A simple and efficient algorithm for solving type 1 intuitionistic fuzzy solid transportation problems

EDITOR-IN-CHIEF
John Wang, Montclair State University, USA
MANAGING EDITOR
Bin Zhou, University of Houston-Downtown, USA
INTERNATIONAL ADVISORY BOARD
Yuval Cohen, Tel-Aviv Afeka College of Engineering, Israel
ASSOCIATE EDITORS
Sungzoon Cho, Seould National University, Korea
Theodore Glickman, George Washington University, USA
Eva K. Lee, Georgia Institute of Technology, USA
Panos Pardalos, University of Florida, USA
Roman Polyak, George Mason University, USA
Jasenkas Rakas, University of California at Berkeley, USA
Kathryn E. Stecke, University of Texas at Dallas, USA
EDITORIAL REVIEW BOARD
Anil K. Aggarwal, University of Baltimore, USA
Sankarshan Basu, Indian Institute of Management Bangalore, India
Melike Baykal-Gursoy, Rutgers University, USA
Kevin Byrnes, Johns Hopkins University, USA
Dean Chatfield, Old Dominion University, USA
Chialin Chen, Queen’s University, Canada
Jagpreet Chhatwal, Harvard Medical School, USA
Wen Chiang, University of Tulsa, USA
Louis Anthony Cox Jr., University of Colorado, USA
Lauren Davis, North Carolina A&T State University, USA
Matt Drake, Duquesne University, USA
Ali Elkamel, University of Waterloo, Canada
Murat Erkoc, University of Miami, USA
Yudi Fernando, Universiti Malaysia Pahang, Malaysia
William P. Fox, Naval Postgraduate School, USA
Hise Gibson, INFORMS, USA
Scott E. Grasman, Rochester Institute of Technology, USA
Nalan Gulpinar, Warwick Business School, UK
Roger Gung, Response Analytics Inc., USA
Zhinling Guo, University of Maryland-Baltimore County, USA
Peter Hahn, University of Pennsylvania, USA
Mohammed Hajeeh, Kuwait Institute for Scientific Research, Kuwait
Steven Harper, James Madison University, USA
Michael J Hirsch, Raytheon Inc., USA
Xinxin Hu, University of Houston - Downtown, USA
Dariusz Jacek Jakóbczak, Koszalin University of Technology, Poland
Manoj K. Jha, Morgan State University, USA
Rex Kincaid, College of William & Mary, USA
Nanda Kumar, University of Texas at Dallas, USA
Hyoung-Gon Lee, Massachusetts Institute of Technology, USA
Loo Hay Lee, National University of Singapore, Singapore
Volume 9 • Issue 3 • July-September 2018 • ISSN: 1947-9328 • eISSN: 1947-9336
An official publication of the Information Resources Management Association
International Journal of Operations Research and Information
Systems

Fei Li, George Mason University, USA
Jian Li, Northeastern Illinois University, USA
Jing Li, Arizona State University, USA
Leonardo Lopes, University of Arizona, USA
Kaye McKinzie, U.S. Army, USA
Yefim Haim Michlin, Israel Institute of Technology, Israel
Somayeh Moazeni, Princeton University, USA
Okesola Moses Olusola, Oludoy Dynamix Consulting Ltd, Nigeria
Josefa Mula, Universitat Politècnica de València, Spain
B.P.S. Murthi, University of Texas at Dallas, USA
Olufemi A Omitaomu, Oak Ridge National Laboratory, USA
Kivanc Ozonat, HP Labs, USA
Dessislava Pachamanova, Babson College, USA
Julia Pahl, University of Hamburg, Germany
Francois Pinet, Irstea, France
Michael Racer, University of Memphis, USA
Marion S. Rauner, University of Vienna, Austria
Enzo Sauma Pontificia, Universidad Catolica de Chile, Chile
Hsu-Shih Shih, Tamkang University, Taiwan
Young-Jun Son, University of Arizona, USA
Huaming Song, Nanjing University of Science & Technology, China
Yang Sun, California State University - Sacramento, USA
Durai Sundaramoorthi, Washington University in St. Louis, USA
Pei-Fang Tsai, State University of New York at Binghamton, USA
M. Ali Ülkü, Dalhousie University, Canada
Bruce Wang, Texas A&M University, USA
Yitong Wang, Tsinghua University, China
Harris Wu, Old Dominion University, USA
Justin Yates, Francis Marion University, USA
Xugang Ye, Johns Hopkins University and Microsoft, USA
Banu Yukse-Ozkaya, Hacettepe University, Turkey
Editorial Review Board
Continued

