A Fuzzy Programming Approach for Bilevel Stochastic Programming

This article presents a fuzzy programming (FP) method for modeling and solving bilevel stochastic decision-making problems involving fuzzy random variables (FRVs) associated with the parameters of the objectives at different hierarchical decision-making units as well as system constraints. In model formulation process, an expectation model is generated first on the basis of the fuzzy random variables involved with the objectives at each level. The problem is then converted into a FP model by considering the fuzzily described chance constraints with the aid of applying chance constrained methodology in a fuzzy context. After that, the model is decomposed on the basis of tolerance ranges of fuzzy numbers associated with the parameters of the problem. To construct the fuzzy goals of the decomposed objectives of both decision-making levels under the extended feasible region defined by the decomposed system constraints, the individual optimal values of each objective at each level are calculated in isolation. Then, the membership functions are formulated to measure the degree of satisfaction of each decomposed objectives in both the levels. In the solution process, the membership functions are converted into membership goals by assigning unity as the aspiration level to each of them. Finally, a fuzzy goal programming model is developed to achieving the highest membership degree to the extent possible by minimizing the under deviational variables of the membership goals of the objectives of the decision makers (DMs) in a hierarchical decision-making environment. To expound the application potentiality of the approach, a numerical example is solved.