Fully Fuzzified Multi-Objective Stochastic Programming

This chapter develops a methodology for solving fully fuzzified multi-objective chance constrained programming (CCP) problems with fuzzy random variables (FRVs) as parameters. In the preceding chapters, it is assumed that the parameters of the multi-objective programming models are uncertain, and these uncertain parameters are expressed through fuzzy numbers (FNs) and FRVs. However, in practical situations, it is also observed that not only the parameters but also the variables of the multi-objective programming problems are uncertain. From that view point, the methodology for solving fully fuzzified multi-objective stochastic programming problems are presented in this chapter. At first the fuzzy probabilistic constraints are modified into fuzzy constraints. Using the defuzzification method of FNs, the different fuzzy parameters and fuzzy variables in the constraints are converted into crisp equivalent parameters and crisp variables. In this chapter, the parameters of the objectives are considered as either symmetric trapezoidal FNs or FRVs whose mean and variances are taken as symmetric trapezoidal FNs. If the parameters of the objectives are FRVs, then expectation model and variance model of the objective is used to find an equivalent form of the objectives whose parameters are only FNs. The ranking function of FNs is then applied to the objectives to convert them into crisp objectives. Then each objective is solved independently under the modified system constraints to construct the membership goals of each objective. Finally, weighted fuzzy goal programming (FGP) model is applied to achieve the most satisfactory solution for the overall benefit of the organization. Two illustrative numerical examples are given to demonstrate the efficiency of the proposed methodology and to compare the solution obtained by the developed methodology with the pre-defined techniques.