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Covering problems with polyellipsoids: A location analysis perspective

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  • Blanco, Víctor
  • Puerto, Justo

Abstract

In this paper we analyze the extension of the classical smallest enclosing disk problem to the case of the location of a polyellipsoid to fully cover a set of demand points in Rd. We prove that the problem is polynomially solvable in fixed dimension and analyze mathematical programming formulations for it. We also consider some geometric approaches for the problem in case the foci of the polyellipsoids are known. Extensions of the classical algorithm by Elzinga-Hearn are also derived for this new problem. Moreover, we address two extensions of the problem, as the case where the foci of the enclosing polyellipsoid are not given and have to be determined among a potential set of points or the induced covering problems when instead of polyellipsoids, one uses ordered median polyellipsoids. For these problems we also present Mixed Integer (Non) Linear Programming strategies that lead to efficient ways to solve it. Extensive computational experiments on different datasets show the usefulness of our solution methods.

Suggested Citation

  • Blanco, Víctor & Puerto, Justo, 2021. "Covering problems with polyellipsoids: A location analysis perspective," European Journal of Operational Research, Elsevier, vol. 289(1), pages 44-58.
    • Handle: RePEc:eee:ejores:v:289:y:2021:i:1:p:44-58
    DOI: 10.1016/j.ejor.2020.06.048
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    References listed on IDEAS

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    1. Nimrod Megiddo, 1983. "The Weighted Euclidean 1-Center Problem," Mathematics of Operations Research, INFORMS, vol. 8(4), pages 498-504, November.
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    6. Andretta, M. & Birgin, E.G., 2013. "Deterministic and stochastic global optimization techniques for planar covering with ellipses problems," European Journal of Operational Research, Elsevier, vol. 224(1), pages 23-40.
    7. 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. Liu, Yanchao, 2023. "An elliptical cover problem in drone delivery network design and its solution algorithms," European Journal of Operational Research, Elsevier, vol. 304(3), pages 912-925.
    2. Blanco, Víctor & Gázquez, Ricardo & Saldanha-da-Gama, Francisco, 2023. "Multi-type maximal covering location problems: Hybridizing discrete and continuous problems," European Journal of Operational Research, Elsevier, vol. 307(3), pages 1040-1054.

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