An efficient genetic algorithm for solving nonlinear optimization problems defined with fuzzy relational equations and max-Lukasiewicz composition

We study a nonlinear optimization problem with a system of fuzzy relational equations as its constraints. We firstly investigate the resolution of the feasible region when it is defined with max-Lukasiewicz composition and present some necessary and sufficient conditions for the feasibility and some procedures for simplifying the problem. Since the feasible solution set of the fuzzy relational equations (FRE) is non-convex and the finding of all minimal solutions is an NP-hard problem, conventional nonlinear programming methods may involve high computational complexity. Based on the theoretical properties of the problem, a genetic algorithm (GA) is presented, which preserves the feasibility of new generated solutions. The proposed GA does not need to initially find the minimal solutions. Also, it does not need to check the feasibility after generating the new solutions. Moreover, we present a method to generate feasible max-Lukasiewicz FREs as test problems for evaluating the performance of our algorithm. The proposed method has been compared with some related works. The obtained results confirm the high performance of the proposed method in solving such nonlinear problems.