Please recommend this publication to your librarian
For a convenient easy-to-use library recommendation form, please visit:
http://www.igi-global.com/IJORIS
Volume 9 • Issue 3 • July-September 2018 • ISSN: 1947-9328 • eISSN: 1947-9336
An official publication of the Information Resources Management Association
All inquiries regarding IJORIS should be directed to the attention of:
John Wang, Editor-in-Chief • IJORIS@igi-global.com
All manuscript submissions to IJORIS should be sent through the online submission system:
http://www.igi-global.com/authorseditors/titlesubmission/newproject.aspx
Computational Intelligence • Computing and information technologies • Continuous and discrete optimization
• Decision analysis and decision support • Education • Engineering management • Environment, energy and
natural resources • Financial engineering • Government • Heuristics • Industrial engineering • Information
management • Information Technology • Inventory Management • Knowledge Management • Logistics and
Supply Chain Management • Maintenance • Manufacturing industries • Marketing engineering • Markov chains
• Mathematics • Military and homeland security • Networks • Operations management • Organizational behavior
• Planning and scheduling • Policy modeling and public sector • Political science • Production management •
Psychology • Queuing theory • Revenue & risk management • Services management • Simulation • Sociology
• Sports • Statistics • Stochastic models • Strategic Management • Systems engineering • Telecommunications
• Transportation
Coverage and major topics
The topics of interest in this journal include, but are not limited to:
The International Journal of Operations Research and Information Systems (IJORIS) aims to present
new and innovative contributions in Operations Research (OR) theories, applications, and case studies, from
a wide spectrum of academics and practitioners. IJORIS spans the traditional functional areas of business,
including management information systems, production/operations management, business processes, quantitative
economics, accounting, finance, marketing, business administration, and international business. IJORIS also
incorporates applications from the related natural and social sciences, including the decision sciences, management
science, statistics, psychology, sociology, political science, and other behavioral sciences. IJORIS encourages
exchange, cooperation, and collaboration among business, industry, and government.
IJORIS encompasses and bridges the following seven channels through theories, applications, and case studies:
Mission
Ideas for Special Theme Issues may be submitted to the Editor(s)-in-Chief
International Journal of Operations Research and Information
Systems
Call for Articles

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

Copyright © 2018, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.

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.

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

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
OnDemand PDF $30.00
Download: List Price: $37.50
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.

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.

118
REFERENCES
Li, D. F., Nan, J. X., Zhang, M. J. (2010). A ranking method of triangular intuitionistic fuzzy numbers and
application to decision making. International Journal of Computational Intelligence Systems, 3(5), 522–530.
Angelov, P. P. (1997). Optimization in an intuitionistic fuzzy environment. Fuzzy Sets and Systems, 86(3), 299–306.
Antony, R. J. P., Savarimuthu, S. J., Pathinathan, T. (2014). Method for solving the transportation problem
using triangular intuitionistic fuzzy number. International Journal of Computing Algorithm, 3, 590–605.
Atanassov, K.T. (1983, June). Intuitionistic fuzzy sets. In VII ITKR’s Session, Sofia (in Bulg.)
Atanassov, K. T. (1995). Ideas for intuitionistic fuzzy equations, inequalities and optimization. Notes on
Intuitionistic Fuzzy Sets, 1(1), 17–24.
Atanassov, K. T. (1999). Intuitionistic fuzzy sets: Theory and applications. Physica -Verlag. Heidelberg, New
York: Springer. doi:10.1007/978-3-7908-1870-3
Ban, A. (2008). Trapezoidal approximations of intuitionistic fuzzy numbers expressed by value, ambiguity,
width and weighted expected value. Notes on Intuitionistic Fuzzy Sets, 14(1), 38–47.
Basu, M., Pal, B. B., Kundu, A. (1994). An algorithm for finding the optimum solution of solid fixed-charge
transportation problem. Optimization, 31(3), 283–291.
Bellman, R. E., Zadeh, L. A. (1970). Decision making in fuzzy environment, Management Science, 17(B),
141-164.
Bit, A. K., Biswal, M. P., Alam, S. S. (1993). Fuzzy programming approach to multiobjective solid transportation
problem. Fuzzy Sets and Systems, 57(2), 183–194.
Burillo, P., Bustince, H., Mohedano, V. (1994). Some definitions of intuitionistic fuzzy number-first properties.
In Proceedings of the First Workshop on Fuzzy Based Expert System, Sofia, Bulgaria (pp. 53-55).
Chakraborty, D., Jana, D. K., Roy, T. K. (2015). Arithmetic operations on generalized intuitionistic fuzzy
number and its applications to transportation problem. Opsearch, 52(3), 431–471. doi:10.1007/s12597-014-0194-1
Charnes, A., Cooper, W. W. (1954). The stepping- stone method for explaining linear programming calculation
in transportation problem. Management Science, 1(1), 49–69.
Cherian, L., Kuriakose, S. (2009). A fuzzy economic production quantity model with capacity constraint:
Intuitionistic fuzzy optimization for linear programming problems. The Journal of Fuzzy Mathematics, 17(1),
139–144.
Dalman, H. (2016). Uncertain programming model for multi-item solid transportation problem. International
Journal of Machine Learning and Cybernetics. doi:10.1007/s13042-016-0538-7
Dalman, H. (2017). Multi-objective Solid Transportation Problem in Uncertain Environment. Iranian Journal of
Science and Technology, Transactions A. Science, 41(2), 505–514. doi:10.1007/s40815-015-0081-9
Dalman, H. (2017). A Fuzzy Logic Based Approach to Solve Interval Multiobjective Nonlinear Transportation
Problem. In Proceedings of the National Academy of Sciences, India Section A: Physical Sciences. doi:10.1007/
s40010-017-0469-z
Dalman, H., Bayram, M. (2017). (in press). Interactive Fuzzy Goal Programming Based on Taylor Series to
Solve Multiobjective Nonlinear Programming Problems with Interval Type 2 Fuzzy Numbers. IEEE Transactions
on Fuzzy Systems. doi:10.1109/TFUZZ.2017.2774191
Dalman, H., Güzel, N., Sivri, M. (2015). A Fuzzy Set-based approach to multi-objective multi-item solid
transportation problem under uncertainty. International Journal of Fuzzy Systems. doi:10.1007/s40815-015-
0081-9
Dalman, H., Güzel, N., Sivri, M. (2016). A fuzzy set-based approach to multi-objective multi-item solid
transportation problem under uncertainty. International Journal of Fuzzy Systems, 18(4), 716–729.

