A Fuzzy Robust Weighted Approach for Multi-Criteria Bilevel Games

This article proposes a fuzzy robust weighted method (FRWM) for a bilevel game model comprising both multiple leaders and followers (named multi-criteria bilevel game). In this game, each decision maker has several competing objectives and plays a noncooperative multi-criteria Nash game with others at his level. Each decision maker is assumed to be uncertain about the exact weights over his objectives, but these weights belong to a given set. Furthermore, each player proposes an FRWM to address the uncertainty, i.e., each player will minimize his maximum weighted sum objective in which the maximization is in regards to the given weight set. Then, a new equilibrium concept called a fuzzy robust weighted Nash equilibrium (FRWNE) is presented. It can be proven at least one this equilibrium is present even though the weights are infinite. When the weight set of multi-criteria bilevel game is polyhedral, we can obtain an FRWNE by solving a group of mathematical programming problems with equilibrium constraints. We illustrate the usefulness and efficiency of our fuzzy robust weighted approach to a supply chain multi-criteria bilevel competition problem.