Neural Computing and Applications

, Volume 29, Issue 9, pp 511–532 | Cite as

Robust bi-objective optimization of uncapacitated single allocation p-hub median problem using a hybrid heuristic algorithm

  • Mohammad Reza Amin-Naseri
  • Amin Yazdekhasti
  • Ali Salmasnia
Original Article


The p-hub median problem aims at locating p-hub facilities in a network and allocating non-hub nodes to the hubs such that the overall transportation cost is minimized. One issue of major importance in this problem remarks the requirement to deal with uncertain factors such as weather conditions and traffic volume. These lead to uncertainty in travel time between origin and destination points. In today’s competitive markets in which customers look for robust delivery services, it is important to minimize the upper bound of uncertainty in the network routes. In this paper, a robust bi-objective uncapacitated single allocation p-hub median problem (RBUSApHMP) is introduced in which travel time has non-deterministic nature. The problem aims to select location of the hubs and allocation of the other nodes to them so that overall transportation cost and maximum uncertainty in network are minimized. To do this, a desirability function-based approach is suggested that ensures both interested objectives to fall within their specification limits. Due to the complexity of the model, a heuristic based on scatter search and variable neighborhood descent is developed. To evaluate the performance of the proposed method a computational analysis on Civil Aeronautics Board and Australian Post data sets was performed. The obtained results using the proposed hybrid metaheuristic are compared to those of the optimum solutions obtained using GAMS. The results indicate excellent performance of the suggested solution procedure to optimize RBUSApHMP.


