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

Document Type: Research Paper

Authors

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.

Keywords


[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.