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A GRASP algorithm for multi container loading problems with practical constraints

  • M. T. AlonsoEmail author
  • R. Alvarez-Valdes
  • F. Parreño
Research Paper


We consider the multicontainer loading problem of a company that has to serve its customers by first putting the products on pallets and then loading pallets onto trucks. When a large number of units of a product have to be shipped, the company requires that homogeneous pallets, with only one product, are built first, then weakly heterogeneous pallets, in which each layer corresponds to a single product, and finally strongly heterogeneous pallets with the remaining units of the products. To be useful in practice, the solutions have to satisfy five types of constraints: geometric constraints, so that pallets are completely inside the trucks and do not overlap; weight constraints, limiting the total weight a truck can bear and the maximum weight supported by each axle; constraints limiting the position of the centre of gravity of the cargo; dynamic stability constraints, to avoid cargo displacement when the truck is moving; and constraints ensuring that the delivery dates of products are respected. We have developed a Greedy Randomized Adaptive Search Procedure, including some improvement methods tailored to the problem, among them an adaptation of ejection chains. The approach has been tested on a benchmark of real problems and it has been shown to be capable of finding high-quality, realistic solutions in short computing times. We also provide a comparison with an integer programming formulation that justifies the use of a metaheuristic algorithm.


Container loading Optimization Heuristics algorithm GRASP 

Mathematics Subject Classification

90B06 (Transportation and logistics) 



This study has been partially supported by the Spanish Ministry of Science and Technology DPI2014-53665-P and by Consejeria de Educacion y Ciencia, Junta de Comunidades de Castilla-La Mancha SBPLY/17/180501/000282.


