IDEAS home Printed from https://ideas.repec.org/a/eee/jomega/v41y2013i4p721-730.html
   My bibliography  Save this article

The maximum dispersion problem

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0305048312001818
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.omega.2012.09.005?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. 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.
    3. Martí, Rafael & Gallego, Micael & Duarte, Abraham, 2010. "A branch and bound algorithm for the maximum diversity problem," European Journal of Operational Research, Elsevier, vol. 200(1), pages 36-44, January.
    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.
    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. 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.
    4. Nickel, Stefan & Velten, Sebastian, 2017. "Optimization problems with flexible objectives: A general modeling approach and applications," European Journal of Operational Research, Elsevier, vol. 258(1), pages 79-88.
    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.

    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. 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.
    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. Zhen, Lu & Gao, Jiajing & Tan, Zheyi & Laporte, Gilbert & Baldacci, Roberto, 2023. "Territorial design for customers with demand frequency," European Journal of Operational Research, Elsevier, vol. 309(1), pages 82-101.
    4. Seizinger, Markus & Brunner, Jens O., 2023. "Optimized planning of nursing curricula in dual vocational schools focusing on the German health care system," European Journal of Operational Research, Elsevier, vol. 304(3), pages 1223-1241.
    5. Johnes, Jill, 2015. "Operational Research in education," European Journal of Operational Research, Elsevier, vol. 243(3), pages 683-696.
    6. Yang, Xiao & Cai, Zonghui & Jin, Ting & Tang, Zheng & Gao, Shangce, 2022. "A three-phase search approach with dynamic population size for solving the maximally diverse grouping problem," European Journal of Operational Research, Elsevier, vol. 302(3), pages 925-953.
    7. Anna Martínez-Gavara & Vicente Campos & Micael Gallego & Manuel Laguna & Rafael Martí, 2015. "Tabu search and GRASP for the capacitated clustering problem," Computational Optimization and Applications, Springer, vol. 62(2), pages 589-607, November.
    8. Seyedmohammadhossein Hosseinian & Dalila B. M. M. Fontes & Sergiy Butenko, 2020. "A Lagrangian Bound on the Clique Number and an Exact Algorithm for the Maximum Edge Weight Clique Problem," INFORMS Journal on Computing, INFORMS, vol. 32(3), pages 747-762, July.
    9. Theresa M. Roeder & Robert M. Saltzman, 2014. "Schedule-Based Group Assignment Using Constraint Programming," INFORMS Transactions on Education, INFORMS, vol. 14(2), pages 63-72, February.
    10. Patricia Domínguez-Marín & Stefan Nickel & Pierre Hansen & Nenad Mladenović, 2005. "Heuristic Procedures for Solving the Discrete Ordered Median Problem," Annals of Operations Research, Springer, vol. 136(1), pages 145-173, April.
    11. Nahid Rezaeinia & Julio César Góez & Mario Guajardo, 2022. "Efficiency and fairness criteria in the assignment of students to projects," Annals of Operations Research, Springer, vol. 319(2), pages 1717-1735, December.
    12. Aringhieri, Roberto & Cordone, Roberto & Grosso, Andrea, 2015. "Construction and improvement algorithms for dispersion problems," European Journal of Operational Research, Elsevier, vol. 242(1), pages 21-33.
    13. Jesus Garcia-Diaz & Jairo Sanchez-Hernandez & Ricardo Menchaca-Mendez & Rolando Menchaca-Mendez, 2017. "When a worse approximation factor gives better performance: a 3-approximation algorithm for the vertex k-center problem," Journal of Heuristics, Springer, vol. 23(5), pages 349-366, October.
    14. Sergey Kovalev & Isabelle Chalamon & Fabio J. Petani, 2023. "Maximizing single attribute diversity in group selection," Annals of Operations Research, Springer, vol. 320(1), pages 535-540, January.
    15. Ricardo M. Lima & Ignacio E. Grossmann, 2017. "On the solution of nonconvex cardinality Boolean quadratic programming problems: a computational study," Computational Optimization and Applications, Springer, vol. 66(1), pages 1-37, January.
    16. Michele Garraffa & Federico Della Croce & Fabio Salassa, 2017. "An exact semidefinite programming approach for the max-mean dispersion problem," Journal of Combinatorial Optimization, Springer, vol. 34(1), pages 71-93, July.
    17. Lingping Kong & Jeng-Shyang Pan & Václav Snášel & Pei-Wei Tsai & Tien-Wen Sung, 2018. "An energy-aware routing protocol for wireless sensor network based on genetic algorithm," Telecommunication Systems: Modelling, Analysis, Design and Management, Springer, vol. 67(3), pages 451-463, March.
    18. R Aringhieri & R Cordone, 2011. "Comparing local search metaheuristics for the maximum diversity problem," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 62(2), pages 266-280, February.
    19. Anton Ovchinnikov & Joseph Milner, 2008. "Spreadsheet Model Helps to Assign Medical Residents at the University of Vermont's College of Medicine," Interfaces, INFORMS, vol. 38(4), pages 311-323, August.
    20. Contreras, Ivan & Fernández, Elena & Reinelt, Gerhard, 2012. "Minimizing the maximum travel time in a combined model of facility location and network design," Omega, Elsevier, vol. 40(6), pages 847-860.

    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:eee:jomega:v:41:y:2013:i:4:p:721-730. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/375/description#description .

    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.