Annals of Operations Research

, Volume 264, Issue 1–2, pp 57–87 | Cite as

Dantzig–Wolfe decomposition approach to the vehicle assignment problem with demand uncertainty in a hybrid hub-and-spoke network

  • Jiyoung Choi
  • Chungmok Lee
  • Sungsoo Park
Original Research


In this article, we investigate the vehicle assignment problem with demand uncertainty in a hybrid hub-and-spoke network with a single hub. The problem is deciding both the transportation routes and the number and types of vehicles to be deployed to minimize the sum of costs to transport all quantities in a hybrid hub-and-spoke network which allows direct transportation between spokes. Daily changes in quantities are reflected with a finite number of scenarios. Regularly scheduled vehicles and temporarily scheduled vehicles are considered to meet the demand variation. We propose a Dantzig–Wolfe decomposition approach which yields a strong LP relaxation bound by introducing a set of feasible direct route patterns. We develop an algorithm which incorporates a column generation procedure at the root node and repeats iteratively a variable fixing and column generation procedure at the non-root nodes until an integral solution is found. Finally, we present computational results using the well-known CAB data sets and real-life data from the Korea Post. The results show that our algorithm can find near optimal solutions very efficiently.


Hybrid hub-and-spoke Vehicle assignment problem Column generation Demand uncertainty 



This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (NRF-2015R1C1A1A01054606). One of the authors (Chungmok Lee) was also supported by Hankuk University of Foreign Studies Research Fund of 2017.


