Optimization using soft computing

Optimization
using
Soft Computing
(Fuzzy Sets)
(07/12/2017)
STTP from 4/12/2017 to 9 /12/2017
on
“Recent Trends in Optimization Techniques & Applications in
Science & Engineering”
Dr. Purnima Pandit
Assistant Professor
(purnima.pandit-appmath@msubaroda.ac.in)
Department of Applied Mathematics
Faculty of Technology and Engineering
The Maharaja Sayajirao University of Baroda

What is soft computing ?
Techniques used in soft computing?

What is Soft Computing ?
(adapted from L.A. Zadeh)
• Soft computing differs from
conventional (hard) computing in that,
unlike hard computing, it is tolerant
of imprecision, uncertainty, partial
truth, and approximation.
In effect, the role model for soft
computing is the human mind.

What is Hard Computing?
• Hard computing, i.e., conventional
computing, requires a precisely stated
analytical model and often a lot of
computation time.
• Many analytical models are valid for
ideal cases.
• Real world problems exist in a non-
ideal environment.

Many contemporary problems do not lend
themselves to precise solutions such as
– Recognition problems (handwriting,
speech, objects, images)
– Mobile robot coordination,
forecasting, combinatorial problems etc.

Guiding Principles of Soft
Computing
• The guiding principle of soft computing
is:
– Exploit the tolerance for imprecision,
uncertainty, partial truth, and
approximation to achieve tractability,
robustness and low solution cost.

Techniques of Soft Computing
• The principal constituents, i.e., tools,
techniques, of Soft Computing (SC) are
– Fuzzy Logic (FL), Neural Networks (NN),
Support Vector Machines (SVM),
Evolutionary Computation (EC), and
– Machine Learning (ML) and Probabilistic
Reasoning (PR)

Properties of Soft computing
• Learning from experimental data
• Soft computing techniques derive their
power of generalization from
approximating or interpolating to produce
outputs from previously unseen inputs by
using outputs from previous learned
inputs
• Generalization is usually done in a high
dimensional space.

Applications using Soft Computing
• Handwriting recognition
• Automotive systems and manufacturing
• Image processing and data compression
• Decision-support systems
• Power systems
• Intelligent systems
• Adaptive control

Future of Soft Computing
(adapted from L.A. Zadeh)
• Soft computing is likely to play an especially
important role in science and engineering, but
eventually its influence may extend much farther.
• Soft computing represents a significant paradigm
shift in the aims of computing
– a shift which reflects the fact that the human
mind, unlike present day computers, possesses a
remarkable ability to store and process
information which is pervasively imprecise,
uncertain and lacking in categorization.

Introduction
• In 1965* Zadeh published his seminal work
"Fuzzy Sets" which described the mathematics
of Fuzzy Set Theory.
• FST has numbers of applications in various
fields- artificial intelligence, automata theory,
computer science, control theory, decision
making, finance etc.
• It is being applied on a major scale in
industries for machine-building (cars, engines,
ships, etc.) through intelligent robots and
controls.
*L. A. ZADEH, Fuzzy Sets, Information Control, 1965, 8, 338-353.

This approach provides a way to translate a linguistic
model of the human thinking process into a mathematical
framework for developing the computer algorithms for
computerized decision-making processes.
crisp
fuzzy
very cold

In general, fuzziness describes
objects or processes that are not
acquiescent to precise definition or
precise measurement. Thus, fuzzy
processes can be defined as processes
that are vaguely defined and have some
uncertainty in their description. The
data arising from fuzzy systems are in
general, soft, with no precise
boundaries.

Fuzziness in Everyday Life
John is tall;
Temperature is hot;
The girl next door is pretty;
The sun is getting relatively hot;
The people living close to Vadodara;
My car is slow.

Characteristic Function in the Case
of Crisp Sets and Fuzzy Sets
P: X  {0,1}
P(x) =
A : X  [0,1]
A = {X, A(x)} if x  X
A Fuzzy Set is a generalized set to which objects can
belongs with various degrees (grades) of memberships
over the interval [0,1].





