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Optimization of fuzzy relation equations with max-product composition. (English) Zbl 1044.90533
Summary: An optimization problem with a linear objective function subject to a system of fuzzy relation equations using max-product composition is considered. Since the feasible domain is non-convex, traditional linear programming methods cannot be applied. We study this problem and capture some special characteristics of its feasible domain and the optimal solutions. Some procedures for reducing the original problem are presented. The problem is transformed into a 0-1 integer program which is then solved by the branch-and-bound method. For illustration purpose, an example of the procedures is provided.

##### MSC:
 90C70 Fuzzy and other nonstochastic uncertainty mathematical programming 90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut
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##### References:
 [1] Adamopoulos, G.I.; Pappis, C.P., Some results on the resolution of fuzzy relation equations, Fuzzy sets and systems, 60, 83-88, (1993) · Zbl 0794.04005 [2] Bellman, R.E.; Zadeh, L.A., Local and fuzzy logics, (), 103-165 · Zbl 0382.03017 [3] Bourke, M.M.; Fisher, D.G., Solution algorithms for fuzzy relational equations with MAX-product composition, Fuzzy sets and systems, 94, 61-69, (1998) · Zbl 0923.04003 [4] Czogala, E.; Drewniak, J.; Pedrycz, W., Fuzzy relation equations on a finite set, Fuzzy sets and systems, 7, 89-101, (1982) · Zbl 0483.04001 [5] Di Nola, A., Relational equations in totally ordered lattices and their complete resolution, J. math. anal. appl., 107, 148-155, (1985) · Zbl 0588.04006 [6] Dubois, D.; Prade, H., New results about properties and semantics of fuzzy set-theoretic operators, (), 59-75 [7] Fang, S.-C.; Li, G., Solving fuzzy relation equations with a linear objective function, Fuzzy sets and systems, 103, 107-113, (1999) · Zbl 0933.90069 [8] Fang, S.-C.; Puthenpura, S., Linear optimization and extensionstheory and algorithms, (1993), Prentice-Hall Englewood Cliffs, NJ [9] Guo, S.Z.; Wang, P.Z.; Di Nola, A.; Sessa, S., Further contributions to the study of finite fuzzy relation equations, Fuzzy sets and systems, 26, 93-104, (1988) · Zbl 0645.04003 [10] Higashi, M.; Klir, G.J., Resolution of finite fuzzy relation equations, Fuzzy sets and systems, 13, 65-82, (1984) · Zbl 0553.04006 [11] G. Li, S.-C. Fang, On the resolution of finite fuzzy relation equations, OR Report No. 322, North Carolina State University, Raleigh, North Carolina, May 1996. [12] Prevot, M., Algorithm for the solution of fuzzy relations, Fuzzy sets and systems, 5, 319-322, (1981) · Zbl 0451.04004 [13] Sanchez, E., Resolution of composite fuzzy relation equations, Inform. and control, 30, 38-48, (1976) · Zbl 0326.02048 [14] Thole, U.; Zimmermann, H.-J.; Zysno, P., On the suitability of minimum and product operators for intersection of fuzzy sets, Fuzzy sets and systems, 2, 167-180, (1979) · Zbl 0408.94030 [15] W.L. Winston, Introduction to Mathematical Programming: Applications and Algorithms, Duxbury Press, Belmont, CA, 1995. · Zbl 0837.90086 [16] Yager, R.R., Some procedures for selecting fuzzy set-theoretic operators, Internat. J. general systems, 8, 235-242, (1982) · Zbl 0488.04005 [17] Zimmermann, H.-J., Fuzzy set theory and its applications, (1991), Kluwer Academic Publishers Boston · Zbl 0719.04002 [18] Zimmermann, H.-J.; Zysno, P., Latent connectives in human decision-making, Fuzzy sets and systems, 4, 37-51, (1980) · Zbl 0435.90009
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