Separable programming problem with bipolar max-T fuzzy relation equation constraints

Abstract

One of the most important classes of nonlinear programming problems is separable programming problem due to its applications in theory and practice. This paper studies this class of the problems subject to a system of bipolar fuzzy relation equations using the max-T composition operator, where T is a continuous and Archimedean t-norm. Its feasible solution set structure is determined by two vectors called lower and upper bound vector of its feasible domain. It is shown that there exists an optimal solution for the problem with max-continuous and Archimedean t-norm such that its components are the corresponding components of either the lower or upper bound vector. Based on the interesting property, some sufficient conditions are proposed to detect some of its optimal components or one of its optimal solutions. These conditions can reduce the dimensions of the original problem when some its optimal components are determined. The objective function of the reduced problem is equivalently rewritten in a specific form. This form guides us to design a value matrix based on the characteristic matrix and ascending or descending of univariate functions constituting the objective function. An approach is extended on the matrix to find the optimal solution of the (reduced) problem. It is important to note that a number of problems can completely be solved using the sufficient conditions.