Xx
Xx
if0
if1

Difference between
(a) crisp set and (b) fuzzy set

• Crisp set:
This is defined in such a way as to dichotomize the individuals in
some given universe of discourse into the two groups- members and
nonmembers.
Full membership and full non-membership in the fuzzy set can still be
indicated by the values 1 and 0, respectively.
• Fuzzy set:
Mathematically, if U is the universe discourse, the fuzzy set is
defined as a pair given as
For each , the value is called a grade of membership of x
in . Here is called membership function of fuzzy set A.
We can consider the concept of a crisp set to be a restricted case of
the more general concept of a fuzzy set.
 ,A  ]1,0[: AA
Ux )(x
 ,A 

Optimization using soft computing

Properties of Crisp sets
( ) ( ) (
Involution
Commutativity
and
Associativity
and
Distrubutivity
and
Idem
) ( )
potence
and
Absor
( ) ( ) ( ) ( ) (
i
)
p
( )
t
A A
A B B A A B B A
A B C A B C A B C A B C
A B C A B A C A B C A B A C
A A A A A A
 
     
         
          






   
 on
and( ) ( )A A B A A A B A     

Absorption by X and Ø
and and
Identity
Law of contradiction
Law of excluded middle
Demorgan’s l
( ) ( )
Out of above
aw
listed laws 'Fu
an
zz
d
A X X A A A
A X A
A A
A A X
A B A B A B A B
  

     
 
 
 
     


  





 y Set' does not satisfy
"Laws of contradiction" and "Laws of excluded middle".

Fuzzy Union :
The union of fuzzy sets that is fuzzy union can be defined in
many ways. The most commonly used method for fuzzy union
is to take the maximum. For two fuzzy sets A and B with
membership functions and
Fuzzy Intersection:
Fuzzy complement:
A A(x)= 1- (x) 
BA
))(),(max()()( xxx BABA  
))(),(min()()( xxx BABA  
Operations on Fuzzy sets

Convex fuzzy set:
A fuzzy set is fuzzy convex set  -cut is convex for all  1,0 .
A fuzzy set is normal if    1, xx  .
Let U is set of real number. A fuzzy number is a convex normal fuzzy set  ,
~
RA 
whose membership function is at least segmentally continuums. If
   babaxx  ,,1, they also fuzzy number call fuzzy interval.

  
  
  
 
If we have any set then,
/ 0
/ 1
/
If height of is 1 then is .
Otherwise it is .
of set = { / }
A
Support of A x A x
Core of A x A x
Height of A x max of A x
A A normal
subnormal
cut A A x A x
 
  
  
 


     , 0,1 . 
Some definitions

Some Concepts
1
0
a b X = [a,b]
core(A)
supp(A)
height(A) = 1 (normal fuzzy set)
Membership
function has a
trapezoidal form

Fuzzy Number
   
 
•A fuzzy set , whose membership function defined
on possess at least following three properties:
i must be a normal fuzzy set.
ii must be a closed interval for every 0,1
i
( )
ii The support
A
R
A
A
 
0
Then, elements of that se
of , must be bounde
ts is known as Fuzzy n
d.
umber.
A A

𝐴1 𝑥 =
0, 𝑥 < 50
𝑥 − 50
5
, 50 < 𝑥 ≤ 55
60 − 𝑥
5
, 55 < 𝑥 ≤ 60
𝛼
𝐴1 = 5𝛼 + 50, 60 − 5𝛼
𝐴2 𝑥 =
0, 𝑥 < 50
𝑥 − 50
2.5
, 50 < 𝑥 ≤ 52.5
1, 52.5 ≤ 𝑥 < 57.5
60 − 𝑥
2.5
, 60 < 𝑥
𝛼
𝐴2 = 2.5𝛼 + 50, 60 − 2.5𝛼

Fuzzy number in parametric form
Parametric representation has advantage of allowing flexible and
easy to control shapes of the fuzzy number.
A fuzzy number in parametric form is an order pair of the form
u = (𝑢 α , 𝑢(α))where 0 ≤ α ≤ 1 satisfying the following
conditions:
 𝑢 α is a bounded left continuous increasing function in the
interval [0, 1]
 𝑢 α is a bounded right continuous decreasing function in the
interval [0, 1]
 𝑢 α ≤ 𝑢(α), α ∈ 0,1 .
If for each α, 𝑢 α = 𝑢 α then 𝑢 is a crisp number.

