An illustration of different concepts of solutions in semivectorial bilevel programming

Bilevel programming deals with hierarchical decision processes with two decision levels, in which the upper level (leader) and the lower level (follower) decision makers control different sets of variables and pursue different objective functions. The problem is even more complicated when multiple objective functions are considered in the lower level (semivectorial bilevel problem) since a set of efficient solutions for each upper level decision exists. This paper presents and illustrates two novel types of solutions (deceiving and rewarding) in addition to the optimistic and pessimistic solutions to the semivectorial bilevel problem. These four solutions represent possible "extreme" outcomes of the decision process, capturing distinct leader's stances and follower's reactions.