Geometric optimization provided to mixed fuzzy relation inequality constraints

In this paper, the problem of minimizing a geometric function subject to the Mixed Fuzzy Relation Inequalities (MFRI) using two operators of max-product and max-average is studied. The structure of feasible domain of the problem is firstly investigated using Mixed Fuzzy Relation Inequality Paths (MFRIP). Its feasible solution set is completely determined by a maximum solution and a finite number of minimal solutions. Then, two cases are considered for the above problem. In each case, the problem is decomposed to two sub-problems. We show that the optimal solutions of the sub-problems are the maximum solution and one of the minimal solutions. Using the fact, it is shown that the optimal solution of the original problem consists of a combination of two above solutions. Also, some rules are presented to simplify the problem using the properties of MFRIP. With regard to the above points, an algorithm is designed to solve the problem.