Location Problems with Multiple Criteria

  • S. Nickel
  • J. PuertoEmail author
  • A. M. Rodríguez-Chía


This chapter analyzes multicriteria continuous, network, and discrete location problems. In the continuous framework, we provide a complete description of the set of weak Pareto, Pareto, and strict Pareto locations for a general Q-criteria location problem based on the characterization of three criteria problems. In the network case, the set of Pareto locations is characterized for general networks as well as for tree networks using the concavity and convexity properties of the distance function on the edges. In the discrete setting, the entire set of Pareto locations is characterized using rational generating functions of integer points in polytopes. Moreover, we describe algorithms to obtain the solutions sets (the different Pareto locations) using the above characterizations. We also include a detailed complexity analysis. A number of references has been cited throughout the chapter to avoid the inclusion of unnecessary technical details and also to be useful for a deeper analysis.


Pareto-optimal Pareto locations Level curves Tree networks Networks Rational functions 



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


  1. Alzorba S, Günther C, Popovici N (2015) A special class of extended multicriteria location problems. Optimization 64(5):1305–1320MathSciNetzbMATHCrossRefGoogle Scholar
  2. Alzorba S, Günther C, Popovici N, Tammer C (2017) A new algorithm for solving planar multiobjective location problems involving the Manhattan norm. Eur J Oper Res 258(1):35–46MathSciNetzbMATHCrossRefGoogle Scholar
  3. Apolinário HCF, Papa Quiroz EA, Oliveira PR (2016) A scalarization proximal point method for quasiconvex multiobjective minimization. J Global Optim 64(1):79–96MathSciNetzbMATHCrossRefGoogle Scholar
  4. Arora S, Arora SR (2010) Multiobjective capacitated plant location problem. Int J Oper Res 7(4):487–505MathSciNetzbMATHCrossRefGoogle Scholar
  5. Barvinok A (1994) A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed. Math Oper Res 19:769–779MathSciNetzbMATHCrossRefGoogle Scholar
  6. Barvinok A, Woods K (2003) Short rational generating functions for lattice point problems. J Amer Math Soc 16:957–979MathSciNetzbMATHCrossRefGoogle Scholar
  7. Bhattacharya U (2018) A mathematical model for locating k-obnoxious facilities on a plane. Int J Oper Res 31(3):384–402MathSciNetCrossRefGoogle Scholar
  8. Blanco V, Puerto J (2012) A new complexity result on multiobjective linear integer programming using short rational generating functions. Optim Lett 6:537–543MathSciNetzbMATHCrossRefGoogle Scholar
  9. Brion M (1988) Points entiers dans les polyèdres convexes. Ann Sci Ecole Norm S Sér 21(4):653–663zbMATHGoogle Scholar
  10. Carrizosa E, Conde E, Fernández FR, Puerto J (1993) Efficiency in Euclidean constrained location problems. Oper Res Lett 14:291–295MathSciNetzbMATHCrossRefGoogle Scholar
  11. Chinchuluun A, Pardalos PM (2007) A survey of recent developments in multiobjective optimization. Ann Oper Res 154:29–50MathSciNetzbMATHCrossRefGoogle Scholar
  12. Colebrook M, Sicilia J (2007a) A polynomial algorithm for the multicriteria cent-dian location problem. Eur J Oper Res 179:1008–1024MathSciNetzbMATHCrossRefGoogle Scholar
  13. Colebrook M, Sicilia J (2007b) Undesirable facility location problems on multicriteria networks. Comput Oper Res 34:1491–1514MathSciNetzbMATHCrossRefGoogle Scholar
  14. Colmenar J, Martí R, Duarte A (2018) Multi-objective memetic optimization for the bi-objective obnoxious p-median problem. Knowl-Based Syst 144:88–101CrossRefGoogle Scholar
  15. De Loera JA, Haws D, Hemmecke R, Huggins P, Sturmfels B, Yoshida R (2004) Short rational functions for toric algebra and applications. J Symb Comput 38:959–973MathSciNetzbMATHCrossRefGoogle Scholar
  16. De Loera JA, Haws D, Hemmecke R, Huggins P, Yoshida R (2005) A computational study of integer programming algorithms based on Barvinok’s rational functions. Discrete Optim 2:135–144MathSciNetzbMATHCrossRefGoogle Scholar
