Interactive Fuzzy Programming for Stochastic Two-level Linear Programming Problems through Probability Maximization

Abstract

ForewordIn this paper, we focus on stochastic two-level linear programming problems involving random variable coefficients both in objective functions and constraints. Using the concept of chance constraints, stochastic constraints are transformed into deterministic ones. Following the probability maximization model, the minimization of each stochastic objective function is replaced with the maximization of the probability that each objective function is less than or equal to a certain value. Under some appropriate assumptions for distribution functions, the formulated stochastic two-level linear programming problems are transformed into deterministic ones. Taking into account vagueness of judgments of the decision makers, we present interactive fuzzy programming. In the proposed interactive method, after determining the fuzzy goals of the decision makers at both levels, a satisfactory solution is derived efficiently by updating the satisfactory degree of the decision maker at the upper level with considerations of overall satisfactory balance among both levels. It should be emphasized here that the transformed deterministic problems for deriving an overall satisfactory solution can be easily solved through the combined use of the bisection method and the phase one of the simplex method. An illustrative numerical example is provided to demonstrate the feasibility of the proposed method.-iii - AbstractThis paper considers stochastic two-level linear programming problems. Using the concept of chance constraints and probability maximization, original problems are transformed into deterministic ones. An interactive fuzzy programming method is presented for deriving a satisfactory solution efficiently with considerations of overall satisfactory balance.

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