclidean distances, homogeneous fleets, hard constraints, fixed service times. . . etc. Unfortunately, most of these assump- tions do not hold in reality. Industrial aspects of vehicle routing have recently started to gain the attention of researchers, and are increasingly being incorporated into models and solution methods that address these problems. Thus, the current trend is towards ‘holistic’ approaches that are capable of solving richer and more realistic VRP models. It is important from a realistic perspective to understand the relationship between theoreti- cal research and commercial applicability in every day business requirements. To this end, we discuss in this chapter some industrial aspects that may be considered when addressing the VRP and its related variants in Section 11.1. Section 11.2 describes the basic compo- nents of commercial software that are adopted in the transportation industry, and explains how a theoretical research, like ours, may be integrated within its framework. Examples of commercial software products and applications in the transportation sector are provided in Section 11.3. Section 11.4 summarizes some future trends in vehicle routing research. Finally, Section 11.5 concludes this discussion with a summary and some brief remarks. Most of the information presented in this chapter is based on the research by SINTEF research organization in Norway2 , published in [72] and [76], unless otherwise indicated. 11.1 Industrial Aspects of Vehicle Routing As mentioned above, research in vehicle routing is now shifting towards solving non- standard and rich VRP models that will facilitate decision making in real-life situations. The advancement in computational power in the last few decades has encouraged resear- chers to consider industrial aspects of vehicle routing, in order to meet the demands of transportation service providers and fleet management companies. Rich VRP models al- low general and more realistic features to be incorporated, as opposed to conventional OR models which are simplistic in nature. Some industrial aspects of vehicle routing are summarized below: • Heterogeneous Fleet: most of the VRP research makes the assumption that the operating fleet is homogeneous, with identical characteristics and operating costs. In reality, though, this is often not the case, since companies usually benefit from 2 http://www.sintef.no/Home/
- 239. 11.1 Industrial Aspectsof Vehicle Routing 215 versatility in their fleet. A rich VRP model should allow non-homogeneous ve- hicles, such that not only the optimum number of vehicles is determined, but also the optimum number of each vehicle type and the optimum cost for vehicle acqui- sition/depreciation. • Drivers’ Working Time: in real life applications, drivers’ working hours are go- verned by certain legislation rules. Hence, a rich VRP model should create working plans that conform to these regulations, such that the allocation and exchange of drivers is also taken into consideration while determining the optimum routing de- cision. • Depots and Service Locations: basic VRP models usually assume that there is only one central depot, such that each vehicle’s journey should start and end at that depot. Nevertheless, this assumption does not hold in many practical situations, since there might be multiple depots, or arbitrary starts and ends for vehicles. In addition, customer locations are sometimes not fixed, with alternative service loca- tions being permitted in the routing plan. • Order Types: in the basic VRP variants, order types are either pickups or delive- ries. In addition, split deliveries are usually not allowed, and each customer can only be visited once. In real life applications, a customer order may be both a pi- ckup and a delivery. Also, some orders may not require the transportation of goods, but only a certain service type (e.g. maintenance). Allowing such variants adds to the complexity of the problem but makes it more realistic from a business point of view. • Distances and Times: the assumption that all distances under consideration are Euclidean distances is not adequate in real-life scenarios. Network characteristics, traffic, vehicle speed, and travel costs should also be taken into consideration. Mo- reover, service times for clients are not fixed in practical applications. Variable service times, depending on order types and volumes, should be allowed in rich VRP models. • Time Windows and Capacity Constraints: most VRP models deal with the time window constraint as a hard constraint, with no violation permitted in the under- lying routing plan. Time windows in reality are not always that rigid. They are often defined by preferred visiting times, with some cost penalty for visiting out- side the specified period. Another extension to the basic model is multiple time windows, where different alternative visiting periods are given. Also, some appli- cations require certain vehicle capacity and loading restrictions. For example, a
- 240. 216 11.2 CommercialTransportation Software specific loading sequence may be enforced in order to facilitate un-loading, or to protect fragile items. • Uncertainty and Dynamic Situations: dynamic vehicle routing refers to the situa- tion where routing decisions are affected by input data arriving in real time. Infor- mation about orders, travel times, service times, vehicle breakdown. . . etc, arriving while the routing plan is being executed should be taken into consideration in most real-life applications, and necessitates an immediate response time. Uncertainty and stochastic variables, for example while planning a certain emergency situation, also add to the complexity of rich VRP models. 11.2 Commercial Transportation Software A commercial software package for decision support in the transportation industry usually integrates an underlying algorithm with an efficient user friendly interface for optimum usage. A survey of commercial vehicle routing software [67] identifies the basic software capabilities as: 1. Geocoding addresses using a digital map database, i.e., determining the coordinates of a location using its address or postal code; 2. Determining the best driving route between pairs of geocoded pints; 3. Solving the VRP, i.e., assigning stops to vehicles and routing vehicles between stops, and 4. Displaying the results in both graphical and tabular forms, such that the dispatcher can communicate the solution to the drivers, and edit these solutions, with a ‘drag- and-drop’ feature if necessary. Step 3 in the above list is where theoretical research, such that presented in this thesis, takes its part. It is in fact the ‘core’ or the ‘engine’ of the software tool, sometimes referred to as aVRP solver. Hasle and Kloster in [72] define a VRP solver as: A software component with functionality for modeling instances of targeted variants of the VRP and finding feasible, optimized solutions to a given instance. The effect of a given routing tool is highly dependent on its VRP solver.