  17. De Loera JA, Hemmecke R, Köppe M (2009) Pareto optima of multicriteria integer linear programs. INFORMS J Comput 21:39–48MathSciNetzbMATHCrossRefGoogle Scholar
  18. Dearing P, Francis R, Lowe T (1976) Convex location problems on tree networks. Oper Res 24:628–642MathSciNetzbMATHCrossRefGoogle Scholar
  19. Drezner Z (1995) Facility location. In: A survey of applications and methods. Springer, New YorkGoogle Scholar
  20. Durier R (1990) On Pareto optima, the Fermat-Weber problem, and polyhedral gauges. Math Program 47:65–79MathSciNetzbMATHCrossRefGoogle Scholar
  21. Durier R, Michelot C (1985) Geometrical properties of the Fermat-Weber problem. Eur J Oper Res 20:332–343MathSciNetzbMATHCrossRefGoogle Scholar
  22. Durier R, Michelot C (1986) Sets of efficient points in a normed space. J Math Anal Appl 117:506–528MathSciNetzbMATHCrossRefGoogle Scholar
  23. Edelsbrunner H (1987) Algorithms in combinatorial geometry. Springer, New YorkzbMATHCrossRefGoogle Scholar
  24. Ehrgott M (2005) Multicriteria optimization. Springer, HeidelbergzbMATHGoogle Scholar
  25. Ehrgott M, Gandibleux X (2000) A survey and annotated bibliography of multiobjective combinatorial optimization. OR Spectr 22:425–460MathSciNetzbMATHCrossRefGoogle Scholar
  26. Ehrgott M, Gandibleux X (2002) Multiple criteria optimization. In: State of the art annotated bibliographic surveys. Kluwer, BostonGoogle Scholar
  27. Elleuch MA, Frikha A (2018) Combining the promethee method and mathematical programming for multi-objective facility location problem. Int J Multicrit Decis Mak 7(3/4):195–216CrossRefGoogle Scholar
  28. Farahani RZ, Steadieseifi M, Asgari N (2010) Multiple criteria facility location problems: a survey. Appl Math Model 34(7):1689–1709MathSciNetzbMATHCrossRefGoogle Scholar
  29. Fernández E, Puerto J (2003) Multiobjective solution of the uncapacitated plant location problem. Eur J Oper Res 145:509–529MathSciNetzbMATHCrossRefGoogle Scholar
  30. Gandibleux X, Jaszkiewicz A, Freville A, Slowinski RE (2000) Special issue ‘multiple objective metaheuristics’. J Heuristics 6:291–431CrossRefGoogle Scholar
  31. Goldman A (1971a) Optimal center location in simple networks. Transport Sci 5:212–221MathSciNetCrossRefGoogle Scholar
  32. Goldman AJ (1971b) Optimal center location in simple networks. Transport Sci 5:212–221MathSciNetCrossRefGoogle Scholar
  33. Hakimi S (1964) Optimum location of switching centers and the absolute centers and medians of a graph. Oper Res 12:450–459zbMATHCrossRefGoogle Scholar
  34. Hamacher H, Nickel S (1996) Multicriteria planar location problems. Eur J Oper Res 94:66–86zbMATHCrossRefGoogle Scholar
  35. Hamacher HW, Labbé M, Nickel S (1999) Multicriteria network location problems with sum objectives. Networks 33:79–92MathSciNetzbMATHCrossRefGoogle Scholar
  36. Hamacher HW, Labbé M, Nickel S, Skriver AJ (2002) Multicriteria semi-obnoxious network location problems (MSNLP) with sum and center objectives. Ann Oper Res 110:33–53MathSciNetzbMATHCrossRefGoogle Scholar
  37. Hansen P, Perreur J, Thisse J (1980) Location theory, dominance and convexity: some further results. Oper Res 28:1241–1250zbMATHCrossRefGoogle Scholar
  38. Hansen P, Labbé M, Thisse JF (1991) From the median to the generalized center. RAIRO 25:73–86MathSciNetzbMATHCrossRefGoogle Scholar
  39. Hershberger J (1989) Finding the upper envelope of n line segments in o(\(n \log n\)) time. Inform Process Lett 33:169–174MathSciNetzbMATHCrossRefGoogle Scholar
  40. Kalcsics J, Nickel S, Pozo MA, Puerto J, Rodríguez-Chía AM (2014) The multicriteria p-facility median location problem on networks. Eur J Oper Res 235(3):484–493MathSciNetzbMATHCrossRefGoogle Scholar
  41. 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
  42. Karatas M, Yakici E (2018) An iterative solution approach to a multi-objective facility location problem. Appl Soft Comput 62:272–287CrossRefGoogle Scholar
