Fuzzy programming for large-scale multiobjective linear programming problems with block angular structure

Abstract

Since Dantzig and Wolfe proposed the decomposition principle for the large-scale linear programming problem with the block angular structure in the early 1960's, there have been made a large number of intensive studies for the large-scale mathematical programming problems with special structures. Considering the fuzziness of the human decision, however, it seems more adequate to assume that the decisionmaker has the fuzzy goals or fuzzy constraints for the objective functions and for the coupled constraints in the problem formulated as the large-scale linear programming problem with the block angular structure. From such a viewpoint, this paper focuses on the large-scale multiobjective linear programming problem and proposes a new method to derive easily the satisficing solution for the decisionmaker considering his/her fuzzy goal. In the proposed method, the fuzzy goal of the decisionmaker for each objective function is specified by the linear membership function through the interaction with the decisionmaker. Fuzzy decision is applied to integrate the fuzzy goals. Then the original problem is decomposed into a master problem and several independent small-scale linear programming subproblems. The Dantzig-Wolfe decomposition principle is applied, where the optimal solution as a whole is obtained through iterations, by updating the goal function of each subproblem according to the solution of the master problem. Thus, it is shown that the satisficing solution is obtained relatively easily considering the fuzzy goals of the decisionmaker.