On bilevel multi-follower decision making: General framework and solutions

Within the framework of any bilevel decision problem, a leader’s decision is influenced by the reaction of his or her follower. When multiple followers who may have had a share in decision variables, objectives and constraints are involved in a bilevel decision problem, the leader’s decision will be affected, not only by the reactions of these followers, but also by the relationships among these followers. This paper firstly identifies nine different kinds of relationships (S1 to S9) amongst followers by establishing a general framework for bilevel multi-follower decision problems. For each of the nine a corresponding bilevel multi-follower decision model is then developed. Also, this paper particularly proposes related theories focusing on an uncooperative decision problem (i.e., S1 model), as this model is the most basic one for bilevel multi-follower decision problems over the nine kinds of relationships. Moreover, this paper extends the Kuhn–Tucker approach for driving an optimal solution from the uncooperative decision model. Finally, a real case study of a road network problem illustrates the application of the uncooperative bilevel decision model and the proposed extended Kuhn–Tucker approach.