PSK Method for Solving Type-1 and Type-3 Fuzzy Transportation Problems

Abbas Al-Refaie, University of Jordan, Jordan
Ahmad Taher Azar, Benha University, Egypt
P. Balasubramaniam, Gandhigram Rural University, India
Zeungnam Bien, UNIST, Korea
Asli Celikyilmaz, University of California-Berkeley, USA
Keeley Crockett, Manchester Metropolitan University, UK
Ali Ebrahimnejad, Islamic Azad University, Iran
K. Honda, Osaka Prefecture University, Japan
Jun-ichi Horiuch, Kitami Institute of Technology, Japan
Richard Jensen, The University of Wales, Aberystwyth, UK
Erich Peter Klement, Johannes Kepler University, Austria
Rudolf Kruse, Otto-von-Guericke-Universität Magdeburg, Germany
Salim Labiod, University of Jijel, Algeria
Yongming Li, Shaanxi Normal University, China
T. Warren Liao, Louisiana State Univeristy, USA
Pawan Lingras, Saint Mary’s University, Canada
Peide Liu, Shandong University, China
Yan-Jun Liu, Liaoning University of Technology, China
Yeh Ching Nee, National University of Singapore, Singapore
Jianbin Qiu, Harbin Institute of Technology, China
Chai Quek, Nanyang Technological University, Singapore
Elisabeth Rakus-Andersson, Blekinge Institute of Technology, Sweden
Soheil Salahshour, Islamic Azad University Mobarakeh Branch, Iran
Ismail Burhan Turksen, TOBB Economy and Technology University, Turkey
Pandian Vasant, Universiti Teknologi PETRONAS, Malaysia
Michael Voskoglou, Graduate Technological Educational Institute (T.E.I.), Greece
Hsiao-Fan Wang, National Tsing Hua University, China
Mao-Jiun J. Wang, National Tsing Hua University, Taiwan
Frank Werner, Otto-von-Guericke University, Germany
Chien-Wei Wu, National Taiwan University of Science and Technology, Taiwan
Tai-Shi Wu, National Taipei University, Taiwan
Zeshui Xu, PLA University of Science and Technology, China
Mesut Yavuz, Shenandoah University, USA
Gaofeng Yu, Sanming University, China
International Editorial Review Board
EDITOR-IN-CHIEF
Deng-Feng Li, Fuzhou University, China
INTERNATIONAL ADVISORY BOARD
Ronald R. Yager, Iona College, USA
Lotfi A. Zadeh, California University at Berkeley, USA
Hans-Jürgen Zimmermann, European Laboratory for Intelligent Techniques Engineering, Inform GmbH, Germany
ASSOCIATE EDITORS
Mark Burgin, UCLA, USA
Mingzhi Chen, Fuzhou University, China
Volume 5 • Issue 4 • October-December 2016 • ISSN: 2156-177X • eISSN: 2156-1761
An official publication of the Information Resources Management Association
International Journal of Fuzzy System Applications

