Solution of Multi Objective Transportation Problem

International Journal of Trend in Scientific Research and Development
Volume 5 Issue 4, May-June
@ IJTSRD | Unique Paper ID – IJTSRD4
Solution of Multi Objective Transportation Problem
1Sheth C. D. Barfiwala College
ABSTRACT
The transportation problem is one of the earliest applications of the linear
programming problems. The basic transportation problem was originally
developed by Hitchcock. Efficient methods of solution derived from the
simplex algorithm were developed in 19
can be modeled as a standard linear programming problem, which can then
be solved by the simplex method. The objective of traditional
transportation is to determine the optimal transportation pattern of a
certain goods from supplier to demand customer so that the transportation
cost become minimum and for this purpose we have different method for
getting initial and optimal solution. We can get an initial basic feasible
solution for the transportation problem by using the Nort
rule, Row minima, Column minima, Matrix minima, or the Vogel
Approximation Method (VAM). To get an optimal solution for the
transportation problem, we use the MODI method (Modified Distribution
Method). In this paper we have developed an algo
programming technique to solve multi objective transportation problem.
We have also compared the result with raw maxima and EMV and show
how the developed approach is more effective than other approaches.
KEYWORDS: Multi- objective, Transportation, Fuzzy Programming, Cost, Time
1. INTRODUCTION
Two types of research work is done for transportation
problem one is formulation of simple and multi objective
transportation problem and second is developed a
solution approach for simple and multi objective
transportation problem. This chapter includes the work
done on multi objective transportation problems as well
as research objective. The transportation prob
formalized by the French Mathematician (Gaspard Monge,
1781). Major advances were made in the field during
World War two by the Sovi- et/Russian mathematician
and economist Leonid Vitaliyevich Kantorovich.
Kantorovich is regarded as the founder of
programming. Consequently, the problem as it is stated is
sometimes known as the Monge
transportation problem.
In 2014, Sudipta Midya and Sankar Kumar Roy [1] solved
single sink, fixed-charge, multiobjective, multi
stochastic transportation problem by using fuzzy
programming approach. A utility function approach to
solve multi objective transportation problem was given by
Gurupada Maity and Sankar Kumar Roy [2] in 2014.
in 2016 Gurupada Maity, Sankar Kumar Roy, and José Luis
Verdegay [3] gave concept of cost reliability in the
transportation cost and they considered supply and
demand as uncertain variables. Sheema Sadia, Neha Gupta
and Qazi M. Ali [4] presented their study on multi
objective capacitated fractional transportatio
2016. They considered mixed linear constraints. Solution
of multi objective transportation problem with non linear
cost and multi choice demand was given by Gurupada
Maity and Sankar Kumar [5] Roy in 2016. Mohammad
Trend in Scientific Research and Development
2021 Available Online: www.ijtsrd.com e-ISSN: 2456
43607 | Volume – 5 | Issue – 4 | May-June 202
f Multi Objective Transportation Problem
Sanjay R. Ahir1, H. M. Tandel2
Sheth C. D. Barfiwala College of Commerce, Surat, Gujarat, India
2Rofel College Vapi, Gujarat, India
The transportation problem is one of the earliest applications of the linear
programming problems. The basic transportation problem was originally
developed by Hitchcock. Efficient methods of solution derived from the
simplex algorithm were developed in 1947. The transportation problem
can be modeled as a standard linear programming problem, which can then
be solved by the simplex method. The objective of traditional
transportation is to determine the optimal transportation pattern of a
upplier to demand customer so that the transportation
cost become minimum and for this purpose we have different method for
getting initial and optimal solution. We can get an initial basic feasible
solution for the transportation problem by using the North-West corner
rule, Row minima, Column minima, Matrix minima, or the Vogel
Approximation Method (VAM). To get an optimal solution for the
transportation problem, we use the MODI method (Modified Distribution
In this paper we have developed an algorithm by fuzzy
programming technique to solve multi objective transportation problem.
We have also compared the result with raw maxima and EMV and show
how the developed approach is more effective than other approaches.
