Linear Programming Approach for Solving Balanced and Unbalanced Intuitionistic Fuzzy Transportation Problems
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This document is an official publication of the International Journal of Operations Research and Information Systems, detailing its editorial board and scope of topics, including logistics, supply chain management, and operations research theories. The journal aims to share innovative contributions from both academics and practitioners, and it supports the exchange of ideas across various sectors. Additionally, it highlights a specific article on linear programming approaches for solving transportation problems in intuitionistic fuzzy environments.
Linear Programming Approach for Solving Balanced and Unbalanced Intuitionistic Fuzzy Transportation Problems
- 2. EDITOR-IN-CHIEF John Wang, Montclair State University, USA MANAGING EDITOR Steve 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 Manoj K. Jha, Morgan State 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 Adedeji B. Badiru, Air Force Institute of Technology, USA Xuegang Jeff Ban, University of Washington, USA Sankarshan Basu, Indian Institute of Management Bangalore, India Melike Baykal-Gursoy, Rutgers University, USA Dirk Briskorn, Universität Siegen, Germany Kevin Byrnes, Johns Hopkins University, USA Gary H. Chao, Kutztown 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 David Ciemnoczolowski, Union Pacific Railroad, USA Barry Cobb, Virginia Military Institute, USA Nagihan Çömez, Bilkent University, Tokelau Louis Anthony Cox Jr., University of Colorado, USA Lauren Davis, North Carolina A&T State University, USA Ivan Derpich, University of Santiago of Chile, Chile Jin Dong, IBM China Research Lab, Chile Matt Drake, Duquesne University, USA Banu Y. Ekren, Izmir University of Economics, Turkey Sandra Eksioglu, Clemson University, USA Ali Elkamel, University of Waterloo, Canada Murat Erkoc, University of Miami, USA Barry Charles Ezell, Old Dominion University, USA Javier Faulin, Public University of Navarre, Spain Yudi Fernando, Universiti Malaysia Pahang, Malaysia William P. Fox, Naval Postgraduate School, USA Hise Gibson, INFORMS, USA Genady Grabarnik, IBM TJ Watson Research, USA Volume 9 • Issue 2 • April-June 2018 • ISSN: 1947-9328 • eISSN: 1947-9336 An official publication of the Information Resources Management Association International Journal of Operations Research and Information Systems
- 3. 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 Ülkü Gürler, Bilkent University, Turkey Alexander Gutfraind, Los Alamos National Laboratory, 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 Samuel Hohmann, University Health System Consortium, USA Xiangling Hu, Grand Valley State University, USA Dariusz Jacek Jakóbczak, Koszalin University of Technology, Poland Manoj K. Jha, Morgan State University, USA Alan W. Johnson, Air Force Institute of Technology, USA Burcu B. Keskin, University of Alabama, USA Adlar Kim, Massachusetts Institute of Technology, USA Rex Kincaid, College of William & Mary, USA Saroj Koul, Jindal Global Business School, India Deepak Kulkarni, NASAAmes Research Center, USA Nanda Kumar, University of Texas at Dallas, USA Chang Won Lee, Hanyang University, Korea Hyoung-Gon Lee, Massachusetts Institute of Technology, USA Loo Hay Lee, National University of Singapore, Singapore Fei Li, George Mason University, USA Feng Li, IBM China Research Laboratory, China Jian Li, Northeastern Illinois University, USA Jing Li, Arizona State University, USA Kunpeng Li, Utah State, USA Xueping Li, University of Tennessee, Knoxville, USA Igor Linkov, US Army Engineer Research & Devel. Center, USA Dengpan Liu, University of Alabama in Huntsville, USA George Liu, Intel Corporation, China Tie Liu, IBM China Research Laboratory, China Leonardo Lopes, University of Arizona, USA Dimitrios Magos, Technological Educational Institute of Athens, Greece Kaye McKinzie, U.S. Army, USA Yefim Haim Michlin, Israel Institute of Technology, Israel Somayeh Moazeni, Princeton University, USA Soumyo Moitra, Carnegie Mellon 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 Nagen Nagarur, Binghamton University, USA Olufemi A Omitaomu, Oak Ridge National Laboratory, USA Mohammad Oskoorouchi, California State University San Marcos, USA Kivanc Ozonat, HP Labs, USA Dessislava Pachamanova, Babson College, USA Julia Pahl, University of Hamburg, Germany Alexander Paz, University of Nevada Las Vegas, USA Francois Pinet, Irstea, France Tania Querido, Linear Options Consulting, LCC, USA Michael Racer, University of Memphis, USA H. Charles Ralph, Clayton State University, USA Marion S. Rauner, University of Vienna, Austria Joe Roise, North Carolina State University, USA Enzo Sauma Pontificia, Universidad Catolica de Chile, Chile Hsu-Shih Shih, Tamkang University, Taiwan Laura Shwartz, IBM T.J. Watson Research Center, USA Sebastian Sitarz, University of Silesia, Poland Young-Jun Son, University of Arizona, USA Huaming Song, Nanjing University of Science & Technology, China Qin Su, Xi’an Jiaotong University, China Editorial Review Board Continued
