Optimization Letters

, Volume 8, Issue 2, pp 407–424 | Cite as

On the Weber facility location problem with limited distances and side constraints

  • Isaac F. Fernandes
  • Daniel Aloise
  • Dario J. Aloise
  • Pierre Hansen
  • Leo Liberti
Original Paper


The objective in the continuous facility location problem with limited distances is to minimize the sum of distance functions from the facility to the customers, but with a limit on each of the distances, after which the corresponding function becomes constant. The problem has applications in situations where the service provided by the facility is insensitive after a given threshold distance. In this paper, we propose a global optimization algorithm for the case in which there are in addition lower and upper bounds on the numbers of customers served.


Facility location Global optimization Reformulation Decomposition 


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  1. 1.
    Aloise D., Hansen P., Liberti L.: An improved column generation algorithm for minimum sum-of-squares clustering. Math. Program. 131, 195–220 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Belotti P., Lee J., Liberti L., Margot F., Wächter A.: Branching and bounds tightening techniques for non-convex MINLP. Optim. Methods Softw. 24, 597–634 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Berman O., Drezner Z., Krass D.: Generalized coverage: new developments in covering location models. Comput. Oper. Res. 37, 1675–1687 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Brimberg J., Chen R., Chen D.: Accelerating convergence in the Fermat–Weber location problem. Oper. Res. Lett. 22, 151–157 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bonami, P., Lee J.: BONMIN user’s manual. Technical report, IBM Corporation (2007)Google Scholar
  6. 6.
    Bonami P., Biegler L., Conn A.R., Cornuéjols G., Grossmann I.E., Laird C.D., Lee J., Lodi A., Margot F., Sawaya N., Wächter A.: An algorithmic framework for convex mixed integer nonlinear programs. Discret. Optim. 5, 186–204 (2008)CrossRefzbMATHGoogle Scholar
  7. 7.
    Boyd S., Vandenberghe L.: Convex Optimization. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  8. 8.
    Church R., ReVelle C.: The maximal covering location problem. Pap. Reg. Sci. Assoc. 32, 101–118 (1974)CrossRefGoogle Scholar
  9. 9.
    Church R., Roberts K.L.: Generalized coverage models and public facility location. Pap. Reg. Sci. Assoc. 53, 117–135 (1983)CrossRefGoogle Scholar
  10. 10.
    Czyzyk J., Mesnier M., Moré J.: The NEOS Server. IEEE J. Comput. Sci. Eng. 5, 68–75 (1998)CrossRefGoogle Scholar
  11. 11.
    Berg M., Krefeld M., Overmars M., Schwarzkopf O.: Computational Geometry: Algorithms and Applications. Springer, Berlin (1997)CrossRefzbMATHGoogle Scholar
  12. 12.
    Drezner Z., Wesolowsky G.O.: A maximin location problem with maximum distance constraints. AIIE Transact. 12, 249–252 (1980)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Drezner Z., Mehrez A., Wesolowsky G.O.: The facility location problem with limited distances. Transp. Sci. 25, 183–187 (1991)CrossRefzbMATHGoogle Scholar
  14. 14.
    Drezner Z., Hamacher H.W.: Facility Location: Applications and Theory. Springer, Berlin (2004)Google Scholar
  15. 15.
    Drezner Z., Wesolowsky G.O., Drezner T.: The gradual covering problem. Nav. Res. Logist. 51, 841–855 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Fekete S.P., Mitchell J.S.B., Beurer K.: On the continuous Fermat–Weber problem. Oper. Res. 53, 61–76 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Gurgel A.M.: Melhoria da segurança pública: Uma proposta para a alocação de unidades policiais utilizando o modelo das p-medianas e do caixeiro viajante. M.Sc. dissertation. Universidade Federal do Rio Grande do Norte (2010)Google Scholar
  18. 18.
    Hansen P., Mladenović N., Mladenović N.: Heuristic solution of the multisource Weber problem as a image-median problem”. Oper. Res. Lett. 22, 55–62 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Liberti L.: Writing global optimization software. In: Liberti, L., Maculan, N. (eds.) Global Optimization: from Theory to Implementation, pp. 211–262. Springer, Berlin (2006)CrossRefGoogle Scholar
  20. 20.
    Liberti L.: Reformulations in mathematical programming: definitions and systematics. RAIRO-RO 43, 55–86 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Pirkul H., Schilling D.A.: The maximal covering location problem with capacities on total workload. Manag. Sci. 37, 233–248 (1991)CrossRefzbMATHGoogle Scholar
  22. 22.
    Schilling D.A., Jayaraman V., Barkhi R.: A review of covering problems in facility location. Locat. Sci. 1, 25–55 (1993)zbMATHGoogle Scholar
  23. 23.
    Smith E., Pantelides C.: A symbolic reformulation/spatial branch-and-bound algorithm for the global optimisation of nonconvex MINLPs. Comput. Chem. Eng. 23, 457–478 (1999)CrossRefGoogle Scholar
  24. 24.
    Smith H.K., Laporte G., Harper P.R.: Locational analysis: highlights of growth to maturity. J. Oper. Res. Soc. 60, 140–148 (2009)CrossRefGoogle Scholar
  25. 25.
    Tawarmalani M., Sahinidis N.V.: A polyhedral branch-and-cut approach to global optimization. Mathe. Program. 103, 225–249 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Wesolowsky G.O.: The Weber problem: history and perspectives. Locat. Sci. 1, 5–23 (1993)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Isaac F. Fernandes
    • 1
  • Daniel Aloise
    • 1
  • Dario J. Aloise
    • 2
  • Pierre Hansen
    • 3
  • Leo Liberti
    • 4
  1. 1.Universidade Federal do Rio Grande do NorteCampus Universitário s/nNatalBrazil
  2. 2.Universidade do Estado do Rio Grande do NorteMossoróBrazil
  3. 3.GERAD and HEC MontréalMontréalCanada
  4. 4.LIX, École PolytechniquePalaiseauFrance

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