IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v246y2016i1d10.1007_s10479-014-1587-5.html
   My bibliography  Save this article

Up- and downgrading the euclidean 1-median problem and knapsack Voronoi diagrams

Author

Listed:
  • Frank Plastria

    (Vrije Universiteit Brussel)

Abstract

We consider the 1-median problem with euclidean distances with uncertainty in the weights, expressed as possible changes within given bounds and a single budget constraint on the total cost of change. The upgrading (resp. downgrading) problem consists of minimizing (resp. maximizing) the optimal 1-median objective value over these weight changes. The upgrading problem is shown to belong to the family of continuous single facility location-allocation problems, whereas the downgrading problem reduces to a convex but highly non-differentiable optimization problem. Several structural properties of the optimal solution are proven for both problems, using novel planar partitions, the knapsack Voronoi diagrams, and lead to polynomial time solution algorithms.

Suggested Citation

  • Frank Plastria, 2016. "Up- and downgrading the euclidean 1-median problem and knapsack Voronoi diagrams," Annals of Operations Research, Springer, vol. 246(1), pages 227-251, November.
    • Handle: RePEc:spr:annopr:v:246:y:2016:i:1:d:10.1007_s10479-014-1587-5
    DOI: 10.1007/s10479-014-1587-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10479-014-1587-5
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10479-014-1587-5?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. Frank Plastria & Mohamed Elosmani, 2013. "Continuous location of an assembly station," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(2), pages 323-340, July.
    2. E. Weiszfeld & Frank Plastria, 2009. "On the point for which the sum of the distances to n given points is minimum," Annals of Operations Research, Springer, vol. 167(1), pages 7-41, March.
    3. Burkard, Rainer E. & Galavii, Mohammadreza & Gassner, Elisabeth, 2010. "The inverse Fermat-Weber problem," European Journal of Operational Research, Elsevier, vol. 206(1), pages 11-17, October.
    4. Zvi Drezner & Atsuo Suzuki, 2004. "The Big Triangle Small Triangle Method for the Solution of Nonconvex Facility Location Problems," Operations Research, INFORMS, vol. 52(1), pages 128-135, February.
    5. Frank Plastria, 2011. "The Weiszfeld Algorithm: Proof, Amendments, and Extensions," International Series in Operations Research & Management Science, in: H. A. Eiselt & Vladimir Marianov (ed.), Foundations of Location Analysis, chapter 0, pages 357-389, Springer.
    6. Plastria, Frank, 1992. "GBSSS: The generalized big square small square method for planar single-facility location," European Journal of Operational Research, Elsevier, vol. 62(2), pages 163-174, October.
    7. Gassner, Elisabeth, 2009. "Up- and downgrading the 1-center in a network," European Journal of Operational Research, Elsevier, vol. 198(2), pages 370-377, October.
    8. Frank Plastria & Mohamed Elosmani, 2008. "On the convergence of the Weiszfeld algorithm for continuous single facility location–allocation problems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 16(2), pages 388-406, December.
    9. Elisabeth Gassner, 2009. "A game-theoretic approach for downgrading the 1-median in the plane with Manhattan metric," Annals of Operations Research, Springer, vol. 172(1), pages 393-404, November.
    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. Esmaeil Afrashteh & Behrooz Alizadeh & Fahimeh Baroughi, 2020. "Optimal approaches for upgrading selective obnoxious p-median location problems on tree networks," Annals of Operations Research, Springer, vol. 289(2), pages 153-172, June.
    2. Murray, Alan T. & Church, Richard L. & Feng, Xin, 2020. "Single facility siting involving allocation decisions," European Journal of Operational Research, Elsevier, vol. 284(3), pages 834-846.
    3. Baldomero-Naranjo, Marta & Kalcsics, Jörg & Marín, Alfredo & Rodríguez-Chía, Antonio M., 2022. "Upgrading edges in the maximal covering location problem," European Journal of Operational Research, Elsevier, vol. 303(1), pages 14-36.

    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. Frank Plastria & Mohamed Elosmani, 2013. "Continuous location of an assembly station," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(2), pages 323-340, July.
    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. 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.
    4. Schöbel, Anita & Scholz, Daniel, 2014. "A solution algorithm for non-convex mixed integer optimization problems with only few continuous variables," European Journal of Operational Research, Elsevier, vol. 232(2), pages 266-275.
    5. Tammy Drezner & Zvi Drezner, 2016. "Sequential location of two facilities: comparing random to optimal location of the first facility," Annals of Operations Research, Springer, vol. 246(1), pages 5-18, November.
    6. T Drezner & Z Drezner, 2008. "Lost demand in a competitive environment," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 59(3), pages 362-371, March.
    7. J. Redondo & J. Fernández & I. García & P. Ortigosa, 2009. "A robust and efficient algorithm for planar competitive location problems," Annals of Operations Research, Springer, vol. 167(1), pages 87-105, March.
    8. Rafael Blanquero & Emilio Carrizosa & Amaya Nogales-Gómez & Frank Plastria, 2014. "Single-facility huff location problems on networks," Annals of Operations Research, Springer, vol. 222(1), pages 175-195, November.
    9. Zvi Drezner & Jack Brimberg & Nenad Mladenović & Said Salhi, 2016. "New local searches for solving the multi-source Weber problem," Annals of Operations Research, Springer, vol. 246(1), pages 181-203, November.
    10. Zvi Drezner & Said Salhi, 2017. "Incorporating neighborhood reduction for the solution of the planar p-median problem," Annals of Operations Research, Springer, vol. 258(2), pages 639-654, November.
    11. Zvi Drezner & Mozart B. C. Menezes, 2016. "The wisdom of voters: evaluating the Weber objective in the plane at the Condorcet solution," Annals of Operations Research, Springer, vol. 246(1), pages 205-226, November.
    12. Daniel Scholz, 2013. "Geometric branch-and-bound methods for constrained global optimization problems," Journal of Global Optimization, Springer, vol. 57(3), pages 771-782, November.
    13. Drezner, Tammy & Drezner, Zvi & Hulliger, Beat, 2014. "The Quintile Share Ratio in location analysis," European Journal of Operational Research, Elsevier, vol. 238(1), pages 166-174.
    14. Plastria, Frank, 2016. "How bad can the centroid be?," European Journal of Operational Research, Elsevier, vol. 252(1), pages 98-102.
    15. Loay Alkhalifa & Jack Brimberg, 2017. "Locating a minisum annulus: a new partial coverage distance model," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(2), pages 373-393, July.
    16. Zvi Drezner & George Wesolowsky, 2014. "Covering Part of a Planar Network," Networks and Spatial Economics, Springer, vol. 14(3), pages 629-646, December.
    17. Zvi Drezner & Carlton Scott, 2013. "Location of a distribution center for a perishable product," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 78(3), pages 301-314, December.
    18. Tammy Drezner & Zvi Drezner, 2011. "A note on equity across groups in facility location," Naval Research Logistics (NRL), John Wiley & Sons, vol. 58(7), pages 705-711, October.
    19. Zvi Drezner & George O. Wesolowsky & Tammy Drezner, 2004. "The gradual covering problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 51(6), pages 841-855, September.
    20. Daniel Scholz, 2012. "Theoretical rate of convergence for interval inclusion functions," Journal of Global Optimization, Springer, vol. 53(4), pages 749-767, August.

    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:spr:annopr:v:246:y:2016:i:1:d:10.1007_s10479-014-1587-5. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.