A Simple Method for Solving Fully Intuitionistic Fuzzy ...

International Journal of Operations Research and Information

Systems

Volume 7 • Issue 2 • April-June 2016 • ISSN: 1947-9328 • eISSN: 1947-9336

An official publication of the Information Resources Management Association

EDITOR-IN-CHIEF John Wang, Montclair State University, USA

ASSOCIATE EDITORS Sungzoon Cho, Seoul National University, Korea Theodore Glickman, The George Washington University, USA Manoj Jha, Morgan State University, USA Eva 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 Ravi Ravindran, Pennsylvania State University, USA Kathryn Stecke, University of Texas at Dallas, USA

International Editorial Review Board Khaled Abdelghany, Southern Methodist University, USA Anil Aggarwal, University of Baltimore, USA Ahad Ali, Lawrence Technological University, USA Mohammad Amini, The University of Memphis, USA Adedeji Badiru, Air Force Institute of Technology, USA Lihui Bai, Valparaiso University, USA Xuegang Ban, Rensselaer Polytechnic Institute, USA Sankarshan Basu, Indian Institute of Management Bangalore, India Melike Baykal-Gursoy, Rutgers University, USA Sherry Borener, Federal Aviation Administration, USA Denis Borenstein, Federal University of Rio Grande do Sul, Brazil Robert Brigantic, Pacific Northwest National Laboratory, USA Dirk Briskorn, Universität Siegen, Germany Kevin Byrnes, Johns Hopkins University, USA Muricio Cabrera Rios, University of Puerto Rico – Mayagüez, Puerto Rico Mei Cao, University of Wisconsin-Superior, USA Gary Chao, Kutztown University, USA Dean Chatfield, Old Dominion University, USA Chialin Chen, Queen’s University, Canada Lijian Chen, University of Louisville, USA Feng Cheng, IBM T.J. Watson Research Center, USA Jagpreet Chhatwal, Merck Research Laboratories, USA Wen Chiang, University of Tulsa, USA David Chin, Federal Aviation Administration, USA David Ciemnoczolowski, Union Pacific Railroad, USA Barry Cobb, Virginia Military Institute, USA Nagihan Çömez, Bilkent University, Tokelau Louis 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

International Editorial Review Board Continued

Parijat Dube, IBM T.J. Watson Research Center, USA Banu Ekren, Izmir University of Economics, Turkey Sandra Eksioglu, Mississippi State University, USA Ali Elkamel, University of Waterloo, Canada Murat Erkoc, University of Miami, USA Barry Ezell, Old Dominion University, USA Javier Faulin, Public University of Navarre, Spain Yudi Fernando, Universiti Sains Malaysia, Malaysia William P. Fox, Naval Postgraduate School, USA Hise Gibson, INFORMS, USA Genady Grabarnik, IBM TJ Watson Research, USA Scott 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 Hirsch, Raytheon Inc., USA Samuel Hohmann, University Health System Consortium, USA Xiangling Hu, Grand Valley State University, USA Dariusz Jakóbczak, Technical University of Koszalin, Poland Manoj Jha, Morgan State University, USA Alan Johnson, Air Force Institute of Technology, USA Burcu Keskin, The 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, NASA Ames Research Center, USA Nanda Kumar, University of Texas at Dallas, USA Chang Won Lee, Hanyang University, Korea, Democratic People’s Republic Of Hyoung-Gon Lee, Massachusetts Institute of Technology, USA Loo 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, Utica College, 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 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 B.P.S. Murthi, University of Texas at Dallas, USA Nagen Nagarur, Binghamton University, USA Olufemi 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 - Clermont Ferrand, France Tania Querido, Linear Options Consulting, LCC, USA

