unit2 linear programming problem in .pdf

Chapter Five
Special Types of Linear
Programming
By;-Dagnaygebaw Goshme (MSc.)

5.1.Transportation problem
One important application of linear programming is
the area of physical distribution (transportation)
of goods from several supply centers (origins) to
several demand centers (destinations).
Transportation problem involves a large number of
variables (transportation/shipping routes) and
constraints, it takes a long time to solve it.
Therefore, other methods (transportation
algorithm) have been developed for this purpose.

Objective: the objective is to determine the
amount of commodities which should be
transported from several sources to different
destinations, at the minimum transportation
cost and /or time.
Sources or origins are the places where goods
originate from (like plants, warehouses etc)
Destinations are places where goods are to be
shipped.
It can also be applied to the maximization of
some total value or utility, in such a way that the
profit is maximized.

5.1.1. GeneralTransportation
Problem Model
The transportation algorithm requires the
assumptions that:
All goods are homogeneous, so that any origin is
capable of supplying to any destination.
Transportation costs are a linear function of (or
directly proportional to) the quantity shipped over
any route.
Each source has a fixed supply of units, where this
entire supply must be distributed to the
destinations.
Similarly, each destination has a fixed demand for
units, where this entire demand must be received
from the sources.

A transportation problem model, which has ‘m’ sending locations
(origins) and ‘n’ receiving locations (destinations), provides a
framework for presenting all relevant data. These are:
Quantity supply of each origin (SSi)
Quantity demand of each destination (DDi)
Unit transportation cost from each origin to each destination (Cij)

Where:
SSi - is total quantity of commodity available at origin I
(total supply of origin i).
ddj - is total quantity of commodities needed at
destinations j (total demand of destination j).
Cij - measures the costs of transporting one unit of
commodity from source i to destination j.
Xij - is the quantity of commodities transported from
ith origin to jth destination.

The Linear Programming Representation of a
Transportation Model

Minimize (total transportation cost)
Z=C11X11+C12X12+C13X13+….+CmnXmn
Subject to:
Capacity constraints (SS constraints)
X11+X12+----+X1n=SS1
X21+X22+----+X2n=SS2
. . . .
. . . .
Xm1+Xm2+.---+Xmn=SSm
Requirements constraints (DD constraint)
X11+X21+----+Xm1=SS1
X21+X22+----+Xm2=SS2
. . . .
. . . .
. . . .
Xn1+Xn2+.---+Xnm=SSn
Xij ≥ 0 for all i and j

Before applying the transportation techniques
(methods) to solve a specific problem, the
problem should satisfy the following
conditions.
Supplies (SS)and requirements (DD) must be
expressed in the same unit.
This condition means that shipments received at any
destination from different sources must be
indistinguishable.
In other words, all shipment must be measured in
homogenous units.

Total supply must equal to total demand ∑SS= ∑DD
The problem satisfying this condition is called
balanced transportation problem; otherwise it is
known as unbalanced transportation problem.
The condition ∑SS= ∑DD is the necessary and
sufficient condition for the existence of feasible
solution to the transportation problem.

5.1.2. Methods of Solving
Transportation Problem
Methods to get the initial solution: even if
there are different methods of such types, the
following three common methods can be used.
The North-West Corner Method
(NWC)
Minimum cost method (MCM), and
The Vogel’s Approximation Method
(VAM).

1. North West Corner method
1st. Balance the problem.That is see whether ΣDD =
ΣSS.
 If not open a dummy column or dummy row
as the case may be and balance the problem.
2nd. Allocate as much as possible to the selected
cell, and adjust the associated amounts of supply
and demand by subtracting the allocated amount.
Since :The method starts at the northwest-corner
cell (route) of the tableau (variable X11).

3rd. Cross out the row or column with zero supply
or demand to indicate that no further
assignments can be made in that row or column.
If both a row and a column net to zero
simultaneously, cross out one only, and leave a zero
supply (demand) in the uncrossed-out row
(column).
4th If exactly one row or column is left uncrossed
out, stop.
Otherwise, move to the cell to the right if a column
has just been crossed out or below if a row has
been crossed out. Go to step 1.

