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On linear bilevel problems with multiple objectives at the lower level

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  • Calvete, Herminia I.
  • Galé, Carmen

Abstract

Bilevel programming problems provide a framework to deal with decision processes involving two decision makers with a hierarchical structure. They are characterized by the existence of two optimization problems in which the constraint region of the upper level problem is implicitly determined by the lower level optimization problem. This paper focuses on bilevel problems for which the lower level problem is a linear multiobjective program and constraints at both levels define polyhedra. This bilevel problem is reformulated as an optimization problem over a nonconvex region given by a union of faces of the polyhedron defined by all constraints. This reformulation is obtained when dealing with efficient solutions as well as weakly efficient solutions for the lower level problem. Assuming that the upper level objective function is quasiconcave, then an extreme point exists which solves the problem. An exact and a metaheuristic algorithm are developed and their performance is analyzed and compared.

Suggested Citation

  • Calvete, Herminia I. & Galé, Carmen, 2011. "On linear bilevel problems with multiple objectives at the lower level," Omega, Elsevier, vol. 39(1), pages 33-40, January.
    • Handle: RePEc:eee:jomega:v:39:y:2011:i:1:p:33-40
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    References listed on IDEAS

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    1. Bard, Jonathan F, 1983. "Coordination of a multidivisional organization through two levels of management," Omega, Elsevier, vol. 11(5), pages 457-468.
    2. Wayne F. Bialas & Mark H. Karwan, 1984. "Two-Level Linear Programming," Management Science, INFORMS, vol. 30(8), pages 1004-1020, August.
    3. Calvete, Herminia I. & Gale, Carmen & Mateo, Pedro M., 2008. "A new approach for solving linear bilevel problems using genetic algorithms," European Journal of Operational Research, Elsevier, vol. 188(1), pages 14-28, July.
    4. Kunsch, P.L. & Kavathatzopoulos, I. & Rauschmayer, F., 2009. "Modelling complex ethical decision problems with operations research," Omega, Elsevier, vol. 37(6), pages 1100-1108, December.
    5. Ankhili, Z. & Mansouri, A., 2009. "An exact penalty on bilevel programs with linear vector optimization lower level," European Journal of Operational Research, Elsevier, vol. 197(1), pages 36-41, August.
    6. H. Bonnel & J. Morgan, 2006. "Semivectorial Bilevel Optimization Problem: Penalty Approach," Journal of Optimization Theory and Applications, Springer, vol. 131(3), pages 365-382, December.
    7. Walker, Warren E., 2009. "Does the best practice of rational-style model-based policy analysis already include ethical considerations?," Omega, Elsevier, vol. 37(6), pages 1051-1062, December.
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    Cited by:

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    2. Kahina Ghazli & Nicolas Gillis & Mustapha Moulaï, 2020. "Optimizing over the properly efficient set of convex multi-objective optimization problems," Annals of Operations Research, Springer, vol. 295(2), pages 575-604, December.
    3. Yunjia Ma & Wei Xu & Lianjie Qin & Xiujuan Zhao, 2019. "Site Selection Models in Natural Disaster Shelters: A Review," Sustainability, MDPI, vol. 11(2), pages 1-24, January.
    4. Xiang Li & Tiesong Hu & Xin Wang & Ali Mahmoud & Xiang Zeng, 2023. "The New Solution Concept to Ill-Posed Bilevel Programming: Non-Antagonistic Pessimistic Solution," Mathematics, MDPI, vol. 11(6), pages 1-13, March.
    5. Dempe, S., 2011. "Comment to "interactive fuzzy goal programming approach for bilevel programming problem" by S.R. Arora and R. Gupta," European Journal of Operational Research, Elsevier, vol. 212(2), pages 429-431, July.
    6. Alves, Maria João & Henggeler Antunes, Carlos, 2022. "A new exact method for linear bilevel problems with multiple objective functions at the lower level," European Journal of Operational Research, Elsevier, vol. 303(1), pages 312-327.
    7. Maria João Alves & Carlos Henggeler Antunes & João Paulo Costa, 2021. "New concepts and an algorithm for multiobjective bilevel programming: optimistic, pessimistic and moderate solutions," Operational Research, Springer, vol. 21(4), pages 2593-2626, December.
    8. Calvete, Herminia I. & Galé, Carmen & Iranzo, José A., 2013. "An efficient evolutionary algorithm for the ring star problem," European Journal of Operational Research, Elsevier, vol. 231(1), pages 22-33.
    9. Lin, Rung-Chuan & Sir, Mustafa Y. & Pasupathy, Kalyan S., 2013. "Multi-objective simulation optimization using data envelopment analysis and genetic algorithm: Specific application to determining optimal resource levels in surgical services," Omega, Elsevier, vol. 41(5), pages 881-892.
    10. Tilahun, Surafel Luleseged, 2019. "Feasibility reduction approach for hierarchical decision making with multiple objectives," Operations Research Perspectives, Elsevier, vol. 6(C).
    11. Li, Deng-Feng, 2011. "Linear programming approach to solve interval-valued matrix games," Omega, Elsevier, vol. 39(6), pages 655-666, December.

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