Fuzzy random bilevel linear programming through expectation optimization using possibility and necessity

In this paper, assuming noncooperative behavior of the decision makers, solution methods for decision-making problems in hierarchical organizations under fuzzy random environments are presented. Taking into account the vagueness of judgments of decision makers, fuzzy goals are introduced into the formulated fuzzy random noncooperative bilevel linear programming problems. Considering the possibility and necessity measure that each objective function fulfills, the corresponding fuzzy goal, the fuzzy random bilevel linear programming problems to minimize each objective function with fuzzy random variables, are transformed into stochastic bilevel programming problems to maximize the degree of possibility and necessity that each fuzzy goal is fulfilled. Through expectation optimization in stochastic programming, which is suitable for risk-neutral decision makers, the transformed stochastic bilevel programming problems can be reduced to deterministic bilevel programming problems. For the transformed problems, extended concepts of Stackelberg solutions are introduced and computational methods are also presented. It is shown that the extended Stackelberg solutions can be obtained through the combined use of the variable transformation method and the Kth best algorithm for bilevel linear programming problems. A numerical example is provided to illustrate the proposed methods.