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Interactive fuzzy programming for two-level linear fractional programming problems with fuzzy parameters. (English) Zbl 0978.90112
Summary: In this paper, we present interactive fuzzy programming for two-level linear fractional programming problems with fuzzy parameters. Using the level sets of fuzzy parameters, the corresponding nonfuzzy two-level linear fractional programming problem is introduced. In our interactive method, after determining fuzzy goals of decision makers at both levels, a satisfactory solution is derived efficiently by updating a minimal satisfactory level of the decision maker at the upper level with considerations of overall satisfactory balance between both levels. The satisfactory solution well-balanced between both levels is easily computed by combined use of the bisection method, the phase one of the simplex method and the variable transformation method by Charnes and Cooper. An illustrative numerical example for two-level linear fractional programming problems with fuzzy parameters is provided to demonstrate the feasibility of the proposed method.

MSC:
90C70Fuzzy programming
90C32Fractional programming
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References:
[1] Bard, J. F.: An efficient point algorithm for a linear two-stage optimization problem. Oper. res. 38, 556-560 (1983) · Zbl 0525.90086
[2] Bard, J. F.: Coordination of a multidivisional firm through two levels of management. Omega 11, 457-465 (1983)
[3] Bard, J. F.; Falk, J. E.: An explicit solution to the multi-level programming problem. Comput. oper. Res. 9, 77-100 (1982)
[4] Bard, J. F.; Plummer, J.; Sourie, J. C.: Determining tax credits for converting nonfood crops to biofuelsan application of bilevel programming. Multilevel optimizationalgorithms and applications, 23-50 (1997) · Zbl 0896.90173
[5] Bellman, R. E.; Zadeh, L. A.: Decision making in a fuzzy environment. Management sci. 17, 141-164 (1970) · Zbl 0224.90032
[6] Bialas, W. F.; Karwan, M. H.: Two-level linear programming. Management sci. 30, 1004-1020 (1984) · Zbl 0559.90053
[7] Bitran, G. R.; Novaes, A. G.: Linear programming with a fractional objective function. Oper. res. 21, 22-29 (1973) · Zbl 0259.90046
[8] Charnes, A.; Cooper, W. W.: Programming with linear fractional functions. Naval res. Logist. quart. 9, 181-186 (1962) · Zbl 0127.36901
[9] Fortuny-Amat, J.; Mccarl, B.: A representation and economic interpretation of a two-level programming problem. J. oper. Res. soc. 32, 783-792 (1981) · Zbl 0459.90067
[10] Kornbluth, J. S. H.; Steuer, R. E.: Goal programming with linear fractional criteria. European J. Oper. res. 8, 58-65 (1981) · Zbl 0486.90077
[11] Kornbluth, J. S. H.; Steuer, R. E.: Multiple objective linear fractional programming. Management sci. 27, 1024-1039 (1981) · Zbl 0467.90064
[12] Lai, Y. J.: Hierarchical optimizationa satisfactory solution. Fuzzy sets and systems 77, 321-335 (1996) · Zbl 0869.90042
[13] Luhandjula, M. K.: Fuzzy approaches for multiple objective linear fractional optimization. Fuzzy sets and systems 13, 11-23 (1984) · Zbl 0546.90094
[14] Marccote, P.: Network design problem with congestion effects: a case of bilevel programming. Math. programming 34, 142-162 (1986) · Zbl 0604.90053
[15] Sakawa, M.: Fuzzy sets and interactive multiobjective optimization. (1993) · Zbl 0842.90070
[16] Sakawa, M.; Nishizaki, I.; Uemura, Y.: Interactive fuzzy programming for multi-level linear programming problems. Comput. math. Appl. 36, 71-86 (1998) · Zbl 0937.90123
[17] Sakawa, M.; Nishizaki, I.; Uemura, Y.: Interactive fuzzy programming for multi-level linear programming problems with fuzzy parameters. Fuzzy sets and systems 109, 3-19 (2000) · Zbl 0956.90063
[18] Sakawa, M.; Yano, H.: An interactive fuzzy satisficing method for multiobjective linear fractional programming problems. Fuzzy sets and systems 28, 129-144 (1988) · Zbl 0654.90089
[19] Sakawa, M.; Yumine, T.: Interactive fuzzy decision-making for multiobjective linear fractional programming problems. Large scale systems 5, 105-113 (1983) · Zbl 0533.90085
[20] Shih, H. S.; Lai, Y. J.; Lee, E. S.: Fuzzy approach for multi-level programming problems. Comput. oper. Res. 23, 73-91 (1996) · Zbl 0838.90140
[21] Shimizu, K.; Ishizuka, Y.; Bard, J. F.: Nondifferentiable and two-level mathematical programming. (1997) · Zbl 0878.90088
[22] Steuer, R. E.: Multiple criteria optimization: theory, computation, and application. (1986) · Zbl 0663.90085
[23] White, D. J.; Anandalingam, G.: A penalty function approach for solving bi-level linear programs. J. global optim. 3, 397-419 (1993) · Zbl 0791.90047
[24] Zimmermann, H. -J.: Fuzzy programming and linear programming with several objective functions. Fuzzy sets and systems 1, 45-55 (1978) · Zbl 0364.90065