119
Dalman, H., Köçken, H. G., Sivri, M. (2013). A solution proposal to indefinite quadratic interval transportation
problem. New Trends in Mathematical Sciences, 1(2), 7-12.
Dantzig, G. B. (1963). Linear programming and extensions. Princeton, New Jersey: Princeton Univ. Press.
Das, S., Guha, D. (2013). Ranking of intuitionistic fuzzy number by centroid point. Journal of Industrial and
Intelligent Information, 1(2), 107–110.
Dinagar, D. S., Palanivel, K. (2009). The transportation problem in fuzzy environment. International Journal
of Algorithms. Computing and Mathematics, 2(3), 65–71.
Dinagar, D. S., Thiripurasundari, K. (2014). A navel method for solving fuzzy transportation problem involving
intuitionistic trapezoidal fuzzy numbers. International Journal of Current Research, 6(6), 7038–7041.
Dubey, D., Mehra, A. (2011). Linear programming with triangular intuitionistic fuzzy number. In Proceedings of
the 7th Conference of the European Society for Fuzzy Logic and Technology (Vol. 1, pp. 563-569). Atlantis Press.
Gen, M., Ida, K., Li, Y., Kubota, E. (1995). Solving bicriteria solid transportation problem with fuzzy numbers
by a genetic algorithm. Computers Industrial Engineering, 29(1), 537–541.
Grzegorzewski, P. (2003). Distance and orderings in a family of intuitionistic fuzzy numbers. In Proceedings of
the Third Conference of the European Society for Fuzzy Logic and Technology, Zittau, Germany (pp. 223-227).
Guha, D., Chakraborty, D. (2010). A theoretical development of distance measure for intuitionistic fuzzy
numbers. International Journal of Mathematics and Mathematical Sciences.
Gupta, G., Kumar, A., Sharma, M. K. (2016). A note on “A new method for solving fuzzy linear programming
problems based on the fuzzy linear complementary problem (FLCP).” International Journal of Fuzzy Systems.
Haley, K. B. (1962). New methods in mathematical programming-The solid transportation problem. Operations
Research, 10(4), 448–463.
Hitchcock, F. L. (1941). The distribution of a product from several sources to numerous localities. Journal of
Mathematics and Physics, 20(2), 224–230. doi:10.1002/sapm1941201224
Hussain, R. J., Kumar, P. S. (2012a). The transportation problem in an intuitionistic fuzzy environment.
International Journal of Mathematics Research, 4(4), 411–420.
Hussain, R. J., Kumar, P. S. (2012b). Algorithmic approach for solving intuitionistic fuzzy transportation
problem. Applied Mathematical Sciences, 6(80), 3981–3989.
Hussain, R. J., Kumar, P. S. (2012c). The transportation problem with the aid of triangular intuitionistic fuzzy
numbers. In Proceedings in International Conference on Mathematical Modeling and Applied Soft Computing
(MMASC-2012), Coimbatore Institute of Technology, Coimbatore (pp. 819-825).
Hussain, R. J., Kumar, P. S. (2013). An optimal more-for-less solution of mixed constraints intuitionistic
fuzzy transportation problems. International Journal of Contemporary Mathematical Sciences, 8(12), 565–576.
doi:10.12988/ijcms.2013.13056
Jana, B., Roy, T. K. (2007). Multi-objective intuitionistic fuzzy linear programming and its application in
transportation model. Notes on Intuitionistic Fuzzy Sets, 13(1), 34–51.
Jimenez, F., Verdegay, J. L. (1996). Interval multiobjective solid transportation problem via genetic algorithms.
In Management of Uncertainty in Knowledge-Based Systems (Vol. 2, pp. 787–792).
Jimenez, F., Verdegay, J. L. (1998). Uncertain solid transportation problems. Fuzzy Sets and Systems, 100(1),
45–57.
Jimenez, F., Verdegay, J. L. (1999). Solving fuzzy solid transportation problems by an evolutionary algorithm
based parametric approach. European Journal of Operational Research, 117(3), 485–510.
Koopmans, T. C. (1949). Optimum utilization of the transportation system. Econometrica, 17, 3–4.
Kumar, A., Kaur, M. (2013). A ranking approach for intuitionistic fuzzy numbers and its application. Journal
of Applied Research and Technology, 11(3), 381–396.

