IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v288y2020i1d10.1007_s10479-019-03473-y.html
   My bibliography  Save this article

A linear ordering problem of sets

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10479-019-03473-y
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10479-019-03473-y?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. Fred Glover & T. Klastorin & D. Kongman, 1974. "Optimal Weighted Ancestry Relationships," Management Science, INFORMS, vol. 20(8), pages 1190-1193, April.
    3. Hudry, Olivier, 2010. "On the complexity of Slater's problems," European Journal of Operational Research, Elsevier, vol. 203(1), pages 216-221, May.
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Labbé, Martine & Landete, Mercedes & Monge, Juan F., 2023. "Bilevel integer linear models for ranking items and sets," Operations Research Perspectives, Elsevier, vol. 10(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Olivier Hudry & Bernard Monjardet, 2010. "Consensus theories: An oriented survey," Documents de travail du Centre d'Economie de la Sorbonne 10057, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    2. 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.
    3. Olivier Hudry, 2015. "Complexity results for extensions of median orders to different types of remoteness," Annals of Operations Research, Springer, vol. 225(1), pages 111-123, February.
    4. Hudry, Olivier, 2012. "On the computation of median linear orders, of median complete preorders and of median weak orders," Mathematical Social Sciences, Elsevier, vol. 64(1), pages 2-10.
    5. Thierry Denœux & Marie-Hélène Masson, 2012. "Evidential reasoning in large partially ordered sets," Annals of Operations Research, Springer, vol. 195(1), pages 135-161, May.
    6. Aliakbari Sani, Sajad & Bahn, Olivier & Delage, Erick, 2022. "Affine decision rule approximation to address demand response uncertainty in smart Grids’ capacity planning," European Journal of Operational Research, Elsevier, vol. 303(1), pages 438-455.
    7. Maher, Stephen J., 2021. "Implementing the branch-and-cut approach for a general purpose Benders’ decomposition framework," European Journal of Operational Research, Elsevier, vol. 290(2), pages 479-498.
    8. William B. Haskell & Wenjie Huang & Huifu Xu, 2018. "Preference Elicitation and Robust Optimization with Multi-Attribute Quasi-Concave Choice Functions," Papers 1805.06632, arXiv.org.
    9. Clavijo López, Christian & Crama, Yves & Pironet, Thierry & Semet, Frédéric, 2024. "Multi-period distribution networks with purchase commitment contracts," European Journal of Operational Research, Elsevier, vol. 312(2), pages 556-572.
    10. Camilo Ortiz-Astorquiza & Ivan Contreras & Gilbert Laporte, 2019. "An Exact Algorithm for Multilevel Uncapacitated Facility Location," Transportation Science, INFORMS, vol. 53(4), pages 1085-1106, July.
    11. Zhu, Xuedong & Son, Junbo & Zhang, Xi & Wu, Jianguo, 2023. "Constraint programming and logic-based Benders decomposition for the integrated process planning and scheduling problem," Omega, Elsevier, vol. 117(C).
    12. Yang, Yongjian & Yin, Yunqiang & Wang, Dujuan & Ignatius, Joshua & Cheng, T.C.E. & Dhamotharan, Lalitha, 2023. "Distributionally robust multi-period location-allocation with multiple resources and capacity levels in humanitarian logistics," European Journal of Operational Research, Elsevier, vol. 305(3), pages 1042-1062.
    13. Moreno, Alfredo & Munari, Pedro & Alem, Douglas, 2019. "A branch-and-Benders-cut algorithm for the Crew Scheduling and Routing Problem in road restoration," European Journal of Operational Research, Elsevier, vol. 275(1), pages 16-34.
    14. Belieres, Simon & Hewitt, Mike & Jozefowiez, Nicolas & Semet, Frédéric & Van Woensel, Tom, 2020. "A Benders decomposition-based approach for logistics service network design," European Journal of Operational Research, Elsevier, vol. 286(2), pages 523-537.
    15. Hassan Zohali & Bahman Naderi & Vahid Roshanaei, 2022. "Solving the Type-2 Assembly Line Balancing with Setups Using Logic-Based Benders Decomposition," INFORMS Journal on Computing, INFORMS, vol. 34(1), pages 315-332, January.
    16. Daniel Baena & Jordi Castro & Antonio Frangioni, 2020. "Stabilized Benders Methods for Large-Scale Combinatorial Optimization, with Application to Data Privacy," Management Science, INFORMS, vol. 66(7), pages 3051-3068, July.
    17. Haodong Yu & Jie Sun & Yanjun Wang, 2021. "A time-consistent Benders decomposition method for multistage distributionally robust stochastic optimization with a scenario tree structure," Computational Optimization and Applications, Springer, vol. 79(1), pages 67-99, May.
    18. Lin, Yun Hui & Wang, Yuan & Lee, Loo Hay & Chew, Ek Peng, 2022. "Omnichannel facility location and fulfillment optimization," Transportation Research Part B: Methodological, Elsevier, vol. 163(C), pages 187-209.
    19. Jean-Paul Doignon & Kota Saito, 2022. "Adjacencies on random ordering polytopes and flow polytopes," Papers 2207.06925, arXiv.org.
    20. Gruson, Matthieu & Cordeau, Jean-François & Jans, Raf, 2021. "Benders decomposition for a stochastic three-level lot sizing and replenishment problem with a distribution structure," European Journal of Operational Research, Elsevier, vol. 291(1), pages 206-217.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:annopr:v:288:y:2020:i:1:d:10.1007_s10479-019-03473-y. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.