- 4. 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 Jiamin Wang, Long Island University, USA Kaibo Wang, ASQ Certified Six Sigma Black Belt, China 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 Donghun Yoon, Keio University, Japan Banu Yukse-Ozkaya, Hacettepe University, Turkey Jun Zhuang, SUNY Buffalo, USA Editorial Review Board Continued
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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 Copyright © 2018, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. 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.
- 9. 26 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/linear-programming-approach-for- solving-balanced-and-unbalanced-intuitionistic-fuzzy- transportation-problems/201579?camid=4v1 This title is available in InfoSci-Operations, Logistics, and Performance Assessment eJournal Collection, InfoSci- Journals, InfoSci-Journal Disciplines Business, Administration, and Management, InfoSci-Journal Disciplines Computer Science, Security, and Information Technology, InfoSci-Journal Disciplines Engineering, Natural, and Physical Science. Recommend this product to your librarian: www.igi-global.com/e-resources/library- recommendation/?id=156 Related Content A Time Dependent Order Level Inventory Model for Beta Deterioration in Two Warehouse Systems Soumendra Kumar Patra, Tapan Kumar Lenka and Er. Purna Chandra Ratha (2015). International Journal of Operations Research and Information Systems (pp. 53-69). www.igi-global.com/article/a-time-dependent-order-level-inventory-model-for- beta-deterioration-in-two-warehouse-systems/125662?camid=4v1a A Unified Classification Ecosystem for Auctions Dimitrios M. Emiris and Charis A. Marentakis (2010). International Journal of Operations Research and Information Systems (pp. 53-74). www.igi-global.com/article/unified-classification-ecosystem- auctions/45763?camid=4v1a
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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 List Price: $37.50 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 Volume 9 • Issue 2 • April-June 2018 91 REFERENCES Aggarwal, S., & Gupta, C. (2014). Algorithm for solving intuitionistic fuzzy transportation problem with generalized trapezoidal intuitionistic fuzzy number via new ranking method. arXiv:1401.3353 Angelov, P. P. (1997). Optimization in an intuitionistic fuzzy environment. Fuzzy Sets and Systems, 86(3), 299–306. doi:10.1016/S0165-0114(96)00009-7 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). Intuitionistic fuzzy sets. In VII ITKR’s Session, Sofia. (in Bulgarian) 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. Basirzadeh, H. (2011). An approach for solving fuzzy transportation problem. Applied Mathematical Sciences, 5(32), 1549–1566. Beauchamp, H., Novoa, C., & Ameri, F. (2015). Supplier selection and order allocation based on integer programming. International Journal of Operations Research and Information Systems, 6(3), 60–79. doi:10.4018/ IJORIS.2015070103 Bharati, S. K., Nishad, A. K., & Singh, S. R. (2014, January). Solution of Multi-Objective Linear Programming Problems in Intuitionistic Fuzzy Environment. In Proceedings of the Second International Conference on Soft Computing for Problem Solving (SocProS 2012), December 28-30 (pp. 161-171). Springer India. doi:10.1007/978- 81-322-1602-5_18 Bit, A. K., Biswal, M. P., & Alam, S. S. (1992). Fuzzy programming approach to multicriteria decision making transportation problem. Fuzzy Sets and Systems, 50(2), 135–141. doi:10.1016/0165-0114(92)90212-M Burillo, P., Bustince, H., & Mohedano, V. (1994, September). 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). A new approach to solve multi-objective multi-choice multiitem Atanassov’s intuitionistic fuzzy transportation problem using chance operator. Journal of Intelligent & Fuzzy Systems, 28(2), 843–865. Chanas,S.,Delgado,M.,Verdegay,J.L.,&Vila,M.A.(1993).Intervalandfuzzyextensionofclassicaltransportation problems. Transportation Planning and Technology, 17(2), 203–218. doi:10.1080/03081069308717511 Chanas, S., Kolodziejczyk, 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. 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 for explaining linear programming calculation in transportation problem. 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. 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.
