This article describes how in solving real-life solid transportation problems (STPs) we often face the state of uncertainty as well as hesitation due to various uncontrollable factors. To deal with uncertainty and hesitation, many authors have suggested the intuitionistic fuzzy (IF) representation for the data. In this article, the author tried to categorise the STP under uncertain environment. He formulates the intuitionistic fuzzy solid transportation problem (IFSTP) and utilizes the triangular intuitionistic fuzzy number (TIFN) to deal with uncertainty and hesitation. The STP has uncertainty and hesitation in supply, demand, capacity of different modes of transport called conveyance and when it has crisp cost it is known as IFSTP of type-1. From this concept, the generalized mathematical model for type-1 IFSTP is explained. To find out the optimal solution to type-1 IFSTPs, a single stage method called intuitionistic fuzzy min-zero min-cost method is presented. A real-life numerical example is presented to clarify the idea of the proposed method. Moreover, results and discussions, advantages of the proposed method, and future works are presented. The main advantage of the proposed method is that the optimal solution of type-1 IFSTP is obtained without using the basic feasible solution and the method of testing optimality.

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5. DOI: 10.4018/IJORIS.2018070105
International Journal of Operations Research and Information Systems
Volume 9 • Issue 3 • July-September 2018

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90
A Simple and Efficient Algorithm for
Solving Type-1 Intuitionistic Fuzzy
Solid Transportation Problems
P. Senthil Kumar, PG and Research Department of Mathematics, Jamal Mohamed College (Autonomous), Tiruchirappalli,
India
ABSTRACT
This article describes how in solving real-life solid transportation problems (STPs) we often face the
state of uncertainty as well as hesitation due to various uncontrollable factors. To deal with uncertainty
and hesitation, many authors have suggested the intuitionistic fuzzy (IF) representation for the data.
In this article, the author tried to categorise the STP under uncertain environment. He formulates
the intuitionistic fuzzy solid transportation problem (IFSTP) and utilizes the triangular intuitionistic
fuzzy number (TIFN) to deal with uncertainty and hesitation. The STP has uncertainty and hesitation
in supply, demand, capacity of different modes of transport celled conveyance and when it has crisp
cost it is known as IFSTP of type-1. From this concept, the generalized mathematical model for
type-1 IFSTP is explained. To find out the optimal solution to type-1 IFSTPs, a single stage method
called intuitionistic fuzzy min-zero min-cost method is presented. A real-life numerical example is
presented to clarify the idea of the proposed method. Moreover, results and discussions, advantages
of the proposed method, and future works are presented. The main advantage of the proposed method
is that the optimal solution of type-1 IFSTP is obtained without using the basic feasible solution and
the method of testing optimality.
Keywords
Intuitionistic Fuzzy Min-Zero Min-Cost Method, Intuitionistic Fuzzy Set, Optimal Solution, Triangular
Intuitionistic Fuzzy Number, Type-1 Intuitionistic Fuzzy Solid Transportation Problem
1. INTRODUCTION
The transportation problem (TP) is a special class of linear programming problem, widely used in
the areas of inventory control, employment scheduling, aggregate planning, communication network,
personal management and so on. In several real life situations, there is a need for shipping the product
from different origins (Factories) to different destinations (Warehouses). The transportation problem
deals with shipping commodities from different origins to various destinations. The objective of the
transportation problem is to determine the optimum amount of a commodity to be transported from
various supply points (origins) to different demand points (destinations) so that the total transportation
cost is minimum or total transportation profit is maximum.
Depending on the nature of objective function, the transportation problem may be classified
into two categories, namely, maximization transportation problems and minimization transportation
problems. A minimization transportation problem involves cost data. The objective of solution is
to minimize the total cost. A maximization transportation problem involves revenue or profit data.

