On the Quadratic Programming Approach for Hub Location Problems

  • Xiaozheng He
  • Anthony Chen
  • Wanpracha Art Chaovalitwongse
  • Henry Liu
Part of the Springer Optimization and Its Applications book series (SOIA, volume 30)


Hub networks play an important role in many real-life network systems such as transportation and telecommunication networks. Hub location problem is concerned with identifying appropriate hub locations in a network and connecting an efficient hub-and-spoke network that minimizes the flow-weighted costs across the network. This chapter is focused on the uncapacitated single allocation p-hub median problem (USApHMP), which arises in many real-world hub networks of logistics operations. There have been many approaches used to solve this problem. We herein focus on a quadratic programming approach, which has been proven very effective and efficient. This approach incorporates the use of the linearization for 0-1 quadratic program. In this chapter, we give a brief review of the linearization techniques for 0-1 quadratic programs and compare the performance of several existing linearization techniques for USApHMP. Toward the end, we discuss some properties, comments and possible developments of these linearization techniques in the real-life USApHMP.


Linearization Technique Access Node Transit Node Quadratic Program Approach Original Quadratic Programming 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Abdinnour-Helm and M. A. Venkataramanan. Solution approaches to hub location problems. Annals of Operations Research, 78:31-50, 1998.MATHCrossRefGoogle Scholar
  2. 2.
    W.P. Adams and R.J. Forrester. A simple recipe for concise mixed 0-1 linearizations. Operations Research Letters, 33:55-61, 2005.MATHCrossRefGoogle Scholar
  3. 3.
    W.P. Adams, R.J. Forrester, and F.W. Glover. Comparison and enhancement strategies for linearizing mixed 0-1 quadratic programs. Discrete Optimization, 1:99-120, 2004.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    W.P. Adams and H.D. Sherali. A tight linearization and an algorithm for zero-one quadratic programming problems. Management Science, 32:1274-1290, 1986.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    T. Aykin. Lagrangian relaxation based approaches to capacitated hub-and-spoke network design problem. European Journal of Operational Research, 79:501-523, 1994.MATHCrossRefGoogle Scholar
  6. 6.
    T. Aykin. Networking policies for hub-and-spoke systems with applications to the air transportation system. Transportation Science, 29(3):201-221, 1995.MATHCrossRefGoogle Scholar
  7. 7.
    N. Boland, A. Ernst, M. Krishnamoorthy, and J. Ebery. Preprocessing and cutting methods for multiple allocation hub location problems. European Journal of Operational Research, 155:638-653, 2004.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    D.L. Bryan and M.E. O’Kelly. Hub-and-spoke network in air transportation: an analytical revew. Journal of Regional Science, 39:275-295, 1999.CrossRefGoogle Scholar
  9. 9.
    J.F. Campbell. Integer programming formulations of discrete hub location problems. European Journal of Operational Research, 72:387-405, 1994.MATHCrossRefGoogle Scholar
  10. 10.
    J.F. Campbell. Hub location and the p-hub median problem. Operations Research, 44:923-935, 1996.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    J.F. Campbell, A. Ernst, and M. Krishnamoorthy. Hub location problems. In H. Hamacher and Z. Drezner, editors. Location Theory: Applications and Theory, pp. 373-406, Springer-Verlag, New York, 2001.Google Scholar
  12. 12.
    W. Chaovalitwongse, P.M. Pardalos, and O.A. Prokoyev. A new linearization technique for multi-quadratic 0-1 programming problems. Operations Research Letter, 32:517-522, 2004.MATHCrossRefGoogle Scholar
  13. 13.
    P. Chardaire and A. Sutter. A decomposition method for quadratic zero-one programming. Management Science, 41:704-712, 1995.MATHCrossRefGoogle Scholar
  14. 14.
    J. Ebery, M. Krishnamoorthy, A. Ernst, and N. Boland. The capacitated multiple allocation hub location problem: Formulations and algorithms. European Journal of Operational Research, 120:614-631, 2000.MATHCrossRefGoogle Scholar
  15. 15.
    S. Elloumi, A. Faye, and E. Soutif. Decomposition and linearization for 0-1 quadratic programming. Annals of Operations Research, 99:79-93, 2000.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    A.T. Ernst and M. Krishnamoorthy. Efficient algorithms for the uncapacitated single allocation p-hub median problem. Location Science, 4:139–154, 1996.MATHCrossRefGoogle Scholar