Fuzzy Arithmetic:
Let 𝐴 and 𝐵 are two fuzzy numbers, then on taking alpha-cut we get,
𝐴 𝛼 = 𝐴, 𝐴 and 𝐵 𝛼 = 𝐵, 𝐵
Then,
𝑖 𝐴 𝛼 + 𝐵 𝛼 = [𝐴 + 𝐵, 𝐴 + 𝐵]
𝑖𝑖 𝐴 𝛼 − 𝐵 𝛼 = [𝐴 − 𝐵, 𝐴 − 𝐵]
𝑖𝑖𝑖 𝐴 𝛼. 𝐵 𝛼 = [min( 𝐴 . 𝐵, 𝐴 . 𝐵, 𝐴 . 𝐵, 𝐴 . 𝐵),
max ( 𝐴. 𝐵, 𝐴 . 𝐵, 𝐴 . 𝐵, 𝐴. 𝐵)]
𝑖𝑣 𝐴 𝛼/ 𝐵 𝛼 = 𝐴 𝛼. 1/𝐵 𝛼

Fuzzy Variables:
• Several fuzzy sets representing linguistic concepts such as low,
medium, high, and so one are often employed to define states of a
variable. Such a variable is usually called a fuzzy variable.
• For example:

Fuzzy Variables:
• Consider three fuzzy sets that represent the concepts of a
young, middle-aged, and old person. The membership functions
are defined on the interval [0,80] as follows:
Find line passing through
(x,y) and (20,1):
1/(35-20) = y/(35-x)

Fuzzy Variables:
• For example:

Fuzzy Variables:
• For example: consider the discrete approximation D2 of fuzzy
set A2

Linear programming Problem (LPP)
The typical linear programming problem is,
𝑀𝑎𝑥 𝑍 = 𝐶𝑋
s.t. 𝐴𝑋 ≤ 𝐵
𝑋 ≥ 0
where, 𝐶 = (𝑐1, 𝑐2, 𝑐3,…, 𝑐 𝑛), 𝐶 is called cost coefficients
𝑋 = (𝑥1, 𝑥2, 𝑥3, … , 𝑥 𝑛) 𝑇 is a vector of variables and
𝐵 = (𝑏1, 𝑏2, 𝑏3, … , 𝑏 𝑚) 𝑇 is called the resource vector.
The vector 𝑋 satisfy all given constraints, is called feasible solution.

In many practical and real life situation, it is may not
be possible to obtain precise value of the coefficients
in the objective functions, coefficients in the LHS of
the constraints or/and the resources.
In such case, we model and solve the problem as
Fuzzy Linear Programming Problem (FLPP).
Fuzzy Linear programming Problem
(FLPP)

Fuzzy LPP
The crisp LPP form changes to following most general FLPP,
max 𝑍 =
𝑗=1
𝑛
𝐶𝑗 𝑋𝑗
𝑗=1
𝑛
𝐴𝑖𝑗 𝑋𝑗 ≤ 𝐵𝑖
𝑋𝑗 ≥ 0, 𝑖, 𝑗 ∈ 𝑁 (1)

Fuzzy LPP (Case I)
In this case, we consider 𝑨𝒊𝒋 and 𝑩𝒊 are fuzzy triangular number.
𝐴𝑖𝑗 = (𝑙𝑖𝑗, 𝑚𝑖𝑗, 𝑟𝑖𝑗)
And,
𝐵𝑖 = (𝐿𝑖, 𝑀𝑖, 𝑅𝑖)
The FLPP is of the form
max 𝑍 =
𝑗=1
𝑛
𝐶𝑗 𝑋𝑗
𝑗=1
𝑛
𝑋𝑗 ≥ 0, 𝑖, 𝑗 ∈ 𝑁 (2)