International Editorial Review Board Continued

Michael Racer, University of Memphis, USA H. Charles Ralph, Clayton State University, USA Marion Rauner, University of Vienna, Austria Joe Roise, North Carolina State University, USA Kedar Sambhoos, CUBRC, 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, The University of Arizona, USA Huaming Song, Nanjing University of Science & Technology, China Qin Su, Xi’an Jiaotong University, 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 Jiamin Wang, Long Island University, USA Kaibo Wang, ASQ Certified Six Sigma Black Belt, China Yitong Wang, Tsinghua University, China Ue-Pyng Wen, National Tsing Hua University, Taiwan Harris Wu, Old Dominion University, USA Changyuan Yan, PNC Bank, USA Justin Yates, Texas A&M University, USA Mesut Yavuz, Shenandoah University, USA Xugang Ye, Johns Hopkins University and Microsoft, USA Donghun Yoon, Keio University, Japan Banu Yukse-Ozkaya, Hacettepe University, Turkey Muhong Zhang, Arizona State University, USA Kangyuan Zhu, CSSI, Inc., USA Yuntao Zhu, Arizona State University, USA Jun Zhuang, SUNY Buffalo, USA

Table of Contents International Journal of Operations Research and Information Systems Volume 7 • Issue 2 • April-June-2016 • ISSN: 1947-9328 • eISSN: 1947-9336

An official publication of the Information Resources Management Association Research Articles 1

Application of SARIMAX Model to Forecast Daily Sales in Food Retail Industry Nari Sivanandam Arunraj, Deggendorf Institute of Technology, Deggendorf, Germany Diane Ahrens, Deggendorf Institute of Technology, Deggendorf, Germany Michael Fernandes, Deggendorf Institute of Technology, Deggendorf, Germany

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AEGISi: Attribute Experimentation Guiding Improvement Searches Inline Framework Michael Racer, Marketing & Supply Chain Management Department, University of Memphis, Memphis, TN, USA Robin Lovgren, Department of Mathematics and Computer Science, Belmont University, Nashville, TN, USA

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A Simple Method for Solving Fully Intuitionistic Fuzzy Real Life Assignment Problem Senthil P. Kumar, PG & Research Department of Mathematics, Jamal Mohamed College (Autonomous), Tiruchirappalli, India Jahir R. Hussain, PG & Research Department of Mathematics, Jamal Mohamed College (Autonomous), Tiruchirappalli, India

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Optimal Transportation and Spatial Integration of Regional Palm Oil Markets in Nigeria L.O.E. Nwauwa, Department of Agricultural Economics, Faculty of Agriculture, University of Ibadan, Ibadan, Nigeria K.O. Adenegan, Department of Agricultural Economics, Faculty of Agriculture, University of Ibadan, Ibadan, Nigeria M.A.Y. Rahji, Department of Agricultural Economics, Faculty of Agriculture, University of Ibadan, Ibadan, Nigeria T.T. Awoyemi, Department of Agricultural Economics, Faculty of Agriculture, University of Ibadan, Ibadan, Nigeria

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Supply Chain Coordination Under Service Level Constraint and Controllable Lead Time Prashant Jindal, Department of Applied Mathematics, Gautam Buddha University, Greater Noida, India Anjana Solanki, Department of Applied Mathematics, Gautam Buddha University, Greater Noida, India

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International Journal of Operations Research and Information Systems Volume 7 • Issue 2 • April-June 2016

A Simple Method for Solving Fully Intuitionistic Fuzzy Real Life Assignment Problem P. Senthil Kumar, PG & Research Department of Mathematics, Jamal Mohamed College (Autonomous), Tiruchirappalli, Tamil Nadu, India R. Jahir Hussain, PG & Research Department of Mathematics, Jamal Mohamed College (Autonomous), Tiruchirappalli, Tamil Nadu, India

ABSTRACT In solving real life assignment problem 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 representations for the data. So, in this paper, the authors consider the assignment problem having uncertainty and hesitation in cost/time/profit. They formulate the problem and utilize triangular intuitionistic fuzzy numbers (TIFNs) to deal with uncertainty and hesitation. The authors propose a new method called PSK (P.Senthil Kumar) method for finding the intuitionistic fuzzy optimal cost/time/profit for fully intuitionistic fuzzy assignment problem (FIFAP). The proposed method gives the optimal object value in terms of TIFN. The main advantage of this method is computationally very simple, easy to understand. Finally the effectiveness of the proposed method is illustrated by means of a numerical example which is followed by graphical representation of the finding. Keywords Fully Intuitionistic Fuzzy Assignment Problem, Intuitionistic Fuzzy Set, Optimal Assignment, PSK Method, Triangular Intuitionistic Fuzzy Number

