Solving a non-convex non-linear optimization problem constrained by fuzzy relational equations and Sugeno-Weber family of t-norms

Sugeno-Weber family of t-norms and t-conorms is one of the most applied one in various fuzzy modelling problems. This family of t-norms and t-conorms was suggested by Weber for modeling intersection and unio‎n of fuzzy sets. Also, the t-conorms were suggested as addition rules by Sugeno for so-called {fuzzy measures. In this paper, we study a nonlinear optimization problem where the feasible region is formed as a system of fuzzy relational equations (FRE) dened by the Sugeno-Weber t-norm. We rstly investigate the resolution of the feasible region when it is dened with max-Sugeno-Weber composition and present some necessary and sucient conditions for determining the feasibility of the problem. Also, two procedures are presented for simplifying the problem. Since the feasible solutions set of FREs is non-convex and the nding of all minimal solutions is an NP-hard problem, conventional nonlinear programming methods may not be directly employed. For these reasons, a genetic algorithm is presented, which preserves the feasibility of new generated solutions. The proposed GA does not need to initially nd the minimal solutions. Also, it does not need to check the feasibility after generating the new solutions. Additionally, we propose a method to generate feasible max-Sugeno-Weber FREs as test problems for evaluating the performance of our algorithm. The proposed method has been compared with some related works. The obtained results conrm the high performance of the proposed method in solving such nonlinear problems.