Ordered Median Location Problems

  • Justo Puerto
  • Antonio M. Rodríguez-ChíaEmail author


This chapter analyzes the ordered median location problem in three different frameworks: continuous, discrete and networks; where some classical but also new results have been collected. For each solution space we study general properties that lead to solution algorithms. In the continuous case, we present two solution approaches for the planar case with polyhedral norms (the most intuitive case) and a novel approach applicable for the general case based on a hierarchy of semidefinite programs that can approximate up to any degree of accuracy the solution of any ordered median problem in finite dimension spaces with polyhedral or p-norms. We also cover the problem on networks deriving finite dominating sets for some particular classes of λ parameters and showing the impossibility of finding a FDS with polynomial cardinality for general lambdas in the multifacility case. Finally, we present a covering based formulation for the capacitated discrete ordered median problem with binary assignment which is rather promising in terms of gap and CPU time for solving this family of problems.


Ordered median function Finite dominating set Mixed integer linear programming 



The authors were partially supported by projects MTM2016-74983-C2-01/02-R (Ministry of Economy and Competitiveness∖FEDER, Spain).


  1. Ben-Israel A, Iyigun C (2010) A generalized Weiszfeld method for the multi-facility location problem. Oper Res Lett 38:207–214MathSciNetzbMATHCrossRefGoogle Scholar
  2. Berman O, Kalcsics J, Krass D, Nickel S (2009) The ordered gradual covering location problem on a network. Discrete Appl Math 157:3689–3707MathSciNetzbMATHCrossRefGoogle Scholar
  3. Blanco V, Ben Ali SEH, Puerto J (2013) Minimizing ordered weighted averaging of rational functions with applications to continuous location. Comput Oper Res 40:1448–1460MathSciNetzbMATHCrossRefGoogle Scholar
  4. Blanco V, Ben Ali SEH, Puerto J (2014) Revisiting several problems and algorithms in continuous location with lp norms. Comput Optim Appl 58:563–595MathSciNetzbMATHCrossRefGoogle Scholar
  5. Blanco V, Puerto J, Ben-Ali SEH (2016) Continuous multifacility ordered median location problems. Eur J Oper Res 250(1):56–64MathSciNetzbMATHCrossRefGoogle Scholar
  6. Blanco V, Puerto J, Salmerón, R (2018) A general framework for locating hyperplanes to fitting set of points. Comput Oper Res 95:172–193MathSciNetzbMATHCrossRefGoogle Scholar
  7. Blanquero R, Carrizosa E (2009) Continuous location problems and big triangle small triangle: constructing better bounds. J Global Optim 45:389–402MathSciNetzbMATHCrossRefGoogle Scholar
  8. Boland N, Domínguez-Marín P, Nickel S, Puerto J (2006) Exact procedures for solving the discrete ordered median problem. Comput Oper Res 33:3270–3300zbMATHCrossRefGoogle Scholar
  9. Brimberg J, Hansen P, Mladenovic N, Taillard ED (2000) Improvement and comparison of heuristics for solving the uncapacitated multisource Weber problem. Oper Res 48:444–460CrossRefGoogle Scholar
  10. Deleplanque S, Labbé M, Ponce D, Puerto J (2019) An extended version of a branch-price-and-cut procedure for the discrete ordered median problem. Informs J Comput.
  11. Domínguez-Marín P, Nickel S, Hansen P, Mladenović N (2005) Heuristic procedures for solving the discrete ordered median problem. Ann Oper Res 136:145–173MathSciNetzbMATHCrossRefGoogle Scholar
  12. Drezner Z (2007) A general global optimization approach for solving location problems in the plane. J Global Optim 37:305–319MathSciNetzbMATHCrossRefGoogle Scholar
  13. Drezner Z, Nickel S (2009a) Constructing a DC decomposition for ordered median problems. J Global Optim 45:187–201MathSciNetzbMATHCrossRefGoogle Scholar
  14. Drezner Z, Nickel S (2009b) Solving the ordered one-median problem in the plane. Eur J Oper Res 195:46–61MathSciNetzbMATHCrossRefGoogle Scholar
  15. Durier R, Michelot C (1985) Geometrical properties of the Fermat-Weber problem. Eur J Oper Res 20:332–343MathSciNetzbMATHCrossRefGoogle Scholar
  16. Edelsbrunner H (1987) Algorithms in combinatorial geometry. Springer, New YorkzbMATHCrossRefGoogle Scholar
  17. Espejo I, Marín A, Puerto J, Rodríguez-Chía AM (2009) A comparison of formulations and solution methods for the minimum-envy location problem. Comput Oper Res 36:1966–1981zbMATHCrossRefGoogle Scholar