  43. Nickel S (1995) Discretization of planar location problems. Fachbereich mathematik, PhD Dissertation, University of KaiserslauternGoogle Scholar
  44. Nickel S (1997) Bicriteria and restricted 2-facility weber problems. Math Method Oper Res 45:167–195MathSciNetzbMATHCrossRefGoogle Scholar
  45. Nickel S, Puerto J (2005) Location theory: a unified approach. Springer, BerlinzbMATHGoogle Scholar
  46. Nickel S, Puerto J, Rodríguez-Chía AM (2005a) MCDM location problems. In: Figueira JA, Greco S, Ehrogott M (eds) Multiple criteria decision analysis: state of the art surveys, international series in operations research & management science, vol 78. Springer, New York, pp 761–787Google Scholar
  47. Nickel S, Puerto J, Rodríguez-Chía AM, Weissler A (2005b) Multicriteria planar ordered median problems. J Optimiz Theory App 126:657–683MathSciNetzbMATHCrossRefGoogle Scholar
  48. Özpeynirci O (2017) On nadir points of multiobjective integer programming problems. J Global Optim 69(3):699–712MathSciNetzbMATHCrossRefGoogle Scholar
  49. Pecci F, Abraham E, Stoianov I (2017) Scalable Pareto set generation for multiobjective co-design problems in water distribution networks: a continuous relaxation approach. Struct Multidiscip Optim 55(3):857–869MathSciNetCrossRefGoogle Scholar
  50. Puerto J, Fernández F (1999) Multi-criteria minisum facility location problems. J Multi-Criteria Decis Anal 8:268–280zbMATHCrossRefGoogle Scholar
  51. Puerto J, Fernández F (2000) Geometrical properties of the symmetrical single facility location problem. J Nonlinear Convex A 1:321–342MathSciNetzbMATHGoogle Scholar
  52. Rockafellar R (1970) Convex analysis. Princeton University Press, PrincetonzbMATHCrossRefGoogle Scholar
  53. Rodríguez-Chía A, Puerto J (2002) Geometrical description of the weakly efficient solution set for multicriteria location problems. Ann Oper Res 111:179–194MathSciNetzbMATHCrossRefGoogle Scholar
  54. Rodríguez-Chía A, Nickel S, Puerto J, Fernández F (2000) A flexible approach to location problems. Math Method Oper Res 51:69–89MathSciNetzbMATHCrossRefGoogle Scholar
  55. Ross GT, Soland RM (1980) A multicriteria approach to the location of public facilities. Eur J Oper Res 4:307–321MathSciNetzbMATHCrossRefGoogle Scholar
  56. Skriver AJ, Andersen KA, Holmberg K (2004) Bicriteria network location (BNL) problems with criteria dependent lengths and minisum objectives. Eur J Oper Res 156:541–549MathSciNetzbMATHCrossRefGoogle Scholar
  57. Ulungu E, Teghem J (1994) Multi-objective combinatorial optimization problems: a survey. J Multi-Criteria Decis Anal 3:83–104zbMATHCrossRefGoogle Scholar
  58. Verdoolaege S (2008) Software barvinok.
  59. Wang SC, Lin CC, Chen TC, Hsiao H (2018) Multi-objective competitive location problem with distance-based attractiveness for two facilities. Comput Electr Eng 71:37–250Google Scholar
  60. Warburton A (1983) Quasiconcave vector maximization : connectedness of the sets of pareto-optimal and weak pareto-optimal alternatives. J Optimiz Theory App 40:537–557MathSciNetzbMATHCrossRefGoogle Scholar
  61. Weissler A (1999) General bisectors and their application in planar location theory. Shaker, AachenGoogle Scholar
  62. Wendell R, Hurter AJ (1973) Location theory, dominance and convexity. Oper Res 21:314–320MathSciNetzbMATHCrossRefGoogle Scholar
  63. Wendell R, Hurter A, Lowe T (1977) Efficient points in location problems. AIIE Trans 9:238–246MathSciNetCrossRefGoogle Scholar
  64. Woods K, Yoshida R (2005) Short rational generating functions and their applications to integer programming. SIAG/OPT Views and News 16:15–19Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • S. Nickel
    • 1
    • 2
  • J. Puerto
    • 3
    Email author
  • A. M. Rodríguez-Chía
    • 4
  1. 1.Institute for Operations ResearchKarlsruhe Institute of Technology (KIT)KarlsruheGermany
  2. 2.Fraunhofer Institute for Industrial Mathematics (ITWM)KaiserslauternGermany
  3. 3.IMUSUniversidad de SevillaSevilleSpain
  4. 4.Universidad de CádizCádizSpain

Personalised recommendations