The International Journal of Fuzzy System Applications is indexed or listed in the following: ACM Digital Library; Bacon’s
Media Directory; Cabell’s Directories; DBLP; Google Scholar; INSPEC; JournalTOCs; MediaFinder; ProQuest Advanced
Technologies & Aerospace Journals; ProQuest Computer Science Journals; ProQuest Illustrata: Technology; ProQuest
SciTech Journals; ProQuest Technology Journals; SCOPUS; The Standard Periodical Directory; Ulrich’s Periodicals Directory
Research Articles
1 A Resource-Constrained Project Scheduling Problem with Fuzzy Activity Times;
Hossein Zoulfaghari, Department of Industrial Engineering, Isfahan University of Technology, Isfahan, Iran
Javad Nematian, Department of Industrial Engineering, University of Tabriz, Tabriz, Iran
Amir Abbas Kanani Nezhad, Department of Industrial Engineering, Khaje Nasir University, Tehran, Iran
18 Contrasting Correlation Coefficient with Distance Measure in Interval Valued Intuitionistic Trapezoidal
Fuzzy MAGDM Problems;
John P. Robinson, Bishop Heber College, Tiruchirappali, India
57 Multi-Fuzzy Complex Nilpotent Matrices;
Asit Dey, Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar
University, Midnapore, India
Madhumangal Pal, Department of Applied Mathematics with Oceanology and Computer Programming,
Vidyasagar University, Midnapore, India
82 Dynamic Tasks Scheduling Algorithm for Distributed Computing Systems under Fuzzy Environment;
Harendra Kumar, Department of Mathematics and Statistics, GurukulaKangri University, Haridwar, India
Nutan Kumari Chauhan, Department of Mathematics and Statistics, GurukulaKangri University, Haridwar,
Uttarakhand, India
Pradeep Kumar Yadav, Department of Research Planning and Business Development, Central Building Research
Institute, Roorkee, India
101 An Improved Second Order Training Algorithm for Improving the Accuracy of Fuzzy Decision Trees;
Swathi Jamjala Narayanan, VIT University, Vellore, India
Rajen B. Bhatt, Robert Bosch Research and Technology Center, Pittsburgh, PA, USA
Ilango Paramasivam, VIT University, Vellore, India
126 PSK Method for Solving Type-1 and Type-3 Fuzzy Transportation Problems;
P. Senthil Kumar, PG & Research Department of Mathematics, Jamal Mohamed College (Autonomous),
Tiruchirappalli, Tamil Nadu, India
152 High Order Time Series Forecasting using Fuzzy Discretization;
Mahua Bose, Department of Computer Science & Engineering, University of Kalyani, Nadia, India
Kalyani Mali, Department of Computer Science & Engineering, University of Kalyani, Nadia, India
170 A Multi-Objective Fuzzy Ant Colony Optimization Algorithm for Virtual Machine Placement;
Boominathan Perumal, VIT University, Vellore, India
Aramudhan M., Department of IT, Perunthalaivar Kamarajar Institute of Engineering and Technology, Puducherry, India
197 Interval-Valued Intuitionistic Fuzzy Sets based Method for Multiple Criteria Decision-Making;
Bhagawati Prasad Joshi, Seemant Institute of Technology, Pithoragarh, India
Copyright
The International Journal of Fuzzy System Applications (IJFSA) (ISSN 2156-177X; eISSN 2156-1761), Copyright © 2016 IGI Global. All rights,
including translation into other languages reserved by the publisher. No part of this journal may be reproduced or used in any form or by any means without
written permission from the publisher, except for noncommercial, educational use including classroom teaching purposes. Product or company names
used in this journal are for identification purposes only. Inclusion of the names of the products or companies does not indicate a claim of ownership by IGI
Global of the trademark or registered trademark. The views expressed in this journal are those of the authors but not necessarily of IGI Global.
Volume 5 • Issue 4 • October-December-2016 • ISSN: 2156-177X • eISSN: 2156-1761
An official publication of the Information Resources Management Association
Table of Contents

DOI: 10.4018/IJFSA.2016100106
Copyright © 2016, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
Volume 5 • Issue 4 • October-December 2016
PSK Method for Solving Type-1 and
Type-3 Fuzzy Transportation Problems
P. Senthil Kumar, PG & Research Department of Mathematics, Jamal Mohamed College (Autonomous), Tiruchirappalli,
Tamil Nadu, India
ABSTRACT
In conventional transportation problem (TP), supplies, demands and costs are always certain. In this
paper, the author tried to categories the TP under the mixture of certain and uncertain environment
and formulates the problem and utilizes the crisp numbers, triangular fuzzy numbers (TFNs) and
trapezoidal fuzzy numbers (TrFNs) to solve the TP. The existing ranking procedure of Liou and Wang
is used to transform the type-1 and type-3 fuzzy transportation problem (FTP) into a crisp one so that
the conventional method may be applied to solve the TP. The solution procedure differs from TP to
type-1 and type-3 FTP in allocation step only. Therefore, the new method called PSK method and
new multiplication operation on TrFN is proposed to find the mixed optimal solution in terms of crisp
numbers, TFNs and TrFNs. The main advantage of this method is computationally very simple, easy
to understand and also the optimum objective value obtained by our method is physically meaningful.
The effectiveness of the proposed method is illustrated by means of a numerical example.
Keywords
Fuzzy Number, Fuzzy Set, Optimal Solution, PSK Method, Type-1 Fuzzy Transportation Problem, Type-3
Fuzzy Transportation Problem
1. INTRODUCTION
The transportation problem is a special class of linear programming problem which deals
with the distribution of single homogeneous product from various origins(sources) to various
destinations(sinks).The objective of the transportation problem is to determine the optimal amount
of a commodity to be transported from various supply points to various demand points so that the
total transportation cost is minimum for a minimization problem or total transportation profit is
maximum for a maximization problem.
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. Efficient algorithms have been developed
for solving transportation problems when the cost coefficients, the demand and supply quantities
are known precisely.
In the history of mathematics, Hitchcock (1941) originally developed the basic transportation
problem. Charnes and Cooper (1954) developed the stepping stone method which provides an
alternative way of determining the simplex method information. Appa (1973) discussed several
variations of the transportation problem. Arsham et al. (1989) proposed a simplex type algorithm
121