120
Kumar, P. S. (2016a). PSK method for solving type-1 and type-3 fuzzy transportation problems. International
Journal of Fuzzy System Applications, 5(4), 121-146, doi:10.4018/IJFSA.2016100106
Kumar, P. S. (2016b). A simple method for solving type-2 and type-4 fuzzy transportation problems. International
Journal of Fuzzy Logic and Intelligent Systems, 16(4), 225–237. doi:10.5391/IJFIS.2016.16.4.225
Kumar, P. S. (2017a). PSK method for solving type-1 and type-3 fuzzy transportation problems. In Fuzzy Systems:
Concepts, Methodologies, Tools, and Applications (pp. 367–392). Hershey, PA: IGI Global; doi:10.4018/978-
1-5225-1908-9.ch017
Kumar, P. S. (2017b). Algorithmic approach for solving allocation problems under intuitionistic fuzzy environment
[PhD thesis]. Jamal Mohamed College, Tiruchirappalli, India.
Kumar, P. S. (2018a). A note on ‘a new approach for solving intuitionistic fuzzy transportation problem of type-
2’, Int. J. Logistics Systems and Management, 29(1), 102–129. doi:10.1504/IJLSM.2018.10009204
Kumar, P. S. (In pressa). Linear programming approach for solving balanced and unbalanced intuitionistic fuzzy
transportation problems. International Journal of Operations Research and Information Systems.
Kumar, P. S. (In pressb). PSK method for solving intuitionistic fuzzy solid transportation problems. International
Journal of Fuzzy System Applications.
Kumar, P. S. (In pressc). 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 in International Conference on Mathematical
Sciences (ICMS-2014), Sathyabama University, Chennai (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. (2014d). A method for solving balanced intuitionistic fuzzy assignment problem.
International Journal of Engineering Research and Applications, 4(3), 897–903.
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
Kumar, P. S., Hussain, R. J. (2016b). A simple method for solving fully intuitionistic fuzzy real life assignment
problem. International Journal of Operations Research and Information Systems, 7(2), 39–61. doi:10.4018/
IJORIS.2016040103
Kumar, P. S., Hussain, R. J. (2016c). An algorithm for solving unbalanced intuitionistic fuzzy assignment
problem using triangular intuitionistic fuzzy number. The Journal of Fuzzy Mathematics, 24(2), 289–302.
Li, Y., Ida, K., Gen, M. (1997b). Improved genetic algorithm for solving multiobjective solid transportation
problem with fuzzy numbers. Computers Industrial Engineering, 33(3), 589–592.
Li, Y., Ida, K., Gen, M., Kobuchi, R. (1997a). Neural network approach for multicriteria solid transportation
problem. Computers Industrial Engineering, 33(3), 465–468.
Liu, S. T. (2006). Fuzzy total transportation cost measures for fuzzy solid transportation problem. Applied
Mathematics and Computation, 174(2), 927–941.
Mahapatra, B. S., Mahapatra, G. S. (2010). Intuitionistic fuzzy fault tree analysis using intuitionistic fuzzy
numbers. International Mathematical Forum, 5(21), 1015-1024.
Mahapatra, G. S., Roy, T. K. (2009). Reliability evaluation using triangular intuitionistic fuzzy numbers,
arithmetic operations. International Scholarly and Scientific Research Innovation, 3(2), 422–429.

