Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

A column generation-based decomposition and aggregation approach for combining orders in inland transportation of containers

  • 5 Accesses


A significant portion of the total cost of the intermodal transportation is generated from the inland transportation of containers. In this paper, we design a mixed integer linear programming (MILP) model for combining orders in the inland, haulage transportation of containers. The pickup and delivery process of both 20 and 40 foot containers from the terminals to the customer locations and vice versa are optimized using heterogeneous fleet consisting of both 20 ft and 40 ft trucks/chasses. Important operational constraints such as the time window at order receivers, the payload weight of containers and the regulation of the working hours are considered. Based on an assignment problem structure, this MILP solves efficiently to optimality for problems with up to 120 orders. To deal with larger instances, a decomposition and aggregation heuristic is designed. The basic idea of this approach is to decompose order locations geographically into fan-shaped subareas based on the angle of the order location to the port baseline, and solve the sub problems using the proposed MILP model. To balance the fleet size amongst all subgroups, column generation is used to iteratively adjust the number of allocated trucks according to the shadow-price of each truck type. Based on decomposed solutions, orders that are “fully” combined with others are removed and an aggregation phase follows to enable wider combination choices across subgroups. The decomposition and aggregation solution process is tested to be both efficient and cost-saving.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7


  1. Braekers K, Janssens G, Caris A (2011) Challenges in managing empty container movements at multiple planning levels. Transp Rev 31(6):681–708

  2. Chung L, Pesenti R, Ukovich W (2006) Minimizing fleet operating costs for a container transportation company. Eur J Oper Res 171(3):776–786

  3. Chung KH, Ko CS, Shin JY, Hwang H, Kim KH (2007) Development of mathematical models for the container road transportation in Korean trucking industries. Comput Ind Eng 53(2):252–262

  4. Chung RK, Shi N, Powell WB, Simao HP (2008) An attribute-decision model for cross-border drayage problem. Transp Res E: Logist Transp Rev 44(2):217–234

  5. Crainic T, Kim K (2006) Intermodal transportation. Transportation 14:467–537

  6. Daganzo C (1984a) The distance traveled to visit n points with a maximum of c stops per vehicle: an analytic model and an application. Transp Sci 18(4):331–350

  7. Daganzo C (1984b) The length of tours in zones of different shapes. Transp Res B: Methodol 18(2):135–145

  8. Fang Z, Tu W, Li Q, Shaw S, Chen S, Chen B (2013) A voronoi neighborhood-based search heuristic for distance/capacity constrained very large vehicle routing problems. Int J Geogr Inf Sci 27(4):741–764

  9. Fisher M, Jaikumar R (1984) A generalized assignment heuristic for vehicle routing. Networks 11(2):109–124

  10. Ford L Jr, Fulkerson D (1958) A suggested computation for maximal multi commodity network flows. Manage Sci 5(1):97–101

  11. Funke J, Kopfer H (2015) A neighborhood search for a multi-size container transportation problem. IFAC-PapersOnLine 48(3):2041–2046

  12. Funke J, Kopfer H (2016) A model for a multi-size inland container transportation problem. Transp Res Part E Logist Transp Rev 89:70–85

  13. Gillett B, Miller L (1974) A heuristic algorithm for the vehicle-dispatch problem. Oper Res 22(2):340–349

  14. Gronalt M, Hartl R, Reimann M (2003) New savings based algorithms for time constrained pickup and delivery of full truckloads. Eur J Oper Res 151(3):520–535

  15. Hajem AD, Yang X, Warnes MK (2017) An efficient mixed integer programming model for pairing containers in inland transportation based on the assignment of orders. J Oper Res Soc 68(6):678–694

  16. Konings R (2005) Foldable containers to reduce the costs of empty transport? A cost-benefit analysis from a chain and multi-actor perspective. Marit Econ Logist 7(3):223–249

  17. Lai M (2013) Models and algorithms for the empty container repositioning and its integration with routing problems. PhD thesis, University of Cagliari

