A fuzzy approach for bi-level integer non-linear programming problem

Bi-level programming, a tool for modeling decentralized decisions, consists of the objective of the leader at its first level and that of the follower at the second level. Integer programming deals with the mathematical programming problems in which some or all the variables are to be integer. This paper studies a bi-level integer non-linear programming problem with linear or non-linear constraints, and in which the non-linear objective function at each level is to maximized. The bi-level integer non-linear programming problem can be thought as a static version of the Stackelberg strategy, which is used leader–follower game in which a Stackelberg strategy is used by the leader, or the higher-level decision-maker, given the rational reaction of the follower, or the lower-level decision-maker. This paper proposes a two-planner integer model and a solution method for solving this problem. This method uses the concept of tolerance membership function and the branch and bound technique to develop a fuzzy Max–Min decision model for generating Pareto optimal solution for this problem; an illustrative numerical example is given to demonstrate the obtained results.