121
Mahapatra, G. S., Roy, T. K. (2013). Intuitionistic fuzzy number and its arithmetic operation with application
on system failure. Journal of Uncertain Systems, 7(2), 92–107.
Mitchell, H. B. (2004). Ranking intuitionistic fuzzy numbers. International Journal of Uncertainty, Fuzziness
and Knowledge-based Systems, 12(3), 377–386.
Mohideen, S. I., Kumar, P. S. (2010). A comparative study on transportation problem in fuzzy environment.
International Journal of Mathematics Research, 2(1), 151–158.
Nayagam, G., Lakshmana, V., Venkateshwari, G., Sivaraman, G. (2008, June). Ranking of intuitionistic
fuzzy numbers. In Proceedings of the IEEE International Conference on Fuzzy Systems FUZZ-IEEE ’08 (pp.
1971-1974). IEEE.
Nehi, H. M. (2010). A new ranking method for intuitionistic fuzzy numbers. International Journal of Fuzzy
Systems, 12(1), 80–86.
Nehi, H. M., Maleki, H. R. (2005, July). Intuitionistic fuzzy numbers and it’s applications in fuzzy optimization
problem. In Proceedings of the Ninth WSEAS International Conference on Systems, Athens, Greece.
Ojha, A., Das, B., Mondal, S., Maiti, M. (2009). An entropy based solid transportation problem for general
fuzzy costs and time with fuzzy equality. Mathematical and Computer Modelling, 50(1), 166–178.
Pandian, P., Anuradha, D. (2010). A new approach for solving solid transportation problems. Applied
Mathematical Sciences, 4(72), 3603–3610.
Pandian, P., Natarajan, G. (2010). A new algorithm for finding a fuzzy optimal solution for fuzzy transportation
problems. Applied Mathematical Sciences, 4(2), 79–90.
Parvathi, R., Malathi, C. (2012). Intuitionistic fuzzy linear programming problems. World Applied Sciences
Journal, 17(12), 1802–1807.
Patel, G., Tripathy, J. (1989). The solid transportation problem and its variants. International Journal of
Management and Systems, 5(5), 17–36.
Shabani, A., Jamkhaneh, E. B. (2014). A new generalized intuitionistic fuzzy number. Journal of Fuzzy Set
Valued Analysis, 24. doi:10.5899/2014/jfsva-00199
Shaw, A. K., Roy, T. K. (2012). Some arithmetic operations on triangular intuitionistic fuzzy number and its
application on reliability evaluation. International Journal of Fuzzy Mathematics and Systems, 2(4), 363–382.
Shell, E. (1955). Distribution of a product by several properties. Directorate of Management Analysis. In
Proceedings of the Second Symposium in Linear Programming (Vol. 2, pp. 615-642).
Singh, S. K., Yadav, S. P. (2014). Efficient approach for solving type-1 intuitionistic fuzzy transportation
problem. Int. J. Syst. Assur. Eng. Manag. doi:10.1007/s13198-014-0274-x
Singh, S. K., Yadav, S. P. (2015). Fuzzy Programming Approach for Solving Intuitionistic Fuzzy Linear
Fractional Programming Problem. International Journal of Fuzzy Systems.
Singh, S. K., Yadav, S. P. (2016). A novel approach for solving fully intuitionistic fuzzy transportation problem.
International Journal of Operational Research, 26(4), 460–472. doi:10.1504/IJOR.2016.077684
Varghese, A., Kuriakose, S. (2012). Centroid of an intuitionistic fuzzy number. Notes on Intuitionistic Fuzzy
Sets, 18(1), 19–24.
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353. doi:10.1016/S0019-9958(65)90241-X

122
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.

A simple and efficient algorithm for solving type 1 intuitionistic fuzzy solid transportation problems

More Related Content

What's hot (19)

Similar to A simple and efficient algorithm for solving type 1 intuitionistic fuzzy solid transportation problems (20)

More from Navodaya Institute of Technology (16)

Recently uploaded (20)

A simple and efficient algorithm for solving type 1 intuitionistic fuzzy solid transportation problems