In this article, two methods are presented, proposed method 1 and proposed method 2. Proposed method 1 is based on linear programming technique and proposed method 2 is based on modified distribution method. Both of the methods are used to solve the balanced and unbalanced intuitionistic fuzzy transportation problems. The ideas of the proposed methods are illustrated with the help of real life numerical examples which is followed by the results and discussion and comparative study is given. The proposed method is computationally very simple when compared to the existing methods, it is shown to be and easier form of evaluation when compared to current methods.

2. EDITOR-IN-CHIEF
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Nalan Gulpinar, Warwick Business School, UK
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Ülkü Gürler, Bilkent University, Turkey
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Steven Harper, James Madison University, USA
Michael J Hirsch, Raytheon Inc., USA
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Dariusz Jacek Jakóbczak, Koszalin University of Technology, Poland
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7. DOI: 10.4018/IJORIS.2018040104
International Journal of Operations Research and Information Systems
Volume 9 • Issue 2 • April-June 2018

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73
Linear Programming Approach for Solving
Balanced and Unbalanced Intuitionistic
Fuzzy Transportation Problems
P. Senthil Kumar, PG and Research Department of Mathematics, Jamal Mohamed College (Autonomous), Tiruchirappalli, India
ABSTRACT
In this article, two methods are presented, proposed method 1 and proposed method 2. Proposed
method 1 is based on linear programming technique and proposed method 2 is based on modified
distribution method. Both of the methods are used to solve the balanced and unbalanced intuitionistic
fuzzy transportation problems. The ideas of the proposed methods are illustrated with the help of real
life numerical examples which is followed by the results and discussion and comparative study is
given. The proposed method is computationally very simple when compared to the existing methods,
it is shown to be and easier form of evaluation when compared to current methods.
Keywords
Balanced and Unbalanced Intuitionistic Fuzzy Transportation Problems, Intuitionistic Fuzzy Set, Optimal
Solution, Triangular Fuzzy Number, Triangular Intuitionistic Fuzzy Number
INTRODUCTION
In several real-life situations, there is need to transport the homogeneous product from numerous
origins (sources) to different destinations and the aim of the decision maker is to find how much
quantity of the product from which source to which destination should be supplied so that all the supply
points are fully used and demand of all the destinations is fulfilled as well as total transportation cost
is minimum. The transportation problems play a vital role in logistics and supply chain management
for reducing cost and improving service. In today’s highly competitive market, the pressure on
companies to find better ways to create and deliver products and services to customers becomes
stronger. How and when to send the products to the customers in the quantities which they want
in a cost-effective manner becomes more challenging. Transportation models provide a powerful
framework to meet this challenge. They ensure the efficient movement and timely availability of raw
materials and finished goods.
Resource allocation is used to assign the available resources in an economic way. When the
resources to be allocated are scarce, a well-planned action is necessary for a decision-maker (DM)
to attain the optimal utility. If the supplying sources and the receiving agents are limited, the best
pattern of the allocation to get the maximum return or the best plan with the least cost, whichever
may be applicable to the problem, is to be found out. This class of problems is termed as ‘allocation

8. International Journal of Operations Research and Information Systems
Volume 9 • Issue 2 • April-June 2018
74
problems’ and is divided into ‘transportation problems’ and ‘assignment problems’. This type of
allocation problems is studied in operations research.
During World War-II, Britain was having very limited military resources; therefore, there was
an urgent need to allocate resources to the various military operations and to the activities within
each operation in an effective manner. Therefore, the British military executives called upon a team
of scientist to apply scientific approach to study the strategic and tactical problems related to air and
land defence of the country. As the team was dealing with research a military operations, the work
of this team of scientist was named as operations research.
The Transportation Problem (TP) is one of the subclasses of Linear Programming Problem (LPP).
The objective of the transportation problem is to transport various quantities of a single homogeneous
product that are initially stored at various origins, to different destinations in such a way that the total
transportation cost is minimum for a minimization problem or total transportation profit is maximum
for a maximization problem. The conventional transportation problem consists in transporting a
certain commodity from each of m origins i m= …1 2 3, , , to any of n destinations j n= …1 2 3, , , .
The origins are factories with respect capacities a a a am1 2 3
, , ,… and the destinations are warehouses
with required levels of demands b b b bn1 2 3
, , ,… . For the transport of a unit of the given commodity
from the ith
origin to the jth
destination a cost cij
is given for which, without loss of generality, we
can assume c i jij
≥ ∀0, , . Hence, one must determine the amounts xij
to be transported from all the
origins a a a am1 2 3
, , ,… to all the destinations b b b bn1 2 3
, , ,… in such a way that the total cost is
minimized.
The conventional transportation problem can be mathematically stated as follows:
Minimize Z c x
i
m
j
n
ij ij
=
= =
∑∑
1 1
subject to:
j
n
ij i
x a for i m
=
∑ ≤ = …
1
1 2, , , , (Row restriction)
i
m
ij j
x b for j n
=
∑ ≥ = …
1
1 2, , , , (Column restriction)
x for i mij
≥ = …0 1 2, , , ,
and:
j n= …1 2, , ,
Hitchcock (1941) developed a basic transportation problem.
The classical transportation problem is a special class of linear programming problem in which
all the constraints are equality type, widely used in the areas of inventory control, communication
network, aggregate planning, employment scheduling, and personnel assignment and so on. Depending
on the nature of the cost function, the transportation problem can be categorized into linear and
nonlinear transportation problem.