- 241. 11.3 Examples ofCommercial Vehicle Routing Tools 217 The efficiency of a VRP solver can be generally attributed to the richness of the VRP model (problem) it is trying to solve, in addition to its algorithmic capabilities measured in terms of the quality of the objective and the processing time needed. To the best of our knowledge, most VRP solvers of today’s commercial software do not adequately handle pickup and delivery problems. Improvements in this area are obviously in demand. 11.3 Examples of Commercial Vehicle Routing Tools Some examples of commercial vehicle routing software are: ILOG Dispatcher, Paragon Routing and Scheduling System, Direct Route, DISC, and JOpt.SDK. Notable installa- tions of these and other commercial software include companies like: Sainsbury’s, Argos, Tesco, Royal Mail, the Home Depot, Samsung, Kraft, Dunkin Donuts, Coca-Cola, BP, TNT, Fujitsu, and many others [68]. One example of a rich generic VRP solver is SPIDER [72], which was developed by SINTEF Applied Mathematics research institution in Norway3 . SPIDER VRP solver is capable of solving a number of vehicle routing problems and its variants, such as the VRPTW, the PDPTW and the multiple-depot variants of these problems. In addition, it takes into consideration many industrial aspects, such as the ones described above. For example, it allows a heterogeneous fleet, multiple time windows, alternative service locations, variable service times, and travel times that vary according to road network topology and information available from Geographic Information Systems (GIS). The algorithmic approach in SPIDER is a unified approach to all problem types and ins- tances. This technique has advantages in terms of simplifying the code and its mainte- nance, but may sometimes suffer in terms of computation time, since some operations will still be performed for problem instances that may not need them. The SPIDER VRP solver is basically a meta-heuristic approach that integrates several features from successful academic research in the VRP field. The algorithm consists of three main components: Construction of Initial Solutions, Tour Depletion and Iterative Improvement. These are briefly explained below: • Construction of Initial Solutions: the construction phase is based on extensions of classical construction heuristics like the Clark and Wright savings heuristic [26], Solomon’s I1 insertion [141], and the regret-based insertion [121]. In addition, an 3 http://www.spidersolutions.no/
- 242. 218 11.4 FutureTrends instance analysis is performed in order to determine if the instance has a heteroge- neous fleet, in which case a special construction heuristic called SPIDER construc- tor is used to build the initial solution. • Tour Depletion: this phase is intended for reducing the number of routes in the initial solution, but is also used as a local search operator during the iterative impro- vement phase. Each route is depleted in turn, and an attempt is made to insert all its requests into other routes. If the attempt was successful, the new solution is ac- cepted. The routes that have been changed by this operation may also be optimized using 2-Opt or 3-Opt improvement heuristics of [103]. • Iterative Improvement: the iterative improvement phase is based on Variable Neighbourhood Descend (VND) [70], using a number of well-known operators wi- thin each route and in-between routes. For example, 2-Opt, 3-Opt, Or-Opt, EX- CHANGE, and CROSS (see Section 3.4 for more details about some of these ope- rators). However, these heuristics have been extended to accommodate the SPIDER rich VRP model which allows heterogeneous fleets and multiple time windows. In addition, several neighbourhood filters have been applied to accelerate the opti- mization, for example by analyzing the current solution and exploring promising moves only. When the VND reaches a local minimum, a diversification mechanism is applied using Large Neighbourhood Search (LNS) [131]. The overall process is a hybrid of VND and Iterated Local Search (ILS). To evaluate its performance, SPIDER was tested on published benchmark data available from the literature and compared favourably with state-of-the-art solution methods (see [72] for some experimental results of SPIDER on published benchmark instances of the PDPTW). 11.4 Future Trends Scientific research in transportation optimization is an indispensable part of any commer- cial software tool, and there is, and will continue to be, a great demand for innovations in research methodologies. Researchers in this field, however, should also be aware of new demands in the industrial sector and try to develop richer models in their research. Issues like heterogeneous fleets, multiple tours, split deliveries, variable service times, soft time windows, special vehicle equipment or driver certificates, dynamic route plan- ning, and many other real-life requirements and constraints should motivate researchers to invest more effort in developing efficient solution techniques that comply with today’s
