A covering-based algorithm for resolution of linear programming problems with max-product bipolar fuzzy relation equation constraints

The linear programming problem provided to bipolar fuzzy relation equation constraints is considered in this paper. The structure of bipolar fuzzy relation equation system is studied with the max-product composition. Two new concepts, called covering and irredundant covering, are introduced in the bipolar fuzzy relation equation system. A covering-based sufficient condition is proposed to check its consistency. The relation between two concepts is discussed. Some sufficient conditions are presented to specify one of its optimal solutions or some its optimal components based on the concepts. Also, some covering-based sufficient conditions are given for uniqueness of its optimal solution. These conditions enable us to design some procedures for simplification and reduction of the problem. Moreover, a matrix-based branch-and-bound method is presented to solve the reduced problem. The sufficient conditions and algorithm are illustrated by some numerical examples. The algorithm is compared to existing methods.