Abstract
When multiple followers are involved in a bilevel programming problem, the leader’s decision will be affected by the reactions of these followers. For actual problems, the leader in general cannot obtain complete information from the followers so that he may be risk-averse. Then he would need a safety margin to bound the damage resulting from the undesirable selections of the followers. This situation is called a pessimistic bilevel multi-follower (PBLMF) programming problem. This research considers a partially-shared linear PBLMF programming in which there is a partially-shared variable among the followers. The concept and solution algorithm of such a problem are developed. As an illustration, the partially-shared linear PBLMF programming model is applied to a company making venture investments.
Highlights
Bilevel programming plays an exceedingly important role in different application fields, such as transportation, economics, ecology, engineering and others; see [ ] and the references therein
Most research on bilevel programming focuses on the optimistic formulation
5 Conclusions This study addresses a partially-shared linear pessimistic bilevel multi-follower (PBLMF) programming problem in which there is a partially-shared variable among followers
Summary
Bilevel programming plays an exceedingly important role in different application fields, such as transportation, economics, ecology, engineering and others; see [ ] and the references therein. Proof For each x ∈ S(X) and fixed ρ > , we have sup d T y + d T y + sT z – ρ πi(x, yi, wi, z) It follows from Assumption (A) that the objective function of the linear programming problem Pρ(x) is bounded from above, and it has at least one solution in V (Z (x)) × V (Z ) × V (Z ). To use the proposed algorithm, we can equivalently transform problem ( ) into the following problem: min x ,x max [– x y ,y ,z s.t. x + x ≤ , x , x ≥ , min y ,z Note that both ( ) and ( ) have the same solutions, and their optimal values are negatives of each other.
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