  1. Alumur, S., & Kara, B. (2008). Network hub location problems: The state of the art. European Journal of Operational Research, 190(1), 1–21.CrossRefGoogle Scholar
  2. Aykin, T. (1995a). The hub location and routing problem. European Journal of Operational Research, 83(1), 200–219.CrossRefGoogle Scholar
  3. Aykin, T. (1995b). Networking policies for hub-and-spoke systems with application to the air transportation system. Transportation Science, 29(3), 201–221.CrossRefGoogle Scholar
  4. Bai, R., Wallace, S. W., Li, J., & Chong, A. Y.-L. (2014). Stochastic service network design with rerouting. Transportation Research Part B: Methodological, 60, 50–65.CrossRefGoogle Scholar
  5. Birge, J. R., & Louveaux, F. (2011). Introduction to stochastic programming. New York: Springer.CrossRefGoogle Scholar
  6. Büther, M., & Briskorn, D. (2012). Reducing the 0–1 knapsack problem with a single continuous variable to the standard 0–1 knapsack problem. International Journal of Operations Research and Information Systems (IJORIS), 3(1), 1–12.CrossRefGoogle Scholar
  7. Campbell, J. (1994). Integer programming formulations of discrete hub location problems. European Journal of Operational Research, 72(2), 387–405.CrossRefGoogle Scholar
  8. Campbell, J., & O’Kelly, M. (2012). Twenty-five years of hub location research. Transportation Science, 46(2), 153–169.CrossRefGoogle Scholar
  9. Chong, L., Kennedy, D., & Chan, W. (2006). Direct shipping logistic planning for a hub-and-spoke network with given discrete intershipment times. International Transactions in Operational Research, 13(1), 17–32.CrossRefGoogle Scholar
  10. Contreras, I., Cordeau, J., & Laporte, G. (2011). Stochastic uncapacitated hub location. European Journal of Operational Research, 212(3), 518–528.CrossRefGoogle Scholar
  11. Contreras, I., Fernández, E., & Marín, A. (2010). The tree of hubs location problem. European Journal of Operational Research, 202(2), 390–400.CrossRefGoogle Scholar
  12. Crainic, T. G., Gendreau, M., Soriano, P., & Toulouse, M. (1993). A tabu search procedure for multicommodity location/allocation with balancing requirements. Annals of Operations Research, 41(4), 359–383.CrossRefGoogle Scholar
  13. Crainic, T. G., Hewitt, M., & Rei, W. (2014a). Scenario grouping in a progressive hedging-based meta-heuristic for stochastic network design. Computers & Operations Research, 43, 90–99.CrossRefGoogle Scholar
  14. Crainic, T. G., Hewitt, M., Toulouse, M., & Vu, D. M. (2014b). Service network design with resource constraints. Transportation Science, 50(4), 1380–1393.CrossRefGoogle Scholar
  15. Crainic, T. G., Hewitt, M., Toulouse, M., & Vu, D. M. (2016). Scheduled service network design with resource acquisition and management. EURO Journal on Transportation and Logistics, 1–33.Google Scholar
  16. Dantzig, G. B., & Wolfe, P. (1960). Decomposition principle for linear programs. Operations Research, 8(1), 101–111.CrossRefGoogle Scholar
  17. Desaulniers, G., Desrosiers, J., & Solomon, M. M. (2005). Column generation (Vol. 5). Berlin: Springer.CrossRefGoogle Scholar
  18. Ebery, J., Krishnamoorthy, M., Ernst, A., & Boland, N. (2000). The capacitated multiple allocation hub location problem: Formulations and algorithms. European Journal of Operational Research, 120(3), 614–631.CrossRefGoogle Scholar
  19. Ghamlouche, I., Crainic, T. G., & Gendreau, M. (2004). Path relinking, cycle-based neighbourhoods and capacitated multicommodity network design. Annals of Operations Research, 131(1), 109–133.CrossRefGoogle Scholar
  20. Gorman, M. F. (1998). An application of genetic and tabu searches to the freight railroad operating plan problem. Annals of Operations Research, 78, 51–69.CrossRefGoogle Scholar
  21. Hadjar, A., Marcotte, O., & Soumis, F. (2006). A branch-and-cut algorithm for the multiple depot vehicle scheduling problem. Operations Research, 54(1), 130–149.CrossRefGoogle Scholar
  22. Hoff, A., Lium, A.-G., Løkketangen, A., & Crainic, T. G. (2010). A metaheuristic for stochastic service network design. Journal of Heuristics, 16(5), 653–679.CrossRefGoogle Scholar
  23. Hult, E. (2011). Stochastic hub and spoke networks. Ph.D. thesis, University of Cambridge.Google Scholar
  24. Kall, P., & Wallace, S. W. (1994). Stochastic programming. New York: John Wiley and Sons Ltd.Google Scholar
  25. Kellerer, H., Pferschy, U., & Pisinger, D. (2004). Knapsack problems. Berlin: Springer.CrossRefGoogle Scholar
  26. Lee, T., Park, S., & Lee, K. (2005). Routing and wavelength assignment in survivable WDM networks without wavelength conversion. International Journal of Management Science, 11(2), 85–103.Google Scholar
  27. Li, X., Wei, K., Aneja, Y., & Tian, P. (2017). Design-balanced capacitated multicommodity network design with heterogeneous assets. Omega, 67, 145–159.CrossRefGoogle Scholar
  28. Liu, J., Li, C., & Chan, C. (2003). Mixed truck delivery systems with both hub-and-spoke and direct shipment. Transportation Research Part E: Logistics and Transportation Review, 39(4), 325–339.CrossRefGoogle Scholar
  29. Lium, A., Crainic, T., & Wallace, S. (2009). A study of demand stochasticity in service network design. Transportation Science, 43(2), 144–157.CrossRefGoogle Scholar
  30. Lübbecke, M. E., & Desrosiers, J. (2005). Selected topics in column generation. Operations Research, 53(6), 1007–1023.CrossRefGoogle Scholar
  31. Lumsden, K., Dallari, F., & Ruggeri, R. (1999). Improving the efficiency of the hub and spoke system for the SKF European distribution network. International Journal of Physical Distribution & Logistics Management, 29(1), 50–66.CrossRefGoogle Scholar
  32. Marianov, V., & Serra, D. (2003). Location models for airline hubs behaving as M/D/c queues. Computers & Operations Research, 30(7), 983–1003.CrossRefGoogle Scholar
  33. O’Kelly, M. (1986). The location of interacting hub facilities. Transportation Science, 20(2), 92–106.CrossRefGoogle Scholar
  34. O’Kelly, M. (1987). A quadratic integer program for the location of interacting hub facilities. European Journal of Operational Research, 32(3), 393–404.CrossRefGoogle Scholar
  35. Santoso, T., Ahmed, S., Goetschalckx, M., & Shapiro, A. (2005). A stochastic programming approach for supply chain network design under uncertainty. European Journal of Operational Research, 167(1), 96–115.CrossRefGoogle Scholar
  36. Sheu, J.-B., & Lin, A. Y.-S. (2012). Hierarchical facility network planning model for global logistics network configurations. Applied Mathematical Modelling, 36(7), 3053–3066.CrossRefGoogle Scholar
  37. Sim, T., Lowe, T., & Thomas, B. (2009). The stochastic p-hub center problem with service-level constraints. Computers & Operations Research, 36(12), 3166–3177.CrossRefGoogle Scholar
  38. Snyder, L. (2006). Facility location under uncertainty: A review. IIE Transactions, 38(7), 547–564.CrossRefGoogle Scholar
  39. Vanderbeck, F. (2005). Implementing mixed integer column generation. In G. Desaulniers, J. Desrosiers, & M. M. Solomon (Eds.), Column generation (pp. 331–358). Heidelberg: Springer.CrossRefGoogle Scholar
  40. Vanderbeck, F., & Savelsbergh, M. W. (2006). A generic view of Dantzig–Wolfe decomposition in mixed integer programming. Operations Research Letters, 34(3), 296–306.CrossRefGoogle Scholar
  41. Wasner, M., & Zäpfel, G. (2004). An integrated multi-depot hub-location vehicle routing model for network planning of parcel service. International Journal of Production Economics, 90(3), 403–419.CrossRefGoogle Scholar
  42. Wieberneit, N. (2008). Service network design for freight transportation: A review. OR Spectrum, 30(1), 77–112.CrossRefGoogle Scholar
  43. Yang, T. (2009). Stochastic air freight hub location and flight routes planning. Applied Mathematical Modelling, 33(12), 4424–4430.CrossRefGoogle Scholar
  44. Zäpfel, G., & Wasner, M. (2002). Planning and optimization of hub-and-spoke transportation networks of cooperative third-party logistics providers. International Journal of Production Economics, 78(2), 207–220.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Broadcasting and Media Research LaboratoryETRIYuseong-guRepublic of Korea
  2. 2.Department of Industrial and Management EngineeringHankuk University of Foreign StudiesYongin-siRepublic of Korea
  3. 3.Department of Industrial and Systems EngineeringKAISTYuseong-guRepublic of Korea

Personalised recommendations