  1. Alonso MT, Alvarez-Valdes R, Parreño F, Tamarit JM (2016) Algorithms for pallet building and truck loading in an inter-depot transportation problem. Math Probl Eng. Article ID 3264214Google Scholar
  2. Alonso MT, Alvarez-Valdes R, Iori M, Parreño F, Tamarit JM (2017) Mathematical models for multicontainer loading problems. Omega 66:106–117CrossRefGoogle Scholar
  3. Baldi MM, Perboli G, Tadei R (2012) The three-dimensional knapsack problem with balancing constraints. Appl Math Comput 218:9802–9818Google Scholar
  4. Bischoff EE, Ratcliff MSW (1995) Issues in the development of approaches to container loading. Omega 23(4):377–390CrossRefGoogle Scholar
  5. Bortfeldt A (2012) A hybrid algorithm for the capacitated vehicle routing problem with three-dimensional loading constraints. Comput Oper Res 39(9):2248–2257CrossRefGoogle Scholar
  6. Bortfeldt A, Wäscher G (2013) Constraints in container loading. A state of the art review. Eur J Oper Res 229(1):1–20CrossRefGoogle Scholar
  7. Contreras I, Tanash M, Vidyarthi N (2017) Exact and heuristic approaches for the cycle hub location problem. Ann Oper Res 258:655–677CrossRefGoogle Scholar
  8. Correcher JF, Alonso MT, Parreño F, Alvarez-Valdes R (2017) Solving a large multicontainer loading problem in the car manufacturing industry. Comput Oper Res 82(1):139–152CrossRefGoogle Scholar
  9. Doerner KF, Fuellerer G, Gronalt M, Hartl RF, Iori M (2007) Metaheuristics for the vehicle routing problem with loading constraints. Networks 49(4):294–307CrossRefGoogle Scholar
  10. Fanslau T, Bortfeldt A (2010) A tree search algorithm for solving the container loading problem. INFORMS J Comput 22(2):222–235CrossRefGoogle Scholar
  11. Feo T, Resende MGC, Smith SH (1994) A greedy randomized adaptive search procedure for maximum independent set. Oper Res 42(5):860–878CrossRefGoogle Scholar
  12. Gendreau M, Iori M, Laporte G, Martello S (2006) A tabu search algorithm for a routing and container loading problem. Transp Sci 40:342–350CrossRefGoogle Scholar
  13. Glover F (1996) Ejection chains, reference structures and alternating path methods for traveling salesman problems. Discrete Appl Math 65:223–253CrossRefGoogle Scholar
  14. Iori M, Martello S (2010) Routing problems with loading constraints. TOP 18(1):4–27CrossRefGoogle Scholar
  15. Iori M, Salazar González JJ, Vigo D (2007) An exact approach for the vehicle routing problem with two-dimensional loading constraints. Transp Sci 41:253–264CrossRefGoogle Scholar
  16. Junqueira L, Morabito R, Yamashita DS (2012) Three-dimensional container loading models with cargo stability and load bearing constraints. Comput Oper Res 39(1):74–85CrossRefGoogle Scholar
  17. Knopp S, Dauzere-Peres S, Yugma C (2017) A batch-oblivious approach for complex job-shop scheduling problems. Eur J Oper Res 263:50–61CrossRefGoogle Scholar
  18. Lim A, Ma H, Qiu C, Zhu W (2013) The single container loading problem with axle weight constraints. Int J Prod Econ 144(1):358–369CrossRefGoogle Scholar
  19. Lopez-Sanchez AD, Hernandez-Diaz AG, Gortazar F, Hinojosa MA (2018) A multiobjective GRASP/VND algorithm to solve the waste collection problem. Int Trans Oper Res 25:545–567CrossRefGoogle Scholar
  20. Moon I, Nguyen TVL (2014) Container packing with balance constraints. OR Spectr 36:837–878CrossRefGoogle Scholar
  21. Morabito R, Morales S (1998) A simple and effective recursive procedure for the manufacturer’s pallet loading problem. J Oper Res Soc 49(8):819–828CrossRefGoogle Scholar
  22. Morabito R, Morales S, Widmer J (2000) Loading optimization of palletized products on trucks. Transp Res Part E Logist Transp Rev 36(4):285–296CrossRefGoogle Scholar
  23. Moura A, Bortfeldt A (2017) A two-stage packing problem procedure. Int Trans Oper Res 24:43–58CrossRefGoogle Scholar
  24. Moura A, Oliveira JF (2005) A GRASP approach to the container-loading problem. IEEE Intell Syst 20(4):50–57CrossRefGoogle Scholar
  25. ORTEC (2018) Company. Accessed 04 Mar 2018
  26. Parreño F, Alvarez-Valdes R, Oliveira JF, Tamarit JM (2010) A hybrid GRASP/VND algorithm for two- and three-dimensional bin packing. Ann Oper Res 179:203–220CrossRefGoogle Scholar
  27. Peng B, Liu M, Lu Z, Kochengber G, Wang H (2016) An ejection chain approach for the quadratic multiple knapsack problem. Eur J Oper Res 253:328–336CrossRefGoogle Scholar
  28. Pisinger D (2000) A minimal algorithm for the bounded knapsack problem. INFORMS J Comput 12(1):75–82CrossRefGoogle Scholar
  29. Pollaris H, Braekers K, Caris A, Janssens G, Limbourg S (2016) Capacitated vehicle routing problem with sequence-based pallet loading and axle weight constraints. EURO J Transp Logist 5:231–255CrossRefGoogle Scholar
  30. Queiroz T, Miyazawa F (2013) Two-dimensional strip packing problem with load balancing, load bearing and multi-drop constraints. Int J Prod Econ 145:511–530CrossRefGoogle Scholar
  31. Queiroz T, Miyazawa F (2014) Order and static stability into the strip packing problem. Ann Oper Res 223:137–154CrossRefGoogle Scholar
  32. Ramos AG, Oliveira JF, Gonçalves JF, Lopes MP (2015) Dynamic stability metrics for the container loading problem. Transp Res Part C Emerg Technol 60:480–497CrossRefGoogle Scholar
  33. Ramos AG, Oliveira JF, Gonçalves JF, Lopes MP (2016) A container loading algorithm with static mechanical equilibrium stability constraints. Transp Res Part B Methodol 91:565–581CrossRefGoogle Scholar
  34. Ramos AG, Silva E, Oliveira JF (2018) A new load balance methodology for container loading problem in road transportation. Eur J Oper Res 266(3):1140–1152CrossRefGoogle Scholar
  35. Resende MGC, Werneck RF (2004) A hybrid heuristic for the p-median problem. J Heuristics 10:59–88CrossRefGoogle Scholar
  36. Sheng L, Hongxia Z, Xisong D, Changjian C (2016) A heuristic algorithm for container loading of pallets with infill boxes. Eur J Oper Res 252:728–736CrossRefGoogle Scholar
  37. Takahara S (2005) Loading problem in multiple containers and pallets using strategic search method. In: Torra V, Narukawa Y, Miyamoto S (eds) Modeling decisions for artificial intelligence, vol 3558. Lecture notes in computer science. Springer, Berlin, pp 448–456CrossRefGoogle Scholar
  38. Toffolo T, Esprit E, Wauters T, Vanden Berghe G (2018) A two-dimensional heuristic decomposition approach to a three-dimensional multiple container loading problem. Eur J Oper Res 257(2):526–538CrossRefGoogle Scholar
  39. Wäscher G, Haußner H, Schumann H (2007) An improved typology of cutting and packing problems. Eur J Oper Res 183(3):1109–1130CrossRefGoogle Scholar
  40. Zachariadis EE, Tarantilis CD, Kiranoudis CT (2012) The pallet-packing vehicle routing problem. Transp Sci 46:341–358CrossRefGoogle Scholar
  41. Zhao X, Bennell J, Betkas T, Dowsland K (2016) A comparative review of 3D container loading algorithms. Int Trans Oper Res 23:287–320CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Castilla-La ManchaAlbaceteSpain
  2. 2.Department of Statistics and Operations ResearchUniversity of ValenciaBurjassot, ValenciaSpain

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