5th Make sure that all the rim conditions are satisfied
and (m+n-1) cells are allocated.
6th Once all the allocations are over, i.e., both rim
requirement (column and row i.e., availability and
requirement constraints) are satisfied, write
allocations and calculate the cost of
transportation.
Example 1.
Let us consider an example at this juncture to
illustrate the application of NWC rule

To
From
D1 D2 D3 Total supply
S1 X11 X12 X13 70
S2 X21 X22 X32 30
S3 X31 X32 X33 50
Total Demand 65 42 43 150
150
5
4
1
6
2
5
7
5
4
ΣDD = ΣSS . The NorthWest Corner cell X11 is chosen for
allocation..

The origin S1 has 70 items and the destination D1
requires only 65 items.
Hence it is enough to allot 65 items from S1 to D1.
The origin S1 which is alive with 5 more items can
supply to the destination to the right is alive with
5 more items can supply to the destination to the
right of D1 namely D2 whose requirement is 42. So,
we supply 5 items to D2 thereby the origin S1 is
exhausted.

D2 requires 37 items more. Now consider the
origin S2 that has 30 items to spare.We allot 30
items to the cell (X22) so that the origin S2 is
exhausted.
Then, move to origin S3 and supply 7 more items
to the destination D2. Now the requirement of the
destination D2 is complete and S3 is left with 43
items and the same can be allotted to the
destination D3.
Now the origin S3 is emptied and the requirement
at the destination D3 is also complete.This
completes the initial solution to the problem

To
From
S1 65 5 70
S2 30 30
S3 7 43 50
150
5
4
1
6
2
5
7
5
4
The total cost of transportation by this method will be:
(65 × 5) + (5 × 6) + (30 × 2) + (7 × 5) + (43× 4) = 325 + 30 + 60 + 35
+ 172 = 622.

Example 2.
To
From
X Y Z Dummy Availability
A 7 3 10
B 8 8
C 1 4 5
D 1 5 6
Requirement 7 12 5 5 29
4
5
6
3
3
6
4
5
2
1
3
4
0
0
0
0
The total cost of transportation by this method will be:
(7x4)+(3x3)+(8x6)+(1x4)+(4x3)+(1x4)+(5x0) = 105

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Example 3
Destination
Supply
Source N S E W
A
16 13 22 17
200
100 100
B
14 13 19 15
350
40 300 10
C
9 20 23 10
150
150
Dummy
0 0 0 0
90
90
Demand 100 140 300 250
Z = 10770

2. Least-Cost Method
Steps in Least-Cost Method:
Step 1: Determine the least cost among all the
rows of the transportation table.
Step 2: Identify the row and allocate the
maximum feasible quantity in the cell
corresponding to the least cost in the row.
Then eliminate that row (column) when an
allocation is made.

Step 3: Repeat steps 1 and 2 for the reduced
transportation table until all the available quantities
are distributed to the required places.
If the minimum cost is not unique, the tie can be
broken arbitrarily.
Step 4: Make sure that all the rim conditions are
satisfied and (m+n-1) cells are allocated.

To
From
S1 70
S2 30
S3 50
150
5
4
6
7
4
7
8
6
7

We examine the rows S1, S2 and S3, 4 is the least
cost element in the cell (S2, D1) and
(S2, D2) and the tie can be broken arbitrarily. Select
(S2, D1).
The origin S2 can supply 30 items to D1 and thus
origin S2 is exhausted.
D1 still requires 35 more units. Hence, shade the row
S2. Shading S2, we observe that 5 is the least element
in the cell (S1, D1) and examine the supply at S1 and
demand at D1.

The destination D1 requires 35 items and this
requirement is satisfied from S1 so that the column D1
is shaded next.
Next, we choose 7 as least element corresponding to
the cell (S1, D2).
We supply 35 units from S1 to D2. Now, only one row
is left behind.
Hence, we allow 7 items from S3 to D2 and 43 items S3
to D3.

