Pareto optimality for bi-level programming problem with fuzzy parameters

Multi-level programming (MLP) approaches are developed to solve decentralized planning problems with multiple decision makers (DMs) in a hierarchical organization. The Bi-level programming (BLP) problem, i.e., a special case of MLP problems with a two level structure, is a set of nested linear optimization problems over polyhedral set of constraints. In this paper, by considering the experts fuzzy understanding of the nature of the parameters in the problem-formulation process, the BLP problems with fuzzy parameters (BLP-FP) are formulated. The fuzzy parameters in the objective functions and in the constraints are characterized by fuzzy numbers. Using the level sets of fuzzy numbers, the corresponding nonfuzzy BLP problems together with an extended Pareto optimality concept are introduced. Also, we propose an algorithm for finding an α-Pareto optimal solution to BLP-FP. In this algorithm, we use the concepts of tolerance membership functions and multiple objective optimization to develop a fuzzy model for solving the BLP-FP. When cooperation is allowed and the two DMs are willing to cooperate, the BLP-FP problem turns into the cooperation problem. In such problem, the concept of cooperative–Pareto optimal solution is introduced. A numerical illustrative example is given to clarify the main results developed in the paper.