Solving linear optimization problems subject to bipolar fuzzy relational equalities defined with max-strict compositions.

There have been several studies in the literature that looked at bipolar fuzzy relation equations using three compositions, max-min, max-product, and max-Lukasiewicz. This paper addresses an optimization problem involving a general class of bipolar fuzzy relation equations. A point to be noted is that the constraints are formed as a system of bipolar fuzzy equations, in which the fuzzy compositions are defined by an arbitrary continuous t-norm. The paper shows that a feasible solution set consists of a finite number of closed convex cells whose lower and upper bounds are determined by corner (extreme) points. In addition, it will be illustrated that, in contrast to the (ordinary) fuzzy relational equations, corner points may neither be the minimum (maximum) nor minimal (maximal) solutions to the feasible region. The generality of the problem leads us to divide the constraints into four classes (each with a different structure) and then investigate the specific characteristics of the equations in each class. Additionally, to evaluate the feasibility of the problem in general, some necessary and sufficient conditions are presented. Due to the high computational complexity of the complete solution of FREs, six rules are introduced to reduce the size of the original problem in such a way that the effort to solve the problem is significantly decreased, followed by an algorithm for solving the current optimization problems. Accordingly, it is shown that the problem has a finite number of local optimal solutions and a global optimal solution can always be obtained by choosing a point having the minimum objective value compared to all the local optimal solutions. In the final section, we will examine an example to demonstrate how the algorithm works. • A generalized bipolar system of fuzzy relation equations programming problem has been introduced which is defined with linear objective function, bipolar fuzzy constraints and an arbitrary continuous strict t-norm. • It is proved that the feasible solution set is formed as a union of the finite number of closed convex cells whose lower and upper bounds are determined by corner (extreme) points that may be neither the minimum (maximum) nor minimal (maximal) solutions. • The constraints of the problem are categorized into four different classes, and the special characteristics of the equations in each class are investigated. • Some necessary and sufficient conditions are presented to conceptualize the feasibility of the problem. • Six rules are introduced with the aim of simplifying the original problem. [ABSTRACT FROM AUTHOR]