- 243. 11.5 Summary andConclusions 219 ever changing domain of industrial vehicle routing. The complexity of these models, ne- vertheless, means that solving the problem to optimality is not an option in most practical situations. In fact, heuristics, meta-heuristics and hybrid approaches dominate scientific research in this field. There is a trend in today’s research towards optimization tools that integrate the whole supply chain rather than individual components. In addition, current concerns over global warming has increased the demand towards rewarding lower carbon emissions and green logistics. More robust planning in dynamic and stochastic situations is also gaining more attention every day. Decision support tools that enable the service provider to choose bet- ween more than one good solution is also gaining more popularity in the current research environment. Another important factor that needs urgent attention is the development of better bench- mark test cases. Most of the benchmark cases available to the research community for the VRP and its related variants are created randomly. In addition, they are often overly simplified and do not reflect real-world cases. Using such test cases increases the risk of over fitting, i.e., a great effort may be invested in developing solution methods that produce good results on published benchmark instances only. Applying these methods on a larger scale, though, may reveal their shortcomings. The research in [76] identifies several features that test cases for the vehicle routing research should have. These are: 1) they should be based on real-world data, 2) they should be as rich as possible, i.e., contain sufficient details, 3) they should have a common format (e.g. XML), 4) solutions (and not just their objective functions) should be published, and 5) test cases and their solutions should be published in the Web. 11.5 Summary and Conclusions This chapter reviewed some industrial aspects of vehicle routing that are currently un- der consideration by the research community to meet the increase in demand for more efficient transportation. The current trend is towards rich VRP models that can be used in commercial decision support tools, in order to achieve better customer service, cost reduction, and efficient resource management in transportation systems. Yet, although scientific research is a major and important step towards a complete real- life applicable solution to routing and scheduling problems, there is often a considerable distance between theoretical research and practical applicability. Integrating theoretical scientific research within a commercially applicable tool is usually done by consulting
- 244. 220 11.5 Summaryand Conclusions companies, routing software vendors, contract research organizations, and large research laboratories or institutions. In addition, it often requires the cooperation of a number of experts in different fields and possibly several years of development effort. These institu- tions, however, do not work in isolation from academic research. They monitor research carried out by the scientific community and incorporate state-of-the-art techniques in their products. Assessing the quality of the different scientific approaches is mostly done by comparing their performance against published test cases available for researchers in the academic field. Hence, the focus of researchers in Computer Science and Operations Re- search should be on developing competitive and robust solution methodologies that can be later integrated within a larger framework for applicability in real-life situations. We have hereby completed a thorough explanation and analysis of the research carried out in this thesis, together with some necessary background information and a literature survey of state-of-the-art techniques in this field. The next chapter will conclude this thesis with a summary of the whole research and its major contributions, in addition to some future research directions.