122
for general transportation problems. An Introduction to Operations Research Taha (2008) deals the
transportation problem.
In today’s real world problems such as in corporate or in industry many of the distribution
problems are imprecise in nature due to variations in the parameters. 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. 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 belongs to the set represent 0.
Due to the applications of fuzzy set theory, several authors like Oheigeartaigh (1982) presented
an algorithm for solving transportation problems where the availabilities and requirements are fuzzy
sets with linear or triangular membership functions. Chanas et al. (1984) presented a fuzzy linear
programming model for solving transportation problems with fuzzy supply, fuzzy demand and crisp
costs. Chanas et al. (1993) formulated the fuzzy transportation problems in three different situations and
proposed method for solving the formulated fuzzy transportation problems. Chanas and Kuchta (1996)
proposed the concept of the optimal solution for the transportation problem with fuzzy coefficients
expressed as fuzzy numbers, and developed an algorithm for obtaining the optimal solution.
Chanas and Kuchta (1998) developed a new method for solving fuzzy integer transportation
problem by representing the supply and demand parameters as L-R type fuzzy numbers. Saad and
Abbas (2003) proposed an algorithm for solving the transportation problems under fuzzy environment.
Liu and Kao (2004) presented a method for solving fuzzy transportation problems based on extension
principle. Chiang (2005) proposed a method to find the optimal solution of transportation problems
with fuzzy requirements and fuzzy availabilities. Gani and Razak (2006) obtained a fuzzy solution
for a two stage cost minimizing fuzzy transportation problem in which availabilities and requirements
are trapezoidal fuzzy numbers using a parametric approach. Das and Baruah (2007) discussed Vogel’s
approximation method to find the fuzzy initial basic feasible solution of fuzzy transportation problem
in which all the parameters (supply, demand and cost) are represented by triangular fuzzy numbers.
Li et al. (2008) proposed a new method based on goal programming approach for solving fuzzy
transportation problems with fuzzy costs.
Chen et al. (2008) proposed the methods for solving transportation problems on a fuzzy network.
Lin (2009) used genetic algorithm for solving transportation problems with fuzzy coefficients.
Dinagar and Palanivel (2009) investigated the transportation problem in fuzzy environment using
trapezoidal fuzzy numbers. De and Yadav (2010) modified the existing method (Kikuchi 2000) by
using trapezoidal fuzzy numbers instead of triangular fuzzy numbers. Pandian et al. (2010) proposed
a new algorithm for finding a fuzzy optimal solution for fuzzy transportation problem where all the
parameters are trapezoidal fuzzy numbers. Mohideen and Kumar (2010) did a comparative study on
transportation problem in fuzzy environment. Sudhakar et al. (2011) proposed a different approach for
solving two stage fuzzy transportation problems in which supplies and demands are trapezoidal fuzzy
numbers. Hadi Basirzadeh (2011) discussed an approach for solving fuzzy transportation problem
where all the parameters are trapezoidal fuzzy numbers. Gani et al. (2011) presented simplex type
algorithm for solving fuzzy transportation problem where all the parameters are triangular fuzzy
numbers. Nasseri and Ebrahimnejad (2011) did sensitivity analysis on linear programming problems
with trapezoidal fuzzy variables.
Biswas and Modak (2012) studied using fuzzy goal programming technique to solve multi-
objective chance constrained programming problems in a fuzzy environment. Saati et al. (2012)
presented a two-fold linear programming model with fuzzy data. Ebrahimnejad (2012) discussed
cost efficiency measures with trapezoidal fuzzy numbers in data envelopment analysis based on
ranking functions: application in insurance organization and hospital. Rani et al. (2014) presented a
method for unbalanced transportation problems in fuzzy environment taking all the parameters are

