A novel approach for solving the fully fuzzy bi-level linear programming problems

Bi-level linear programming (BLP) is a problem with two decision makers and two levels: the Leader in the upper and the Follower in the lower levels. Decision on one level affects the other one. In this respect, finding an optimal solution for BLP problems with inexact parameters and variables (proposed in many real-world applications) is non-convex and very hard to solve regarding its structure. In the present study, Multi-Objective Linear Programming (MOLP) is applied to offer a new approach is proposed to find an optimal fuzzy solution for the BLP problems, in which all parameters and variables have fuzzy nature. The main contribution of this research can be described as follows. First based on lexicographic ordering and using triangular fuzzy numbers, the given fully fuzzy BLP problem is converted into its equivalent multi-objective BLP problem. Then, the lexicographic method is used to solve the obtained model in the previous step. Subsequently, the optimal solution of the multi-objective BLP problem is obtained. However the answer to the main question is given in Theorem 1 if the optimal solution of the multi-objective BLP problem can be considered an optimal solution of the fully fuzzy BLP problem. Finally, to demonstrate the applicability of the proposed approach, it is run to solve some examples, and its results are compared with one of the existing methods.