- 245. 221 Chapter 12 Conclusions andFuture Directions In this research we investigated some heuristic and meta-heuristic algorithms for solving selected pickup and delivery problems, namely the PDPTW and the 1-PDP. Innovations in solution techniques that handle these and similar vehicle routing problems are in conti- nuous demand, since they can be used in decision support tools and may help reduce transport costs and optimize resource consumption. Having explained in detail our research and its findings, in addition to reviewing impor- tant related work in this field, we present in this chapter a summary of the research and its main achievements in Section 12.1. Section 12.2 emphasizes parts of this research where further work may be carried out and summarizes some future research suggestions. Section 12.3 afterwards concludes this chapter and the whole thesis with some brief final remarks. 12.1 Research Summary and Contribution Advances in computational power in the last few decades has contributed to the emer- gence of a trend among researchers towards powerful algorithms for solving optimization problems. A common phenomenon that existed as a result of this trend, though, is that solution algorithms have tended to became increasingly complex, often with many sophis- ticated and intertwined components. As a consequence, recently published results can be difficult to replicate, and some algorithms are indeed challenging to implement. Additio- nally, it may be difficult to assess the contribution of the different algorithmic components to the overall performance. In our view, it is good practice to provide a thorough analysis of all components of an optimization algorithm, to ensure that all are making a valuable contribution. Those that are not, should be removed and the algorithm simplified. Another weakness that we perceive in much of the VRP literature is an overemphasis on beating best published results for benchmark data, at the expense of algorithm robustness. If solution algorithms are finely tuned for special benchmark instances, they may not work
- 246. 222 12.1 ResearchSummary and Contribution at all well on unseen data or real world instances. Robustness is a key requirement, if an approach is to be eventually useful in practice. The main philosophy in our research is the development of simple heuristic or meta- heuristic frameworks that can be easily understood and implemented. We concentrate here on some very challenging variants of vehicle routing problems involving pickup and delivery. The difficulty in dealing with pickup and delivery problems stems mainly from the existence of several problem constraints that must be dealt with during the construction and improvement phases of a solution process. Keeping this in mind, we directed our attention towards establishing effective ways to deal adequately with problem constraints. Two main aspects have been focused on to achieve this goal: the solution representation and the neighbourhood moves. Our solution representation for the PDPTW tried to overcome the precedence constraint, between the pickup and the delivery, by assigning the same code to both locations, and always considering the first occurrence as the pickup. This simple technique removed the burden of having to check and correct the precedence infeasibility at each step of the solution process. This representation was used for both the single and multiple vehicle cases of the problem. In addition to handling the precedence issue, our representation also overcomes the coupling constraint that must be enforced when multiple vehicles are used, since both the pickup and its delivery should be served by the same vehicle. Hence, this simple approach further reduced the number of hard constraints that the solution algorithm has to deal with during the search. In addition, neighbourhood moves played a central role in our research. For the PDPTW, a simple neighbourhood move that is guided by the time window proved successful on several occasions. It was used as a mutation operator and as a solution improvement tool within the different heuristic and meta-heuristics techniques applied. In addition, a neighbourhood move used for the first time within the VNS approach to the 1-PDP, na- mely the displacement and inversion move, has demonstrated its effectiveness in helping the search escape local optima. We also introduced some adaptive moves. For example, different bounds within the time window interval were used to direct the neighbourhood move at different stages during the search, when solving the PDPTW. Similarly, the in- crease in neighbourhood size within the VNS approach for the 1-PDP was controlled by the current search progress, such that a larger neighbourhood size is only attempted when such increase seems fruitful to the search process. Our research methodology has made it possible for us to accomplish a number of achie- vements that we believe are significant to scientific research in this area. The main achie- vements of this study are:
- 247. 12.1 Research Summaryand Contribution 223 1. We used our solution representation and neighbourhood moves to develop new simple routing heuristics for the SV-PDPTW. The results obtained in this part of the research were impressive, both in terms of the solution quality and processing time. One particular approach appears to have the best potential for the SV-PDPTW, the 3-stage SA routing heuristic. This approach was able to obtain results better than those previously published in all test cases and also performed significantly better than all other algorithms implemented in this part of our research. The ex- perimental results and comparison with other heuristics indicate that the success of this approach was mostly due to the guided neighbourhood moves that were adop- ted to overcome the difficult time window constraint. Another heuristic which also showed potential, especially in terms of processing time, is the Hill Climbing (HC) routing heuristic. The HC algorithm also employed the same time window guided neighbourhood moves during the search. The routing heuristics developed for the SV-PDPTW can be easily integrated within real-world optimization tools that deal with this problem. Two publications were made out of the results obtained, one late breaking conference paper [79] and another journal paper [84]. 