24 more pages are available in the full version of this
document, which may be purchased using the "Add to Cart"
button on the product's webpage:
www.igi-global.com/article/psk-method-for-solving-type-1-
and-type-3-fuzzy-transportation-
problems/170556?camid=4v1
This title is available in InfoSci-Journals, InfoSci-Journal
Disciplines Computer Science, Security, and Information
Technology, InfoSci-Select. Recommend this product to your
librarian:
www.igi-global.com/e-resources/library-
recommendation/?id=2
Related Content
Applications of DEC-MDPs in Multi-Robot Systems
Aurélie Beynier and Abdel-Illah Mouaddib (2012). Decision Theory Models for
Applications in Artificial Intelligence: Concepts and Solutions (pp. 361-384).
www.igi-global.com/chapter/applications-dec-mdps-multi-
robot/60936?camid=4v1a
Mining Matrix Pattern from Mobile Users
John Goh and David Taniar (2006). International Journal of Intelligent Information
Technologies (pp. 37-67).
www.igi-global.com/article/mining-matrix-pattern-mobile-
users/2396?camid=4v1a
The Role of Brand Loyalty on CRM Performance: An Innovative Framework
for Smart Manufacturing
Kijpokin Kasemsap (2014). Smart Manufacturing Innovation and Transformation:
Interconnection and Intelligence (pp. 252-284).
www.igi-global.com/chapter/the-role-of-brand-loyalty-on-crm-
performance/102110?camid=4v1a

The Design and Evaluation of the Persuasiveness of e-Learning Interfaces
Eric Brangier and Michel C. Desmarais (2013). International Journal of Conceptual
Structures and Smart Applications (pp. 38-47).
www.igi-global.com/article/the-design-and-evaluation-of-the-persuasiveness-
of-e-learning-interfaces/100452?camid=4v1a

Reference to this paper should be made as follows:
MLA
Kumar, P. Senthil. "PSK Method for Solving Type-1 and Type-3 Fuzzy
Transportation Problems." IJFSA 5.4 (2016): 121-146. Web. 16 Mar. 2017.
doi:10.4018/IJFSA.2016100106
APA
Kumar, P. S. (2016). PSK Method for Solving Type-1 and Type-3 Fuzzy
Transportation Problems. International Journal of Fuzzy System Applications
(IJFSA), 5(4), 121-146. doi:10.4018/IJFSA.2016100106
Chicago
Kumar, P. Senthil. "PSK Method for Solving Type-1 and Type-3 Fuzzy
Transportation Problems," International Journal of Fuzzy System Applications
(IJFSA) 5 (2016): 4, accessed (March 16, 2017),
doi:10.4018/IJFSA.2016100106