Fuzzy LPP (Case I)
The FLPP becomes
max 𝑍 =
𝑗=1
𝑛
𝐶𝑗 𝑋𝑗
𝑗=1
𝑛
(𝑙𝑖𝑗, 𝑚𝑖𝑗, 𝑟𝑖𝑗) 𝑋𝑗 ≤ (𝐿𝑖, 𝑀𝑖, 𝑅𝑖) 𝑖 ∈ 𝑁 𝑚
𝑋𝑗 ≥ 0, 𝑗 ∈ 𝑁 𝑛,

Fuzzy LPP (Case I)
The FLPP (2) becomes
max 𝑍 =
𝑗=1
𝑛
𝐶𝑗 𝑋𝑗
𝑗=1
𝑛
𝑙𝑖𝑗 𝑋𝑗 ≤ 𝐿𝑖
𝑗=1
𝑛
𝑚𝑖𝑗 𝑋𝑗 ≤ 𝑀𝑖
𝑗=1
𝑛
𝑟𝑖𝑗 𝑋𝑗 ≤ 𝑅𝑖
𝑋𝑗 ≥ 0, 𝑗 ∈ 𝑁 𝑛, (3)

Example-1:
Consider the following fuzzy LPP,
max 𝑍 = 5𝑥1 + 4 𝑥2
(2,4,5) 𝑥1+(2,5,6) 𝑥2 ≤ (19, 24,32)
(3,4,6) 𝑥1+(0.5, 1,2) 𝑥2 ≤ (6,12,15)
𝑥1, 𝑥2 ≥ 0

By using equation (3), we can rewrite equations,
max 𝑍 = 5𝑥1 + 4𝑥2
s.t.
4𝑥1 + 5𝑥2 ≤ 24
4𝑥1 + 𝑥2 ≤ 12
2𝑥1 + 2𝑥2 ≤ 19
3𝑥1 + 0.5𝑥2 ≤ 6
5𝑥1 + 6𝑥2 ≤ 32
6𝑥1 + 2𝑥2 ≤ 15
𝑥1, 𝑥2 ≥ 0

Solving graphically:
Max z is obtained at X* = (1.2,3.8) and z *= 21.2

When RHS is fuzzy (case-2)
In problem (3), when only 𝐵𝑖 is fuzzy we get:
𝑚𝑎𝑥
𝑗=1
𝑛
𝐶𝑗 𝑋𝑗
𝑗=1
𝑛
𝑋𝑗≥ 0, 𝑖, 𝑗 ∈ 𝑁 (4)

In this case 𝐵𝑖 is typically have form given below,
𝐵𝑖 𝑥 =
1 𝑤ℎ𝑒𝑛 𝑥 ≤ 𝑏𝑖
𝑏𝑖 + 𝑝𝑖 − 𝑥
𝑝𝑖
𝑤ℎ𝑒𝑛 𝑏𝑖 < 𝑥 < 𝑏𝑖 + 𝑝𝑖
0 𝑤ℎ𝑒𝑛𝑏𝑖 < 𝑥
where 𝑥 ∈ 𝑅, for each vector 𝑥 = (𝑥1, 𝑥2, … , 𝑥 𝑛)
We first calculate the degree 𝐷𝑖 𝑥 to which x satisfies the ith
constraint by this formula 𝐷𝑖 𝑥 = 𝐵𝑖( 𝑗=1
𝑛
𝑎𝑖𝑗 𝑥𝑗). These
degrees are fuzzy sets on 𝑅 𝑛 and their integration, 𝑖=1
𝑚
𝐷𝑖 is a
fuzzy feasible set.
𝑏𝑖 𝑏𝑖 + 𝑝𝑖
0
1

Next, we determine fuzzy set of optimal values, this is done
by calculating the lower and upper bounds of the optimal
values first.
The lower bound of the optimal values 𝑍𝑙, is obtained by
solving the standard LPP
max 𝑍 = 𝑗=1
𝑛
𝐶𝑗 𝑋𝑗
𝑗=1
𝑛
𝑎𝑖𝑗 𝑋𝑗 ≤ 𝑏𝑖, 𝑗 ∈ 𝑁 𝑚
𝑋𝑗≥ 0, 𝑖, 𝑗 ∈ 𝑁 𝑛 (5)
and upper bound 𝑍 𝑢 is obtained by replacing, 𝑏𝑖 to 𝑏𝑖 +
𝑝𝑖, 𝑎𝑠
max 𝑍 = 𝑗=1
𝑛
𝐶𝑗 𝑋𝑗
𝑗=1
𝑛
𝑎𝑖𝑗 𝑋𝑗 ≤ 𝑏𝑖 + 𝑝𝑖, 𝑗 ∈ 𝑁 𝑚
𝑋𝑗 ≥ 0, 𝑖, 𝑗 ∈ 𝑁𝑛 (6)