2. Since there are no standard benchmark instances for the SV-PDPTW, we were able in this research to create test cases that can be used as benchmark data and used by researchers for testing their algorithms. 3. Based on the simple routing heuristics developed for the SV-PDPTW, we designed and compared several solution construction methods for the MV-PDPTW. These construction methods, especially the sequential construction approach (SEQ), can be used in any heuristic or meta-heuristic that deals with the PDPTW and in the re- lated dial-a-ride problem as well. The construction heuristics are distinguished by their simplicity and ease in coding and application, compared to classical construc- tion methods from the literature. This part of the research was published in [81]. 4. We developed new problem-specific genetic operators and neighbourhood moves for the MV-PDPTW. These operators use techniques developed in the first parts of our research, i.e., for the SV-PDPTW and for the solution construction of the MV-PDPTW. Our operators are characterized by the ability to create feasible solu- tions throughout the search. The operators developed here can be easily adopted by other GA approaches for different vehicle routing problems. For example, a cros- sover that ranks routes to guide inheritance is applicable to any routing problem. These operators can also be adapted and employed within different heuristics and meta-heuristics for solving the problem, as done in this research by using similar operators in both the GA and the SA for solving the MV-PDPTW. The research
- 248. 224 12.1 ResearchSummary and Contribution dealing with the GA approach in this part of the thesis was published in [80]. 5. We developed an interesting adaptive hybrid VNS-SA approach for the 1-PDP. The algorithm is distinguished by adopting, for the first time, the traditional displace- ment and inversion move from previous vehicle routing research as a ‘shaking’ pro- cedure within the VNS. In addition, adaptation is introduced in a novel way within the VNS meta-heuristic. Traditional VNS approaches usually increase the neigh- bourhood size up to a certain pre-defined limit. In our approach, we adapted this requirement, such that a further increase in the neighbourhood size is only applied when it seems beneficial from a search perspective. The algorithm introduced here proved its potential by beating previously best published results for 90% of the large problem instances solved. Two conference papers covering this part of the research have now been accepted for publication in GECCO2010 conference (as a late brea- king abstract) [82], and the PPSN2010 conference [85]. In addition, a third journal paper has been submitted to the Journal of Heuristics and is currently under review [83]. The above achievements indicate that the techniques we developed to guide our heuristic and meta-heuristic approaches were successful to a large extent in accomplishing the ob- jectives of the research. Nevertheless, we can also identify a few parts of the research that did not meet the anticipated standards. Three main areas have shown some shortcomings and need further investigation. These are: 1. The improvement heuristics of the MV-PDPTW (explained in Chapter 8), since the results obtained by both the GA and the SA algorithms were in general of lesser quality than the best known results. 2. The evolutionary perturbation heuristic of the 1-PDP (explained in Chapter 9). Des- pite its success on other VRPs, this technique did not obtain high quality solutions for the 1-PDP. However, the approach did show promise in removing infeasibility. 3. The processing time needed by the VNS approach to the 1-PDP (explained in Chap- ter 10), since the final solution obtained, albeit having a very good quality, needed quite a long processing time. The possible reasons behind these shortcomings have been addressed in their respective chapters. Nevertheless, we will further elaborate on some future research directions that may be pursued to remedy these shortcomings. To this end, Section 12.2 includes a brief critical analysis of some parts of the current research and suggestions of additional work that may be performed to complement the research carried out in this thesis.
- 249. 12.2 Critical Analysisand Future Work 225 12.2 Critical Analysis and Future Work Some aspects of this thesis where further work can be done are summarized below. 12.2.1 The SV-PDPTW To complement the research done so far for this problem, more problem instances of different sizes and different characteristics need to be created. For example, similar to the benchmark instances of the VRPTW and the MV-PDPTW, a certain distribution of nodes may be enforced, such as having some instances with clustered or partially clustered locations. Also, problem instances may differ according to the width of the time window, by having short or long schedule horizons. In addition, a more thorough investigation of the approaches developed in this part of the research may be undertaken by testing the algorithms on a larger number of test cases and performing an in-depth statistical analysis of the results. 12.2.2 Solution Construction for the MV-PDPTW Further investigation of the solution construction heuristics developed in this part of the research may be carried out by implementing one or more ‘traditional’ construction me- thods, such as Solomon’s I1 insertion heuristic [141], after adapting it for the PDPTW. This should then be followed by comparing the performance of the traditional heuristic(s) with the new construction algorithms developed in this research, on benchmark instances. 12.2.3 Solution Improvement for the MV-PDPTW Our selection of GAs as a candidate solution improvement method for the MV-PDPTW was based on the general attractive features that GAs possess, such as simplicity and ro- bustness. Moreover, we were also encouraged by the success of GAs in solving closely related vehicle routing problems, like the VRPTW. As previously discussed during the course of this thesis, though, GAs usually suffer when dealing the MV-PDPTW. This is mostly due to the number of underlying constraints that must be dealt with when desi- gning problem-specific genetic operators. Another important factor is the difficulty in handling the grouping and the routing aspects of the problem simultaneously, in the solu- tion encoding as well as the genetic operators. Similar difficulties also seem to apply to