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11. Linear Programming Approach for Solving Balanced and Unbalanced Intuitionistic
Fuzzy Transportation Problems
P. Senthil Kumar (PG and Research Department of Mathematics, Jamal Mohamed College
(Autonomous), Tiruchirappalli, India)
Source Title: International Journal of Operations Research and Information Systems (IJORIS)
9(2)
Copyright: © 2018 |Pages: 28
DOI: 10.4018/IJORIS.2018040104
OnDemand PDF Download:
$30.00
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Reference to this paper should be made as follows:
MLA
Kumar, P. Senthil. "Linear Programming Approach for Solving Balanced and Unbalanced
Intuitionistic Fuzzy Transportation Problems." International Journal of Operations Research
and Information Systems (IJORIS) 9.2 (2018): 73-100.
APA
Kumar, P. S. (2018). Linear Programming Approach for Solving Balanced and Unbalanced
Intuitionistic Fuzzy Transportation Problems. International Journal of Operations Research
and Information Systems (IJORIS), 9(2), 73-100.
Chicago
Kumar, P. Senthil. "Linear Programming Approach for Solving Balanced and Unbalanced
Intuitionistic Fuzzy Transportation Problems." International Journal of Operations Research
and Information Systems (IJORIS) 9, no. 2 (2018): 73-100.
Harvard
Kumar, P.S., 2018. Linear Programming Approach for Solving Balanced and Unbalanced
Intuitionistic Fuzzy Transportation Problems. International Journal of Operations Research
and Information Systems (IJORIS), 9(2), pp.73-100.
Vancouver
Kumar PS. Linear Programming Approach for Solving Balanced and Unbalanced
Intuitionistic Fuzzy Transportation Problems. International Journal of Operations Research
and Information Systems (IJORIS). 2018 Apr 1;9(2):73-100.

12. International Journal of Operations Research and Information Systems
Volume 9 • Issue 2 • April-June 2018
89
3. Assuming that m is a membership value and n is a non-membership value at c. Then 100m%
experts are in favour and 100n% experts are opposing but 100 1− −( )m n % are in confusion
that the transportation cost is c .
Values of m cZI ( ) and n cZI ( ) at different values of c can be determined using equations given
as follows:
,
,
,m c
forc
c
for c
forc
ZI ( )=
<
−
≤ ≤
=
0 108
108
194
108 302
1 302
341−−
≤ ≤
>





c
for c
forc
39
302 341
0 341
,
,
,
,
,n c
forc
c
for c
for c
c
ZI ( )=
<
−
≤ ≤
=
−
1 83
302
219
83 302
0 302
3022
85
302 387
1 387
,
,
for c
forc
≤ ≤
>





Advantages of the Proposed Method
By using the proposed method, a decision maker has the following advantages:
1. The optimum objective value of the unbalanced IFTP is non-negative triangular intuitionistic
fuzzy number i.e., there is no negative part in the obtained triangular intuitionistic fuzzy number;
2. The proposed method is computationally very simple and easy to understand.
CONCLUSION
On the basis of the present study, it can be concluded that the IFTP and UIFTP which can be
solved by the existing methods (Hussain and Kumar (2012a), Gani and Abbas (2012), Antony et
al. (2014), Dinagar and Thiripurasundari (2014)) can also be solved by the proposed methods.
However, it is much easier to apply the proposed methods as compared to all the existing methods.
The solution obtained by this method the objective value of the unbalanced IFTP remains always
positive i.e., there is no negative part in the TIFN. Hence the proposed method is physically
meaningful and computationally very simple when compared to all the existing methods. In
feature, the proposed method may be modified to find intuitionistic fuzzy optimal solution of solid
intuitionistic fuzzy transportation problems and solid assignment problems with intuitionistic
fuzzy numbers. This method can help decision makers in the logistics related issues of real life
problems by aiding them in the decision-making process and providing an optimal solution in
a simple and effective manner.