1. INTRODUCTION Assignment Problem (AP) is used worldwide in solving real world problems. An assignment problem plays an important role in assigning of persons to jobs, drivers to trucks, trucks to routes, operators to machines, or problems to research teams, etc. The assignment problem is a special kind of linear programming problem (LPP) in which the aim of the decision maker (DM) is to assign n number of jobs to n number of machines (persons) at a minimum cost/minimum time/ maximum profit. In literature, to find the solution to assignment problems, Kuhn (1955) proposed the Hungarian method for solving the assignment problem. Thompson (1981) discussed a recursive method for solving assignment problem. Avis and Devroye (1985) presented an analysis of a decomposition heuristic for the assignment problem. Balinski (1986) did a competitive (dual) simplex method for the assignment problem. Paparrizos (1991) developed an efficient exterior point simplex type algorithm for the assignment problem. Barr et al. (1977) gave the alternating basis algorithm for assignment problems. Ping et al. (1997) discussed a new algorithm for the assignment problem which they also called an alternative to the Hungarian Method. Their assignment algorithm is based on a 2n*2n matrix where operators are performed on the matrix until an optimal solution is found. Lin and Wen DOI: 10.4018/IJORIS.2016040103 Copyright © 2016, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.

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International Journal of Operations Research and Information Systems Volume 7 • Issue 2 • April-June 2016

(2004) proposed an efficient algorithm based on a labeling method for solving the linear fractional programming case. Singh (2012) discussed note on assignment algorithm with easy method of drawing lines to cover all zeros. However, in real life situations, the parameter of assignment problem is in imprecise instead of fixed real numbers because time/cost/profit for doing a job by a facility (machine/person) might vary due to different reasons. 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 whereas in crisp set the element belongs to the set that represents 1 and the element not in the set that represents 0. Therefore the applications of fuzzy set theory enabled many authors to solve assignment, transportation and linear programming problems by using fuzzy representation for data. Kumar et al. (2009) proposed a method for solving fully fuzzy assignment problems using triangular fuzzy numbers. Mukherjee and Basu (2010) presented an application of fuzzy ranking method for solving assignment problems with fuzzy costs. Kumar and Gupta (2012) investigated assignment and travelling salesman problems with cost coefficients as LR fuzzy parameters. De and Yadav (2012) evolved a general approach for solving assignment problems involving with fuzzy costs coefficients. Thorani and Shankar (2013) did fuzzy assignment problem with generalized fuzzy numbers. Kumar and Kaur (2011) presented methods for solving fully fuzzy transportation problems based on classical transportation methods. Ebrahimnejad et al. (2011) proposed bounded primal simplex algorithm for bounded linear programming with fuzzy cost coefficients. Nasseri and Ebrahimnejad (2011) did sensitivity analysis on linear programming problems with trapezoidal fuzzy variables. Pattnaik (2015) presented decision making approach to fuzzy linear programming problems with post optimal analysis. In the assignment problem, the performing time of each job to the workers is not known exactly. This may be due to lack of experience, interest, capacity, understanding, etc. In such situation the DM cannot predict performing time exactly. Hence the decision maker may hesitate. The fuzzy set deals with the belongingness of an element in the set but it does not consider the non-belongingness (rejections level) of an element in the set. So, to counter these uncertainties with hesitation, Atanassov (1983) proposed the intuitionistic fuzzy set (IFS) which is more reliable than the fuzzy set proposed by Zadeh (1965). The major advantage of intuitionistic fuzzy set over fuzzy set is that IFS separates the degree of membership (belongingness) and the degree of non membership (non belongingness) of an element in the set. With the help of IFS theory, decision maker can decide about the degree of acceptance, degree of non acceptance and degree of hesitation for some quantity. In case of assignment problem, the DM can decide about the level of acceptance and non-acceptance for the assignment cost/profit/time. Due to this, the application of IFS theory becomes very popular in project schedules, transportation problems, decision making theory and network flow problems etc. In literature, due to the lack of uncertainty of the parameter of the fuzzy assignment problem, many authors have solved assignment problem with intuitionistic fuzzy version. Mukherjee and Basu (2012) presented the solution of a class of intuitionistic fuzzy assignment problem by using similarity measures. Jose and Kuriakose (2013) discussed algorithm for solving an assignment model in intuitionistic fuzzy context. Kumar and Hussain (2014) presented a method for finding an optimal solution of an assignment problem under mixed intuitionistic fuzzy environment. Kumar and Bajaj (2014) evolved on solution of interval valued intuitionistic fuzzy assignment problem using similarity measure and score function. Kumar and Hussain (2014) did a method for solving balanced intuitionistic fuzzy assignment problem. Dinagar and Thiripurasundari (2014) found a new method for finding the cost of fuzzy assignment problem using genetic algorithm of artificial intelligence. Prabakaran and Ganesan (2014) presented fuzzy Hungarian method for solving intuitionistic fuzzy 40

