Interactive fuzzy random two-level linear programming through fractile criterion optimization

In this paper, assuming cooperative behavior of the decision makers, solution methods for decisionmaking problems in hierarchical organizations under fuzzy randomenvironments are considered. To deal with the formulated two-level linear programming problems involving fuzzy random variables, α-level sets of fuzzy random variables are introduced and an α-stochastic two-level linear programming problem is defined for guaranteeing the degree of realization of the problem. Taking into account the vagueness of judgments of decision makers, fuzzy goals are introduced and the α-stochastic two-level linear programmingproblem is transformed into the problem tomaximize the satisfaction degree for each fuzzy goal. Through the use of the fractile criterion optimization model, the transformed stochastic two-level programming problem can be reduced to a deterministic one. Interactive fuzzy programming to obtain a satisfactory solution for the decisionmaker at the upper level in consideration of the cooperative relation between decision makers is presented. It is shown that all of the problems to be solved in the proposed interactive fuzzy programming can be easily solved by the simplex method, the sequential quadratic programming or the combined use of the bisection method and the sequential quadratic programming. An illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed method. © 2011 Elsevier Ltd. All rights reserved.