Skip to main content
SpringerLink
Log in

A tree search algorithm for solving the multi-dimensional strip packing problem with guillotine cutting constraint

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

The article presents a tree search algorithm (TRSA) for the strip packing problem in two and three dimensions with guillotine cutting constraint. In the 3D-SPP a set of rectangular items (boxes) and a container with fixed width and height but variable length are given. An arrangement of all boxes within the container has to be determined so that the required length is minimised. The 2D-SPP is analogously defined. The proposed TRSA is based on a tree search algorithm for the container loading problem by Fanslau and Bortfeldt (INFORMS J. Comput. 22:222–235, 2010). The TRSA generates guillotine packing patterns throughout. In a comparison with all recently proposed 3D-SPP methods the TRSA performs very competitive. Fine results are also achieved for the 2D-SPP.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Allen, S. D., Burke, E. K., & Kendall, G. (2011). A hybrid placement strategy for the three-dimensional strip packing problem. European Journal of Operational Research, 209, 219–227.

    Article  Google Scholar 

  • Alvarez-Valdes, R., Parreno, F., & Tamarit, J. M. (2008). Reactive GRASP for the strip-packing problem. Computers and Operations Research, 35, 1065–1083.

    Article  Google Scholar 

  • Baker, B. S., Coffmann, E. G., & Rivest, R. L. (1980). Orthogonal packings in two dimensions. SIAM Journal on Computing, 9, 846–855.

    Article  Google Scholar 

  • Bekrar, A., & Kacem, I. (2009). An exact method for the 2D guillotine strip packing problem. Advances in Operations Research, 2009, 732010.

    Article  Google Scholar 

  • Bekrar, A., Kacem, I., & Chu, C. (2007). A comparative study of exact algorithms for the two dimensional strip packing problem. Journal of Industrial and Systems Engineering, 1, 151–170.

    Google Scholar 

  • Belov, G., Scheithauer, G., & Mukhacheva, E. A. (2008). One-dimensional heuristics adapted for two-dimensional rectangular strip packing. The Journal of the Operational Research Society, 59, 823–832.

    Article  Google Scholar 

  • Berkey, J. O., & Wang, P. Y. (1987). Two dimensional finite bin packing algorithms. The Journal of the Operational Research Society, 38, 423–429.

    Google Scholar 

  • Bischoff, E. E., & Mariott, M. D. (1990). A comparative evaluation of heuristics for container loading. European Journal of Operational Research, 44, 267–276.

    Article  Google Scholar 

  • Bischoff, E. E., & Ratcliff, M. S. W. (1995). Issues in the development of approaches to container loading. Omega, 23, 377–390.

    Article  Google Scholar 

  • Bortfeldt, A. (2006). A genetic algorithm for the two-dimensional strip packing problem with rectangular pieces. European Journal of Operational Research, 172, 814–837.

    Article  Google Scholar 

  • Bortfeldt, A., & Gehring, H. (1999). Two metaheuristics for strip packing problems. In D. K. Despotis & C. Zopounidis (Eds.), Proceedings of the fifth international conference of the decision sciences institute, Athens 1999 (Vol. 2, pp. 1153–1156).

    Google Scholar 

  • Bortfeldt, A., & Mack, D. (2007). A heuristic for the three-dimensional strip packing problem. European Journal of Operational Research, 183, 1267–1279.

    Article  Google Scholar 

  • Burke, E. K., Kendall, G., & Whitwell, G. (2004). A new placement heuristic for the orthogonal stock-cutting problem. INFORMS Journal on Computing, 52, 655–671.

    Google Scholar 

  • Burke, E. K., Kendall, G., & Whitwell, G. (2009). A simulated annealing enhancement of the best-fit heuristic for the orthogonal stock cutting problem. INFORMS Journal on Computing, 21, 505–516.

    Article  Google Scholar 

  • Cui, Y., Yang, Y., Cheng, X., & Song, P. (2008). A recursive branch-and-bound algorithm for the rectangular guillotine strip packing problem. Computers & Operations Research, 35, 1281–1291.

    Article  Google Scholar 

  • Davies, A. P., & Bischoff, E. E. (1998). Weight distribution considerations in container loading (Technical Report). European Business Management School, University of Wales, Swansea, Statistics and OR Group.

  • Fanslau, T., & Bortfeldt, A. (2010). A tree search algorithm for solving the container loading problem. INFORMS Journal on Computing, 22, 222–235.

    Article  Google Scholar 

  • Fekete, S. P., & Schepers, J. (1997). On more-dimensional packing III: Exact algorithms (Technical Report ZPR97-290). Mathematisches Institut, Universität zu Köln.

  • Fekete, S. P., Schepers, J., & van der Veen, J. C. (2007). An exact algorithm for higherdimensional orthogonal packing. Operations Research, 55, 569–587.

    Article  Google Scholar 

  • Hopper, E. (2000). Two-dimensional packing utilising evolutionary algorithms and other meta-heuristic methods. Ph.D. Thesis, University of Wales.

  • Hopper, E., & Turton, B. C. H. (2000). An empirical investigation of meta-heuristic and heuristic algorithms for a 2D packing problem. European Journal of Operational Research, 128, 34–57.