To
From
S1 35 35 70
S2 30 30
S3 7 43 50
150
5
4
6
7
4
7
8
6
7
The cost of the allocation by the least cost method is
(35 x 5) + (35 x 7) + (30 x 4) + (7 x 7) + (43 x 7) = 890

© 2011 Pearson Education
Example 2
To (A)
Albuquerque
(B)
Boston
(C)
Cleveland
(D) Des Moines
(E) Evansville
(F) Fort Lauder
Warehouse
requirement 300 200 200
Factory
capacity
300
300
100
700
$5
$5
$4
$4
$3
$3
$9
$8
$7
From
100
First, $3 is the lowest cost cell so ship 100 units from Des
Moines to Cleveland and cross off the first row as Des
Moines is satisfied
Figure C.4

To (A)
Albuquerque
(B)
Boston
(C)
Cleveland
(D) Des Moines
(E) Evansville
(F) Fort Lauderdale
Warehouse
Factory
capacity
300
300
100
700
$5
$5
$4
$4
$3
$3
$9
$8
$7
From
100
100
Second, $3 is again the lowest cost cell so ship 100 units
from Evansville to Cleveland and cross off column C as
Cleveland is satisfied
Figure C.4

To (A)
Albuquerque
(B)
Boston
(C)
Cleveland
(D) Des Moines
(E) Evansville
(F) Fort Lauderdale
Warehouse
Factory
capacity
300
300
100
700
$5
$5
$4
$4
$3
$3
$9
$8
$7
From
100
100
200
Third, $4 is the lowest cost cell so ship 200 units from
Evansville to Boston and cross off column B and row E as
Evansville and Boston are satisfied
Figure C.4

To (A)
Albuquerque
(B)
Boston
(C)
Cleveland
(D) Des Moines
(E) Evansville
(F) Fort Lauderdale
Warehouse
Factory
capacity
300
300
100
700
$5
$5
$4
$4
$3
$3
$9
$8
$7
From
100
100
200
300
Finally, ship 300 units from Albuquerque to Fort Lauder
as this is the only remaining cell to complete the
allocations
Figure C.4

To (A)
Albuquerque
(B)
Boston
(C)
Cleveland
(D) Des Moines
(E) Evansville
(F) Fort Lauderdale
Warehouse
Factory
capacity
300
300
100
700
$5
$5
$4
$4
$3
$3
$9
$8
$7
From
100
100
200
300
Total Cost = $3(100) + $3(100) + $4(200) + $9(300)
= $4,100
Figure C.4

3.Vogel Approximation Method
Steps inVAM:
Step 1: For each row (column), determine a penalty measure
by subtracting the smallest unit cost element in the row (column)
from the next smallest unit cost element in the same row
(column).
Step 2: Identify the row or column with the largest penalty. If
there is a tie (equal penalty) it can be broken by selecting the
cell where maximum allocation can be made.
Allocate as much as possible to the variable with the least unit
cost in the selected row or column.
Adjust the supply and demand, and cross out the satisfied row or
column.
If a row and a column are satisfied simultaneously, only one of
the two is crossed out, and the remaining row (column) is
assigned zero supply (demand).

Step 3:
(a) If exactly one row or column with zero supply
or demand remains uncrossed out, stop.
(b) If one row (column) with positive supply
(demand) remains uncrossed out, determine the
basic variables in the row (column) by the least-
cost method. Stop.
(c) If all the uncrossed out rows and columns have
(remaining) zero supply and demand, determine
the zero basic variables by the least-cost method.
Stop.
(d) Otherwise, go to step 1.
Step 4: Make sure that all the rim conditions are
satisfied and cells are allocated.

To
From
D1 D2 D3 Total
supply
Row penalty
S1 65 5 70 2 2(ii) 1(iii) 0
S2 30 30 0 0 0 0
S3 7 43 50 1 1 0 0
Total
Demand
65 42 43 150
Column
penalty
1
1
0
3(i)
0
0
1
1
1
5
4
6
7
4
7
8
6
7
The cost of allocation (i.e., the associated objective value) by Vogel's
Approximation Method will be: (65× 5) + (5× 7) + (30 × 4) + (7 × 7) + (43× 7) =
325 + 35 +120 + 49 + 301 = 830.

The difference between the smallest and next to the
smallest element in each row and in each column is
calculated.
We choose the maximum from among the
differences.
The first individual allocation will be to the
smallest cost of a row or column with the
largest difference.
So we select the column D2 (penalty = 3) for the first
individual allocation, and allocate to (S2, D2) as much
as we can, since this cell has the least cost location.
Thus 30 units from S2 are allocated to D2.This exhausts
the supply from S2. However, there is still a demand of
12 units from D2.

