Mathematical Methods of Operations Research

, Volume 71, Issue 1, pp 125–163

An extended covering model for flexible discrete and equity location problems

Original Article

Abstract

To model flexible objectives for discrete location problems, ordered median functions can be applied. These functions multiply a weight to the cost of fulfilling the demand of a customer which depends on the position of that cost relative to the costs of fulfilling the demand of the other customers. In this paper a reformulated and more compact version of a covering model for the discrete ordered median problem (DOMP) is considered. It is shown that by using this reformulation better solution times can be obtained. This is especially true for some objectives that are often employed in location theory. In addition, the covering model is extended so that ordered median functions with negative weights are feasible as well. This type of modeling weights has not been treated in the literature on the DOMP before. We show that several discrete location problems with equity objectives are particular cases of this model. As a result, a mixed-integer linear model for this type of problems is obtained for the first time.

Keywords

Equity Flexible model Discrete location Integer programming 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Boland N, Domí nguez-Marí n P, Nickel S, Puerto J (2005) Exact procedures for solving the discrete ordered median problem. Comput Oper Res 33: 3270–3300CrossRefGoogle Scholar
  2. Cánovas L, García S, Labbé M, Marín A (2007) A strengthened formulation for the simple plant location problem with order. Oper Res Lett 35(2): 141–150MATHCrossRefMathSciNetGoogle Scholar
  3. Daskin M (1995) Network and discrete location. Wiley, New YorkMATHGoogle Scholar
  4. Domínguez-Marín P (2003) The discrete ordered median problem: models and solution methods. Kluwer, PhD ThesisGoogle Scholar
  5. Domínguez-Marín P, Nickel S, Mlandenović N, Hansen P (2005) Heuristic procedures for solving the discrete ordered median problem. Ann Oper Res 136: 145–173MATHCrossRefMathSciNetGoogle Scholar
  6. Drezner, Z, Hamacher, H (eds) (2002) Facility location: applications and theory. Springer, BerlinMATHGoogle Scholar
  7. Drezner Z, Thisse J-F, Wesolowsky G (1986) The minmax-min location problem. J Reg Sci 26: 87–101CrossRefGoogle Scholar
  8. Eiselt H, Laporte G (1995) Objectives in location. In: Drezner Z(eds) Facility location: a survey of applications and methods. Springer, New York, pp 151–180Google Scholar
  9. Erkut E (1993) Inequality measures for location problems. Locat Sci 1: 199–217MATHGoogle Scholar
  10. Francis R, Lowe T, Tamir A (2002) Aggregation error bounds for a class of location models. Oper Res 48: 294–307CrossRefMathSciNetGoogle Scholar
  11. Hakimi S (1964) Optimal locations of switching centers and the absolute centers and medians of a graph. Oper Res 12: 450–459MATHCrossRefGoogle Scholar
  12. Halpern J (1978) Finding minimal center-median convex combination (cent-dian) of a graph. Manage Sci 24: 353–544CrossRefGoogle Scholar
  13. Kalcsics J (2006) Unified approaches to territory design and facility location. Shaker, AachenGoogle Scholar
  14. Kariv O, Hakimi S (1979) An algorithmic approach to network location problems. II: the p-medians. SIAM J Appl Math 37: 539–560MATHCrossRefMathSciNetGoogle Scholar
  15. Marín A (2007) Lower bounds for the two-stage uncapacitated facility location problem. Eur J Oper Res 179: 1126–1142MATHCrossRefGoogle Scholar
  16. Marín A, Nickel S, Puerto J, Velten S (2009) A flexible model and efficient solution strategies for discrete location problems. Discrete Appl Math 157: 1128–1145MATHCrossRefMathSciNetGoogle Scholar
  17. Marsh M, Schilling D (1994) Equity measurement in facility location analysis: a review and framework. Eur J Oper Res 74: 1–17MATHCrossRefGoogle Scholar
  18. Melo T, Nickel S, Saldanha da Gama F (2006) Dynamic multi-commodity facility location: a mathematical framework for strategic supply chain planning. Comput Oper Resh 33: 181–208MATHCrossRefGoogle Scholar
  19. Mirchandani, P, Francis, R (eds) (1990) Discrete location theory.. Wiley, New YorkMATHGoogle Scholar
  20. Nickel S (2001) Discrete ordered weber problems. In: Operations research proceedings 2000. Springer, Heidelberg, pp 71–76Google Scholar
  21. Nickel S, Puerto J (1999) A unified approach to network location problems. Networks 34: 283–290MATHCrossRefMathSciNetGoogle Scholar
  22. Nickel S, Puerto J (2005) Facility location: a unified approach. Springer, BerlinGoogle Scholar
  23. Puerto J, Fernández F (1995) The symmetrical single facility location problem. Technical Report, Prepublicación de la Facultaf de Mathemáticas, Universidad de SevillaGoogle Scholar
  24. Rodríguez-Chía A, Nickel S, Puerto J, Fernández F (2000) A flexible approach to location problems. Math Methods Oper Res 51(1): 69–89MATHCrossRefMathSciNetGoogle Scholar
  25. Slater P (1978) Structure of the k-centra in a tree. In: Proceedings of the 9th Southwest conference on combinatorics, Graph Theory and Computing, pp 663–670Google Scholar
  26. van Roy T, Erlenkotter D (1982) A dual-based procedure for dynamic facility location. Manage Sci 28((10): 1091–1105MATHGoogle Scholar
  27. Wesolowsky G (1993) The Weber problem: history and perspectives. Locat Sci 1: 5–23MATHGoogle Scholar
  28. Yager R (1988) On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Trans Syst Man Cybernat 18: 183–190MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Alfredo Marín
    • 1
  • Stefan Nickel
    • 2
  • Sebastian Velten
    • 3
  1. 1.Departamento de Estadística e Investigación OperativaUniversidad de MurciaMurciaSpain
  2. 2.Lehrstuhl für Operations Research und LogistikUniversität des SaarlandesSaarbrückenGermany
  3. 3.Abteilung OptimierungFraunhofer Institut für Techno- und Wirtschaftsmathematik (ITWM)KaiserslauternGermany

Personalised recommendations