Networks and Spatial Economics

, Volume 14, Issue 1, pp 1–25 | Cite as

Multi-product Capacitated Single-Allocation Hub Location Problems: Formulations and Inequalities

  • Isabel Correia
  • Stefan Nickel
  • Francisco Saldanha-da-Gama


In this paper we extend the classical capacitated single-allocation hub location problem by considering that multiple products are to be shipped through the network. We propose a unified modeling framework for the situation in which no more than one hub can be located in each node. In particular, we consider the case in which all hubs are dedicated to handling a single-product as well as the case in which all hubs can handle all products. The objective is to minimize the total cost, which includes setup costs for the hubs, setup costs for each product in each hub and flow routing costs. Hubs are assumed to be capacitated. For this problem several models are proposed which are based on existing formulations for the (single-product) capacitated single-allocation hub location problem. Additionally, several classes of inequalities are proposed in order to strengthen the models in terms of the lower bound provided by the linear relaxation. We report results of a set of computational experiments conducted to test the proposed models and their enhancements.


Hub location Single-allocation Multiple products MILP formulations 



This work was supported by National Funding from FCT - Fundação para a Ciência e a Tecnologia, under the projects PEst-OE/MAT/UI0152 (CIO/FCUL) and PEst-OE/MAT/UIO297 (CMA/FCT/UNL).The authors would like to express their gratitude to the Associate Editor and to the two anonymous reviewers for the constructive criticism, comments and suggestions which helped to improve the paper.


  1. Alumur S, Kara BY (2008) Network hub location problems: the state of the art. Eur J Oper Res 190(1):1–21CrossRefGoogle Scholar
  2. Alumur S, Kara BY (2009) A hub covering network design problem for cargo applications in Turkey. J Oper Res Soc 60(10):1349–1359CrossRefGoogle Scholar
  3. Alumur S, Kara BY, Karasan OE (2009) The design of single allocation incomplete hub networks. Transp Res B 43(10):936–951CrossRefGoogle Scholar
  4. Aykin T (1994) Lagrangean relaxation based approaches to capacitated hub-and-spoke network design problem. Eur J Oper Res 79(3):501–523CrossRefGoogle Scholar
  5. Aykin T (1995) Networking policies for hub-and-spoke systems with application to the air transportation system. Transp Sci 29(3):201–221CrossRefGoogle Scholar
  6. Calik H, Alumur SA, Kara BY, Karasan OE (2009) A tabu-search based heuristic for the hub covering problem over incomplete hub networks. Comput Oper Res 36(12):3088–3096CrossRefGoogle Scholar
  7. Campbell JF (1994) Integer programming formulations of discrete hub location problems. Eur J Oper Res 72(2):387–405CrossRefGoogle Scholar
  8. Campbell JF, Ernst AT, Krishnamoorthy M (2002) Hub location problems. In: Drezner Z, Hamacher HW (eds) Facility location: applications and theory. Springer, New York, pp 373–407CrossRefGoogle Scholar
  9. Contreras I, Díaz J, Fernández E (2009a) Lagrangean relaxation for the capacitated hub location problem with single assignment. OR Spectrum 31(3):483–505CrossRefGoogle Scholar
  10. Contreras I, Fernández E, Marín A (2009b) Tight bounds from a path based formulation for the tree of hubs location problem. Comput Oper Res 36(12):3117–3127CrossRefGoogle Scholar
  11. Contreras I, Fernández E, Marín A (2010) The tree of hubs location problem. Eur J Oper Res 202(2):390–400CrossRefGoogle Scholar
  12. Contreras I, Días JA, Fernández E (2011) Branch and price for large-scale capacitated hub location problems with single assignment. INFORMS J Comput 23(1):41–55CrossRefGoogle Scholar
  13. Correia I, Nickel S, Saldanha-da-Gama F (2010a) Single-assignment hub location problems with multiple capacity levels. Transp Res B 44(8–9):1047–1066CrossRefGoogle Scholar
  14. Correia I, Nickel S, Saldanha-da-Gama F (2010b) The capacitated single-allocation hub location problem revisited: a note on a classical formulation. Eur J Oper Res 207(1):92–96CrossRefGoogle Scholar
  15. Costa MG, Captivo ME, Clímaco J (2008) Capacitated single allocation hub location problem—a bi-criteria approach. Comput Oper Res 35(11):3671–3695CrossRefGoogle Scholar
  16. Ernst AT, Krishnamoorthy M (1996) Efficient algorithms for the uncapacitated single allocation p-hub median problem. Locat Sci 4(3):139–154CrossRefGoogle Scholar
  17. Ernst AT, Krishnamoorthy M (1999) Solution algorithms for the capacitated single allocation hub location problem. Ann Oper Res 86:141–159CrossRefGoogle Scholar
  18. Klincewicz JG (1998) Hub location in backbone/tributary network design: a review. Locat Sci 6:307–335CrossRefGoogle Scholar
  19. Labbé M, Yaman H, Gourdin E (2005) A branch and cut algorithm for the hub location problems with single assignment. Math Program 102(2):371–405CrossRefGoogle Scholar
  20. Nickel S, Schobel A, Sonneborn T (2001) Hub location problems in urban traffic networks . In: Niittymaki J, Pursula M (eds) Mathematics methods and optimization in transportation systems. Kluwer Academic Publishers, The Netherlands, pp 95–107CrossRefGoogle Scholar
  21. O’Kelly M (2010) Routing traffic at hub facilities. Netw Spat Econ 10:173–191CrossRefGoogle Scholar
  22. Skorin-Kapov D, Skorin-Kapov J, O’Kelly M (1996) Tight linear programming relaxations of uncapacitated p-hub median problems. Eur J Oper Res 94(3):582–593CrossRefGoogle Scholar
  23. Vidović M, Zečević S, Kilibarda M, Vlajić J, Bjelić N, Tadić S (2011) The p-hub model with hub-catchment areas, existing hubs, and simulation: a case study of serbian intermodal terminals. Netw Spat Econ 11:295–314CrossRefGoogle Scholar
  24. Yaman H, Carello G (2005) Solving the hub location problem with modular link capacities. Comput Oper Res 32(12):3227–3245CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Isabel Correia
    • 1
  • Stefan Nickel
    • 2
    • 3
  • Francisco Saldanha-da-Gama
    • 4
  1. 1.Faculdade de Ciências e Tecnologia, Departamento de Matemática - CMAUniversidade Nova LisboaCaparicaPortugal
  2. 2.Institute of Operations ResearchKarlsruhe Institute of Technology (KIT)KarlsruheGermany
  3. 3.Fraunhofer Institute for Industrial Mathematics (ITWM)KaiserslauternGermany
  4. 4.DEIO-CIO, Faculdade de CiênciasUniversidade de LisboaLisboaPortugal

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