Transportation, Fuzzy Programming, Cost, Time
How to cite this paper
H. M. Tandel "Solu
Transportation
Problem" Published
in International
Journal of Trend in
Scientific Research
and Development
(ijtsrd), ISSN: 2456
6470, Volume
Issue-4, June 2021, pp.1331
www.ijtsrd.com/papers/ijtsrd43607.pdf
Copyright © 20
International Journal
Scientific Research and Development
Journal. This is an Open Access article
distributed under
the terms
Creative Commons
Attribution License
(http: //creativecommons.org/licenses/by/4.0
Two types of research work is done for transportation
problem one is formulation of simple and multi objective
transportation problem and second is developed a
solution approach for simple and multi objective
transportation problem. This chapter includes the work
done on multi objective transportation problems as well
as research objective. The transportation problem was
formalized by the French Mathematician (Gaspard Monge,
1781). Major advances were made in the field during
et/Russian mathematician
and economist Leonid Vitaliyevich Kantorovich.
Kantorovich is regarded as the founder of linear
programming. Consequently, the problem as it is stated is
sometimes known as the Monge–Kantorovich
In 2014, Sudipta Midya and Sankar Kumar Roy [1] solved
charge, multiobjective, multi-index
portation problem by using fuzzy
programming approach. A utility function approach to
multi objective transportation problem was given by
Gurupada Maity and Sankar Kumar Roy [2] in 2014. Then,
in 2016 Gurupada Maity, Sankar Kumar Roy, and José Luis
erdegay [3] gave concept of cost reliability in the
transportation cost and they considered supply and
demand as uncertain variables. Sheema Sadia, Neha Gupta
and Qazi M. Ali [4] presented their study on multi
objective capacitated fractional transportation problem in
2016. They considered mixed linear constraints. Solution
of multi objective transportation problem with non linear
cost and multi choice demand was given by Gurupada
Maity and Sankar Kumar [5] Roy in 2016. Mohammad
Asim Nomani, Irfan Ali an
algorithm of proposed method in 2017. In proposed
method they have used weighted sum method based on
goal programming. In 2017 only, Sankar Kumar Roy,
Gurupada Maity, Gerhard Wilhelm Weber and Sirma
Zeynep Alparslan Gök [7] solved
transportation problem by using conic scalarization
approach with interval goal. Then, by using utility
approach with goals Sankar Kumar Roy, Gurupada Maity
and Gerhard-Wilhelm Weber [8] solved multi objective
two stage grey transportation
inspired from Zimmermann’s fuzzy programming and the
neutrosophic set terminology recently in 2018, Rizk M.
Rizk-Allah, Aboul Ella Hassanien and Mohamed Elhoseny
[9] proposed a model under neutrosophic environment. In
this model for each objective functions, they considered
three membership functions namely, truth membership,
indeterminacy membership and falsity membership.
Srikant Gupta, Irfan Ali and Aquil Ahmed [10] presented
their study on multi objective capaciated transportation
problem with uncertain supply and demand. They
formulated deterministic form of the problem by using
solution procedure of multi choice and fuzzy numbers.
Then they used goal programming approach to solve
fractional objective function.
Many researchers have done tremendous work with this
method, which is not mentioned all but some of the
research is summarized over here. It is just a brief
summary of fuzzy programming technique based
optimization to provide comprehensive knowledge of
fuzzy optimization and solutions.
Trend in Scientific Research and Development (IJTSRD)
ISSN: 2456 – 6470
2021 Page 1331
f Multi Objective Transportation Problem
f Commerce, Surat, Gujarat, India
How to cite this paper: Sanjay R. Ahir |
H. M. Tandel "Solution of Multi Objective
Transportation
Problem" Published
in International
Journal of Trend in
Scientific Research
and Development
(ijtsrd), ISSN: 2456-
6470, Volume-5 |
4, June 2021, pp.1331-1337, URL:
www.ijtsrd.com/papers/ijtsrd43607.pdf
Copyright © 2021 by author (s) and
International Journal of Trend in
Scientific Research and Development
Journal. This is an Open Access article
distributed under
the terms of the
Creative Commons
Attribution License (CC BY 4.0)
//creativecommons.org/licenses/by/4.0)
Asim Nomani, Irfan Ali and A. Ahmed [6] presented
method in 2017. In proposed
method they have used weighted sum method based on
goal programming. In 2017 only, Sankar Kumar Roy,
Gurupada Maity, Gerhard Wilhelm Weber and Sirma
Zeynep Alparslan Gök [7] solved multi objective
transportation problem by using conic scalarization
approach with interval goal. Then, by using utility
approach with goals Sankar Kumar Roy, Gurupada Maity
Wilhelm Weber [8] solved multi objective
two stage grey transportation problem in 2017. By
inspired from Zimmermann’s fuzzy programming and the
neutrosophic set terminology recently in 2018, Rizk M.
