Advertisement

Annals of Operations Research

, Volume 199, Issue 1, pp 51–75 | Cite as

MIP-based approaches for the container loading problem with multi-drop constraints

  • Leonardo Junqueira
  • Reinaldo Morabito
  • Denise Sato Yamashita
Article

Abstract

In this paper, we present approaches based on a mixed integer linear programming model (MIP) for the problem of packing rectangular boxes into a container or truck, considering multi-drop constraints. We assume that the delivery route of the container is already known in advance and that the volume of the cargo is less than or equal to the container volume. Considering the sequence that the boxes should be unloaded, the aim is to avoid additional handling when each drop-off point of the route is reached, as well as ensuring that the boxes do not overlap each other and the cargo loading is stable. Computational tests with the proposed model and the approaches were performed with randomly generated instances and instances from the literature using an optimization solver embedded into a modeling language. The results validate the model and the approaches, but indicate that they are able to handle only problems of a moderate size. However, the model and the approaches can be useful to motivate future research to solve larger problems, as well as to solve more general problems considering integrated vehicle routing and container loading problems.

Keywords

Three-dimensional container loading Multi-drop constraints Cutting and packing problems Combinatorial optimization Mathematical modeling 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Beasley, J. E. (1985). An exact two-dimensional non-guillotine cutting tree search procedure. Operations Research, 33(1), 49–64. CrossRefGoogle Scholar
  2. Bischoff, E. E., & Marriott, M. D. (1990). A comparative evaluation of heuristics for container loading. European Journal of Operational Research, 44(2), 267–276. CrossRefGoogle Scholar
  3. Bischoff, E. E., & Ratcliff, M. S. W. (1995). Issues in the development of approaches to container loading. Omega, 23(4), 377–390. CrossRefGoogle Scholar
  4. Christensen, S. G., & Rousøe, D. M. (2009). Container loading with multi-drop constraints. International Transactions in Operational Research, 16(6), 727–743. CrossRefGoogle Scholar
  5. Christofides, N., & Whitlock, C. (1977). An algorithm for two-dimensional cutting problems. Operations Research, 25(1), 30–44. CrossRefGoogle Scholar
  6. Dowsland, W. B. (1991). Three-dimensional packing—solution approaches and heuristic development. International Journal of Production Research, 29(8), 1673–1685. CrossRefGoogle Scholar
  7. Eley, M. (2002). Solving container loading problems by block arrangement. European Journal of Operational Research, 141(2), 393–409. CrossRefGoogle Scholar
  8. Fuellerer, G., Doerner, K. F., Hartl, R. F., & Iori, M. (2010). Metaheuristics for vehicle routing problems with three-dimensional loading constraints. European Journal of Operational Research, 201(3), 751–759. CrossRefGoogle Scholar
  9. Gendreau, M., Iori, M., Laporte, G., & Martello, S. (2006). A tabu search algorithm for a routing and container loading problem. Transportation Science, 40(3), 342–350. CrossRefGoogle Scholar
  10. George, J. A., & Robinson, D. F. (1980). A heuristic for packing boxes into a container. Computers and Operations Research, 7(3), 147–156. CrossRefGoogle Scholar
  11. Haessler, R. W., & Talbot, F. B. (1990). Load planning for shipments of low density products. European Journal of Operational Research, 44(2), 289–299. CrossRefGoogle Scholar
  12. Han, C. P., Knott, K., & Egbelu, P. J. (1989). A heuristic approach to the three-dimensional cargo-loading problem. International Journal of Production Research, 27(5), 757–774. CrossRefGoogle Scholar
