Linear programming problem subject to bipolar max-product fuzzy relation equations with product negation

In this paper, we study the linear optimization problem with bipolar max-product fuzzy relation equations and product negation. First, we investigate the structure of the solution set for its feasible domain and specify the complete solution set without explicit computation of all of its maximal and minimal solutions under a sufficient condition. In a general case, the complete solution set cannot be completely determined by a finite number of pairs of minimal and maximal solution. Hence, lower and upper bound vectors are presented for its solution set. Some sufficient conditions are also proposed to reduce the dimensions of its feasible domain. We show that each binding variable from a feasible vector can be expressed in terms of the corresponding component of the lower or upper bound vectors. Some sufficient conditions are proposed so an optimal solution for the problem exists such that each of its components is the component corresponding to the lower and upper bound vectors. A value matrix is created based on the characteristic matrix, coefficients of the objective function, and recent property. Some rules are presented to reduce the dimensions of the matrix. A modified branch-and-bound algorithm is then applied to the matrix to find the optimal solution of the problem without detecting the complete solution set of its feasible domain and comparing them. The algorithm decreases the computational complexity considerably compared with the existing algorithms for solving the problem.