144
REFERENCES
Appa, G. M. (1973). The transportation problem and its variants. The Journal of the Operational Research
Society, 24(1), 79–99. doi:10.1057/jors.1973.10
Arsham, H., & Kahn, A. B. (1989). A simplex-type algorithm for general transportation problems: An alternative
to stepping-stone. The Journal of the Operational Research Society, 40(6), 581–590. doi:10.1057/jors.1989.95
Basirzadeh, H. (2011). An approach for solving fuzzy transportation problem. Applied Mathematical Sciences,
5(32), 1549-1566.
Bellman, R. E., & Zadeh, L. A. (1970). Decision-making in a fuzzy environment. Management science, 17,:
B 141 – B 164.
Biswas, A., & Modak, N. (2012). Using Fuzzy Goal Programming Technique to Solve Multiobjective Chance
Constrained Programming Problems in a Fuzzy Environment. International Journal of Fuzzy System Applications,
2(1), 71–80. doi:10.4018/IJFSA.2012010105
Chanas,S.,Delgado,M.,Verdegay,J.L.,&Vila,M.A.(1993).Intervalandfuzzyextensionsofclassicaltransportation
problems. Transportation Planning and Technology, 17(2), 203–218. doi:10.1080/03081069308717511
Chanas, S., Kołodziejczyk, W., & Machaj, A. (1984). A fuzzy approach to the transportation problem. Fuzzy
Sets and Systems, 13(3), 211–221. doi:10.1016/0165-0114(84)90057-5
Chanas, S., & Kuchta, D. (1996). A concept of the optimal solution of the transportation problem with fuzzy
cost coefficients. Fuzzy Sets and Systems, 82(3), 299–305. doi:10.1016/0165-0114(95)00278-2
Chanas, S., & Kuchta, D. (1998). Fuzzy integer transportation problem. Fuzzy Sets and Systems, 98(3), 291–298.
doi:10.1016/S0165-0114(96)00380-6
Charnes, A., & Cooper, W. W. (1954). The stepping stone method of explaining linear programming calculations
in transportation problems. Management Science, 1(1), 49–69. doi:10.1287/mnsc.1.1.49
Chen, M., Ishii, H., & Wu, C. (2008). Transportation problems on a fuzzy network. International Journal of
Innovative Computing, Information, & Control, 4(5), 1105–1109.
Chen, S. H., & Hsieh, C. H. (1999). Graded mean integration representation of generalized fuzzy numbers.
Journal of Chinese Fuzzy System, 5, 1-7.
Chiang, J. (2005). The optimal solution of the transportation problem with fuzzy demand and fuzzy product.
J. Inf. Sci. Eng, 21, 439–451.
Das, M., & Baruah, H. K. (2007). Solution of the transportation problem in fuzzified form. Journal of Fuzzy
Mathematics, 15(1), 79–95.
De, P. K., & Yadav, B. (2010). Approach to defuzzify the trapezoidal fuzzy number in transportation problem.
Internat. J. Comput. Cognit, 8, 64–67.
Dinagar, D. S., & Palanivel, K. (2009). The transportation problem in fuzzy environment. Int. Journal of
Algorithm, computing and mathematics, 2:65-71.
Ebrahimnejad, A. (2012). Cost Efficiency Measures with Trapezoidal Fuzzy Numbers in Data Envelopment
Analysis Based on Ranking Functions: Application in Insurance Organization and Hospital.[IJFSA]. International
Journal of Fuzzy System Applications, 2(3), 51–68. doi:10.4018/ijfsa.2012070104
Gani, A. N., & Razak, K. A. (2006). Two stage fuzzy transportation problem. The Journal of Physiological
Sciences; JPS, 10, 63–69.
Gani, A. N., Samuel, A. E., & Anuradha, D. (2011). Simplex type algorithm for solving fuzzy transportation
problem. Tamsui Oxford Journal of Information and Mathematical Sciences., 27, 89–98.
Hasan, M. K. (2012). Direct methods for finding optimal solution of a transportation problem are not always
reliable. International Refereed Journal of Engineering and Science, 1(2), 46–52.

