SAFI, M., MALEKI, H., ZAEIMAZAD, E. (2007). A NOTE ON THE ZIMMERMANN METHOD FOR SOLVING FUZZY LINEAR PROGRAMMING PROBLEMS. Iranian Journal of Fuzzy Systems, 4(2), 31-45. doi: 10.22111/ijfs.2007.369

MOHAMMADREZA SAFI; HAMIDREZA MALEKI; EFFAT ZAEIMAZAD. "A NOTE ON THE ZIMMERMANN METHOD FOR SOLVING FUZZY LINEAR PROGRAMMING PROBLEMS". Iranian Journal of Fuzzy Systems, 4, 2, 2007, 31-45. doi: 10.22111/ijfs.2007.369

SAFI, M., MALEKI, H., ZAEIMAZAD, E. (2007). 'A NOTE ON THE ZIMMERMANN METHOD FOR SOLVING FUZZY LINEAR PROGRAMMING PROBLEMS', Iranian Journal of Fuzzy Systems, 4(2), pp. 31-45. doi: 10.22111/ijfs.2007.369

SAFI, M., MALEKI, H., ZAEIMAZAD, E. A NOTE ON THE ZIMMERMANN METHOD FOR SOLVING FUZZY LINEAR PROGRAMMING PROBLEMS. Iranian Journal of Fuzzy Systems, 2007; 4(2): 31-45. doi: 10.22111/ijfs.2007.369

A NOTE ON THE ZIMMERMANN METHOD FOR SOLVING FUZZY LINEAR PROGRAMMING PROBLEMS

^{1}DEPARTMENT OF MATHEMATICES, UNIVERSITY OF SHAHID-BAHONAR KERMAN, KERMAN, IRAN

^{2}DEPARTMENT OF BASIC SCIENCES, SHIRAZ UNIVERSITY OF TECHNOLOGY, SHIRAZ, IRAN

Abstract

There are several methods for solving fuzzy linear programming (FLP) problems. When the constraints and/or the objective function are fuzzy, the methods proposed by Zimmermann, Verdegay, Chanas and Werners are used more often than the others. In the Zimmerman method (ZM) the main objective function cx is added to the constraints as a fuzzy goal and the corresponding linear programming (LP) problem with a new objective (λ ) is solved. When this new LP has alternative optimal solutions (AOS), ZM may not always present the "best" solution. Two cases may occur: cx may have different bounded values for the AOS or be unbounded. Since all of the AOS have the same λ , they have the same values for the new LP. Therefore, unless we check the value of cx for all AOS, it may be that we do not present the best solution to the decision maker (DM); it is possible that cx is unbounded but ZM presents a bounded solution as the optimal solution. In this note, we propose an algorithm for eliminating these difficulties.

[1] R. E. Bellman and L. A. Zadeh, Decision making in a fuzzy environment, Management Science, 17 (1970), 141-164. [2] J. M. Cadenas and J. L. Verdegay, A Primer on fuzzy optimization models and methods, Iranian Journal of Fuzzy Systems (to appear). [3] J. M. Cadenas and J. L. Verdegay, Using ranking functions in multi-objective fuzzy linear programming, Fuzzy sets and systems, 111 (2000), 47-53. [4] L. Campus and J. L. Verdegay, Linear programming problem and ranking of fuzzy numbers, Fuzzy Sets and Systems, 32 (1989), 1-11. [5] S. Chanas, The use of parametric programming in fuzzy linear programming, Fuzzy Sets and Systems, 11 (1983), 243-251. [6] M. Delgado, J. L Verdegay and M. A. Vila, A general model for fuzzy linear programming, Fuzzy Sets and Systems, 29 (1989), 21-29. [7] D. Dubois, H. Fargier and H. Prade, Refinements of the maximum approach to decision making in a fuzzy environment, Fuzzy Sets and Systems, 81 (1996), 103-122. [8] S. M. Guu and Y. K. Wu, Two phase approach for solving the fuzzy linear programming problems, Fuzzy Sets and Systems, 107 (1999), 191-195. [9] Y. J. Lai and C. L. Hwang, Fuzzy mathematical programming methods and applications, Springer-Verlag, Berlin, 1992. [10] Y. J. Lai and C. L. Hwang, Interactive fuzzy linear programming, Fuzzy Sets and Systems, 45 (1992), 169-183. [11] X. Li, B. Zhang and H. Li, Computing efficient solution to fuzzy multiple objective linear programming problems, Fuzzy Sets and Systems, 157 (2006), 1328-1332.

[12] H. R. Maleki, Ranking functions and their applications to fuzzy linear programming, Far East Journal of Mathematical Sciences, 4(3) (2003), 283-301. [13] H. R. Maleki, M. Tata and M. Mashinchi, Linear programming with fuzzy variables, Fuzzy Set and Systems, 109 (2000), 21-33. [14] H. R. Maleki, M. Tata and M. Mashinchi, Fuzzy number linear programming, in: C. Lucas (Ed), Proc. Internat. Conf. on Intelligent and Cognitive System FSS ’96, sponsored by IEE ISRF, Tehran, Iran, 1996, 145-148. [15] WinQSB 1, Yih-Long Chang and Kiran Desai, John wiley & Sons, Inc. [16] J. Ramik and J. Raminak, Inequality relation between fuzzy numbers and its use in fuzzy optimization, Fuzzy Sets and Systems, 16 (1985), 123-138. [17] H. Tanaka, T. Okuda and K. Asai, On fuzzy mathematical programming, Journal of Cybernetics, 3(4) (1974), 37-46. [18] R. N. Tiwari, S. Deharmar and J. R. Rao, Fuzzy goal programming – an additive model, Fuzzy Sets and Systems, 24 (1987), 27-34. [19] J. L. Verdegay, Fuzzy mathematical programming, in: M. M. Gupta and E. Sanchez, Eds., Fuzzy Information and Decision Processes, North-Holland, Amsterdam, 1982, 231- 236. [20] B. Werners, An interactive fuzzy programming system, Fuzzy Sets and Systems, 23 (1987), 131-147. [21] E. Zaeimazad, Fuzzy linear programming: a geometric approach, Msc thesis, University of Shahid–Bahonar, Kerman, Iran, 2005. [22] H. J. Zimmermann, Description and optimization of fuzzy systems, International Journal of General Systems, 2 (1976), 209- 215. [23] H. J. Zimmermann, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems, 1 (1978), 45-55.