## Duality in linear programming with fuzzy parameters and matrix games with fuzzy pay-offs.(English)Zbl 1061.90120

Summary: A dual for linear programming problems with fuzzy parameters is introduced and it is shown that a two person zero sum matrix game with fuzzy pay-offs is equivalent to a primal-dual pair of such fuzzy linear programming problems. Further certain difficulties with similar studies reported in the literature are discussed.

### MSC:

 90C70 Fuzzy and other nonstochastic uncertainty mathematical programming 91A80 Applications of game theory 90C05 Linear programming 90C46 Optimality conditions and duality in mathematical programming

### Keywords:

Fuzzy numbers; Fuzzy matrix game; Fuzzy duality
Full Text:

### References:

 [1] Bector, C.R.; Chandra, S., On duality in linear programming under fuzzy environment, Fuzzy sets and systems, 125, 317-325, (2002) · Zbl 1014.90117 [2] Campos, L., Fuzzy linear programming models to solve fuzzy matrix games, Fuzzy sets and systems, 32, 275-289, (1989) · Zbl 0675.90098 [3] Dubois, D.; Prade, H., Ranking fuzzy numbers in the setting of possibility theory, Inform. sci., 30, 183-224, (1983) · Zbl 0569.94031 [4] Hamacher, H.; Leberling, H.; Zimmermann, H.-J., Sensitivity analysis in fuzzy linear programming, Fuzzy sets and systems, 1, 269-281, (1978) · Zbl 0408.90051 [5] Inuiguchi, M.; Ichihashi, H.; Kume, Y., Relationship between modality constrained programming problems and various fuzzy mathematical programming problems, Fuzzy sets and systems, 49, 243-259, (1992) · Zbl 0786.90090 [6] Inuiguchi, M.; Ichihashi, H.; Kume, Y., Some properties of extended fuzzy preference relations using modalities, Inform. sci., 61, 187-209, (1992) · Zbl 0763.90001 [7] Inuiguchi, M.; Ichihashi, H.; Kume, Y., Modality constrained programming problemsa unified approach to fuzzy mathematical programming problems in the setting of possibility theory, Inform. sci., 67, 93-126, (1993) · Zbl 0770.90078 [8] Inuiguchi, M.; Ramik, J.; Tanino, T.; Vlach, M., Satisficing solutions and duality in interval and fuzzy linear programming, Fuzzy sets and systems, 135, 151-177, (2003) · Zbl 1026.90105 [9] Nishizaki, I.; Sakawa, M., Fuzzy and multiobjective games for conflict resolution, (2001), Physica-Verleg Heidelberg · Zbl 0973.91001 [10] Owen, G., Game theory, (1995), Academic Press San Diego · Zbl 0159.49201 [11] Rödder, W.; Zimmermann, H.-J., Duality in fuzzy linear programming, (), 415-429 [12] Werner, B., Interactive multiple objective programming subject to flexible constraints, European J. oper. res., 31, 342-349, (1987) · Zbl 0636.90085 [13] R.R. Yager, Ranking fuzzy subsets over the unit interval, Proc. CDC (1978) 1435-1437. [14] Zimmermann, H.-J., Fuzzy programming and linear programming with several objective function, Fuzzy sets and systems, 1, 45-55, (1978) · Zbl 0364.90065 [15] Zimmermann, H.-J., Fuzzy set theory—its application, (1991), Kluwer Academic Publishers Dordrecht · Zbl 0719.04002
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