- 15. International Journal of Operations Research and Information Systems Volume 9 • Issue 2 • April-June 2018 92 Chiang, J. (2005). The optimal solution of the transportation problem with fuzzy demand and fuzzy product. Journal of Information Science and Engineering, 21(2), 439–451. Dantzig, G. B. (1963). Linear programming and extensions. Princeton, New Jersey: Princeton Univ. Press. doi:10.1515/9781400884179 Das, M., & Baruah, H. K. (2007). Solution of the transportation problem in fuzzified form. Journal of Fuzzy Mathematics, 15(1), 79–95. 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 (pp. 563-569). Atlantis Press. doi:10.2991/eusflat.2011.78 Ebrahimnejad, A., Nasseri, S. H., & Mansourzadeh, S. M. (2011). Bounded primal simplex algorithm for bounded linear programming with fuzzy cost coefficients. International Journal of Operations Research and Information Systems, 2(1), 96–120. doi:10.4018/joris.2011010105 Gani, A. N., & Abbas, S. (2012). Intuitionistic fuzzy transportation problem. In Proceedings of the Heber International Conference on Applications of Mathematics and Statistics (HICAMS) (pp. 528-535). Grzegorzewski, P. (2003a). 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). Grzegorzewski, P. (2003b), Intuitionistic fuzzy numbers. In Proceedings of the IFSA 2003 World Congress. Guha, D., & Chakraborty, D. (2010). A theoretical development of distance measure for intuitionistic fuzzy numbers. International Journal of Mathematics and Mathematical Sciences. Hasan, M. K., & Ban, X. (2013). A link-node nonlinear complementarity model for a multiclass simultaneous transportation dynamic user equilibria. International Journal of Operations Research and Information Systems, 4(1), 27–48. doi:10.4018/joris.2013010102 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. Klir, G. J., & Yuan, B. (2003). Fuzzy sets and fuzzy logic: Theory and applications. New York: Prentice Hall. Koopmans, T. C. (1949). Optimum utilization of the transportation system. Econometrica, Supplement: Report of the Washington Meeting, 17, 136-146. doi:10.2307/1907301 Kumar, A., & Kaur, A. (2011). Methods for solving fully fuzzy transportation problems based on classical transportation methods. International Journal of Operations Research and Information Systems, 2(4), 52–71. doi:10.4018/joris.2011100104
- 16. International Journal of Operations Research and Information Systems Volume 9 • Issue 2 • April-June 2018 93 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/ s10852-010-9147-8 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). 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 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, 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. doi:10.1080/18756891.2010.9727719 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 Proceedings of the 2008 International Symposiums on Information Processing (ISIP) (pp. 3-8). IEEE. doi:10.1109/ISIP.2008.9 Lin, F. T. (2009, August). Solving the transportation problem with fuzzy coefficients using genetic algorithms. In Proceedings of the IEEE International Conference on Fuzzy Systems FUZZ-IEEE ’09 (pp. 1468-1473). IEEE. doi:10.1109/FUZZY.2009.5277202 Mahapatra, G. S., & Roy, T. K. (2009). Reliability evaluations using triangular intuitionistic fuzzy numbers, arithmetic operations. International Scholarly and Scientific Research & Innovation, 3(2), 422–429. 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. doi:10.1142/S0218488504002886