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REFERENCES Aggarwal, S., & Gupta, C. (2014). Algorithm for solving intuitionistic fuzzy transportation problem with generalized trapezoidal intuitionistic fuzzy number via new ranking method. arXiv preprint arXiv:1401.3353. 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. (1983, June). Intuitionistic fuzzy sets. VII ITKR’s Session, Sofia. Atanassov, K. (1999). Intuitionistic fuzzy sets: theory and applications. Springer. doi:10.1007/978-3-7908-1870-3 Avis, D., & Devroye, L. (1985). An analysis of a decomposition heuristic for the assignment problem. Operations Research Letters, 3(6), 279–283. doi:10.1016/0167-6377(85)90001-X Balinski, M. L. (1986). A competitive (dual) simplex method for the assignment problem. Mathematical Programming, 34(2), 125–141. doi:10.1007/BF01580579 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. Barr, R. S., Glover, F., & Klingman, D. (1977). The alternating basis algorithm for assignment problems. Mathematical Programming, 13(1), 1–13. doi:10.1007/BF01584319 Bellman, R. E., & Zadeh, L. A. (1970). Decision-making in a fuzzy environment. Management Science, 17(B), 141-164. Burillo, P., Bustince, H., & Mohedano, V. (1994, September). Some definitions of intuitionistic fuzzy numberfirst properties. 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. Das, S., & Guha, D. (2013). Ranking of intuitionistic fuzzy number by centroid point. Journal of Industrial and Intelligent Information, 1(2), 107–110. doi:10.12720/jiii.1.2.107-110 De, P. K., & Yadav, B. (2012). A general approach for solving assignment problems involving with fuzzy cost coefficients. Modern Applied Science, 6(3), 2. doi:10.5539/mas.v6n3p2 Dinagar, D. S., & Thiripurasundari, K. (2014). A new method for finding the cost of fuzzy assignment problem using genetic algorithm of artificial intelligence. International Electronic Journal of Pure and Applied Mathematics, 8(4). 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. 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. Proceedings of the Heber International Conference on Applications of Mathematics and Statistics (HICAMS) (pp. 528-535). 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)3. Guha, D., & Chakraborty, D. (2010). A theoretical development of distance measure for intuitionistic fuzzy numbers. International Journal of Mathematics and Mathematical Sciences, 2010. Jose, S., & Kuriakose, S. (2013). Algorithm for solving an assignment model in intuitionistic fuzzy context. International Journal of Fuzzy Mathematics and Systems, 3(5), 345–349. Kuhn, H. W. (1955). The Hungarian method for the assignment problem. Naval Research Logistics Quarterly, 2(1‐2), 83–97. doi:10.1002/nav.3800020109

