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Using Block Norms for Location Modeling

Author

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  • James E. Ward

    (Purdue University, West Lafayette, Indiana)

  • Richard E. Wendell

    (University of Pittsburgh, Pittsburgh, Pennsylvania)

Abstract

In formulating a continuous location model with facilities represented as points in R n (e.g., typically in the plane), one must characterize the distance between two points as a function of their coordinates. Two criteria in selecting a distance function are (1) to obtain good approximations of actual distances, and (2) to obtain a mathematical model of the location problem that is easy to solve. In this paper, we show how a class of norms with polygonal contours, called block norms, can yield attractive choices as distance functions with respect to these criteria. In particular, we consider the following relevant properties of block norms: they generalize the concepts of rectilinear or city-block travel; they are dense in the set of all norms; they have interesting travel interpretations; in the plane, they can be expressed as a sum of the absolute values of linear functions; they often give better approximations to actual highway distances than the most frequently used family of norms, the l p norms; and, finally, they yield linear programming formulations of certain facility location problems (i.e., the Weber problem and the Rawls problem).

Suggested Citation

  • James E. Ward & Richard E. Wendell, 1985. "Using Block Norms for Location Modeling," Operations Research, INFORMS, vol. 33(5), pages 1074-1090, October.
  • Handle: RePEc:inm:oropre:v:33:y:1985:i:5:p:1074-1090
    DOI: 10.1287/opre.33.5.1074
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    Citations

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    Cited by:

    1. 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.
    2. Simin Huang & Rajan Batta & Kathrin Klamroth & Rakesh Nagi, 2005. "The K-Connection Location Problem in a Plane," Annals of Operations Research, Springer, vol. 136(1), pages 193-209, April.
    3. 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.
    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. Carrizosa, Emilio & Goerigk, Marc & Schöbel, Anita, 2017. "A biobjective approach to recoverable robustness based on location planning," European Journal of Operational Research, Elsevier, vol. 261(2), pages 421-435.
    6. Nickel, Stefan, 1998. "Restricted center problems under polyhedral gauges," European Journal of Operational Research, Elsevier, vol. 104(2), pages 343-357, January.
    7. P. Dearing & K. Klamroth & R. Segars, 2005. "Planar Location Problems with Block Distance and Barriers," Annals of Operations Research, Springer, vol. 136(1), pages 117-143, April.
    8. J. Fliege, 1997. "Nondifferentiability Detection and Dimensionality Reduction in Minisum Multifacility Location Problems," Journal of Optimization Theory and Applications, Springer, vol. 94(2), pages 365-380, August.
    9. Enrique R. Venta & Francis J. Nourie, 1989. "Facility location on a grid with a diagonal line," Naval Research Logistics (NRL), John Wiley & Sons, vol. 36(5), pages 709-717, October.
    10. Sönke Behrends & Anita Schöbel, 2020. "Generating Valid Linear Inequalities for Nonlinear Programs via Sums of Squares," Journal of Optimization Theory and Applications, Springer, vol. 186(3), pages 911-935, September.
    11. Mark-Christoph Körner & Jack Brimberg & Henrik Juel & Anita Schöbel, 2011. "Geometric fit of a point set by generalized circles," Journal of Global Optimization, Springer, vol. 51(1), pages 115-132, September.
    12. Jianlin Jiang & Su Zhang & Yibing Lv & Xin Du & Ziwei Yan, 2020. "An ADMM-based location–allocation algorithm for nonconvex constrained multi-source Weber problem under gauge," Journal of Global Optimization, Springer, vol. 76(4), pages 793-818, April.
    13. Anita Schöbel, 2014. "Generalized light robustness and the trade-off between robustness and nominal quality," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 80(2), pages 161-191, October.
    14. Víctor Blanco, 2019. "Ordered p-median problems with neighbourhoods," Computational Optimization and Applications, Springer, vol. 73(2), pages 603-645, June.
    15. Yan Gu & Jianlin Jiang & Shun Zhang, 2023. "Distributionally robust Weber problem with uncertain demand," Computational Optimization and Applications, Springer, vol. 85(3), pages 705-752, July.
    16. AltInel, I. Kuban & Durmaz, Engin & Aras, Necati & ÖzkIsacIk, Kerem Can, 2009. "A location-allocation heuristic for the capacitated multi-facility Weber problem with probabilistic customer locations," European Journal of Operational Research, Elsevier, vol. 198(3), pages 790-799, November.
    17. Blanco, Víctor & Gázquez, Ricardo & Ponce, Diego & Puerto, Justo, 2023. "A branch-and-price approach for the continuous multifacility monotone ordered median problem," European Journal of Operational Research, Elsevier, vol. 306(1), pages 105-126.
    18. Phipps Arabie, 1991. "Was euclid an unnecessarily sophisticated psychologist?," Psychometrika, Springer;The Psychometric Society, vol. 56(4), pages 567-587, December.
    19. Blanco, Víctor & Puerto, Justo, 2021. "Covering problems with polyellipsoids: A location analysis perspective," European Journal of Operational Research, Elsevier, vol. 289(1), pages 44-58.
    20. 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.
    21. Brimberg, Jack & Drezner, Zvi & Mladenović, Nenad & Salhi, Said, 2014. "A new local search for continuous location problems," European Journal of Operational Research, Elsevier, vol. 232(2), pages 256-265.
    22. Kafer, Barbara & Nickel, Stefan, 2001. "Error bounds for the approximative solution of restricted planar location problems," European Journal of Operational Research, Elsevier, vol. 135(1), pages 67-85, November.
    23. Jack Brimberg & Robert F. Love, 1991. "Estimating travel distances by the weighted lp norm," Naval Research Logistics (NRL), John Wiley & Sons, vol. 38(2), pages 241-259, April.
    24. 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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