p-Hub median problem Uncertainty Desirability function Scatter search Variable neighborhood descent 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Alumur SA, Kara BY (2008) Network hub location problems: the state of the art. Eur J Oper Res 190(1):1–21MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alumur SA, Nickel S, Saldanha-da-Gama F (2012) Hub location under uncertainty. Transp Res Part B 46(4):529–543CrossRefGoogle Scholar
  3. 3.
    Aykin T (1994) Lagrangian relaxation based approaches to capacitated hub-and-spoke network design problem. Eur J Oper Res 79:501–523CrossRefzbMATHGoogle Scholar
  4. 4.
    Ben-Tal A, Nemirovski A (1998) Robust convex optimization. Math Oper Res 23:769–805MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ben-Tal A, Nemirovski A (1999) Robust solutions of uncertain linear programs. Oper Res Lett 25:1–13MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ben-Tal A, Nemirovski A (2000) Robust solutions of linear programming problems contaminated with uncertain data. Math Program 88:411–424MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bertsimas D, Sim M (2003) Robust discrete optimization and network flows. Math Program 98:49–71MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Brimberg J, Mladenović N, Todosijević R et al (2016) General variable neighborhood search for the uncapacitated single allocation p-hub center problem. Optim Lett. doi: 10.1007/s11590-016-1004-x zbMATHGoogle Scholar
  9. 9.
    Campbell JF (1991) Hub location problems and the p-hub median problem. Center for Business and Industrial Studies, University of Missouri–St. Louis, St. LouiszbMATHGoogle Scholar
  10. 10.
    Campbell JF (1992) Location and allocation for distribution systems with transshipments and transportation economies of scale. Ann Oper Res 40:77–99CrossRefzbMATHGoogle Scholar
  11. 11.
    Campbell JF (1994) Integer programming formulations of discrete hub location problems. Eur J Oper Res 72:387–405CrossRefzbMATHGoogle Scholar
  12. 12.
    Campbell JF (1996) Hub location and p-hub median problem. Oper Res 44(6):923–935MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Campbell JF, Ernst AT, Krishnamoorthy M (2002) Hub location problems. In: Drezner Z, Hammacher H (eds) Facility location: applications and theory. Springer, BerlinGoogle Scholar
  14. 14.
    Campbell AM, Lowe TJ, Zhang L (2007) The p-hub center allocation problem. Eur J Oper Res 176(2):819–835MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chou CC (2010) Application of FMCDM model to selecting the hub location in the marine transportation: a case study in southeastern Asia. Math Comput Model 51:791–801CrossRefzbMATHGoogle Scholar
  16. 16.
    Contreras I, Cordeau JF, Laporte G (2011) Stochastic uncapacitated hub location. Eur J Oper Res 212:518–528MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cunha CB, Silva MR (2007) A genetic algorithm for the problem of configuring a hub-and-spoke network for a LTL trucking company in Brazil. Eur J Oper Res 179:747–758CrossRefzbMATHGoogle Scholar
  18. 18.
    Diaz JA, Fernandez E (2006) Hybrid scatter search and path relinking for the capacitated p-median problem. Eur J Oper Res 169:570–585MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ebery J (2001) Solving large single allocation p-hub problems with two or three hubs. Eur J Oper Res 128:447–458CrossRefzbMATHGoogle Scholar
  20. 20.
    Elhedhli S, Hu FX (2005) Hub-and-spoke network design with congestion. Comput Oper Res 32:1615–1632CrossRefzbMATHGoogle Scholar
  21. 21.
    Ernst AT, Hamacher H, Jiang H, Krishnamoorthy M, Woeginger G (2009) Uncapacitated single and multiple allocation p-hub center problems. Comput Oper Res 36:2230–2241MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ernst A, Hamacher H, Jiang H, Krishnamoorthy M, Woeginger G (2002) Uncapacitated single and multiple allocation \( p \)-hub center problems. Unpublished Report, CSIRO Mathematical and Information Sciences, AustraliaGoogle Scholar
  23. 23.
    Ernst AT, Krishnamoorthy M (1996) Efficient algorithms for the uncapacitated single allocation p-hub median problem. Locat Sci 4:139–154CrossRefzbMATHGoogle Scholar
  24. 24.
    Garfinkel RS, Sundararaghavan PS, Noon C, Smith DR (1996) Optimal use of hub facilities: a two-hub model with fixed arc costs. Top 4:331–343MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Geyik F, Dosdoğru AT (2013) Process plan and part routing optimization in a dynamic flexible job shop scheduling environment: an optimization via simulation approach. Neural Comput Appl 23:1631–1641CrossRefGoogle Scholar
  26. 26.
    Ge W, Zhu J-F (2012) Research on robust optimization model of capacitated hub-and-spoke network design problem. Adv Inf Sci Serv Sci 4(13):379–386Google Scholar
  27. 27.
    Ghaffari-nasab N, Ghazangari M, Teimouri E (2015) Robust optimization approach to the design of hub-and-spoke networks. Int J Adv Manuf Technol 76:1091–1110CrossRefGoogle Scholar
  28. 28.
    Ghaoui LE, Lebret H (1997) Robust solutions to least-squares problems with uncertain data. SIAM J Matrix Anal Appl 18:1035–1064MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Glover F (1977) Heuristics for integer programming using surrogate constraints. Decis Sci 8:156–166CrossRefGoogle Scholar
  30. 30.
    Hamacher HW, Meyer T (2006) Hub cover and hub center problems. Working paper. Department of Mathematics, University of Kaiserslautern, Gottlieb-Daimler-Strasse, 67663 KaiserslauternGoogle Scholar
  31. 31.
    Hwang HY, Lee HY (2012) Uncapacitated single allocation p-hub maximal covering problem. Comput Ind Eng 63(2):382–389CrossRefGoogle Scholar
  32. 32.
    Ilić A, Urošević D, Brimberg J, Mladenovic N (2010) A general variable neighborhood search for solving the uncapacitated single allocation p-hub median problem. Eur J Oper Res 206:289–300MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Junior GDM, Chamargo RSD, Pinto LR, Conceic SV, Ferreira RPM (2011) Hub location under hub congestion and demand uncertainty: the Brazilian study. Pesqui Oper 31:319–349CrossRefGoogle Scholar
  34. 34.
    Jun Z, Yu-An T, Xue-Lan Z, Jun L (2010) An improved dynamic structure-based neural networks determination approaches to simulation optimization problems. Neural Comput Appl 19:883–901CrossRefGoogle Scholar
  35. 35.
    Kara BY, Tansel BC (2000) On the single assignment p-hub center problem. Eur J Oper Res 125(3):648–655CrossRefzbMATHGoogle Scholar