6. International Journal of Operations Research and Information Systems
Volume 9 • Issue 3 • July-September 2018
91
The objective of solution is maximization of total profit. These problems are occurring either in
corporations or in industry.
The unit costs, that is, the cost of transporting one unit from a particular supply point to a
particular demand point, the amounts available at the supply points and the amounts required at the
demand points are the parameters of the transportation problem.
In literature, Hitchcock (1941) developed a basic transportation problem. Koopmans (1949)
presented optimum utilization of the transportation system. Charnes and Cooper (1954) developed
the Stepping Stone Method (SSM), which provides an alternative way of determining the simplex
method information. The transportation algorithm for solving transportation problems with equality
constraints introduced by Dantzig (1963) is the simplex method specialized to the format of a table
called transportation table. It involves two steps. First, we compute an initial basic feasible solution
for the transportation problem and then, we test optimality and look at improving the basic feasible
solution to the transportation problem. Dalman et al. (2013) designed a solution proposal to indefinite
quadratic interval transportation problem.
The solid transportation problem is a generalization of the classical transportation problem in
which three-dimensional properties are taken into account in the objective and constraint set instead
of source (origin) and destination. Shell (1955) stated an extension of well-known transportation
problem is called a solid transportation problem in which bounds are given on three items, namely,
supply, demand and conveyance. In many industrial problems, a homogeneous product is transported
from an origin to a destination by means of different modes of transport called conveyances, such as
trucks, cargo flights, goods trains, ships and so on. Haley (1962) proposed the solution procedure for
solving solid transportation problem, which is an extension of the modified distribution method. Patel
and Tripathy (1989) presented a computationally superior method for a solid transportation problem
with mixed constraints. Basu et al. (1994) studied an algorithm for finding the optimum solution of
a solid fixed charge linear transportation problem.
For finding an optimal solution, the solid transportation problem requires m n l 2 non-
negative values of the decision variables to start with a basic feasible solution. Jimenez and Verdegay
(1996) investigated interval multiobjective solid transportation problem via genetic algorithms. Li
et al. (1997a) designed a neural network approach for a multicriteria solid transportation problem.
Efficient algorithms have been developed for solving transportation problems when the coefficient
of the objective function, demand, supply and conveyance values are known precisely.
Many of the distribution problems are imprecise in nature in today’s world such as in corporate
or in industry due to variations in the parameters. The TP is a distribution-type problem, the main
objective of which is to decide how to transfer goods from various sending locations to various
receiving locations with least costs or maximum profit. The sending locations is also known as origins
or sources or factories. Similarly, the receiving locations is also known as destinations or warehouses
or retail stores. In classical transportation problem it is assumed that the transportation costs and values
of supplies and demands are exactly known. But in real life, the transportation parameters may not
be precise always due to lack of information, environmental factors, changing weather, imprecision
in judgment, social, or economic conditions and so on. Therefore, it is very interesting to deal with
TPs under uncertainty. The best way to denote the imprecise data is fuzzy number. In literature, to
deal quantitatively with imprecise information in making decision, Zadeh (1965) introduced the fuzzy
set theory and has applied it successfully in various fields. The use of fuzzy set theory becomes very
rapid in the field of optimization after the pioneering work done by Bellman and Zadeh (1970). The
fuzzy set deals with the degree of membership (belongingness) of an element in the set but it does
not consider the non-membership (non-belongingness) of an element in the set. In a fuzzy set the
membership value (level of acceptance or level of satisfaction) lies between 0 and 1 where as in crisp
set the element belongs to the set represent 1 and the element not in the set represent 0.
Due to the lack of certainty in the parameters of a crisp transportation problem, several authors
Dinagar and Palanivel (2009), Mohideen and Kumar (2010), Pandian and Natarajan (2010) have