The allocations to other cells in that column are 0.The
next step is to cross out row S2 (as it is exhausted).
The next largest unit difference corresponds to the
row S1.This leads to an allocation in the corresponding
minimum cost location in row S1, namely cell (S1, D1).
The maximum possible allocation is only 65 as required
byW1 from S1 and allocation of 0 to others in the row
S1. Column D1 is thus crossed out.
Maximum difference is 1 in row S3 and in column D3.
Select arbitrarily S3 and allot the least cost cell (S1, D2)
5 units. Cross out row S1 for it is already exhausted.
Now, we have only one row S3 and two columns D2
and D3 indicating that the entire available amount from
S3 has to be moved to D2 and D3 as per their
requirements.

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Vogel’s Method (1): calculate differences
Destination
Supply diff
Source N S E W
A
16 13 22 17
200 3
B
14 13 19 15
350 1
C
9 20 23 10
150 1
Dummy
0 0 0 0
90 0
Demand 100 140 300 250
diff 9 13 19 10

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Vogel’s Method (2): select xDummyE as basic
variable
Destination
Supply diff
Source N S E W
A
16 13 22 17
200 3
B
14 13 19 15
350 1
C
9 20 23 10
150 1
Dummy
0 0 0 0
90 0
Demand 100 140 300 250
diff 9 13 19 10
90

Tra
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tion
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Vogel’s Method (3): update supply, demand
and differences
Destination
Supply diff
Source N S E W
A
16 13 22 17
200 3
B
14 13 19 15
350 1
C
9 20 23 10
150 1
Dummy
0 0 0 0
--- ---
Demand 100 140 210 250
diff 5 0 3 5
90

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tion
-40
Vogel’s Method (4): select xCN as basic variable
Destination
Supply diff
Source N S E W
A
16 13 22 17
200 3
B
14 13 19 15
350 1
C
9 20 23 10
150 1
Dummy
0 0 0 0
--- ---
Demand 100 140 210 250
diff 5 0 3 5
90
100

Tra
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tion
-41
and differences
Destination
Supply diff
Source N S E W
A
16 13 22 17
200 4
B
14 13 19 15
350 2
C
9 20 23 10
50 10
Dummy
0 0 0 0
--- ---
Demand --- 140 210 250
diff --- 0 3 5
90
100

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tion
-42
Vogel’s Method (6): select xCW as basic
variable
Destination
Supply diff
Source N S E W
A
16 13 22 17
200 4
B
14 13 19 15
350 2
C
9 20 23 10
50 10
Dummy
0 0 0 0
--- ---
Demand --- 140 210 250
diff --- 0 3 5
90
100 50

Tra
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tion
-43
and differences
Destination
Supply diff
Source N S E W
A
16 13 22 17
200 4
B
14 13 19 15
350 2
C
9 20 23 10
--- ---
Dummy
0 0 0 0
--- ---
Demand --- 140 210 200
diff --- 0 3 2
90
100 50

Tra
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tion
-44
Vogel’s Method (8): select xAS as basic
variable
Destination
Supply diff
Source N S E W
A
16 13 22 17
200 4
B
14 13 19 15
350 2
C
9 20 23 10
--- ---
Dummy
0 0 0 0
--- ---
Demand --- 140 210 200
diff --- 0 3 2
90
100 50
140

Tra
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tion
-45
and differences
Destination
Supply diff
Source N S E W
A
16 13 22 17
60 5
B
14 13 19 15
350 4
C
9 20 23 10
--- ---
Dummy
0 0 0 0
--- ---
Demand --- --- 210 200
diff --- --- 3 2
90
100 50
140

Tra
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tion
-46
Vogel’s Method (10): select xAW as basic
variable
Destination
Supply diff
Source N S E W
A
16 13 22 17
60 5
B
14 13 19 15
350 4
C
9 20 23 10
--- ---
Dummy
0 0 0 0
--- ---
Demand --- --- 210 200
diff --- --- 3 2
90
100 50
140 60

Tra
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tion
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and differences
Destination
Supply diff
Source N S E W
A
16 13 22 17
--- ---
B
14 13 19 15
350 4
C
9 20 23 10
--- ---
Dummy
0 0 0 0
--- ---
Demand --- --- 210 140
diff --- ---
90
100 50
140 60