13. International Journal of Operations Research and Information Systems
Volume 9 • Issue 2 • April-June 2018
90
ACKNOWLEDGMENT
The author sincerely thanks the anonymous reviewers and Editor-in-Chief Professor John Wang for
their careful reading, constructive comments and fruitful suggestions. The author would also like to
acknowledge Dr.S.Ismail Mohideen, Additional Vice Principal, My Guide and Associate Professor
Dr.R.Jahir Hussain, Dr.A.Nagoor Gani, Associate Professor, Dr.K.Ramanaiah, Associate Professor
(retired), Mr.N.Shamsudeen, Associate Professor (retired), Jamal Mohamed College (Autonomous),
Tiruchirappalli, Tamil Nadu, India for their motivation and kind support.

14. International Journal of Operations Research and Information Systems
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Kumar, A., Kaur, A., & Gupta, A. (2011). Fuzzy linear programming approach for solving fuzzy transportation
problems with transshipment. Journal of Mathematical Modelling and Algorithms, 10(2), 163–180. doi:10.1007/
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Kumar, P. S. (2016a). PSK method for solving type-1 and type-3 fuzzy transportation problems. International
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Kumar, P. S. (2017b). A note on ‘a new approach for solving intuitionistic fuzzy transportation problem of type-
2’. International Journal of Logistics Systems and Management.
Kumar, P. S. (2017c). Intuitionistic fuzzy zero point method for solving type-2 intuitionistic fuzzy transportation
problem. International Journal of Operational Research.
Kumar, P. S., & Hussain, R. J. (2014a). A systematic approach for solving mixed intuitionistic fuzzy transportation
problems. International Journal of Pure and Applied Mathematics, 92(2), 181–190. doi:10.12732/ijpam.v92i2.4
Kumar, P. S., & Hussain, R. J. (2014b) A method for finding an optimal solution of an assignment problem
under mixed intuitionistic fuzzy environment. In Proceedings of the International Conference on Mathematical
Sciences (ICMS-2014) (pp. 417-421). Elsevier.
Kumar, P. S., & Hussain, R. J. (2014c). New algorithm for solving mixed intuitionistic fuzzy assignment problem.
In Elixir Applied Mathematics (pp. 25971–25977). Salem, Tamilnadu, India: Elixir publishers.
Kumar, P. S., & Hussain, R. J. (2015). A method for solving unbalanced intuitionistic fuzzy transportation
problems. Notes on Intuitionistic Fuzzy Sets, 21(3), 54–65.
Kumar, P. S., & Hussain, R. J. (2016a). Computationally simple approach for solving fully intuitionistic fuzzy
real life transportation problems. International Journal of System Assurance Engineering and Management,
7(1), 90–101. doi:10.1007/s13198-014-0334-2
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problem. International Journal of Operations Research and Information Systems, 7(2), 39–61. doi:10.4018/
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problem using triangular intuitionistic fuzzy number. The Journal of Fuzzy Mathematics, 24(2), 289–302.
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Mohideen, S. I., & Kumar, P. S. (2010). A comparative study on transportation problem in fuzzy environment.
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problem. In Proceedings of the Ninth WSEAS International Conference on Systems, Athens, Greece.
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fuzzy environment. International Journal of System Assurance Engineering and Management, 6(2), 206–215.
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P. Senthil Kumar has been working as an Assistant Professor in PG and Research Department of Mathematics at
Jamal Mohamed College (Autonomous), Tiruchirappalli, Tamil Nadu, India. His research interests include operations
research, fuzzy optimization, intuitionistic fuzzy optimization, numerical analysis and graph theory, etc. He was
born in Tamil Nadu, India in the year 1985. He graduated B.Sc., M.Sc., M.Phil degrees from Jamal Mohamed
College, Tiruchirappalli, in 2006, 2008, 2010 respectively. He completed B.Ed in 2009 at Jamal Mohamed College
of Teacher Education. He completed PGDCA in 2011 in the Bharathidasan University and PGDAOR in 2012 in
the Annamalai University, Tamil Nadu, India. He has done his Ph.D in the area of Intuitionistic Fuzzy Optimization
Technique at Jamal Mohamed College in 2017. He has published many research papers in referred national and
international journals like Springer, IGI Global, Inderscience, etc. He also presented his research papers in Elsevier
Conference Proceedings (ICMS-2014), MMASC-2012, etc.
Figure 1. Graphical representation of FIFTC