IDEAS home Printed from https://ideas.repec.org/a/eee/ejores/v233y2014i1p34-42.html
   My bibliography  Save this article

A geometric characterisation of the quadratic min-power centre

Author

Listed:
  • Brazil, M.
  • Ras, C.J.
  • Thomas, D.A.

Abstract

For a given set of nodes in the plane the min-power centre is a point such that the cost of the star centred at this point and spanning all nodes is minimised. The cost of the star is defined as the sum of the costs of its nodes, where the cost of a node is an increasing function of the length of its longest incident edge. The min-power centre problem provides a model for optimally locating a cluster-head amongst a set of radio transmitters, however, the problem can also be formulated within a bicriteria location model involving the 1-centre and a generalised Fermat-Weber point, making it suitable for a variety of facility location problems. We use farthest point Voronoi diagrams and Delaunay triangulations to provide a complete geometric description of the min-power centre of a finite set of nodes in the Euclidean plane when cost is a quadratic function. This leads to a new linear-time algorithm for its construction when the convex hull of the nodes is given. We also provide an upper bound for the performance of the centroid as an approximation to the quadratic min-power centre. Finally, we briefly describe the relationship between solutions under quadratic cost and solutions under more general cost functions.

Suggested Citation

  • Brazil, M. & Ras, C.J. & Thomas, D.A., 2014. "A geometric characterisation of the quadratic min-power centre," European Journal of Operational Research, Elsevier, vol. 233(1), pages 34-42.
    • Handle: RePEc:eee:ejores:v:233:y:2014:i:1:p:34-42
    DOI: 10.1016/j.ejor.2013.09.004
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0377221713007406
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.ejor.2013.09.004?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. Duin, C.W. & Volgenant, A., 2012. "On weighting two criteria with a parameter in combinatorial optimization problems," European Journal of Operational Research, Elsevier, vol. 221(1), pages 1-6.
    2. Leon F. McGinnis & John A. White, 1978. "A Single Facility Rectilinear Location Problem with Multiple Criteria," Transportation Science, INFORMS, vol. 12(3), pages 217-231, August.
    3. Durier, Roland & Michelot, Christian, 1985. "Geometrical properties of the Fermat-Weber problem," European Journal of Operational Research, Elsevier, vol. 20(3), pages 332-343, June.
    4. Montemanni, Roberto & Leggieri, Valeria & Triki, Chefi, 2008. "Mixed integer formulations for the probabilistic minimum energy broadcast problem in wireless networks," European Journal of Operational Research, Elsevier, vol. 190(2), pages 578-585, October.
    5. Roland Durier & Christian Michelot, 1994. "On the Set of Optimal Points to the Weber Problem: Further Results," Transportation Science, INFORMS, vol. 28(2), pages 141-149, May.
    6. Jack Elzinga & Donald Hearn & W. D. Randolph, 1976. "Minimax Multifacility Location with Euclidean Distances," Transportation Science, INFORMS, vol. 10(4), pages 321-336, November.
    7. 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.
    8. Ohsawa, Yoshiaki, 1999. "A geometrical solution for quadratic bicriteria location models," European Journal of Operational Research, Elsevier, vol. 114(2), pages 380-388, April.
    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. Lina Mallozzi & Justo Puerto & Moisés Rodríguez-Madrena, 2019. "On Location-Allocation Problems for Dimensional Facilities," Journal of Optimization Theory and Applications, Springer, vol. 182(2), pages 730-767, August.

    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. H. Martini & K.J. Swanepoel & G. Weiss, 2002. "The Fermat–Torricelli Problem in Normed Planes and Spaces," Journal of Optimization Theory and Applications, Springer, vol. 115(2), pages 283-314, November.
    2. M. Hakan Akyüz & Temel Öncan & İ. Kuban Altınel, 2019. "Branch and bound algorithms for solving the multi-commodity capacitated multi-facility Weber problem," Annals of Operations Research, Springer, vol. 279(1), pages 1-42, August.
    3. 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.
    4. Ndiaye, M. & Michelot, C., 1998. "Efficiency in constrained continuous location," European Journal of Operational Research, Elsevier, vol. 104(2), pages 288-298, January.
    5. Díaz-Báñez, J.M. & Korman, M. & Pérez-Lantero, P. & Ventura, I., 2013. "The 1-median and 1-highway problem," European Journal of Operational Research, Elsevier, vol. 225(3), pages 552-557.
    6. Carrizosa, Emilio & Rodriguez-Chia, Antonio M., 1997. "Weber problems with alternative transportation systems," European Journal of Operational Research, Elsevier, vol. 97(1), pages 87-93, February.
    7. Jack Brimberg & Robert Love & Nenad Mladenović, 2009. "Extension of the Weiszfeld procedure to a single facility minisum location model with mixed ℓ p norms," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 70(2), pages 269-283, October.
    8. Zvi Drezner & G. O. Wesolowsky, 1991. "Facility location when demand is time dependent," Naval Research Logistics (NRL), John Wiley & Sons, vol. 38(5), pages 763-777, October.
    9. Valeria Leggieri & Paolo Nobili & Chefi Triki, 2008. "Minimum power multicasting problem in wireless networks," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 68(2), pages 295-311, October.
    10. Avella, P. & Benati, S. & Canovas Martinez, L. & Dalby, K. & Di Girolamo, D. & Dimitrijevic, B. & Ghiani, G. & Giannikos, I. & Guttmann, N. & Hultberg, T. H. & Fliege, J. & Marin, A. & Munoz Marquez, , 1998. "Some personal views on the current state and the future of locational analysis," European Journal of Operational Research, Elsevier, vol. 104(2), pages 269-287, January.
    11. Emilio Carrizosa & Frank Plastria, 2008. "Optimal Expected-Distance Separating Halfspace," Mathematics of Operations Research, INFORMS, vol. 33(3), pages 662-677, August.
    12. Pawel Kalczynski & Atsuo Suzuki & Zvi Drezner, 2023. "Obnoxious facility location in multiple dimensional space," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(2), pages 331-354, July.
    13. Hamacher, H. W. & Nickel, S., 1996. "Multicriteria planar location problems," European Journal of Operational Research, Elsevier, vol. 94(1), pages 66-86, October.
    14. 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.
    15. Cavalier, Tom M. & Conner, Whitney A. & del Castillo, Enrique & Brown, Stuart I., 2007. "A heuristic algorithm for minimax sensor location in the plane," European Journal of Operational Research, Elsevier, vol. 183(1), pages 42-55, November.
    16. Frank Plastria, 2009. "Asymmetric distances, semidirected networks and majority in Fermat–Weber problems," Annals of Operations Research, Springer, vol. 167(1), pages 121-155, March.
    17. Carrizosa, Emilio & Conde, Eduardo, 2002. "A fractional model for locating semi-desirable facilities on networks," European Journal of Operational Research, Elsevier, vol. 136(1), pages 67-80, January.
    18. Nickel, Stefan, 1998. "Restricted center problems under polyhedral gauges," European Journal of Operational Research, Elsevier, vol. 104(2), pages 343-357, January.
    19. E. Carrizosa & J. B. G. Frenk, 1998. "Dominating Sets for Convex Functions with Some Applications," Journal of Optimization Theory and Applications, Springer, vol. 96(2), pages 281-295, February.
    20. Frank Plastria, 2021. "Using the power of ideal solutions: simple proofs of some old and new results in location theory," 4OR, Springer, vol. 19(3), pages 449-467, September.

    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:ejores:v:233:y:2014:i:1:p:34-42. 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/locate/eor .

    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.