A Logic-Based Benders Decomposition Approach to Improve Coordination of Inland Vessels for Inter-Terminal Transport

  • Shijie LiEmail author
  • Rudy R. Negenborn
  • Gabriel Lodewijks
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9855)


Large seaports usually contain multiple terminals serving container vessels, railways, trucks and other modes of hinterland transportation. Every time an inland vessel enters a seaport, it visits several terminals for loading and unloading containers. A vessel rotation is the sequence in which a vessel visits the different terminals in a large seaport. Currently, in a seaport like the port of Rotterdam, around 40 % of the inland vessels have to spend a longer time in the port area than originally planned, due to the low utilization of terminal quay resources and uncertainty of waiting times at different terminals. To better utilize the terminal resources in the ports, as well as to reduce the amount of time inland vessels spend in the port area, this paper first proposes a new model in which inland vessels coordinate with each other with respect to the arrival, departure time and the number of inter-terminal containers carried, besides their conventional hinterland containers, with the aim to prevent possible conflicts of their rotations. Then, a logic-based Benders’ decomposition approach is proposed to minimize the total time the inland vessels spent in the port. We compare the performance of the proposed approach with the performance of a centralized approach on the aspects of the runtime, solution quality, and three logistical performance indicators. Simulation results show that the proposed approach generates both faster optimal and faster high-quality solutions than the centralized approach in both small and large problem instance.


Master Problem Port Area Centralize Approach Bender Decomposition Total Waiting Time 
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.



This research is supported by the China Scholarship Council under Grant 201206680009.


  1. 1.
    Benders, J.: Partitioning procedures for solving mixed-variables programming problems. Numer. Math. 4(1), 238–252 (1962)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Chu, Y., Xia, Q.: Generating benders cuts for a general class of integer programming problems. In: Régin, J.-C., Rueher, M. (eds.) CPAIOR 2004. LNCS, vol. 3011, pp. 127–141. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  3. 3.
    Douma, A., Schutten, M., Schuur, P.: Waiting profiles: an efficient protocol for enabling distributed planning of container barge rotations along terminals in the port of Rotterdam. Transp. Res. Part C Emerg. Technol. 17(2), 133–148 (2009)CrossRefGoogle Scholar
  4. 4.
    Douma, A.M.: Aligning the operations of barges and terminals through distributed planning. Ph.D. thesis, University of Twente, Enschede, The Netherlands, December 2008Google Scholar
  5. 5.
    Duinkerken, M., Dekker, R., Kurstjens, S., Ottjes, J., Dellaert, N.: Comparing transportation systems for inter-terminal transport at the maasvlakte container terminals. In: Kim, K., Gunther, H.O. (eds.) Container Terminals and Cargo Systems, pp. 37–61. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  6. 6.
    Fazel-Zarandi, M.M., Beck, J.C.: Using logic-based benders decomposition to solve the capacity- and distance-constrained plant location problem. INFORMS J. Comput. 24(3), 387–398 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Hooker, J., Ottosson, G.: Logic-based benders decomposition. Math. Program. 96(1), 33–60 (2003)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Hooker, J.: Logic-Based Methods for Optimization: Combining Optimization and Constraint Satisfaction. Wiley, New York (2000)CrossRefzbMATHGoogle Scholar
  9. 9.
    Moonen, H., Van de Rakt, B., Miller, I., Van Nunen, J., Van Hillegersberg, J.: Agent technology supports inter-organizational planning in the port. Technical report, ERIM Report Series Reference No. ERS-2005-027-LIS (2005)Google Scholar
  10. 10.
    Nieuwkoop, F., Corman, F., Negenborn, R., Duinkerken, M., van Schuylenburg, M., Lodewijks, G.: Decision support for vehicle configuration determination in inter terminal transport system design. In: Proceedings of the 2014 IEEE International Conference on Networking, Sensing, and Control (ICNSC 2014), Miami, Florida, pp. 613–618, April 2014Google Scholar
  11. 11.
    Port of Rotterdam Authority: Container terminals and depots in port of Rotterdam (2011).
  12. 12.
    Rossi, F., Beek, P.V., Walsh, T.: Handbook of Constraint Programming. Elsevier (2006)Google Scholar
  13. 13.
    Schroer, H., Corman, F., Duinkerken, M., Negenborn, R., Lodewijks, G.: Evaluation of inter terminal transport configurations at Rotterdam Maasvlakte using discrete event simulation. In: Proceedings of the Winter Simulation Conference 2014 (WSC 2014), Savannah, Georgia, pp. 1771–1782, December 2014Google Scholar
  14. 14.
    Tierney, K., Voß, S., Stahlbock, R.: A mathematical model of inter-terminal transportation. Eur. J. Oper. Res. 235(2), 448–460 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Wheatley, D., Fatma, G., Jewkes, E.: Logic-based benders decomposition for an inventory-location problem with service constraints. Omega 55, 10–23 (2015)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Shijie Li
    • 1
    Email author
  • Rudy R. Negenborn
    • 1
  • Gabriel Lodewijks
    • 1
  1. 1.Department of Maritime and Transport TechnologyDelft University of TechnologyDelftThe Netherlands

Personalised recommendations