A new matrix-based-algorithm for solving latticized linear programming subject to max-min-product fuzzy relation inequalities

This paper introduces a system of Fuzzy Relation Inequalities (FRIs) with the max-min-product composition operator. To determine the structure of solution set of the system, we firstly focus on a single inequality and study its solution set and properties. Then, the structure of solution set of the system is determined by the points. The necessary and sufficient conditions are proposed for its consistency. Some useful properties of the system of the max-min-product FRIs are presented to determine the structure of its minimal solutions. A latticized linear programming problem is proposed with constraints as FRIs using the max-min-product composition. It is shown that one of its optimal solutions can be given in terms of a closed form. Based on the closed form, a matrix-based-algorithm with a polynomial computational complexity is designed to find one of its optimal solutions. A practical example is presented to illustrate the system and the optimization problem in the area of data transmission mechanism.