Fuzzy Linear Multi-Objective Stochastic Programming Models

Abstract

In this chapter, fuzzy goal programming (FGP) technique is presented to solve fuzzy multi-objective chance constrained programming (CCP) problems having parameters associated with the system constrains following different continuous probability distributions. Also, the parameters of the models are presented in the form of crisp numbers or fuzzy numbers (FNs) or fuzzy random variables (FRVs). In model formulation process, the imprecise probabilistic problem is converted into an equivalent fuzzy programming model by applying CCP methodology and the concept of cuts of FNs, successively. If the parameters of the objectives are in the form of FRVs then expectation model of the objectives are employed to remove the probabilistic nature from multiple objectives. Afterwards, considering the fuzzy nature of the parameters involved with the problem, the model is converted into an equivalent crisp model using two different approaches. The problem can either be decomposed on the basis of tolerance values of the parameters; alternatively, an equivalent deterministic model can be obtained by applying different defuzzification techniques of FNs. In the solution process, the individual optimal value of each objective is found in isolation to construct the fuzzy goals of the objectives. Then the fuzzy goals are transformed into membership goals on the basis of optimum values of each objective. Then priority-based FGP under different priority structures or weighted FGP is used for achievement of the highest membership degree to the extent possible to achieve the ideal point dependent solution in the decision-making context. Finally, several numerical examples considering different types of probability distributions and different forms of FNs are considered to illustrate the developed methodologies elaborately.