  17. 17.
    A.T. Ernst and M. Krishnamoorthy. Exact and heuristic algorithm for the uncapacitated multiple allocation p-hub median problem. European Journal of Operational Research, 104:100-112, 1998.MATHCrossRefGoogle Scholar
  18. 18.
    A.T. Ernst and M. Krishnamoorthy. An exact solution approach based on shortest-paths for p-hub median problem. INFORMS Journal on Computing, 10:149-162, 1998.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    A.T. Ernst and M. Krishnamoorthy. Solution algorithms for the capacitated single allocation hub location problem. Annals of Operations Research, 86:141-159, 1999.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    A. Frieze and J. Yadeger. On the quadratic assignment problem. Discrete Applied Mathematic, 5:89-98, 1983.MATHCrossRefGoogle Scholar
  21. 21.
    F. Glover. Improved linear integer programming formulations of nonlinear integer programs. Management Science, 22:455-460, 1975.CrossRefMathSciNetGoogle Scholar
  22. 22.
    F. Glover and E. Woolsey. Further reduction of zero-one polynomial programming problems to zero-one linear programming problems. Operations Research, 21:156-161, 1973.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    F. Glover and E. Woolsey. Converting the 0-1 polynomial programming problem to a 0-1 linear program. IRMIS working paper, 9304, School of Business, Indiana University, Bollomington, Indiana, 1974.Google Scholar
  24. 24.
    S.A. Helm and M.A. Venkataramanan. Using simulated annealing to solve the p-hub location problem. Operations Research, 22:180-182, 1993.Google Scholar
  25. 25.
    L. Kaufman and F. Broeckx. An algorithm for the quadratic assignment problem using Benders’ decomposition. European Journal of Operational Research, 2:204-211, 1978.CrossRefGoogle Scholar
  26. 26.
    J.G. Klincewicz. Dual algorithms for the uncapacitated hub location problem. Location Science, 4:173-184, 1996.MATHCrossRefGoogle Scholar
  27. 27.
    R.F. Love, J.G. Morris, and G.O. Wesolowsky. Facilities Location, North Holland, Amsterdam, 1988.Google Scholar
  28. 28.
    M. O’Kelly. A quadratic integer program for the location of interacting hub facilities. European Journal of Operational Research, 32:393-404, 1987.MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    M. O’Kelly. Hub facility location with fixed costs. Papers in Regional Science: The Journal of the RSAI, 71:293-306, 1992.CrossRefGoogle Scholar
  30. 30.
    M. O’Kelly, D. Skorin-Kapov, and J. Skorin-Kapov. Lower bounds for the hub location problem. Management Science, 41:713-721, 1995.MATHCrossRefGoogle Scholar
  31. 31.
    H.D. Sherali, J. Desai, and H. Rakha. A discrete optimization approach for locating automatic vehicle identification readers for the provision of roadway travel times. Transportation Research Part B: Methodological, 40:857-871, 2006.CrossRefGoogle Scholar
  32. 32.
    H.D. Sherali and J.C. Smith. An improved linearization strategy for zero-one quadratic programming problems. Optimization Letters, 1:33-47, 2007.MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    D. Skorin-Kapov and J. Skorin-Kapov. On tabu search for the location of interacting hub facilities. European Journal of Operational Research, 73:502-509, 1994.MATHCrossRefGoogle Scholar
  34. 34.
    D. Skorin-Kapov, J. Skorin-Kapov, and M. O’Kelly. Tight linear programming relaxations of uncapacitated p-hub median problems. European Journal of Operational Research, 94:582-593, 1996.MATHCrossRefGoogle Scholar
  35. 35.
    L. Watters. Reduction of integer polynomial programming problems to zero-one linear programming problems. Operations Research, 15:1171-1174, 1967.CrossRefGoogle Scholar
  36. 36.
    W.I. Zangwill. Media selection by decision programming. Journal of Advertising Research, 5:30-36, 1965.Google Scholar

Copyright information

© Springer-Verlag US 2009

Authors and Affiliations

  • Xiaozheng He
    • 1
  • Anthony Chen
    • 2
  • Wanpracha Art Chaovalitwongse
    • 3
  • Henry Liu
    • 4
  1. 1.Department of Civil EngineeringUniversity of MinnesotaMinnesota
  2. 2.Department of Civil EngineeringUtah State UniversityLogan
  3. 3.Department of Industrial and Systems EngineeringRutgers UniversityPiscataway
  4. 4.Department of Civil EngineeringUniversity of MinnesotaMinnesota

Personalised recommendations