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Discrete representation of non-dominated sets in multi-objective linear programming

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  • Shao, Lizhen
  • Ehrgott, Matthias

Abstract

In this paper we address the problem of representing the continuous but non-convex set of non-dominated points of a multi-objective linear programme by a finite subset of such points. We prove that a related decision problem is NP-complete. Moreover, we illustrate the drawbacks of the known global shooting, normal boundary intersection and normal constraint methods concerning the coverage error and uniformity level of the representation by examples. We propose a method which combines the global shooting and normal boundary intersection methods. By doing so, we overcome their limitations, but preserve their advantages. We prove that our method computes a set of evenly distributed non-dominated points for which the coverage error and the uniformity level can be guaranteed. We apply this method to an optimisation problem in radiation therapy and present illustrative results for some clinical cases. Finally, we present numerical results on randomly generated examples.

Suggested Citation

  • Shao, Lizhen & Ehrgott, Matthias, 2016. "Discrete representation of non-dominated sets in multi-objective linear programming," European Journal of Operational Research, Elsevier, vol. 255(3), pages 687-698.
    • Handle: RePEc:eee:ejores:v:255:y:2016:i:3:p:687-698
    DOI: 10.1016/j.ejor.2016.05.001
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    1. S. Ruzika & M. M. Wiecek, 2005. "Approximation Methods in Multiobjective Programming," Journal of Optimization Theory and Applications, Springer, vol. 126(3), pages 473-501, September.
    2. Lizhen Shao & Matthias Ehrgott, 2008. "Approximately solving multiobjective linear programmes in objective space and an application in radiotherapy treatment planning," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 68(2), pages 257-276, October.
    3. Stacey Faulkenberg & Margaret Wiecek, 2012. "Generating equidistant representations in biobjective programming," Computational Optimization and Applications, Springer, vol. 51(3), pages 1173-1210, April.
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    3. Angelo Aliano Filho & Antonio Carlos Moretti & Margarida Vaz Pato & Washington Alves Oliveira, 2021. "An exact scalarization method with multiple reference points for bi-objective integer linear optimization problems," Annals of Operations Research, Springer, vol. 296(1), pages 35-69, January.
    4. Oylum S¸eker & Mucahit Cevik & Merve Bodur & Young Lee & Mark Ruschin, 2023. "A Multiobjective Approach for Sector Duration Optimization in Stereotactic Radiosurgery Treatment Planning," INFORMS Journal on Computing, INFORMS, vol. 35(1), pages 248-264, January.
    5. Bazgan, Cristina & Jamain, Florian & Vanderpooten, Daniel, 2017. "Discrete representation of the non-dominated set for multi-objective optimization problems using kernels," European Journal of Operational Research, Elsevier, vol. 260(3), pages 814-827.
    6. Eichfelder, Gabriele & Warnow, Leo, 2023. "Advancements in the computation of enclosures for multi-objective optimization problems," European Journal of Operational Research, Elsevier, vol. 310(1), pages 315-327.
    7. Breedveld, Sebastiaan & Craft, David & van Haveren, Rens & Heijmen, Ben, 2019. "Multi-criteria optimization and decision-making in radiotherapy," European Journal of Operational Research, Elsevier, vol. 277(1), pages 1-19.
    8. Ali Najmi & Taha H. Rashidi & James Vaughan & Eric J. Miller, 2020. "Calibration of large-scale transport planning models: a structured approach," Transportation, Springer, vol. 47(4), pages 1867-1905, August.
    9. E. Filatovas & O. Kurasova & J. L. Redondo & J. Fernández, 2020. "A reference point-based evolutionary algorithm for approximating regions of interest in multiobjective problems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(2), pages 402-423, July.
    10. Kuan-Min Lin & Matthias Ehrgott & Andrea Raith, 2017. "Integrating column generation in a method to compute a discrete representation of the non-dominated set of multi-objective linear programmes," 4OR, Springer, vol. 15(4), pages 331-357, December.
    11. Doğan, Ilgın & Lokman, Banu & Köksalan, Murat, 2022. "Representing the nondominated set in multi-objective mixed-integer programs," European Journal of Operational Research, Elsevier, vol. 296(3), pages 804-818.

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