145
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
Kikuchi, S. (2000). A method to defuzzify the fuzzy number: Transportation problem application. Fuzzy Sets
and Systems, 116(1), 3–9. doi:10.1016/S0165-0114(99)00033-0
Kumar, A., Kaur, A., & Gupta, A. (2011). Fuzzy linear programming approach for solving fuzzy transportation
problems with transhipment. Journal of Mathematical Modelling and Algorithms, 10(2), 163–180. doi:10.1007/
s10852-010-9147-8
Li, L., Huang, Z., Da, Q., & Hu, J. (2008, May). A new method based on goal programming for solving
transportation problem with fuzzy cost. In Information Processing (ISIP), 2008 International Symposiums on
(pp. 3-8). IEEE. doi:10.1109/ISIP.2008.9
Liao, X. (2015). Decision Method of Optimal Investment Enterprise Selection under Uncertain Information
Environment. International Journal of Fuzzy System Applications, 4(1), 33–42. doi:10.4018/IJFSA.2015010102
Lin, F. T. (2009, August). Solving the transportation problem with fuzzy coefficients using genetic algorithms.
Proceedings of the IEEE International Conference on Fuzzy Systems FUZZ-IEEE ‘09 (pp. 1468-1473). IEEE.
doi:10.1109/FUZZY.2009.5277202
Liou, T. S., & Wang, M. J. J. (1992). Ranking fuzzy numbers with integral value. Fuzzy Sets and Systems, 50(3),
247–255. doi:10.1016/0165-0114(92)90223-Q
Mohideen, S. I., & Kumar, P. S. (2010). A comparative study on transportation problem in fuzzy environment.
International Journal of Mathematics Research, 2, 151–158.
Nasseri, S. H., & Ebrahimnejad, A. (2011). Sensitivity Analysis on Linear Programming Problems with
Trapezoidal Fuzzy Variables. International Journal of Operations Research and Information Systems, 2(2),
22–39. doi:10.4018/joris.2011040102
Oheigeartaigh, M. (1982). A fuzzy transportation algorithm. Fuzzy Sets and Systems, 8(3), 235–243. doi:10.1016/
S0165-0114(82)80002-X
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.
Pattnaik, M. (2015). Decision making approach to fuzzy linear programming (FLP) problems with post optimal
analysis. International Journal of Operations Research and Information Systems, 6(4), 75–90. doi:10.4018/
IJORIS.2015100105
Ping, J. I., & Chu, K. F. (2002). A dual-matrix approach to the transportation problem. Asia-Pacific Journal of
Operational Research, 19(1), 35–45.
Rani, D., Gulati, T. R., & Kumar, A. (2014). A method for unbalanced transportation problems in fuzzy
environment. Indian Academy of Sciences, Sadhana, 39(3), 573-581.
Saad, O. M., & Abass, S. A. (2003). A Parametric Study on Transportation Problem under Fuzzy Environment.
Journal of Fuzzy Mathematics, 11(1), 115–124.
Saati, S., Hatami-Marbini, A., Tavana, M., & Hajiahkondi, E. (2012). A Two-Fold Linear Programming Model
with Fuzzy Data. International Journal of Fuzzy System Applications, 2(3), 1–12. doi:10.4018/ijfsa.2012070101
Shankar, N. R., Saradhi, B. P., & Babu, S. S. (2013). Fuzzy Critical Path Method Based on a New Approach
of Ranking Fuzzy Numbers using Centroid of Centroids. International Journal of Fuzzy System Applications,
3(2), 16–31. doi:10.4018/ijfsa.2013040102
Solaiappan, S., & Jeyaraman, K. (2014). A new optimal solution method for trapezoidal fuzzy transportation
problem. International Journal of Advanced Research, 2(1), 933–942.
Sudhakar, V. J., & Kumar, V. N. (2011). A Different Approach for Solving Two Stage Fuzzy Transportation
Problems. Int. J. Contemp. Math. Sciences, 6(11), 517–526.
Taha, H. A. (2008). Operations Research: An Introduction (8th ed.). Pearson Education India.
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353. doi:10.1016/S0019-9958(65)90241-X

146
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 six years of teaching experience. He received
his BSc, MSc and MPhil from Jamal Mohamed College, Tiruchirappalli in 2006, 2008, 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 has
submitted his PhD thesis in the area of intuitionistic fuzzy optimisation technique to the Bharathidasan University
in 2015. He has published many research papers in referred national and international journals like Springer, IGI
Global, etc. He also presented his research in Elsevier Conference Proceedings (ICMS-2014), MMASC-2012, etc.
His areas of interest include operations research, fuzzy optimisation, intuitionistic fuzzy optimisation, numerical
analysis and graph theory, etc.

PSK Method for Solving Type-1 and Type-3 Fuzzy Transportation Problems

More Related Content

Similar to PSK Method for Solving Type-1 and Type-3 Fuzzy Transportation Problems

More from Navodaya Institute of Technology

Recently uploaded

PSK Method for Solving Type-1 and Type-3 Fuzzy Transportation Problems