Stackelberg solutions for fuzzy random bilevel linear programming through level sets and probability maximization

Abstract

This paper considers Stackelberg solutions for bilevel linear programming problems under fuzzy random environments. To deal with the formulated fuzzy random bilevel linear programming problem, α-level sets of fuzzy random variables are introduced and an α-stochastic bilevel linear programming problem is defined for guaranteeing the degree of realization of the problem. Taking into account vagueness of judgments of decision makers, fuzzy goals are introduced and the α-stochastic bilevel linear programming problem is transformed into the problem to maximize the satisfaction degree for each fuzzy goal. Through probability maximization in stochastic programming, the transformed stochastic bilevel programming problem can be reduced to a deterministic bilevel programming problem. An extended concept of Stackelberg solution is introduced and a computational method is also presented. A numerical example is provided to illustrate the proposed method.