Allah, Aboul Ella Hassanien and Mohamed Elhoseny
[9] proposed a model under neutrosophic environment. In
h objective functions, they considered
three membership functions namely, truth membership,
indeterminacy membership and falsity membership.
Srikant Gupta, Irfan Ali and Aquil Ahmed [10] presented
their study on multi objective capaciated transportation
oblem with uncertain supply and demand. They
formulated deterministic form of the problem by using
solution procedure of multi choice and fuzzy numbers.
Then they used goal programming approach to solve
done tremendous work with this
method, which is not mentioned all but some of the
research is summarized over here. It is just a brief
summary of fuzzy programming technique based
optimization to provide comprehensive knowledge of
lutions. In real situation the
IJTSRD43607

International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
@ IJTSRD | Unique Paper ID – IJTSRD43607 | Volume – 5 | Issue – 4 | May-June 2021 Page 1332
objective parameter are decided according to the
requirement of decision maker. Many times he is unable to
give such kind of information and to deal with these
imprecision the parameter are formulated as fuzzy
number, especially as triangular fuzzy numbers. Means,
the objective function is fuzzified and leverage is provided
to the decision maker to operate. Zadeh [11] first
introduced the concept of fuzzy set theory. Then
Zimmermann [12] first applied the fuzzy set theory
concept with some suitable membership functions to solve
linear programming problem with several objective
functions. He showed that solutions obtained by fuzzy
linear programming are always efficient. Bit et al. [13]
applied the fuzzy programming technique with linear
membership function to solve the multi-objective
transportation problem.
In this paper we have find the solution of multi objective
transportation problems by fuzzy programming technique
using linear membership function.
2. Multi objective Transportation Problem:
Let us consider that any company has m production
centres, say , , , … … and n warehouses or
markets, say , , , … … . Let supply capacities of
each production centers be , , , … … respectively
and demand levels of each destinations be
, , , … … respectively. The decision maker or
manager of company wants to optimize r number of
penalties ( , ℎ = 1,2,3 … … . ) like minimize the
transportation cost, maximize the profit, minimize the
transportation time, minimize the risk, etc. Now, if be
the cost associated with objective to transport a unit
product from production center to warehouse and
! be the unknown quantity to be transported from
production center to warehouse. Then to solve this
type of problem the multi objective transportation
problem is defined as shown in model (1.2). [3]
Model: Multi objective transportation problem
Minimize: Z*
= + + C-.
*
X-., r =
1
.2
1,2,3, … … … K
4
-2
;
Subject to the constraints + X-.
1
.2
= a-,
i = 1,2,3, … … … m;
+ X-.
4
-2
= b., j = 1,2,3, … … … n;
X-. ≥ 0, ∀ i, j.
Where, m = Number of sources;
n = Number of destinations;
a- = Available supplies at iKL
source;
b. = Demand level of jKL
destination;
=Cost associated with rKL
objective for
transporting a unit of product from iKL
source
to jKL
destination;
X-. = The quantity of product to be
transported from iKL
source to jKL
destination.
To find optimal solution of any multi objective
transportation problem, so many approaches are there
like, goal programming, fuzzy approach, genetic algorithm,
etc. In most of the solution approaches one general
objective function is defined by considering each single
objective function as the constraints. Solution given by
general objective function may or may not give optimal
solution to each objective function but this will give us
compromise solution.
3. Fuzzy Programming Technique to Solve Multi-
Objective Problems
Most of the entrepreneur now a day’s do not have a aim of
single objective but they wish to target multi objective i.e.`
they not only try to minimize cost but try to minimize
some recourse so that their business can grow in best of
manner. In competitive world entrepreneur need to be
aware of competition and should monopolized business.
Their important objective could be to minimize risk using
the same set of constraints. Such general multi objective
linear programming problem can be defined as under
[14,15]
Minimize
1
, 1,2,3,4.....,
i n
k
k i i
i
z c x k r
=
=
= =
∑
Subject to the constraints,
1
( , , ) , 1,2,3......
n
i i j
i
a x B j m
=
≤ = ≥ =
∑ ,
0
i
x ≥ .
In fuzzy programming technique following procedure
applied to solve the multi objective optimization problem
[12]:
The formulated multi objective linear programming
problem first solve by using single objective function and
derive optimal solution say 1 1 2 3
( , , .......... )
n
f x x x x for first
objective 11
z and then obtain other objective value with
the same solution say 21
z 31
z 41
z … 1
k
z . Procedure repeats
same for 2........ r
z z objectives.
Step 2: Corresponding to above data we can construct a
pay off matrix which can give various alternate optimal
value.