Now the fuzzy set of optimal values, G , which is fuzzy subset of
𝑅 𝑛
is defined by,
G 𝑥 =
1
𝐶𝑋−𝑍 𝑙
𝑍 𝑢−𝑍 𝑙
0
𝑤ℎ𝑒𝑛 𝑍 𝑢 ≤ 𝐶𝑋
𝑤ℎ𝑒𝑛 𝑍𝑙 ≤ 𝐶𝑋 ≤ 𝑍 𝑢
𝑤ℎ𝑒𝑛 𝐶𝑋 < 𝑍𝑙
Now the problem (4) becomes,
max 𝜆
𝜆(𝑍 𝑢- 𝑍𝑙)-𝐶𝑋 ≤ − 𝑍𝑙
𝜆 𝑝𝑖 + 𝑗=1
𝑛
𝑎𝑖𝑗 𝑋𝑗 ≤ 𝑏𝑖 + 𝑝𝑖
𝜆 , 𝑋𝑗 ≥ 0, 𝑖, 𝑗 ∈ 𝑁 (7)

Example-2:
Assume that company makes two products, 𝑃1 & 𝑃2. Product
𝑃1 has $0.40 profit and 𝑃2 has $0.30 profit. Each unit of
product 𝑃1 requires twice as many labor hours of product 𝑃2.
The total available labor hours are at least 500 hours per day
and may possibly be extended to 600 hours per day, due to
arrangement of overtime work. The supply of material is at
least sufficient for 400 units for products 𝑃1 & 𝑃2 per day and
may possibly be extended to 500 units per day according to
previous experience.
The problem is how many units of products 𝑃1 & 𝑃2 should be
made per day to maximize profit?

The given problem can be formulated as
following fuzzy LPP,
max 𝑍 = 0.4𝑥1 + 0.3 𝑥2
𝑥1+𝑥2 ≤ 𝐵1
2𝑥1+ 𝑥2 ≤ 𝐵2
𝑥1, 𝑥2 ≥ 0
where, 𝐵1 and 𝐵2 are fuzzy and are is defined as,

𝐵1 𝑥 =
1 𝑤ℎ𝑒𝑛 𝑥 ≤ 400
500 − 𝑥
100
𝑤ℎ𝑒𝑛 400 < 𝑥 ≤ 500
0 𝑤ℎ𝑒𝑛 500 < 𝑥
𝐵2 𝑥 =
1 𝑤ℎ𝑒𝑛 𝑥 ≤ 400
600 − 𝑥
100
𝑤ℎ𝑒𝑛 500 < 𝑥 ≤ 600
0 𝑤ℎ𝑒𝑛 600 < 𝑥

By using equation (5) and (6), we can calculate 𝑍𝑙 and 𝑍 𝑢 for the
example 2 by solving
max 𝑍 = 0.4𝑥1 + 0.3 𝑥2
𝑥1+𝑥2 ≤ 400
2𝑥1+ 𝑥2 ≤ 500
𝑥1, 𝑥2 ≥ 0 (8)
And
max 𝑍 = 0.4𝑥1 + 0.3 𝑥2
𝑥1+𝑥2 ≤ 500
2𝑥1+ 𝑥2 ≤ 600
𝑥1, 𝑥2 ≥ 0 (9)
giving us, 𝑍𝑙 = 130 and 𝑍 𝑢 = 160.
Now the example (2) becomes,

Then the solution of fuzzy LPP can now be
obtained by solving the following LPP:
max 𝜆
30𝜆-(0.4𝑥1 + 0.3𝑥2)≤ −130
100𝜆 +𝑥1+𝑥2 ≤ 500
100 𝜆+2𝑥1+ 𝑥2 ≤ 600
𝜆 , 𝑥1, 𝑥2 ≥ 0
Using Simplex Algorithm we get:
𝑥1 = 100, 𝑥2 = 350, 𝜆=0.5 and 𝑍 = 145.

