Minimization of linear objective functions under the constraints expressed by a system of fuzzy relation equations

This paper considers the following optimization problem: minimization of linear objective functions under the constraints expressed by a system of fuzzy relation equations using max-product composition. We first show some properties of minimal solutions of a system of fuzzy relation equations with max-product composition and present some rules for reducing the original problem. Then we derive an algorithm which enables us to find all the optimal solutions for solving the optimization problem and provide some numerical examples to illustrate.