  36. 36.
    Kara BY, Tansel BC (2001) The latest arrival hub location problem. Manag Sci 47:1408–1420CrossRefzbMATHGoogle Scholar
  37. 37.
    Kara BY, Tansel BC (2003) The single-assignment hub covering problem: models and linearizations. J Oper Res Soc 54(1):59–64CrossRefzbMATHGoogle Scholar
  38. 38.
    Karimi H, Bashiri M (2011) Hub covering location problems with different coverage types. Sci Iran 18(6):1571–1578CrossRefGoogle Scholar
  39. 39.
    Kim KJ, Lin DKJ (2006) Optimization of multiple responses considering both location and dispersion effects. Eur J Oper Res 169:133–145MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Köksalan M, Soylu B (2010) Bicriteria p-Hub location problems and evolutionary algorithms. INFORMS J Comput 22:528–542MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Kratica J (2013) An electromagnetism-like metaheuristic for the uncapacitated multiple allocation p-hub median problem. Comput Ind Eng 66:1015–1024CrossRefGoogle Scholar
  42. 42.
    Love RF, Morris JG, Wesolowsky G (1996) Facility location: models and methods. Publications in Operations Research, New YorkGoogle Scholar
  43. 43.
    Laguna M, Marti R (2003) Scatter search: methodology and implementations in C. Kluwer, BostonCrossRefzbMATHGoogle Scholar
  44. 44.
    Marianov V, Serra D (2003) Location models for airline hubs behaving as M/D/c queues. Comput Oper Res 30:983–1003CrossRefzbMATHGoogle Scholar
  45. 45.
    Marti R, Corberan Á, Perio J (2015) Scatter search for an uncapacitated p-hub median problem. Comput Oper Res 58:53–66MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Meyer T, Ernst AT, Krishnamoorthy M (2009) A 2-phase algorithm for solving the single allocation p-hub center problem. Comput Oper Res 36:3143–3151CrossRefzbMATHGoogle Scholar
  47. 47.
    Mohammadi M, Jolai F, Rostami H (2011) An M/M/c queue model for hub covering location problem. Math Comput Model 54:2623–2638MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    O’Kelly ME (1987) A quadratic integer program for the location of interacting hub facilities. Eur J Oper Res 32:393–404MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    O’Kelly ME (1992) Hub facility location with fixed costs. Pap Reg Sci 71:292–306Google Scholar
  50. 50.
    Peiro J, Corberan Á, Marti R (2014) GRASP for the uncapacitated r-allocation p-hub median problem. Comput Oper Res 43:50–60MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Peker M, Kara BY (2015) The P-Hub maximal covering problem and extensions for gradual decay functions. Omega 54:158–172CrossRefGoogle Scholar
  52. 52.
    Sadeghi M, Jolai F, Tavakkoli-Moghaddam R, Rahimi Y (2015) A new stochastic approach for a reliable p-hub covering location problem. Comput Ind Eng 90:371–380CrossRefGoogle Scholar
  53. 53.
    Salmasnia A, Bashiri M, Salehi M (2013) A robust interactive approach to optimize correlated multiple responses. Int J Adv Manuf Technol 67:1923–1935CrossRefGoogle Scholar
  54. 54.
    Salmasnia A, Bashiri M (2015) A new desirability function-based method for correlated multiple response optimization. Int J Adv Manuf Technol 76:1047–1062CrossRefGoogle Scholar
  55. 55.
    Sender J, Clausen U (2013) Heuristics for solving a capacitated multiple allocation hub location problem with application in German wagonload traffic. Electron Notes Discrete Math 41:13–20CrossRefGoogle Scholar
  56. 56.
    Silva MR, Cunha CB (2009) New simple and efficient heuristics for the uncapacitated single allocation hub location problem. Comput Oper Res 36:3152–3165CrossRefzbMATHGoogle Scholar
  57. 57.
    Sim T, Lowe TJ, Thomas BW (2009) The stochastic p-hub center problem with service-level constraints. Comput Oper Res 36:3166–3177CrossRefzbMATHGoogle Scholar
  58. 58.
    Soyster AL (1973) Convex programming with set-inclusive constraints and applications to inexact linear programming. Oper Res 21:1154–1157MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Staojevic P, Marić M, Stanimirovic Z (2015) A hybridization of an evolutionary algorithm and a parallel branch and bound for solving the capacitated single allocation hub location problem. Appl Soft Comput 33:24–36CrossRefGoogle Scholar
  60. 60.
    Talbi EG (2009) Metaheuristics from design to implementation. Wiley, HobokenzbMATHGoogle Scholar
  61. 61.
    Topcuoglu H, Corut F, Ermis M, Yilmaz G (2005) Solving the uncapacitated hub location using genetic algorithms. Comput Oper Res 32:467–984CrossRefzbMATHGoogle Scholar
  62. 62.
    Wagner B (2004) Model formulations for hub covering problems. Working paper, Institute of Operations Research, Darmstadt University of Technology, Hochschulstrasse 1, 64289 DarmstadtGoogle Scholar
  63. 63.
    Wolf S, Merz P (2007) Evolutionary local search for the super-peer selection problem and the p-hub median problem. In: Bartz-Beielstein et al. (eds) Proceedings of the 4th international workshop on hybrid metaheuristics—HM2007, Lecture Notes in Computer Science, Springer, Berlin, pp 1–15Google Scholar
  64. 64.
    Yaman H, Elloumi S (2012) Star p-hub center problem and star p-hub median problem with bounded path lengths. Comput Oper Res 39(11):2725–2732MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Yang K, Liu Y, Yang G (2014) Optimizing fuzzy p-hub center problem with generalized value-at-risk criterion. Appl Math Model 38:3987–4005MathSciNetCrossRefGoogle Scholar
  66. 66.
    Yang K, Liu Y-K, Yang G-Q (2013) Solving fuzzy p-hub center problem by genetic algorithm incorporating local search. Appl Soft Comput 13:2624–2632CrossRefGoogle Scholar
  67. 67.
    Yang TH (2009) Stochastic air freight hub location and flight routes planning. Appl Math Model 33:4424–4430CrossRefzbMATHGoogle Scholar
  68. 68.
    Zanjirani Farahani R, Hekmatfar H, Boloori Arabani A, Nikbakhsh E (2013) Hub location problems: a review of models, classification, solution techniques, and applications. Comput Ind Eng 64(4):1096–1109CrossRefGoogle Scholar

Copyright information

© The Natural Computing Applications Forum 2016

Authors and Affiliations

  • Mohammad Reza Amin-Naseri
    • 1
  • Amin Yazdekhasti
    • 1
  • Ali Salmasnia
    • 2
  1. 1.Department of Industrial Engineering, Faculty of EngineeringTarbiat Modares UniversityTehranIran
  2. 2.Department of Industrial Engineering, Faculty of Engineering and TechnologyUniversity of QomQomIran

Personalised recommendations