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Using the power of ideal solutions: simple proofs of some old and new results in location theory

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  • Frank Plastria

    (Vrije Universiteit Brussel)

Abstract

When all objectives have a common minimum the existence of this ideal solution directly yields all efficient solutions of the corresponding multi-objective problem, as well as all minimising solutions of any positively weighted sum of these objectives. Some of the classical results in location theory are easy consequences of this simple property, while the same methodology also leads to some lesser known or new results.

Suggested Citation

  • Frank Plastria, 2021. "Using the power of ideal solutions: simple proofs of some old and new results in location theory," 4OR, Springer, vol. 19(3), pages 449-467, September.
    • Handle: RePEc:spr:aqjoor:v:19:y:2021:i:3:d:10.1007_s10288-020-00457-8
    DOI: 10.1007/s10288-020-00457-8
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    References listed on IDEAS

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    1. P. L. Papini & J. Puerto, 2005. "Location Problems with Different Norms for Different Points," Journal of Optimization Theory and Applications, Springer, vol. 125(3), pages 673-695, June.
    2. Simone Görner & Christian Kanzow, 2016. "On Newton’s Method for the Fermat–Weber Location Problem," Journal of Optimization Theory and Applications, Springer, vol. 170(1), pages 107-118, July.
    3. Pelegrin, Blas & Michelot, Christian & Plastria, Frank, 1985. "On the uniqueness of optimal solutions in continuous location theory," European Journal of Operational Research, Elsevier, vol. 20(3), pages 327-331, June.
    4. Durier, Roland & Michelot, Christian, 1985. "Geometrical properties of the Fermat-Weber problem," European Journal of Operational Research, Elsevier, vol. 20(3), pages 332-343, June.
    5. Frank Plastria, 2009. "Asymmetric distances, semidirected networks and majority in Fermat–Weber problems," Annals of Operations Research, Springer, vol. 167(1), pages 121-155, March.
    6. Frank Plastria, 2011. "The Weiszfeld Algorithm: Proof, Amendments, and Extensions," International Series in Operations Research & Management Science, in: H. A. Eiselt & Vladimir Marianov (ed.), Foundations of Location Analysis, chapter 0, pages 357-389, Springer.
    7. F. Plastria, 2014. "Improved fixed point optimality conditions for mixed norms minisum location," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(1), pages 170-184, April.
    8. Tsang, Eric W. K., 2014. "Old and New," Management and Organization Review, Cambridge University Press, vol. 10(03), pages 390-390, November.
    9. Matthias Ehrgott, 2005. "Multicriteria Optimization," Springer Books, Springer, edition 0, number 978-3-540-27659-3, June.
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