A PSO Approach to Semivectorial Bilevel Programming: Pessimistic, Optimistic and Deceiving Solutions


In a bilevel programming problem, the upper level decision maker (leader) decides first, but he must incorporate the reaction of the lower level decision maker (follower) in his decision. The existence of multiple objectives at the lower level gives rise to a set of lower level efficient solutions for each leader's decision, which poses additional difficulties for the leader to anticipate the follower's reaction. The optimistic approach assumes that the follower accepts any efficient solution, while the pessimistic approach considers that the leader prepares for the worst case. In this work, we first discuss the assumptions and implications of optimistic vs. pessimistic approaches in bilevel problems with multiple objective functions at the lower level (semivectorial bilevel problems) or at both levels. Three types of solutions are analyzed: the optimistic, pessimistic and deceiving solutions, the latter being a new solution concept introduced herein, which represents the worst outcome of a failed optimistic approach (i.e., when the leader believes that the follower will pursue his own interests but the follower does not react accordingly). Then we propose a particle swarm optimization algorithm to semivectorial bilevel problems, which aims to approximate these three types of solutions in a single run. Some experimental results of the algorithm are presented.