  13. Iori, M., Gonzalez, J. S., & Vigo, D. (2007). An exact approach for the vehicle routing problem with two-dimensional loading constraints. Transportation Science, 41(2), 253–264. CrossRefGoogle Scholar
  14. Jin, Z., Ohno, K., & Du, J. (2004). An efficient approach for the three-dimensional container packing problem with practical constraints. Asia-Pacific Journal of Operational Research, 21(3), 279–295. CrossRefGoogle Scholar
  15. Junqueira, L., Morabito, R., & Yamashita, D. S. (2012). Three-dimensional container loading models with cargo stability and load bearing constraints. Computers and Operations Research, 31(1), 74–85. CrossRefGoogle Scholar
  16. Lai, K. K., Xue, J., & Xu, B. (1998). Container packing in a multi-customer delivering operation. Computers & Industrial Engineering, 35(1–2), 323–326. CrossRefGoogle Scholar
  17. Lin, J. L., Chang, C. H., & Yang, J. Y. (2006). A study of optimal system for multiple-constraint multiple-container packing problems. In Proceedings of the 19th international conference on industrial, engineering and other applications of applied intelligent systems, Annecy (Vol. 4031, pp. 1200–1210). Google Scholar
  18. Lins, L., Lins, S., & Morabito, R. (2002). An n-tet graph approach for non-guillotine packing of n-dimensional boxes into an n-container. European Journal of Operational Research, 141(2), 421–439. CrossRefGoogle Scholar
  19. Miyazawa, F. K., & Wakabayashi, Y. (1999). Approximation algorithms for the orthogonal Z-oriented three-dimensional packing problem. SIAM Journal on Computing, 29(3), 1008–1029. CrossRefGoogle Scholar
  20. Morabito, R., & Arenales, M. (1994). An and/or-graph approach to the container loading problem. International Transactions in Operational Research, 1(1), 59–73. CrossRefGoogle Scholar
  21. Moura, A., & Bortfeldt, A. (2009). A packing and routing application for a Portuguese trading company. In Book of abstracts of the 6th ESICUP meeting, Valencia. Google Scholar
  22. Moura, A., & Oliveira, J. F. (2005). A GRASP approach to the container-loading problem. IEEE Intelligent Systems, 4(20), 50–57. Google Scholar
  23. Moura, A., & Oliveira, J. F. (2009). An integrated approach to the vehicle routing and container loading problems. OR-Spektrum, 31(4), 775–800. CrossRefGoogle Scholar
  24. Parreño, F., Alvarez-Valdes, R., Oliveira, J. F., & Tamarit, J. M. (2010). A hybrid GRASP/VND algorithm for two- and three-dimensional bin packing. Annals of Operation Research, 179(1), 203–220. CrossRefGoogle Scholar
  25. Scheithauer, G., Terno, J., Riehme, J., & Sommerweiss, U. (1996). A new heuristic approach for solving the multi-pallet packing problem. Technical report, MATH-NM-03-1996. Technische Universität Dresden, Dresden. Google Scholar
  26. Silva, J. L. C., Soma, N. Y., & Maculan, N. (2003). A greedy search for the three-dimensional bin packing problem: the packing static stability case. International Transactions in Operational Research, 10(2), 141–153. CrossRefGoogle Scholar
  27. Tarantilis, C. D., Zachariadis, E. E., & Kiranoudis, C. T. (2009). A hybrid metaheuristic algorithm for the integrated vehicle routing and three-dimensional container-loading problem. IEEE Transactions on Intelligent Transportation Systems, 10(2), 255–271. CrossRefGoogle Scholar
  28. Terno, J., Scheithauer, G., Sommerweiss, U., & Riehme, J. (2000). An efficient approach for the multi-pallet loading problem. European Journal of Operational Research, 123(2), 372–381. CrossRefGoogle Scholar
  29. Tetris (2009). 25th anniversary. Available at: http://www.tetris.com. Accessed in: 14 out 2009.

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Leonardo Junqueira
    • 1
  • Reinaldo Morabito
    • 1
  • Denise Sato Yamashita
    • 1
  1. 1.Departamento de Engenharia de ProduçãoUniversidade Federal de São CarlosSão CarlosBrazil

Personalised recommendations