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Kumar, A., & Gupta, A. (2012). Assignment and travelling salesman problems with coefficients as LR fuzzy parameters. International Journal of Applied Science and Engineering, 10(3), 155–170. Kumar, A., Gupta, A., & Kaur, A. (2009). Method for solving fully fuzzy assignment problems using triangular fuzzy numbers. International Journal of Computer and Information Engineering, 3, 231–234. 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 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. doi:10.1016/S1665-6423(13)71548-7 Kumar, G., & Bajaj, R. K. (2014). On Solution of Interval Valued Intuitionistic Fuzzy Assignment Problem Using Similarity Measure and Score Function. International Journal of Mathematical, Computational. Physical and Quantum Engineering, 8(4), 713–718. Kumar, P. S., & Hussain, R. J. (2014). 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. (2014). A method for finding an optimal solution of an assignment problem under mixed intuitionistic fuzzy environment. Proceedings of International Conference on Mathematical Sciences (ICMS-2014), Sathyabama University (pp. 417-421). Kumar, P. S., & Hussain, R. J. (2015). Computationally simple approach for solving fully intuitionistic fuzzy real life transportation problems. International Journal of System Assurance Engineering and Management, 2015(1). doi:10.1007/s13198-014-0334-2 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 Lin, C. J., & Wen, U. P. (2004). A labeling algorithm for the fuzzy assignment problem. Fuzzy Sets and Systems, 142(3), 373–391. doi:10.1016/S0165-0114(03)00017-4 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. 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 Mukherjee, S., & Basu, K. (2010). Application of fuzzy ranking method for solving assignment problems with fuzzy costs. International Journal of Computational and Applied Mathematics, 5(3), 359–368. Mukherjee, S., & Basu, K. (2012). Solution of a class of intuitionistic fuzzy assignment problem by using similarity measures. Knowledge-Based Systems, 27, 170–179. doi:10.1016/j.knosys.2011.09.007 Nasseri, S. H., & Ebrahimnejad, A. (2011). Sensitivity Analysis on Linear Programming Problems with Trapezoidal Fuzzy Variables. [IJORIS]. 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. 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. Proceedings of the Ninth WSEAS International Conference on Systems, Athens, Greece (pp. 1-5).

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Paparrizos, K. (1991). An infeasible (exterior point) simplex algorithm for assignment problems. Mathematical Programming, 51(1-3), 45–54. doi:10.1007/BF01586925 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. Prabakaran, K., & Ganesan, K. (2014). Fuzzy Hungarian method for solving intuitionistic fuzzy assignment problems. International Journal of Scientific and Engineering Research, 5(9), 11–17. 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. 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. (2014). Efficient approach for solving type-1 intuitionistic fuzzy transportation problem. International Journal of System Assurance Engineering and Management, 1-9. doi:.10.1007/s13198014-0274-x Srinivas, B., & Ganesan, G. (2015). A method for solving intuitionistic fuzzy assignment problem using Branch and Bound Method, International Journal of Engineering Technology. Management and Applied Sciences, 3(2), 227–237. 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 Thompson, G. L. (1981). A recursive method for solving assignment problems. Journal of Science Direct- NorthHolland Mathematics Studies, 59, 319-343. Thorani, Y. L. P., & Shankar, N. R. (2013). Fuzzy assignment problem with generalized fuzzy numbers. Applied Mathematical Sciences, 7(71), 3511–3537. 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

P. Senthil Kumar is 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. He received his BSc., MSc., MPhil degrees from Jamal Mohamed College, Tiruchirappalli, in 2006, 2008, 2010 respectively. He completed his BEd 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 is now pursuing his PhD (Part Time) in the area of Intuitionistic Fuzzy Optimization Technique. He has published research papers in referred journals like Springer. He also presented his research in ELSEVIER conference proceedings. R. Jahir Hussain received his MSc from AVC College (Autonomous), Mayiladudurai, MPhil and PhD from Bharathidasan University, Tiruchirappalli, Tamilnadu. In 1996, he joined Jamal Mohamed College, Tiruchirappalli as Lecturer in PG & Research Department of Mathematics. Now he is an Associate Professor. His activities currently focus on Applications of Graph Theory. His research areas include Fuzzy Graph Theory and Fuzzy Optimization. 57