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An integer programming approach for solving the p-dispersion problem

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  • Sayyady, Fatemeh
  • Fathi, Yahya

Abstract

Given a collection of n items (elements) and an associated symmetric distance dij between each pair of items i and j, we seek a subset P of these items (with a given cardinality p) so that the minimum pairwise distance among the selected items is maximized. This problem is known as the max–min diversity problem or the p-dispersion problem, and it is shown to be np-hard. We define a collection of node packing problems associated with each instance of this problem and employ a binary search among these node packing problems to devise an effective procedure for solving the original problem. We employ existing integer programming techniques, i.e., branch-and-bound and strong valid inequalities, to solve these node packing problems. Through a computational experiment we show that this approach can be used to solve relatively large instances of the p-dispersion problem, i.e., instances with more than 1000 items. We also discuss an application of this problem in the context of locating traffic sensors in a highway network.

Suggested Citation

  • Sayyady, Fatemeh & Fathi, Yahya, 2016. "An integer programming approach for solving the p-dispersion problem," European Journal of Operational Research, Elsevier, vol. 253(1), pages 216-225.
    • Handle: RePEc:eee:ejores:v:253:y:2016:i:1:p:216-225
    DOI: 10.1016/j.ejor.2016.02.026
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    References listed on IDEAS

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    1. Prokopyev, Oleg A. & Kong, Nan & Martinez-Torres, Dayna L., 2009. "The equitable dispersion problem," European Journal of Operational Research, Elsevier, vol. 197(1), pages 59-67, August.
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    Cited by:

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    2. Rafael Martí & Francisco Parreño & Jorge Mortes, 2024. "Mathematical models and solving methods for diversity and equity optimization," Journal of Heuristics, Springer, vol. 30(5), pages 291-323, December.
    3. Juan F. Gomez & Anna Martínez-Gavara & Javier Panadero & Angel A. Juan & Rafael Martí, 2024. "A Forward–Backward Simheuristic for the Stochastic Capacitated Dispersion Problem," Mathematics, MDPI, vol. 12(6), pages 1-22, March.
    4. Spiers, Sandy & Bui, Hoa T. & Loxton, Ryan, 2023. "An exact cutting plane method for the Euclidean max-sum diversity problem," European Journal of Operational Research, Elsevier, vol. 311(2), pages 444-454.
    5. Martí, Rafael & Martínez-Gavara, Anna & Pérez-Peló, Sergio & Sánchez-Oro, Jesús, 2022. "A review on discrete diversity and dispersion maximization from an OR perspective," European Journal of Operational Research, Elsevier, vol. 299(3), pages 795-813.
    6. Parreño, Francisco & Álvarez-Valdés, Ramón & Martí, Rafael, 2021. "Measuring diversity. A review and an empirical analysis," European Journal of Operational Research, Elsevier, vol. 289(2), pages 515-532.
    7. Juan F. Gomez & Javier Panadero & Rafael D. Tordecilla & Juliana Castaneda & Angel A. Juan, 2022. "A Multi-Start Biased-Randomized Algorithm for the Capacitated Dispersion Problem," Mathematics, MDPI, vol. 10(14), pages 1-20, July.

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