  18. Espejo I, Rodríguez-Chía AM, Valero C (2009) Convex ordered median problem with lp-norms. Comput Oper Res 36:2250–2262MathSciNetzbMATHCrossRefGoogle Scholar
  19. Francis R, Lowe T, Tamir A (2000) Aggregation error bounds for a class of location models. Oper Res 48:294–307MathSciNetzbMATHCrossRefGoogle Scholar
  20. Grzybowski J, Nickel S, Pallaschke D, Urbański R (2011) Ordered median functions and symmetries. Optimization 60:801–811MathSciNetzbMATHCrossRefGoogle Scholar
  21. Hakimi S (1964) Optimal location of switching centers and the absolute centers and medians of a graph. Oper Res 12:450–459zbMATHCrossRefGoogle Scholar
  22. Hakimi S, Labbé M, Schmeichel E (1992) The Voronoi partition of a network and its applications in location theory. Orsa J Comput 4:412–417MathSciNetzbMATHCrossRefGoogle Scholar
  23. Hardy GH, Littlewood JE, Pólya G (1952) Inequalities, 2nd ed. Cambridge University Press, Cambridge,zbMATHGoogle Scholar
  24. Hooker J, Garfinkel R, Chen C (1991) Finite dominating sets for network location problems. Oper Res 39:100–118MathSciNetzbMATHCrossRefGoogle Scholar
  25. Jibetean D, de Klerk E (2006) Global optimization of rational functions: a semidefinite programming approach. Math Program 106:93–109MathSciNetzbMATHCrossRefGoogle Scholar
  26. Kalcsics J, Nickel S, Puerto J, Tamir A (2002) Algorithmic results for ordered median problems. Oper Res Lett 30:149–158MathSciNetzbMATHCrossRefGoogle Scholar
  27. Kalcsics J, Nickel S, Puerto J (2003) Multifacility ordered median problems on networks: a further analysis. Networks 41:1–12MathSciNetzbMATHCrossRefGoogle Scholar
  28. Kalcsics J, Nickel S, Puerto J, Rodríguez-Chía AM (2010a) Distribution systems design with role dependent objectives. Eur J Oper Res 202:491–501MathSciNetzbMATHCrossRefGoogle Scholar
  29. Kalcsics J, Nickel S, Puerto J, Rodríguez-Chía AM (2010b) The ordered capacitated facility location problem. TOP 18:203–222MathSciNetzbMATHCrossRefGoogle Scholar
  30. Kalcsics J, Nickel S, Puerto J, Rodríguez-Chía AM (2015) Several 2-facility location problems on networks with equity objectives. Networks 65(1):1–9MathSciNetCrossRefGoogle Scholar
  31. Kim-Chuan T, Todd MJ, Tutuncu RH (2006) On the implementation and usage of SDPT3–a matlab software package for semidefinite-quadratic-linear programming, version 4.0. Optimization software.
  32. Labbé M, Ponce D, Puerto J (2017) A comparative study of formulations and solution methods for the discrete ordered p-median problem. Comput Oper Res 78:230–242MathSciNetzbMATHCrossRefGoogle Scholar
  33. Lasserre J (2009) Moments, positive polynomials and their applications. Imperial College Press, LondonCrossRefGoogle Scholar
  34. López-de-los-Mozos M, Mesa JA, Puerto J (2008) A generalized model of equality measures in network location problems. Comput Oper Res 35:651–660MathSciNetzbMATHCrossRefGoogle Scholar
  35. Marín A, Nickel S, Puerto J, Velten S (2009) A flexible model and efficient solution strategies for discrete location problems. Discrete Appl Math 157:1128–1145MathSciNetzbMATHCrossRefGoogle Scholar
  36. Marín A, Nickel S, Velten S (2010) An extended covering model for flexible discrete and equity location problems. Math Method Oper Res 71:125–163MathSciNetzbMATHCrossRefGoogle Scholar
  37. Martínez-Merino LI, Albareda-Sambola M, Rodríguez-Chía AM (2017) The probabilistic p-center problem: planning service for potential customers. Eur J Oper Res 262:509–520MathSciNetzbMATHCrossRefGoogle Scholar
  38. McCormick S (2005) Submodular function minimization. In: Discrete optimization. Elsevier, Amsterdam, pp 321–391CrossRefGoogle Scholar
  39. Nickel S (2001) Discrete ordered weber problems. In: Operations research proceedings 2000. Selected papers of the symposium, Dresden, OR 2000, September 9–12, 2000. Springer, Berlin, pp 71–76Google Scholar
  40. Nickel S, Puerto J (1999) A unified approach to network location problems. Networks 34:283–290MathSciNetzbMATHCrossRefGoogle Scholar
  41. Nickel S, Puerto J (2005) Location theory: A unified approach. Springer, BerlinzbMATHGoogle Scholar
  42. Nickel S, Puerto J, Rodríguez-Chía AM, Weissler A (2005) Multicriteria planar ordered median problems. J Optimiz Theory App 126:657–683MathSciNetzbMATHCrossRefGoogle Scholar
  43. Okabe A, Boots B, Sugihara K (1992) Spatial tessellations: concepts and applications of Voronoı̆ diagrams. In: Wiley series in probability and mathematical statistics: applied probability and statistics. Wiley, Chichester. With a foreword by D. G. KendallGoogle Scholar