The fully fuzzy linear programming problems are given
as:
Maximize Z =
𝑗=1
𝑛
𝑐𝑗 𝑥 𝑗
s. t.
𝑗=1
𝑛
𝑎𝑖𝑗 𝑥𝑗 ≤ 𝑏𝑖, 𝑖 = 1, … , 𝑚
𝑥𝑗 ≥ 0, 𝑗 = 1, … , 𝑛
(10)
where, 𝑐𝑗, 𝑎𝑖𝑗 and 𝑏𝑖 are fuzzy numbers and 𝑥𝑗 are fuzzy
variables whose states are fuzzy numbers.
Fully fuzzy LPP (case-3)

Considering the fuzzy parameters as triangular fuzzy numbers the
problem (10) can be given as
Maximize Z
where, Z =
𝑗=1
𝑛
(𝑐𝑙, 𝑐𝑚, 𝑐𝑟) 𝑗 𝑥 𝑗 ,
s. t.
𝑗=1
𝑛
(𝑎𝑙, 𝑎𝑚, 𝑎𝑟)𝑖𝑗 𝑥𝑗 ≤ (𝑏𝑙, 𝑏𝑚, 𝑏𝑟)𝑖, 𝑖 = 1, … , 𝑚
𝑥𝑗 ≥ 0, 𝑗 = 1, … , 𝑛
where, (𝑐𝑙, 𝑐𝑚, 𝑐𝑟) 𝑗 is the jth fuzzy coefficient in the objective
function, (𝑎𝑙, 𝑎𝑚, 𝑎𝑟)𝑖𝑗 is the fuzzy coefficient of jth variable in the
ith constraint, (𝑏𝑙, 𝑏𝑚, 𝑏𝑟)𝑖 is the ith fuzzy resource.

Thakre et al. solved such problem by converting (10) into to
equivalent crisp multi-objective linear problem as given below
)Maximize (Z1, Z2, Z3
where, Z1 =
𝑗=1
𝑛
𝑐𝑙𝑗 𝑥 𝑗 , Z2 =
𝑗=1
𝑛
𝑐𝑚𝑗 𝑥 𝑗 , Z3 =
𝑗=1
𝑛
𝑐𝑟𝑗 𝑥 𝑗
s. t.
𝑗=1
𝑛
𝑎𝑙𝑖𝑗 𝑥𝑗 ≤ 𝑏𝑙𝑖, 𝑖 = 1, … , 𝑚
𝑗=1
𝑛
𝑎𝑚𝑖𝑗 𝑥 𝑗 ≤ 𝑏𝑚𝑖 𝑖 = 1, … , 𝑚
𝑗=1
𝑛
𝑎𝑟𝑖𝑗 𝑥 𝑗 ≤ 𝑏𝑟𝑖 𝑖 = 1, … , 𝑚
𝑥𝑗 ≥ 0, 𝑗 = 1, … , 𝑛
(11)

Maximize (𝑍1
1
, 𝑍2
1
, 𝑍3
1
, 𝑍1
2
, … , 𝑍3
𝑘
𝑍1
𝑘
=
𝑗=1
𝑛
𝑐𝑙𝑗
𝑘
𝑥𝑗 ; 𝑍2
𝑘
=
𝑗=1
𝑛
𝑐𝑚𝑗
𝑘
𝑥𝑗; 𝑍3
𝑘
=
𝑗=1
𝑛
𝑐𝑟𝑗
𝑘
𝑥𝑗 𝑘 = 1, … , 𝐾
s. t.
𝑗=1
𝑛
𝑎𝑙𝑖𝑗 𝑥𝑗 ≤ 𝑏𝑠𝑖, 𝑖 = 1, … , 𝑚
𝑗=1
𝑛
𝑎𝑚𝑖𝑗 𝑥𝑗 ≤ 𝑏𝑚𝑖, 𝑖 = 1, … , 𝑚
𝑗=1
𝑛
𝑎𝑙𝑖𝑗 𝑥𝑗 ≤ 𝑏𝑙𝑖, 𝑖 = 1, … , 𝑚
𝑥𝑗 ≥ 0, 𝑗 = 1, … , 𝑛
(12)
Multi-objective Fully fuzzy LPP
(case-4)