- 250. 226 12.2 CriticalAnalysis and Future Work our SA approach, which was attempted for solving this problem, since it employs similar operators to those adopted by our GA approach. Nevertheless, we may still be able to improve the results obtained by both the GA and SA in the improvement phase of the MV-PDPTW by augmenting the solution approach with some local search method. For example, a 2-Opt or a 3-Opt heuristic may be used to improve individual routes at various stages during the search. In retrospect, given the opportunity to repeat the project, a new meta-heuristic technique would have probably been chosen for the improvement phase of the MV-PDPTW. For example, VNS would be one option that would definitely be considered. In fact, it was notable that a significant number of papers presented on VRPs at the recent MIC20091 conference, used VNS, most of them achieving impressive results. This was indeed one of the reasons that encouraged us to apply VNS to the 1-PDP, after the ‘disappointing’ results of the EPS heuristic. Applying VNS to the PDPTW would also seem an attrac- tive prospect, since many neighbourhood moves that have been previously used for the VRPTW and the PDPTW can easily be incorporated into the VNS framework. In addi- tion, to the best of our knowledge, VNS has not been previously tried on the PDPTW. It would be interesting to test the routing and construction heuristics developed in this research within a VNS meta-heuristic for solving the MV-PDPTW. 12.2.4 The Evolutionary Perturbation Heuristic for the 1-PDP As previously mentioned in Chapter 9, the EPS heuristic was not very successful in achie- ving the anticipated results. Both the quality of the results obtained and the processing time were inferior to previous heuristics in this field. The main suggestion in terms of the solution quality would be to modify the construction algorithm or experiment with a new construction technique. As previously discussed in Section 9.5, one possibility for impro- vement is to modify the construction algorithm such that rather than halting the insertion process when infeasibility is encountered, the algorithm would continue to insert nodes despite infeasibility. If this is done, correcting infeasibility would probably be easier for the perturbation heuristic, and the overall quality of generated solutions would improve. Moreover, this technique has demonstrated its potential within the AVNS-SA approach to the problem as explained in Chapter 10. In terms of processing time, a possible option for improvement is to optimize the per- turbed coordinates using a faster heuristic or meta-heuristic. For example, a simulated 1 VIII Metaheuristic International conference, Hamburg, Germany - July 13-16, 2009 http://www.smartframe.de/mic09/Home.html
- 251. 12.3 Final Remarks227 annealing or a hill climbing algorithm may be used instead of a GA to improve the per- turbed problem coordinates. Together, with an appropriate and fast construction heuristic the overall scheme may be considerably enhanced. 12.2.5 The AVNS-SA Approach to the 1-PDP The AVNS-SA approach was in fact very successful in dealing with the 1-PDP, especially for large problem sizes. As previously mentioned, the algorithm was able to obtain new best results for 90% of the large problem instances. The only drawback of the algorithm seems to be the long processing time. As previously discussed in Chapter 10, an improve- ment in this area can be achieved if we direct our attention to the local search component of the algorithm. The 2-Opt heuristic used in the local search phase is the most time consuming part of the algorithm. This heuristic adopts a best improvement strategy and is invoked to optimize every new generated solution. Several suggestions may help in this respect. One option is to reduce the number of calls to the 2-Opt algorithm based on some solution quality. For example, the 2-Opt procedure can be restricted only to solutions that look promising from a search perspective (depending on the objective function value or the number of constraint violations, for instance). In addition, the 2-Opt algorithm may be applied to selected edges of the solution and not to the whole solution. Some faster implementations of the 2-Opt algorithm may also be tried to reduce the time requirement of this component (e.g. [13]). Finally, the stopping condition of the overall AVNS-SA algorithm also needs further in- vestigation. To improve the processing time of the algorithm, it is necessary to find an appropriate criterion to stop repeating the AVNS-SA procedure, without sacrificing the quality of the final solution returned. However, obtaining a balance between solution quality and processing time may not be very straightforward for such hard problem. 12.3 Final Remarks To sum up, we believe that the work done in this thesis contributes positively to scien- tific research on vehicle routing and scheduling. This study provided simple ideas on constraint handling mechanisms that can be used in designing effective and robust heuris- tic and meta-heuristic algorithms. The techniques developed in this research can be easily integrated within a larger framework and used in optimization tools to help improve trans- portation and logistic planing.
- 252. 228 12.3 FinalRemarks In this final chapter of the thesis we highlighted the most significant findings and the most promising areas of our research. We also tried to identify some research areas where further investigation and some elaboration are still needed. We aspire to carry out in the near future some of the outlined suggestions to complement the work done in this thesis and advance the research to its uttermost standards. Additional publications of the findings of this thesis are also underway.
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