7. A Simple and Efficient Algorithm for Solving Type-1 Intuitionistic Fuzzy Solid
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(3)
Copyright: © 2018 |Pages: 33
DOI: 10.4018/IJORIS.2018070105
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Reference to this paper should be made as follows:
MLA
Kumar, P. Senthil. A Simple and Efficient Algorithm for Solving Type-1 Intuitionistic
Fuzzy Solid Transportation Problems. IJORIS 9.3 (2018): 90-122. Web. 2 Jun. 2018.
doi:10.4018/IJORIS.2018070105
APA
Kumar, P. S. (2018). A Simple and Efficient Algorithm for Solving Type-1 Intuitionistic
Fuzzy Solid Transportation Problems. International Journal of Operations Research and
Information Systems (IJORIS), 9(3), 90-122. doi:10.4018/IJORIS.2018070105
Chicago
Kumar, P. Senthil. A Simple and Efficient Algorithm for Solving Type-1 Intuitionistic
Fuzzy Solid Transportation Problems, International Journal of Operations Research and
Information Systems (IJORIS) 9 (2018): 3, accessed (June 02, 2018),
doi:10.4018/IJORIS.2018070105
Harvard
Kumar, P.S., 2018. A Simple and Efficient Algorithm for Solving Type-1 Intuitionistic Fuzzy
Solid Transportation Problems. International Journal of Operations Research and
Information Systems (IJORIS), 9(3), pp.90-122.
Vancouver
Kumar PS. A Simple and Efficient Algorithm for Solving Type-1 Intuitionistic Fuzzy Solid
Transportation Problems. International Journal of Operations Research and Information
Systems (IJORIS). 2018 Jun 2;9(3):90-122.

8. International Journal of Operations Research and Information Systems
Volume 9 • Issue 3 • July-September 2018
117
6.2. Advantages of the Proposed Method
By using the proposed method a decision maker has the following advantages:
1. We need not find out the basic feasible solution and we need not apply the optimality test because
the solution obtained by proposed method is always optimal;
2. The proposed method is a single step method. So, the use of intuitionistic fuzzy modified
distribution is not required.
7. CONCLUSION AND FUTURE WORK
In this article, new problem called type-1 IFSTP is introduced. Different types of IFSTPs is identified.
Intuitionistic fuzzy min-zero min-cost method for finding the intuitionistic fuzzy optimal solution to
type-1 IFSTP is proposed. The proposed methodology is illustrated with the help of real life numerical
example and the optimal solution is obtained in terms of triangular intuitionistic fuzzy numbers.
Results, discussion and advantages of the proposed method is also devoted.
The type-1 IFSTPs are solved by the proposed method, which differs from the existing methods
namely, the extended version of modified distribution method (Haley (1962)) and intuitionistic fuzzy
modified distribution method (Kumar and Hussain (2015)). The extended version of modified
distribution method and intuitionistic fuzzy modified distribution method both are depends on its
basic feasible solution. But, the main advantage of the proposed method is that the obtained solution
is always optimal. To apply this method, there is no necessity to have m n l+ + −( )2 number of
non-negative allotted entries (i.e., basic feasible solution). Also, we need not test the optimality
condition. It is applicable to type-1, type-2, type-3 and type-4 IFSTPs. The proposed 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. Further, it can
be served as an important tool for a decision-maker when he/she handles various types of logistic
problems having different types of parameters. In future this approach can be applied in solving solid
transportation problems having uncertainty and hesitation in costs. In future our algorithm can be
extended for solving solid transportation problems having all parameters as TIFNs. In addition, we
will attempt to develop a linear programming problem approach for solving type-1 IFSTP to reduce
both the computation time and computation complexity of the proposed method.
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.

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P. Senthil Kumar is an Assistant Professor in PG and Research Department of Mathematics at Jamal Mohamed
College (Autonomous), Tiruchirappalli, Tamil Nadu, India. He has seven years of teaching experience. He received
his BSc, MSc and MPhil degrees from Jamal Mohamed College, Tiruchirappalli in 2006, 2008 and 2010, respectively.
He completed his BEd in Jamal Mohamed College of Teacher Education in 2009. He completed PGDCA in 2011 in
the Bharathidasan University and PGDAOR in 2012 in the Annamalai University, Tamil Nadu, India. He completed his
PhD 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, Korean Institute of Intelligent
Systems (KIIS), IGI Global, Inderscience, etc. He also presented his research papers in Elsevier Conference
Proceedings (ICMS-2014), MMASC-2012, etc. His areas of interest include operations research, fuzzy optimization,
intuitionistic fuzzy optimization, numerical analysis and graph theory, etc.