Consider the Multi objective fuzzy Linear Problem
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑍1
, 𝑍2
where, 𝑍1
= (7,10,14)𝑥1 + (20,25,35)𝑥2
𝑍2
= (10,14,25)𝑥1 + (25,35,40)𝑥2
subject to the constraints
3,2,1 𝑥1 + 6,4,1 𝑥2 ≤ (13,5,2)
4,1,2 𝑥1 + 6,5,4 𝑥2 ≤ (7,4,2)
(13)
Example-3:

It is equivalent to solving the MOLPP
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒
7𝑥1 + 20𝑥2, 10𝑥1 + 25𝑥2,
14𝑥1 + 35𝑥2, 25𝑥1 + 40𝑥2
3𝑥1 + 6𝑥2 ≤ 13
𝑥1 + 2𝑥2 ≤ 8
4𝑥1 + 7𝑥2 ≤ 15
4𝑥1 + 6𝑥2 ≤ 7
3𝑥1 + 𝑥2 ≤ 3
6𝑥1 + 10𝑥2 ≤ 9
𝑥1, 𝑥2 ≥ 0
(12)

That is, solving
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒
𝑤1 7𝑥1 + 20𝑥2 + 𝑤2 10𝑥1 + 25𝑥2
+𝑤3 14𝑥1 + 35𝑥2 + 𝑤4(25𝑥1 + 40𝑥2)
3𝑥1 + 6𝑥2 ≤ 13
𝑥1 + 2𝑥2 ≤ 8
4𝑥1 + 7𝑥2 ≤ 15
4𝑥1 + 6𝑥2 ≤ 7
3𝑥1 + 𝑥2 ≤ 3
6𝑥1 + 10𝑥2 ≤ 9
𝑥1, 𝑥2 ≥ 0
such that wi = S = 2

Sr. No 𝒘 𝟏 𝒘 𝟐 𝒘 𝟑 𝒘 𝟒 𝒙 𝟏
∗
, 𝒙 𝟐
∗
1 0 1 1 0 (0, 0.9)
2 0 1 0.5 0 (0, 0.9)
3 0.2 0.4 0.5 0.2 (0, 0.9)
4 0.1 0.2 0.3 0.4 (0, 0.9)
5 0 0.3 0 0.4 (0, 0.9)
6 0.2 0.4 0.6 0.8 (0, 0.9)
7 0.5 0 0.5 0 (0, 0.9)
8 0 1 1 0 (0, 0.9)
9 0 0 0 0.5 (0, 0.9)
10 0.3 0.1 1 1 (0, 0.9)
11 0.5 0.5 0.5 0.5 (0, 0.9)
12 0 0 0.5 0.5 (0, 0.9)
13 0.2 0.5 0.5 0.5 (0, 0.9)
14 0.1 0.2 0.3 0.4 (0, 0.9)
15 0 0.2 0 0.2 (0, 0.9)

1. Klir G, Yuan B, Fuzzy Sets and Fuzzy Logic Theory and Applications, Prentice
Hall, (1997).
2. Pandit P, Multi-objective Linear Programming Problems involving Fuzzy
Parameters, International Journal of Soft Computing and Engineering (IJSCE)
ISSN: 2231-2307, Volume-3, Issue-2, May (2013).
3. Tanaka H, Ichihashi H, Asai K, Formulation of fuzzy linear programming problem
based on comparison of fuzzy numbers, Control Cybernetics 3 (3): 185-194. (1991).
4. Thakre P A, Shelar D S, Thakre S P, Solving Fuzzy Linear Programming Problem
as Multi Objective Linear Programming Problem, Proceedings of the World
Congress on Engineering, Vol II WCE, July 1 - 3, (2009).
5. Zadeh LA, Fuzzy sets as a basic for theory of possibility, FSSI, 3-28,(1978) .
References:

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