A Matrix Approach to Latticized Linear Programming With Fuzzy-Relation Inequality Constraints

This paper investigates the latticized linear programming that is subject to the fuzzy-relation inequality (FRI) constraints with the max-min composition by using the semi-tensor product method, and proposes a matrix approach to this problem. First, the resolution of the FRI is studied, and it is proved that all the minimal solutions and the unique maximum solution are within the finite parameter set solutions. Based on this and using the semi-tensor product, solving FRIs is converted to solving a set of algebraic inequalities, and some new results on the resolution of FRIs are presented. Second, the latticized linear programming that is subject to the FRI constraints is solved by taking the following two key steps: 1) the optimal value is obtained by calculating the minimum value of the objective function among all the minimal solutions to the FRI constraints; and 2) the optimal solution set is obtained by solving the fuzzy-relation equation that is generated by letting the objective function equal to the optimal value. The study of illustrative examples shows that the new results that are obtained in this paper are very effective in solving the latticized linear programming subject to the FRI constraints.