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A fresh view on the Discrete Ordered Median Problem based on partial monotonicity

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  • Marín, Alfredo
  • Ponce, Diego
  • Puerto, Justo

Abstract

This paper presents new results for the Discrete Ordered Median Problem (DOMP). It exploits properties of k-sum optimization to derive specific formulations for the monotone DOMP (MDOMP), that arises when the λ weights are non-decreasing monotone, and new formulations for the general non-monotone DOMP. The main idea in our approach is to express ordered weighted averages as telescopic sums whose terms are k-sums, with positive and negative coefficients. Formulations of k-sums with positive coefficients derive from the linear programming representations obtained by Ogryczack and Tamir (2003) and Blanco, Ali, and Puerto (2014). Valid formulations for k-sums with negative coefficients are more elaborated and we present 4 different approaches, all of them based on mixed integer programming formulations. An extensive computational experience based on a collection of well-known instances shows the usefulness of the new formulations to solve difficult problems such as trimmed and anti-trimmed mean.

Suggested Citation

  • Marín, Alfredo & Ponce, Diego & Puerto, Justo, 2020. "A fresh view on the Discrete Ordered Median Problem based on partial monotonicity," European Journal of Operational Research, Elsevier, vol. 286(3), pages 839-848.
    • Handle: RePEc:eee:ejores:v:286:y:2020:i:3:p:839-848
    DOI: 10.1016/j.ejor.2020.04.023
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    References listed on IDEAS

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    1. Espejo, Inmaculada & Marín, Alfredo & Rodríguez-Chía, Antonio M., 2012. "Closest assignment constraints in discrete location problems," European Journal of Operational Research, Elsevier, vol. 219(1), pages 49-58.
    2. J. Puerto & A. M. Rodríguez-Chía & A. Tamir, 2009. "Minimax Regret Single-Facility Ordered Median Location Problems on Networks," INFORMS Journal on Computing, INFORMS, vol. 21(1), pages 77-87, February.
    3. Nickel, Stefan & Velten, Sebastian, 2017. "Optimization problems with flexible objectives: A general modeling approach and applications," European Journal of Operational Research, Elsevier, vol. 258(1), pages 79-88.
    4. Justo Puerto, 2008. "A New Formulation of the Capacitated Discrete Ordered Median Problems with {0, 1}-Assignment," Operations Research Proceedings, in: Jörg Kalcsics & Stefan Nickel (ed.), Operations Research Proceedings 2007, pages 165-170, Springer.
    5. Samuel Deleplanque & Martine Labbé & Diego Ponce & Justo Puerto, 2020. "A Branch-Price-and-Cut Procedure for the Discrete Ordered Median Problem," INFORMS Journal on Computing, INFORMS, vol. 32(3), pages 582-599, July.
    6. Jörg Kalcsics & Stefan Nickel & Justo Puerto & Antonio Rodríguez-Chía, 2010. "The ordered capacitated facility location problem," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 18(1), pages 203-222, July.
    7. Víctor Blanco, 2019. "Ordered p-median problems with neighbourhoods," Computational Optimization and Applications, Springer, vol. 73(2), pages 603-645, June.
    8. Alfredo Marín & Stefan Nickel & Sebastian Velten, 2010. "An extended covering model for flexible discrete and equity location problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 71(1), pages 125-163, February.
    9. Olender, Paweł & Ogryczak, Włodzimierz, 2019. "A revised Variable Neighborhood Search for the Discrete Ordered Median Problem," European Journal of Operational Research, Elsevier, vol. 274(2), pages 445-465.
    10. Sourour Elloumi & Martine Labbé & Yves Pochet, 2004. "A New Formulation and Resolution Method for the p-Center Problem," INFORMS Journal on Computing, INFORMS, vol. 16(1), pages 84-94, February.
    11. J. Puerto, 2020. "An exact completely positive programming formulation for the discrete ordered median problem: an extended version," Journal of Global Optimization, Springer, vol. 77(2), pages 341-359, June.
    12. Victor Blanco & Justo Puerto & Safae El Haj Ben Ali, 2014. "Revisiting several problems and algorithms in continuous location with $$\ell _\tau $$ ℓ τ norms," Computational Optimization and Applications, Springer, vol. 58(3), pages 563-595, July.
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    Cited by:

    1. Calvino, José J. & López-Haro, Miguel & Muñoz-Ocaña, Juan M. & Puerto, Justo & Rodríguez-Chía, Antonio M., 2022. "Segmentation of scanning-transmission electron microscopy images using the ordered median problem," European Journal of Operational Research, Elsevier, vol. 302(2), pages 671-687.
    2. Blanco, Víctor & Gázquez, Ricardo & Ponce, Diego & Puerto, Justo, 2023. "A branch-and-price approach for the continuous multifacility monotone ordered median problem," European Journal of Operational Research, Elsevier, vol. 306(1), pages 105-126.
    3. Francesco Cesarone & Justo Puerto, 2024. "New approximate stochastic dominance approaches for Enhanced Indexation models," Papers 2401.12669, arXiv.org.

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