    Article  Google Scholar 

  • Hopper, E., & Turton, B. C. H. (2001). A review of the application of meta-heuristic algorithms to 2D strip packing problems. Artificial Intelligence Review, 16, 257–300.

    Article  Google Scholar 

  • Iori, M., Martello, S., & Monaci, M. (2002). Metaheuristic algorithms for the strip packing problem. In P. Pardalos & V. Korotkich (Eds.), Optimization and industry: new frontieres. Norwell: Kluwer Academic.

    Google Scholar 

  • Karabulut, K., & Inceoglu, M. M. (2004). A hybrid genetic algorithm for packing in 3D with deepest bottom left with fill method. In Lecture notes in computer science (Vol. 3261, pp. 441–450). Berlin: Springer.

    Google Scholar 

  • Kenmochi, M., Imamichi, T., Nonobe, K., Yagiura, M., & Nagamochi, H. (2009). Exact algorithms for the 2-dimensional strip packing problem with and without rotations. European Journal of Operational Research, 198, 73–83.

    Article  Google Scholar 

  • Kröger, B. (1993). Parallele genetische Algorithmen zur Lösung eines zweidimensionalen Bin Packing Problems. Ph.D. Thesis, Fachbereich Mathematik and Informatik, Universität Osnabrück.

  • Kröger, B. (1995). Guillotine bin packing: A genetic approach. European Journal of Operational Research, 84, 645–661.

    Article  Google Scholar 

  • Lesh, N., Marks, J., McMahon, A., & Mitzenmacher, M. (2004). Exhaustive approaches to 2D rectangular perfect packings. Information Processing Letters, 90, 7–14.

    Article  Google Scholar 

  • Lesh, N., Marks, J., McMahon, A., & Mitzenmacher, M. (2005). New heuristic and interactive approaches to 2D rectangular strip packing. ACM Journal of Experimental Algorithmics, 10, 1–18.

    Google Scholar 

  • Lodi, A., Martello, S., & Vigo, D. (1999). Heuristic and metaheuristic approaches for a class of two-dimensional bin packing problems. INFORMS Journal on Computing, 11, 345–357.

    Article  Google Scholar 

  • Martello, S., & Vigo, D. (1998). Exact solution of the two-dimensional finite bin packing problem. Management Science, 44, 388–399.

    Article  Google Scholar 

  • Martello, S., Monaci, M., & Vigo, D. (2003). An exact approach to the strip-packing problem. INFORMS Journal on Computing, 15, 310–319.

    Article  Google Scholar 

  • Mumford-Valenzuela, C. L., Vick, J., & Wang, P. Y. (2004). Heuristics for large strip packing problems with guillotine patterns: an empirical study. In: Metaheuristics: computer decision-making (pp. 501–522). Norwell: Kluwer Academic.

    Google Scholar 

  • Ortmann, F. G., Nthabiseng, N., & van Vuuren, J. H. (2010). New and improved level heuristics for the rectangular strip packing and variable-sized bin packing problems. European Journal of Operational Research, 203, 306–315.

    Article  Google Scholar 

  • Pisinger, D. (2002). Heuristics for the container loading problem. European Journal of Operational Research, 141, 382–392.

    Article  Google Scholar 

  • Riff, M. C., Bonnaire, X., & Neveu, B. (2009). A revision of recent approaches for two dimensional strip-packing problems. Engineering Applications of Artificial Intelligence, 22, 823–827.

    Article  Google Scholar 

  • Schnecke, V. (1996). Hybrid genetic algorithms for solving constrained packing and placement problems. Ph.D. Thesis, Fachbereich Mathematik und Informatik, Universität Osnabrück.

  • Sixt, M. (1996). Dreidimensionale Packprobleme. Lösungsverfahren basierend auf den Metaheuristiken Simulated Annealing und Tabu-Suche. Ph.D. Thesis, Frankfurt am Main, Peter Lang, Europäischer Verlag der Wissenschaften.

  • Wäscher, G., Haussner, H., & Schumann, H. (2007). An improved typology of cutting and packing problems. European Journal of Operational Research, 183, 1109–1130.

    Article  Google Scholar 

  • Wei, L., Zhang, D., & Chen, Q. (2009). A least wasted first heuristic algorithm for the rectangular packing problem. Computers & Operations Research, 36, 1608–1614.

    Article  Google Scholar 

  • Zhang, D., Liu, Y., Chen, S., & Xie, X. (2005). A meta-heuristic algorithm for the strip rectangular packing problem. In Lecture notes in computer science, part III (Vol. 3612, pp. 1235–1241). Berlin: Springer.

    Google Scholar 

  • Zhang, D., Kang, Y., & Deng, A. (2006). A new heuristic recursive algorithm for the strip rectangular packing problem. Computers & Operations Research, 33, 2209–2217.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Information Systems, University in Hagen, 58084, Hagen, Germany

    Andreas Bortfeldt & Sabine Jungmann

Authors
  1. Andreas Bortfeldt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sabine Jungmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andreas Bortfeldt.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bortfeldt, A., Jungmann, S. A tree search algorithm for solving the multi-dimensional strip packing problem with guillotine cutting constraint. Ann Oper Res 196, 53–71 (2012). https://doi.org/10.1007/s10479-012-1084-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-012-1084-7

Keywords

Search

Navigation