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The maximum dispersion problem

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  • Fernández, Elena
  • Kalcsics, Jörg
  • Nickel, Stefan

Abstract

In the maximum dispersion problem, a given set of objects has to be partitioned into a number of groups. Each object has a non-negative weight and each group has a target weight, which may be different for each group. In addition to meeting the target weight of each group, all objects assigned to the same group should be as dispersed as possible with respect to some distance measure between pairs of objects. Potential applications for this problem come from such diverse fields as the problem of creating study groups or the design of waste collection systems. We develop and compare two different (mixed-) integer linear programming formulations for the problem. We also study a specific relaxation that enables us to derive tight bounds that improve the effectiveness of the formulations. Thereby, we obtain an upper bound by finding in an auxiliary graph subsets of given size with minimal diameter. A lower bound is derived based on the relation of the optimal solution of the relaxation to the chromatic number of a series of auxiliary graphs. Finally, we propose an exact solution scheme for the maximum dispersion problem and present extensive computational experiments to assess its efficiency.

Suggested Citation

  • Fernández, Elena & Kalcsics, Jörg & Nickel, Stefan, 2013. "The maximum dispersion problem," Omega, Elsevier, vol. 41(4), pages 721-730.
    • Handle: RePEc:eee:jomega:v:41:y:2013:i:4:p:721-730
    DOI: 10.1016/j.omega.2012.09.005
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    References listed on IDEAS

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    1. Z P Fan & Y Chen & J Ma & S Zeng, 2011. "A hybrid genetic algorithmic approach to the maximally diverse grouping problem," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 62(1), pages 92-99, January.
    2. Dmitry Krass & Anton Ovchinnikov, 2006. "The University of Toronto’s Rotman School of Management Uses Management Science to Create MBA Study Groups," Interfaces, INFORMS, vol. 36(2), pages 126-137, April.
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    4. 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.
    5. Z P Fan & Y Chen & J Ma & S Zeng, 2011. "Erratum: A hybrid genetic algorithmic approach to the maximally diverse grouping problem," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 62(7), pages 1423-1430, July.
    6. Jörg Kalcsics & Stefan Nickel & Michael Schröder, 2005. "Towards a unified territorial design approach — Applications, algorithms and GIS integration," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 13(1), pages 1-56, June.
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    Cited by:

    1. Lei, Ting L. & Church, Richard L., 2015. "On the unified dispersion problem: Efficient formulations and exact algorithms," European Journal of Operational Research, Elsevier, vol. 241(3), pages 622-630.
    2. Arne Schulz, 2022. "A new mixed-integer programming formulation for the maximally diverse grouping problem with attribute values," Annals of Operations Research, Springer, vol. 318(1), pages 501-530, November.
    3. Ríos-Mercado, Roger Z. & Bard, Jonathan F., 2019. "An exact algorithm for designing optimal districts in the collection of waste electric and electronic equipment through an improved reformulation," European Journal of Operational Research, Elsevier, vol. 276(1), pages 259-271.
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    5. Gliesch, Alex & Ritt, Marcus, 2021. "A hybrid heuristic for the maximum dispersion problem," European Journal of Operational Research, Elsevier, vol. 288(3), pages 721-735.

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