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tion
-48
Vogel’s Method (12): select xBW and xBE as
basic variables
Destination
Supply diff
Source N S E W
A
16 13 22 17
--- ---
B
14 13 19 15
---
C
9 20 23 10
--- ---
Dummy
0 0 0 0
--- ---
Demand --- --- --- ---
diff --- ---
90
100 50
140 60
140
210
Z = 10330

5.2.Assignment problem
In many business situations, management
needs to assign - personnel to jobs, - jobs
to machines, - machines to job locations,
or - salespersons to territories.
Consider the situation of assigning n jobs to n
machines.
When a job i (=1,2,....,n) is assigned to
machine j (=1,2, .....n) that incurs a cost Cij.
The objective is to assign the jobs to
machines at the least possible total cost.

This situation is a special case of the
transportation model and it is known as the
assignment problem.
Here, jobs represent “sources” and
machines represent “destinations.”
The supply available at each source is 1 unit and
demand at each destination is 1 unit.
Formulation/construction of the
model

The assignment model can be expressed mathematically as
follows:
Xij= 0, if the job j is not assigned to machine i
1, if the job j is assigned to machine I
Now the problem is which work is to be assigned to whom
so that the cost of completion of work will be minimum.

To minimize z (cost)
where xij =
1; if ith person is assigned jth work
0; if ith person is not assigned the jth work
i.e., ith person will do only one work.
i.e., jth work will be done only by one person.

Example
A farm produce different agricultural products and that
Products are manufactured on five different assembly lines
(1,2,3,4,5).
When manufacturing is finished, products are transported from
the assembly lines to one of the five different potential customers
(A,B,C,D,E).
Transporting products from five assembly lines to five inspection
areas requires different times (in minutes)
potential customers

Methods of solving AP
(The Hungarian Method)
 In order to find the proper assignment it is
essential for us to know the Hungarian method.
Step I
(A) Row reduction:
Select the smallest value in each row.
Subtract this value from each value in that row in
the cost matrix.
(B) Column reduction:
After completion of row reduction, subtract the
minimum entry of each column from all the
entries of the respective column.

Step II
Zero assignment:
(A) Starting with first row of the matrix received in
first step, examine the rows one by one until a
row containing exactly one zero is found.
 Then an experimental assignment indicated
by ‘ ’ is marked to that zero.
 Now cross all the zeros in the column in which
the assignment is made.
 Cross out zero, if there are other zero in ether
column or row.
 This procedure should be adopted for each
row assignment.

(B) When the set of rows has been
completely examined, an identical
procedure is applied successively to
columns.
Starting with column 1, examine all
columns until a column containing
exactly one zero is found.
Then make an experimental assignment in that
position and
Cross other zeros in the row in which the
assignment was made.

Continue these successive operations on rows and
columns until all zero’s have either been
assigned or corssed-out.
Now there are two possibilities:
(a) Either all the zeros are assigned or crossed
out, i.e., we get the maximal assignment.
or
(b) At least two zeros are remained by
assignment or by crossing out in each row or
column.
In this situation we try to exclude some of the
zeros by trial and error method.
This completes the second step.

After this step we can get two situations.
(i)Total assigned zero’s = n
The assignment is optimal.
(ii)Total assigned zero’s < n
Use step III and onwards.
Since n= # of assignments
Step III
Draw of minimum lines to cover zero’s :
In order to cover all the zero’s at least
once you may adopt the following
procedure.

(i) Marks (√) to all rows in which the assignment
has not been done.
(ii) See the position of zero in marked (√) row and then
mark (√) to the corresponding column.
(iii) See the marked (√) column and find the
position of assigned zero’s and then mark (√) to
the corresponding rows which are not marked till now.
(iv) Repeat the procedure (ii) and (iii) till the completion
of marking.
(v) Draw the lines through unmarked rows and
marked columns.
Note: If the above method does not work then
make an arbitrary assignment and then follow step IV.