  18. Lai M, Crainic T, Di Francesco M, Zuddas P (2013) An heuristic search for the routing of heterogeneous trucks with single and double container loads. Transp Res Part E Logist Transp Rev 56:108–118

  19. Lun YV, Lai K, Cheng T (2010) Shipping and logistics management. Springer

  20. Lübbecke M, Zimmermann U (2003) Engine routing and scheduling at industrial in-plant railroads. Transp Sci 37:183–197

  21. Morlok E, Spasovic L (1994) Redesigning rail-truck intermodal drayage operations for enhanced service and cost performance. J Transp Res Forum 34(1):16–31

  22. Namboothiri R, Erera A (2008) Planning local container drayage operations given a port access appointment system. Transp Res E Logist Transp Rev 44:185–202

  23. Nordsieck N, Buer T, Schönberger, J (2016) Potential of improving truck-based drayage operations of marine terminals through street turns. Dyn Logist, pp 433–443

  24. Notteboom TE, Rodrigue J-P (2005) Port regionalization: towards a new phase in port development. Marit Policy Manage 32(3):297–313

  25. Parragh S, Doerner KF, Hartl RF (2008) A survey on pickup and delivery problems part I: transportation between customers and depot. J Betriebswirtschaft 58(1):21–51

  26. Parragh S, Doemer KF, Hartl RF (2008) A survey on pickup and delivery problems part II: transportation between pickup and delivery locations. J Betriebswirtschaft 58:81–117

  27. Popović D, Vidović M, Nikolić M (2014) The variable neighborhood search heuristic for the containers drayage problem with time windows. Soft computing in industrial applications, pp 351–364

  28. Reinhardt LB, Spoorendonk S, Pisinger D (2012) Solving vehicle routing with full container load and time windows. ICCL 2012: Computational Logistics, pp 120–128

  29. Reinhardt LB, Pisinger D, Spoorendonk S, Sigurd MM (2016) Optimization of the drayage problem using exact methods. Inf Syst Oper Res 54(1):33–51

  30. Shiri S, Huynh N (2016) Optimization of drayage operations with time-window constraints. Int J Prod Econ 176:7–20

  31. Shiri S, Huynh N (2018) Assessment of us chassis supply models on drayage productivity and air emissions. Transp Res D: Transp Environ 61:174–203

  32. Smilowitz K (2006) Multi-resource routing with flexible tasks: an application in drayage operations. IIE Trans 38(7):577–590

  33. Stahlbock R, VoB S (2008) Operations research at container terminals: a literature update. OR Spect 30(1):1–52

  34. Steenken D, Voß S, Stahlbock R (2004) Container terminal operation and operations research - a classification and literature review. OR spectrum 26(1):3–49

  35. UNCTAD (2015) Review of maritime transport. United Nations publication, New York and Geneva, Available at

  36. Vidović M, Radivojević G, Raković B (2011) Vehicle routing in containers pickup up and delivery processes. Proc Soc Behav Sci 20:335–343

  37. Vidović M, Nikolić M, Popović D (2012) Two mathematical formulations for the containers drayage problem with time windows. Int J Bus Sci Appl Manag 7(3):23–32

  38. Wang X, Regan CA (2002) Local truckload pickup and delivery with hard time window constraints. Transp Res B: Methodol 36:97–112

  39. Wen S, Zhou P (2007) A container vehicle routing model with variable traveling time. In: IEEE international conference on automation and logistics, 2007, pp 2243–2247

  40. Wren A, Holliday A (1972) Computer scheduling of vehicles from one or more depots to a number of delivery points. J Oper Res Soc 23(3):333–344

  41. Zhang R, Yun WY, Kopfer H (2015) Multi-size container transportation by truck: modeling and optimization. Flex Serv Manufact J 27(2–3):403–430

Download references

Author information

Correspondence to Hajem A. Daham.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yang, X., Daham, H.A. A column generation-based decomposition and aggregation approach for combining orders in inland transportation of containers. OR Spectrum (2020).

Download citation


  • Combining orders
  • Inland transportation
  • MILP model
  • Heuristic decomposition
  • Column generation