optimization of linear objective function subject to fuzzy relation inequalities constraints with max-product composition

in this paper, we study the finitely many constraints of the fuzzyrelation inequality problem and optimize the linear objectivefunction on the region defined by the fuzzy max-product operator.simplification operations have been given to accelerate theresolution of the problem by removing the components having noeffect on the solution process. also, an algorithm and somenumerical and applied examples are presented to abbreviate andillustrate the steps of the problem resolution.