A novel approach to pessimistic bilevel problems. An application to the rank pricing problem with ties

Abstract

This paper introduces a novel method to address the pessimistic approach to the bilevel problem. It consists of considering a lexicographic biobjective optimization problem at the lower level. To emphasize the significance of this approach, we implement it in the context of the Rank Pricing Problem with Ties. This problem can be formulated as a bilevel problem that inherently demands the use of the pessimistic approach. Considering the properties of the lexicographic biobjective problem involved, we formulate this problem as a single level mixed integer optimization problem, deriving also valid values for the big-Ms involved and valid inequalities for this formulation. The computational experiment carried out confirms the relevance of the proposed method.

Keywords:

• Bilevel optimization
• pessimistic approach
• lexicographic biobjective optimization
• rank pricing problems

2020 Mathematics Subject Classifications:

• 90C27
• 90C46
• 90C90

Acknowledgments

We are grateful to the referees whose comments have improved the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research has been funded by the Spanish Ministry of Science and Innovation under grants PID2019-104263RB-C43, PID2022-139543OB-C43 and TED2021-130961B-I00, and by the Gobierno de Aragón under grants E41-20R and E41-23R.