An algorithm for optimizing the linear function with fuzzy relation equation constraints regarding max-prod composition

Fuzzy sets as the feasible region for optimization problems is an interesting and on-going research topic [S.C. Fang, G. Li, Solving fuzzy relations equations with a linear objective function, Fuzzy Sets Syst. 103 (1999) 107-113 [7]; J. Lu, S.C. Fang, Solving nonlinear optimization problems with fuzzy relation constraints, Fuzzy Sets Syst. 119 (2001) 1-20 [16]; E. Khorram, A. Ghodpusian, Linear objective function optimization with fuzzy relation constraints regarding max-av composition, Appl. Math. Comput., in press, doi:10.1016/j.amc.2005.04.021]. In this paper, we focus on these kind problems in which the solutions region is the fuzzy relation equation with max-prod composition and the objective function is linear. Whereas, one of the major difficulties in such problems is non-convexity of the feasible region, it is preferable to study these regions in the first step. Hence, we have primarily investigated two methods and their relationship and then we have determined the feasible region via them. After determining the feasible set, we have given an algorithm to optimize the linear objective function on such these regions. Finally, we have presented two examples to illustrate the methods and algorithms.