  44. Papini P, Puerto J (2004) Averaging the k largest distances among n: k-centra in Banach spaces. J Math Anal Appl 291:477–487MathSciNetzbMATHCrossRefGoogle Scholar
  45. Puerto J (2008) A new formulation of the capacitated discrete ordered median problems with {0,  1} assignment. In: Operations research proceedings 2007. Selected papers of the annual international conference of the German Operations Research Society (GOR), Saarbrücken, September 5–7, 2007. Springer, Berlin, pp 165–170Google Scholar
  46. Puerto J, Fernández F (2000) Geometrical properties of the symmetric single facility location problem. J Nonlinear Convex Anal 1:321–342MathSciNetzbMATHGoogle Scholar
  47. Puerto J, Rodríguez-Chía AM (2005) On the exponential cardinality of FDS for the ordered p-median problem. Oper Res Lett 33:641–651MathSciNetzbMATHCrossRefGoogle Scholar
  48. Puerto J, Tamir A (2005) Locating tree-shaped facilities using the ordered median objective. Math Program 102:313–338MathSciNetzbMATHCrossRefGoogle Scholar
  49. Puerto J, Ramos AB, Rodríguez-Chía AM (2011) Single-allocation ordered median hub location problems. Comput Oper Res 38:559–570MathSciNetzbMATHCrossRefGoogle Scholar
  50. Puerto J, Ramos AB, Rodríguez-Chía AM (2013) A specialized branch & bound & cut for single-allocation ordered median hub location problems. Discrete Appl Math 161:2624–2646MathSciNetzbMATHCrossRefGoogle Scholar
  51. Puerto J, Pérez-Brito D, García-González C (2014) A modified variable neighborhood search for the discrete ordered median problem. Eur J Oper Res 234:61–76MathSciNetzbMATHCrossRefGoogle Scholar
  52. Puerto J, Ricca F, Scozzari A (2018) Extensive facility location problems on networks: an updated review. TOP 26(2):187–226MathSciNetzbMATHCrossRefGoogle Scholar
  53. Redondo JL, Marín A, Ortigosa PM (2016) A parallelized Lagrangean relaxation approach for the discrete ordered median problem. Ann Oper Res 246(1–2):253–272MathSciNetzbMATHCrossRefGoogle Scholar
  54. Rodríguez-Chía AM, Nickel S, Puerto J, Fernández FR (2000) A flexible approach to location problems. Math Method Oper Res 51:69–89MathSciNetzbMATHCrossRefGoogle Scholar
  55. Rodríguez-Chía AM, Puerto J, Pérez-Brito D, Moreno JA (2005) The p-facility ordered median problem on networks. TOP 13:105–126MathSciNetzbMATHCrossRefGoogle Scholar
  56. Rodríguez-Chía AM, Espejo I, Drezner Z (2010) On solving the planar k-centrum problem with Euclidean distances. Eur J Oper Res 207:1169–1186MathSciNetzbMATHCrossRefGoogle Scholar
  57. Rosenbaum R (1950) Subadditive functions. Duke Math J 17:227–247MathSciNetzbMATHCrossRefGoogle Scholar
  58. Rozanov M, Tamir A (2018) The nestedness property of location problems on the line. TOP 26:257–282MathSciNetzbMATHCrossRefGoogle Scholar
  59. Ruszczynski A, Syski W (1986) On convergence of the stochastic subgradient method with on-line stepsize rules. J Math Anal Appl 114(2):512–527MathSciNetzbMATHCrossRefGoogle Scholar
  60. Schnepper T (2017) Location problems with k-max functions-modelling and analysing outliers in center problems. In: PhD dissertation, Universität Wuppertal, GermanyGoogle Scholar
  61. Schnepper T, Klamroth K, Stiglmayr M, Puerto J (2019) Exact algorithms for handling outliers in center location problems on networks using k-max functions. Eur J Oper Res 273(2):441–451MathSciNetzbMATHCrossRefGoogle Scholar
  62. Schöbel A, Scholz D (2010) The big cube small cube solution method for multidimensional facility location problems. Comput Oper Res 37:115–122MathSciNetzbMATHCrossRefGoogle Scholar
  63. Turner L, Hamacher HW (2011) On universal shortest paths. In: Operations research proceedings 2010, pp 313–318zbMATHCrossRefGoogle Scholar
  64. Turner L, Ehrgott M, Hamacher HW (2015) On the generality of the greedy algorithm for solving matroid base problems. Discrete Appl Math 195:114–128MathSciNetzbMATHCrossRefGoogle Scholar
  65. Ward J, Wendell R (1985) Using block norms for location modeling. Oper Res 33:1074–1090MathSciNetzbMATHCrossRefGoogle Scholar
  66. Yager R (1988) On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Trans Syst Man Cybern 18:183–190zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.IMUSUniversidad de SevillaSevilleSpain
  2. 2.Dpto. Estadística e Investigación OperativaUniversidad de CadizCadizSpain

Personalised recommendations