Nonlinear optimization problem subjected to fuzzy relational equations defined by Dubois-Prade family of t-norms

Abstract

Abstract In fuzzy set theory, triangular norms (t-norm for short) and triangular co-norms (t-conorm for short) play a key role by providing generic models for intersection and union operations on fuzzy sets. Various continuous and discontinuous t-norms have been proposed by many authors. Despite variation in the t-norms, most of the well-known continuous t-norms are Archimedean (for example, Frank, Yager, Hamacher, Sugeno-Weber and Schweizer-Sklar family). An interesting family of non-Archimedean continuous t-norms was introduced by Dubois and Prade. This paper is an attemp to study a nonlinear optimization problem whose constraints are formed as a special system of fuzzy relational equations (FRE). In this type of constraint, FREs are defined with max-Dubois-Prade composition. Firstly, we investigate the resolution of the feasible solutions set. Then, some necessary and sufficient conditions are presented to determine the feasibility or infeasibility of the solutions set. Also, some procedures are introduced for simplifying the problem. Since the feasible solutions sets of FREs are non-convex, conventional nonlinear programming methods may not be directly employed to solve the problem. Therefore, in order to overcome this difficulty, a genetic algorithm (GA) is designed based on some theoretical properties of the problem. It is shown that the proposed algorithm preserves the feasibility of new generated solutions. Moreover, a method is presented to generate feasible max-Dubois-Prade FREs as test problems. These test problems are used to evaluate the performance of our algorithm. Finally, the algorithm are compared with some related works. The obtained results confirm the high performance of the proposed algorithm in solving such nonlinear problems.