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On the Set of Optimal Points to the Weber Problem: Further Results

Author

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  • Roland Durier

    (Laboratoire d'Analyse Numérique, Université de Bourgogne, Dijon, France)

  • Christian Michelot

    (CERMSEM, Université de Paris I, Panthéon-Sorbonne, Paris, France)

Abstract

In a recent paper, Z. Drezner and A. J. Goldman address the problem of determining the smallest set among those containing at least one optimal solution to every Weber problem based on a set of demand points in the plane. In the case of arbitrary mixed gauges (i.e., possibly nonsymmetric norms), the authors have shown that the set of strictly efficient points which are also intersection points always meets the set of Weber solutions. As shown by Drezner and Goldman, this set is optimal with the ℓ 1 or ℓ x distance but is not optimal with the ℓ p distance, 1 p p distance case, we disprove a conjecture of Drezner and Goldman about the possibility of extending their result to more than two dimensions. The paper contains a different view of the problem: whereas Drezner and Goldman use algebraic-analytical approach, the authors use a geometrical approach which permits us to obtain more general results and also clarifies the geometric nature of the problem.

Suggested Citation

  • Roland Durier & Christian Michelot, 1994. "On the Set of Optimal Points to the Weber Problem: Further Results," Transportation Science, INFORMS, vol. 28(2), pages 141-149, May.
    • Handle: RePEc:inm:ortrsc:v:28:y:1994:i:2:p:141-149
    DOI: 10.1287/trsc.28.2.141
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    Citations

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    Cited by:

    1. H. Martini & K.J. Swanepoel & G. Weiss, 2002. "The Fermat–Torricelli Problem in Normed Planes and Spaces," Journal of Optimization Theory and Applications, Springer, vol. 115(2), pages 283-314, November.
    2. Sándor P. Fekete & Joseph S. B. Mitchell & Karin Beurer, 2005. "On the Continuous Fermat-Weber Problem," Operations Research, INFORMS, vol. 53(1), pages 61-76, February.
    3. Brazil, M. & Ras, C.J. & Thomas, D.A., 2014. "A geometric characterisation of the quadratic min-power centre," European Journal of Operational Research, Elsevier, vol. 233(1), pages 34-42.
    4. Ndiaye, M. & Michelot, C., 1998. "Efficiency in constrained continuous location," European Journal of Operational Research, Elsevier, vol. 104(2), pages 288-298, January.
    5. M. Hakan Akyüz & Temel Öncan & İ. Kuban Altınel, 2019. "Branch and bound algorithms for solving the multi-commodity capacitated multi-facility Weber problem," Annals of Operations Research, Springer, vol. 279(1), pages 1-42, August.

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