Step IV
Select the smallest element from the uncovered
elements :
(i) Subtract this smallest element from all those
elements which are not covered.
(ii) Add this smallest element to all those
elements which are at the intersection of two
lines.
Step V
Thus we have increased the number of zero’s.
Now, modify the matrix with the help of step II and
find the required assignment.
Finally calculate the total minimized cost by
summing up numbers from the original table

Example 1.
Four persons A,B,C and D are to be assigned four
jobs I, II, III and IV. The cost matrix is given as
under, find the proper assignment.
Jobs
Man
A B C D
I 8 10 17 9
II 3 8 5 6
III 10 12 11 9
IV 6 13 9 7
Solution :
In order to find the proper assignment we apply the
Hungarian algorithm as follows:

I (A) Row reduction
Jobs
Man
A B C D
I 0 2 9 1
II 0 5 2 3
III 1 3 2 0
IV 0 7 3 1
Jobs
Man
A B C D
I 0 0 7 1
II 0 3 0 3
III 1 1 0 0
IV 0 5 1 1
I (B) Column reduction

II Zero assignment:
Jobs
Man
A B C D
I 0 0 7 1
II 0 3 0 3
III 1 1 0 0
IV 0 5 1 1
In this way all the zero’s are either crossed out or assigned.
Also total assigned zero’s = 4 (i.e., number of rows or columns).
Thus, the assignment is optimal.
From the table we get:
I B;
II  C:
III  D and
IV  A. Total minimized cost = 10+5+9+6= 30

Example 2:
There are five machines and five jobs are to be
assigned and the associated cost matrix is as
follows. Find the proper assignment.
machines
Jobs
I II III IV V
A 6 12 3 11 15
B 4 2 7 1 10
C 8 11 10 7 11
D 16 19 12 23 21
E 9 5 7 6 10

Solution:
In order to find the proper assignment, we apply the
Hungarian method as follows:
I A (Row reduction)
machines
Jobs
I II III IV V
A 3 9 0 8 12
B 3 1 6 0 9
C 1 4 3 0 4
D 4 7 0 11 9
E 4 0 2 1 5

machines
Jobs
I II III IV V
A 2 9 0 8 8
B 2 1 6 0 5
C 0 4 3 0 0
D 3 7 0 11 5
E 3 0 2 1 1
machines
Jobs
I II III IV V
A 2 9 0 8 8
B 2 1 6 0 5
C 0 4 3 0 0
D 3 7 0 11 5
E 3 0 2 1 1
II (Zero assignment)
IB (Column reduction)

From the last table we see that all the zeros are
either assigned or crossed out, but the total number
of assignment, i.e., 4<5 (number of jobs to be assigned
to machines).
Therefore, we have to follow step III and onwards
as follows:
machines
Jobs
I II III √ 2nd IV V
A 2 9 0 8 8 √ 3rd
B 2 1 6 0 5
C 0 4 3 0 0
D 3 7 0 11 5 √ 1st
E 3 0 2 1 1

Step IV: Here, the smallest element among the uncovered
elements is 2.
(i) Subtract 2 from all those elements which are not
covered.
(ii) Add 2 to those entries which are at the
junction(intersection ) of two lines.
Make as it is rows that pass single line .
machines
Jobs
I II III IV V
A 0 7 0 6 6
B 2 1 8 0 5
C 0 4 5 0 0
D 1 5 0 9 3
E 3 0 4 1 1

Step V. using step II again
machines
Jobs
I II III IV V
A 0 7 0 6 6
B 2 1 8 0 5
C 0 4 5 0 0
D 1 5 0 9 3
E 3 0 4 1 1
Thus, we have got five assignments as required by the
problem.
The assignment is as follows:
A I, B  IV, C V, D  III and E  II.
Thus from
the cost matrix the minimum cost = 6+1+11+12+5=35.

START
Construct the effectiveness matrix if not already given
Row reduction
Column reduction
Is zero assignment
possible
(i) Draw minimum number of lines to cover
all the zeros
(ii) Choose the least uncovered element
(iii)Subtract this from the uncovered
elements and add it to the elements at
intersection of the lines.
ASSIGNMENT
Put square over the zero and
cross out all zeros (if any) of
the corresponding column.
SOLUTION
Add the elements of the given
matrix correspond to each
square from original table
STOP
Is Zero
assignment
Possible
Yes
No
No
Yes

unit2 linear programming problem in .pdf

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unit2 linear programming problem in .pdf