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A linear ordering problem of sets

Author

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  • Juan Aparicio

    (University Miguel Hernandez of Elche)

  • Mercedes Landete

    (University Miguel Hernandez of Elche)

  • Juan F. Monge

    (University Miguel Hernandez of Elche)

Abstract

Given a ranking of elements of a set and given a disjoint partition of the same set, the ranking does not generally imply a total order of the partition. In this paper, we introduce the Kendall-$$\tau $$τpartition ranking, a linear order of the subsets of the partition which follows from the given ranking. We prove that, under certain assumptions, the Kendall-$$\tau $$τ partition ranking is robust, in the sense that it remains the same when removing subsets of the partition. Then, we give several results (properties) concerning the adequacy of the ranking for ordering the partition and we prove that the integrality gap of the 0–1 problem associated to the Kendall-$$\tau $$τ partition ranking tends to 8/7. Additionally, we provide a comparison of the new ranking with respect to the mean and median based scores from a theoretical and empirical point of view. Finally, a real application with data from the Programme for International Student Assessment is presented, where countries are ordered based on their school rankings.

Suggested Citation

  • Juan Aparicio & Mercedes Landete & Juan F. Monge, 2020. "A linear ordering problem of sets," Annals of Operations Research, Springer, vol. 288(1), pages 45-64, May.
    • Handle: RePEc:spr:annopr:v:288:y:2020:i:1:d:10.1007_s10479-019-03473-y
    DOI: 10.1007/s10479-019-03473-y
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    References listed on IDEAS

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    1. Fred Glover & T. Klastorin & D. Kongman, 1974. "Optimal Weighted Ancestry Relationships," Management Science, INFORMS, vol. 20(8), pages 1190-1193, April.
    2. Hudry, Olivier, 2010. "On the complexity of Slater's problems," European Journal of Operational Research, Elsevier, vol. 203(1), pages 216-221, May.
    3. Rahmaniani, Ragheb & Crainic, Teodor Gabriel & Gendreau, Michel & Rei, Walter, 2017. "The Benders decomposition algorithm: A literature review," European Journal of Operational Research, Elsevier, vol. 259(3), pages 801-817.
    4. Irène Charon & Olivier Hudry, 2010. "An updated survey on the linear ordering problem for weighted or unweighted tournaments," Annals of Operations Research, Springer, vol. 175(1), pages 107-158, March.
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    Cited by:

    1. Manuel V. C. Vieira, 2024. "A linear ordering problem with weighted rank," Journal of Combinatorial Optimization, Springer, vol. 47(2), pages 1-24, March.
    2. Labbé, Martine & Landete, Mercedes & Monge, Juan F., 2023. "Bilevel integer linear models for ranking items and sets," Operations Research Perspectives, Elsevier, vol. 10(C).

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