Bilevel linear programming with lower-level fuzzy objective function

Bilevel linear programming (BLP) is a solution method for linear optimization problem with two sequential decision steps of the leader and the follower. In this paper, we assume that the follower's objective function is imprecise and can be represented by a fuzzy function, the BLP with the follower's fuzzy objective function (BLPwFFO). We apply the approach of necessity measure optimization to obtain the global optimal solution for the leader. This solution is not only secure but comprehensively reflects the follower's rational reaction. In the case that the follower's coefficient vector is defined by the convex polyhedron fuzzy set, our proposed BLPwFFO is formulated as a special kind of three-level programming problem. Because an optimal solution exists at a vertex of feasible region, we use the k-th best method to search for the global optimal solution. The numerical example is used to demonstrate our computational method.