Z1 Z2 ......... Zr
1 1 2 3
( , , .......... )
n
f x x x x Z11 Z21 ......... Zr1
2 1 2 3
( , , .......... )
n
f x x x x Z12 Z22 ......... Zr2
...... .........
1 2 3
( , , .......... )
n n
f x x x x Z1n Z2n ......... Zrn
Table: 1- Pay-off matrix for MOLPP
Here,
ki
z : indicated optimal solution of ‘kth objective using
solution of ‘ith objective, 1,2,3,4.....,
k r
= and
1,2,3......
i n
= .
Or

Find out the positive ideal solution (PIS) and negative
ideal solution (NIS) for each objective function of the
model
Now, by using pay-off matrix or positive ideal solution
(PIS) and negative ideal solution (NIS) define a
membership function S(UV)for the W objective function.
Here two different membership function are utilized to
find efficient solution of this multi-objective resource
allocation problem and by using this membership function
convert the MOLPP into the following model
Model -1:
Maximum λ ,
Subject to the constraints
( )
k
Z
λ µ
≤ ,
1
( , , ) , 1,2,3......
m
i i j
i
a x B j n
=
≤ = ≥ =
∑
0
i
x ≥ ,
When we utilize Fuzzy linear membership function [12]
then model structure is as follows
Model- 2:
Maximum λ ,
( )
k k k k
z U L U
λ
+ − ≤ ,
1
( , , ) , 1,2,3......
m
i i j
i
a x B j n
=
≤ = ≥ =
∑
0
i
x ≥ .
Solution of this model will give you an efficient solution
4. Algorithm to solve Multi-Objective Linear
Programming Problem
Input: Parameters: 1 2
( , ,..., , )
k
Z Z Z n
Output: Solution of multi-objective programming problem
Solve multi-objective programming problem ( ,
k
Z X
↓ ↑ )
begin
read: problem
while problem = multi-objective programming problem
do
for k=1 to m do
enter matrix k
Z
end
-| determine pay-off matrix
Or
-| the positive ideal solution and negative ideal
solution for each objective.
for k=1 to m do
( )
0
PIS
min
ij i
z z
=
Under given constraints
end
for k=1 to m do
( )
0
NIS
max
ij i
z z
=
end
- find single objective optimization models under
given constraints from multi-objective optimization
models.
fork=1 to m do
max λ
Subject to the constraints:
( )
ij
E
Z x
λ µ
≤
End
|- find the solution SOPs using Lingo software.
5. Numerical Examples
This section considers several numerical examples of
transportation problem and finds their solution by fuzzy
programming technique
Numerical Illustration 1:
Transportation problem with some demand and supply
are given below [21].
D1 D2 D3 D4 Supply
S1 1 2 7 7 8
S2 1 9 3 4 19
S3 8 9 4 6 17
Demand 11 3 14 16
Table 2 showing objective function 1
D1 D2 D3 D4 Supply
S1 4 4 3 3 8
S2 5 8 9 10 19
S3 6 2 5 1 17
Demand 11 3 14 16
Table 3 showing objective function 2
Mathematical Formulation of this problem can be written
as
Min Z1 = x11 + 2x12 + 7x13 + 7x14 + x21 + 9x22 + 3x23 + 4x24+
8x31+ 9x32+ 4x33 + 6x34
Min Z2 = 4x11 + 4x12 + 3x13 + 3x14 + 5x21 + 8x22 + 9x23 +
10x24+ 6x31+ 2x32+ 5x33 + x34
x11+x12+x13+x14=8;
x21+x22+x23+x24=19;
x31+x32+x33+x34=17;
x11+x21+x31=11;
x12+x22+x32=3;
x13+x23+x33=14;
x14+x24+x34=16;
x11 0; x12 0; x13 0; x14 0;

x21 0; x22 0; x23 0; x24 0; x31 0; x32 0; x33 0; x34 0;
PIS and NIS value of first objective function is given by
PIS = 143, NIS = 265
PIS and NIS value of second objective function is given by
PIS = 167, NIS = 310
Hence,
U1=265, L1=143, U2= 310, L2=167
U1- L1= 122
U2 - L2 = 143
Applying fuzzy linear membership function, we get the following model
When we solve this problem with computational software like LINGO then the solution of the model is as follows:
The allocations are,
X11 = 3.000000 X21 = 8.000000 X33 = 1.000000
X12 = 3.000000 X23 = 11.00000 X34 = 16.00000
X13 = 2.000000
The values of objective functions are as follows: Z1 = 164.0000, Z2 = 190.0000
Using these allocations we have Z1=164, Z2= 190 with degree of satisfaction=0.8278689

The table given below shows the comparison of the given transportation problem with other approaches
Method Numerical Example
Row maxima method [19] Z1= 172,Z2=213.
