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Finding Minimal Center-Median Convex Combination (Cent-Dian) of a Graph

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  • Jonathan Halpern

    (University of Calgary, Canada)

Abstract

The graph median and center problems are well known with numerous possible applications. The first is suitable for locating a facility providing a routine service, by means of minimizing the average distance of customers to it. The second is appropriate for emergency services where the objective is to have the furthest customer as near as possible to the center. In reality a combination of both, usually antagonistic, goals is common. This paper presents a procedure to locate a facility on a graph, such that a convex combination of the median and the center objective functions is minimized. The term "cent-dian" is coined for this point of the graph. Since it is usually difficult to assign precise weights to the two objectives, when they are convexly combined, the procedure generates the cent-dians for all possible combinations. Finally, an equivalent median problem on an expanded graph is presented.

Suggested Citation

  • Jonathan Halpern, 1978. "Finding Minimal Center-Median Convex Combination (Cent-Dian) of a Graph," Management Science, INFORMS, vol. 24(5), pages 535-544, January.
    • Handle: RePEc:inm:ormnsc:v:24:y:1978:i:5:p:535-544
    DOI: 10.1287/mnsc.24.5.535
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    Cited by:

    1. Dieperink, H. & Nijkamp, P., 1987. "A multiple criteria location model for innovative firms in a communication network," Serie Research Memoranda 0072, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.
    2. Li, Hongmei & Luo, Taibo & Xu, Yinfeng & Xu, Jiuping, 2018. "Minimax regret vertex centdian location problem in general dynamic networks," Omega, Elsevier, vol. 75(C), pages 87-96.
    3. Adel A. Aly & Boubekeur Rahali, 1990. "Analysis of a bicriteria location model," Naval Research Logistics (NRL), John Wiley & Sons, vol. 37(6), pages 937-944, December.
    4. Liying Kang & Jianjie Zhou & Erfang Shan, 2018. "Algorithms for connected p-centdian problem on block graphs," Journal of Combinatorial Optimization, Springer, vol. 36(1), pages 252-263, July.
    5. Becker, Ronald I. & Lari, Isabella & Scozzari, Andrea, 2007. "Algorithms for central-median paths with bounded length on trees," European Journal of Operational Research, Elsevier, vol. 179(3), pages 1208-1220, June.
    6. Richard Francis & Timothy Lowe, 2014. "Comparative error bound theory for three location models: continuous demand versus discrete demand," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(1), pages 144-169, April.
    7. Welch, S. B. & Salhi, S., 1997. "The obnoxious p facility network location problem with facility interaction," European Journal of Operational Research, Elsevier, vol. 102(2), pages 302-319, October.
    8. Yoshiaki Ohsawa, 2000. "Bicriteria Euclidean location associated with maximin and minimax criteria," Naval Research Logistics (NRL), John Wiley & Sons, vol. 47(7), pages 581-592, October.
    9. Igor Averbakh & Oded Berman, 2000. "Minmax Regret Median Location on a Network Under Uncertainty," INFORMS Journal on Computing, INFORMS, vol. 12(2), pages 104-110, May.
    10. Ogryczak, Wlodzimierz, 2000. "Inequality measures and equitable approaches to location problems," European Journal of Operational Research, Elsevier, vol. 122(2), pages 374-391, April.
    11. Carrizosa, E. J. & Puerto, J., 1995. "A discretizing algorithm for location problems," European Journal of Operational Research, Elsevier, vol. 80(1), pages 166-174, January.
    12. Ohsawa, Yoshiaki, 1999. "A geometrical solution for quadratic bicriteria location models," European Journal of Operational Research, Elsevier, vol. 114(2), pages 380-388, April.
    13. Soudabeh Seyyedi Ghomi & Fahimeh Baroughi, 2024. "Robust vertex centdian facility location problem on tree networks," Annals of Operations Research, Springer, vol. 341(2), pages 1135-1149, October.
    14. Colebrook, Marcos & Sicilia, Joaquin, 2007. "A polynomial algorithm for the multicriteria cent-dian location problem," European Journal of Operational Research, Elsevier, vol. 179(3), pages 1008-1024, June.
    15. R. L. Francis & T. J. Lowe & Arie Tamir, 2000. "Aggregation Error Bounds for a Class of Location Models," Operations Research, INFORMS, vol. 48(2), pages 294-307, April.
    16. Justo Puerto & Federica Ricca & Andrea Scozzari, 2018. "Extensive facility location problems on networks: an updated review," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(2), pages 187-226, July.
    17. Dionisio Brito & José Moreno Pérez, 2000. "The generalizedp-Centdian on network," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 8(2), pages 265-285, December.
    18. Martínez-Merino, Luisa I. & Albareda-Sambola, Maria & Rodríguez-Chía, Antonio M., 2017. "The probabilistic p-center problem: Planning service for potential customers," European Journal of Operational Research, Elsevier, vol. 262(2), pages 509-520.
    19. Sune Lauth Gadegaard & Andreas Klose & Lars Relund Nielsen, 2018. "A bi-objective approach to discrete cost-bottleneck location problems," Annals of Operations Research, Springer, vol. 267(1), pages 179-201, August.
    20. Alfredo Marín & Stefan Nickel & Sebastian Velten, 2010. "An extended covering model for flexible discrete and equity location problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 71(1), pages 125-163, February.

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