- 17. International Journal of Operations Research and Information Systems Volume 9 • Issue 2 • April-June 2018 94 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. 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 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. Nishad, A. K., & Singh, S. R. (2015). Solving multi-objective decision making problem in intuitionistic fuzzy environment. International Journal of System Assurance Engineering and Management, 6(2), 206–215. doi:10.1007/s13198-014-0331-5 Nwauwa, L., Adenegan, K., Rahji, M., & Awoyemi, T. (2016). Optimal transportation and spatial integration of regional Palm Oil markets in Nigeria. International Journal of Operations Research and Information Systems, 7(2), 62–83. doi:10.4018/IJORIS.2016040104 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. Parvathi, R., & Malathi, C. (2012). Intuitionistic fuzzy linear programming problems. World Applied Sciences Journal, 17(12), 1802–1807. 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 Pramila, K., & Uthra, G. (2014). Optimal solution of an intuitionistic fuzzy transportation problem. Annals of Pure and Applied Mathematics, 8(2), 67–73. Rabbani, M., Mamaghani, M. G., Farshbaf-Geranmayeh, A., & Mirzayi, M. (2016). A novel mixed integer programming formulation for selecting the best renewable energies to invest: A fuzzy goal programming approach. International Journal of Operations Research and Information Systems, 7(3), 1–22. doi:10.4018/ IJORIS.2016070101 Rani, D., Gulathi, T. R., & Kumar, A. (2014). A method for unbalanced transportation problems in fuzzy environment. Indian Academy of Sciences. Sadhana, 39(3), 573–581. doi:10.1007/s12046-014-0243-8 Ruiz-Torres, A. J., Paletta, G., & Perez-Roman, E. (2015). Maximizing the percentage of on-time jobs with sequence dependent deteriorating process times. International Journal of Operations Research and Information Systems, 6(3), 1–18. doi:10.4018/IJORIS.2015070101 Saad, O. M., & Abbas, S. A. (2003). A parametric study on transportation problem under fuzzy environment. Journal of Fuzzy Mathematics, 11(1), 115–124. Sauma, E. (2013). A survey and comparison of optimization methods for solving multi-stage stochastic programs with recourse. International Journal of Operations Research and Information Systems, 4(2), 22–35. doi:10.4018/ joris.2013040102 Shabani, A., & Jamkhaneh, E. B. (2014). A new generalized intuitionistic fuzzy number. Journal of Fuzzy Set Valued Analysis, 24, 1–10. 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.
- 18. International Journal of Operations Research and Information Systems Volume 9 • Issue 2 • April-June 2018 95 Singh, S. (2012). Note on assignment algorithm with easy method of drawing lines to cover all zeros. International Journal of Operations Research and Information Systems, 3(3), 87–97. doi:10.4018/joris.2012070106 Singh, S. K., & Yadav, S. P. (2015). Efficient approach for solving type-1 intuitionistic fuzzy transportation problem. International Journal of System Assurance Engineering and Management, 6(3), 259–267. doi:10.1007/ s13198-014-0274-x Singh, S. K., & Yadav, S. P. (2016). A new approach for solving intuitionistic fuzzy transportation problem of type-2. Annals of Operations Research, 243(1-2), 349–363. doi:10.1007/s10479-014-1724-1 Solaiappan, S., & Jeyaraman, K. (2014). A new optimal solution method for trapezoidal fuzzy transportation problem. International Journal of Advanced Research, 2(1), 933–942. Srinivas, B., & Ganesan, G. (2015). Optimal solution for intuitionistic fuzzy transportation problem via Revised Distribution Method. International Journal of Mathematics Trends and Technology, 19(2), 150–161. doi:10.14445/22315373/IJMTT-V19P519 Sudhakar, V. J., & Kumar, V. N. (2011). A different approach for solving two stage fuzzy transportation problems. International Journal of Contemporary Mathematical Sciences, 6(11), 517–526. Taha, H. A. (2008). Operations Research: An Introduction (8th ed.). Pearson Education India. Varghese, A., & Kuriakose, S. (2012). Centroid of an intuitionistic fuzzy number. Notes on Intuitionistic Fuzzy Sets, 18(1), 19–24. Ye, X., Han, S., & Lin, A. (2010). A note on the connection between the Primal-Dual and the A* algorithm. InternationalJournalofOperationsResearchandInformationSystems,1(1),73–85.doi:10.4018/joris.2010101305 Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353. doi:10.1016/S0019-9958(65)90241-X
- 19. International Journal of Operations Research and Information Systems Volume 9 • Issue 2 • April-June 2018 100 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