EMV method [20 ] Z1=143, Z2=265
fuzzy Multiobjective programming
Z1= 164,
Z2= 190.
Numerical illustration -2:
A supplier, supply a product to different destination from different sources. The supplier has to take decision in this TP so
that the transportation cost and transportation time should be minimum[21]. The data for the cost and time is as follows:
Destination
sources↓
D1 D2 D3 Supply
S1 16 19 12 14
S2 22 13 19 16
S3 14 28 8 12
Demand 10 15 17 42
Table 4 Data for time
Destination
sources↓
D1 D2 D3 SUPPLY
S1 9 14 12 14
S2 16 10 14 16
S3 8 20 6 12
Demand 10 15 17 42
Table 5 Data for cost
Method Objectives values
Row maxima method [19] Z1= 518, Z2=374
EMV method [20] Z1= 518, Z2=374
fuzzy Multi objective programming Z1= 518, Z2=374
The data is collected by a person, who supplies product to different companies after taking it from different origins. There
are four different suppliers named as A, B, C and D and four demand destinations namely E, F, G and H. How much amount
of material is supplied from different origins to all other demand destinations so that total cost of transportation and
product impairment is minimum [17].
Supplies: a1 = 21, a2 = 24, a3 = 18, a4 = 30.
Demands: b1 = 15, b2 = 22, b3 = 26, b4 = 30.
Table 6 Transportation cost for TP
E F G H Supply
A 24 29 18 23 21
B 33 20 29 32 24
C 21 42 12 20 18
D 25 30 1 24 30
Demand 15 22 26 30
Table 7 Product Impairment for TP
E F G H Supply
A 24 29 18 23 21
B 33 20 29 32 24
C 21 42 12 20 18
D 25 30 1 24 30
Demand 15 22 26 30
Objectives
Osuji et. al.
(LMF) [17]
Osuji et. al.
(HMF) [17]
Jignash G Patel
GSDMT Method [21]
Fuzzy Programming
Approach
Cost 1900 1898 1902 1902
Product impairment 1279 1286 1198 1198

Illustration 4: A company has four origins A, B, C and D with production capability of 5, 4, 2 and 9 units of manufactured
goods, respectively. These units are to be transported to fivewarehouses E, F, G, H and I with necessity of 4, 4, 6, 2 and 4
units, respectively. The transportation cost, risk and product impairment between companies to warehouses are given
below [18].
E F G H I Supply
A 9 12 9 6 9 5
B 7 3 7 7 5 4
C 6 5 9 11 3 2
D 6 8 11 2 2 9
Demand 4 4 6 2 4
E F G H I Supply
A 2 9 8 1 4 5
B 1 9 9 5 2 4
C 8 1 8 4 5 2
D 2 8 6 9 8 9
Demand 4 4 6 2 4
Objectives Abo-Elnaga et. al. [18]. Jignash G Patel GSDMT Method [21] Fuzzy Programming Approach
Cost 144 155 129
Risk 104 90 97
Product
impairment
173 80 83
The comparison of four shows that the solution obtained by fuzzy programming technique gives better solution of multi
objective transportation problem
Conclusion
This paper discussed a fuzzy programming technique for
solution of multi objective transportation problem. With
numerical illustrations, we conclude that fuzzy
programming technique is a good alternative approach for
find the solution of multi objective transportation problem
References
[1] Sudipta Midya and Sankar Kumar Roy (2014),
Single sink fixed charge multi objective multi index
stochastic transportation problem, American
Journal of Mathematical and Management Sciences,
33:4, 300-314, DOI: 10. 1080/01966324. 2014.
942474.
[2] Gurupada Maity and Sankar Kumar Roy (2014),
Solving multi-choice multi-objective transportation
problem: a utility function approach, Maity and Roy
Journal of Uncertainty Analysis and Applications
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[3] Gurupada Maity, Sankar Kumar Roy, and José Luis
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Multi objective capaciated fractional transportation
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Solving a multi objective transportation problem
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TRANSPORTATION PROBLEMS AND THEIR
SOLUTIONS, Ph. D. Thesis, Applied Mathematics and
Humanities Department, S. V. National Institute of
Technology, pp: 1-360.

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