Development of Fuzzy Multi-Objective Stochastic Fractional Programming Models

In this chapter, two methodologies for solving multi-objective linear fractional stochastic programming problems containing fuzzy numbers (FNs) and fuzzy random variables (FRVs) associated with the system constraints are developed. In the model formulation process, the fuzzy probabilistic constraints are converted into equivalent fuzzy constraints by applying chance constrained programming (CCP) technique in a fuzzily defined probabilistic decision-making situation. Then two techniques, -cut and defuzzification methods, are used to convert the model into the corresponding deterministic model. In the method of using -cut for FNs, the tolerance level of FNs is considered, and the constraints are reduced to constraints with interval coefficients. Alternatively, in using defuzzification method, FNs are replaced by their defuzzified values. Consequently, the constraints are modified into constraints in deterministic form. In the next step, the constraints with interval coefficients are customized into its equivalent form by using the convex combination of each interval. If the parameters of the objectives are triangular FNs, then on the basis of their tolerance ranges each objective is decomposed into three objectives with crisp coefficients. Then each objective is solved independently to find their best and worst values and those values are used to construct membership function of each objective. Finally, the compromise solution of multi-objective linear fractional CCP problems is obtained by applying any of the approaches: priority-based fuzzy goal programming (FGP) method, Zimmermann's approach, -connective process, or minimum bounded sum operator technique. To demonstrate the efficiency of the above-described techniques, two illustrative examples, studied previously, are solved, and the solutions are compared with the existing methodology.