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Some Properties of Location Problems with Block and Round Norms

Author

Listed:
  • J.-F. Thisse

    (Université Catholique de Louvain, Louvain-la-Nueve, Belgium)

  • J. E. Ward

    (Purdue University, West Lafayette, Indiana)

  • R. E. Wendell

    (University of Pittsburgh, Pittsburgh, Pennsylvania)

Abstract
The point-objective problem and the Weber problem are two well-known formulations for locating a new facility with respect to a set of fixed facilities. When locations are represented as points on a plane, the point-objective problem is a multiple objective formulation of minimizing the distance from a variable point to each of the fixed points. Similarly, the Weber problem is a single objective formulation of minimizing the sum of transportation costs between the variable point and the fixed points, where transportation cost is a function of distance. Generalizing solution properties for these problems from distance measures given by the Euclidean, rectilinear, I p , and one-infinity norms; this paper develops solution properties under the broad classes of distance measures given by block and round norms. For the point-objective problem, we show that (i) the efficient set for all round norms is the convex hull of the set of fixed points and (ii) the efficient set under a block norm tends to the convex hull for a sequence of block norms approaching a round norm. For the Weber problem, we prove that (i) an optimal location for any block norm may be found in a finite set of intersection points belonging to the convex hull and (ii) this set tends to the convex hull for a sequence of block norms approaching a round norm. Finally, we use these results to propose a synthesis of some of the main properties in continuous and network location theory.

Suggested Citation

  • J.-F. Thisse & J. E. Ward & R. E. Wendell, 1984. "Some Properties of Location Problems with Block and Round Norms," Operations Research, INFORMS, vol. 32(6), pages 1309-1327, December.
    • Handle: RePEc:inm:oropre:v:32:y:1984:i:6:p:1309-1327
    DOI: 10.1287/opre.32.6.1309
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    Cited by:

    1. Stefan Nickel & Justo Puerto & Antonio M. Rodriguez-Chia, 2003. "An Approach to Location Models Involving Sets as Existing Facilities," Mathematics of Operations Research, INFORMS, vol. 28(4), pages 693-715, November.
    2. B. Pelegrin & F. R. Fernandez, 1988. "Determination of efficient points in multiple‐objective location problems," Naval Research Logistics (NRL), John Wiley & Sons, vol. 35(6), pages 697-705, December.
    3. 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.
    4. H Younies & G O Wesolowsky, 2007. "Planar maximal covering location problem under block norm distance measure," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 58(6), pages 740-750, June.
    5. Christian Günther & Christiane Tammer, 2016. "Relationships between constrained and unconstrained multi-objective optimization and application in location theory," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 84(2), pages 359-387, October.
    6. 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.
    7. 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.
    8. M. Akyüz & İ. Altınel & Temel Öncan, 2014. "Location and allocation based branch and bound algorithms for the capacitated multi-facility Weber problem," Annals of Operations Research, Springer, vol. 222(1), pages 45-71, November.

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