Abstract
This paper investigates the two-dimensional strip packing problem considering the case in which items should be arranged to form a physically stable packing satisfying a predefined item unloading order from the top of the strip. The packing stability analysis is based on conditions for the static equilibrium of rigid bodies, differing from others strategies which are based on area and percentage of support. We consider an integer linear programming model for the strip packing problem with the order constraint, and a cutting plane algorithm to handle stability, leading to a branch-and-cut approach. We also present two heuristics: the first is based on a stack building algorithm; and, the last is a slight modification of the branch-and-cut approach. The computational experiments show that the branch-and-cut model can handle small and medium-sized instances, whereas the heuristics found almost optimal solutions quickly for several instances. With the combination of heuristics and the branch-and-cut algorithm, many instances are solved to near optimality in a few seconds.
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.
Alvarez-Valdes, R., Parreño, F. P., & Tamarit, J. M. (2009). A branch and bound algorithm for the strip packing problem. OR Spectrum, 31(2), 431–459.
Alvarez-Valdes, R., Parreño, F., & Tamarit, J. M. (2008). Reactive grasp for the strip-packing problem. Computers and Operations Research, 35, 1065–1083.
Arahori, Y., Imamichi, T., & Nagamochi, H. (2012). An exact strip packing algorithm based on canonical forms. Computers and Operations Research, 39(12), 2991–3011.
Armas, J., León, C., Miranda, G., & Segura, C. (2010). Optimisation of a multi-objective two-dimensional strip packing problem based on evolutionary algorithms. International Journal of Production Research, 48(7), 2011–2028.
Asık, O. B., & Özcan, E. (2009). Bidirectional best-fit heuristic for orthogonal rectangular strip packing. Annals of Operations Research, 172, 405–427.
Beasley, J. E. (1990). OR-Library: distributing test problems by electronic mail. Journal of the Operational Research Society, 41(11), 1069–1072.
Beer, F. P., Johnston, E. R., DeWolf, J., & Mazurek, D. (2008). Mechanics of Materials (Vol. 5). Ohio: McGraw-Hill Science.
Belov, G., Scheithauer, G., & Mukhacheva, E. A. (2008). One-dimensional heuristics adapted for two-dimensional rectangular strip packing. Journal of the Operational Research Society, 59(6), 823–832.
Bischoff, E. E., & Ratcliff, M. S. W. (1995). Issues in the development of approaches to container loading. Omega, 23(4), 377–390.
Bortfeldt, A., & Jungmann, S. (2012). A tree search algorithm for solving the multi-dimensional strip packing problem with guillotine cutting constraint. Annals of Operations Research, 196, 53–71.
Cintra, G. F., Miyazawa, F. K., Wakabayashi, Y., & Xavier, E. C. (2008). Algorithms for two-dimensional cutting stock and strip packing problems using dynamic programming and column generation. European Journal of Operational Research, 191, 59–83.
Coffman, E. G, Jr, Johnson, D. S., Shor, P. W., & Weber, R. R. (1997). Bin packing with discrete item sizes, part II: Tight bounds on first fit. Random Structures and Algorithms, 10, 69–101.
Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: a guide to the theory of NP-completeness. San Francisco: Freeman.
Gavin, H. P. (2009). The Three-Moment Equation for Continuous Beam. Technical Report Department of Civil and Environmental Engineering - Duke Unversity.
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.
Gendreau, M., Iori, M., Laporte, G., & Martello, S. (2008). A tabu search heuristic for the vehicle routing problem with two-dimensional loading constraints. Networks, 51, 14–18.
Halliday, D., Resnick, R., & Walker, J. (2007). Fundamentals of Physics (8th ed.). Cleveland: Wiley.
Hibbeler, R. C. (2010). Statics and mechanics of materials (3rd ed.). New York: Prentice Hall.
Hifi, M. (1998). Exact algorithms for the guillotine strip cutting/packing problem. Computers and Operations Research, 25, 925–940.
Iori, M., Salazar-González, J., & Vigo, D. (2007). An exact approach for the vehicle routing problem with two-dimensional loading constrains. Transportation Science, 41(2), 253–264.
Junqueira, L., Morabito, R., & Yamashita, D. S. (2012a). Mip-based approaches for the container loading problem with multi-drop constraints. Annals of Operations Research, 199, 51–75.
Junqueira, L., Morabito, R., & Yamashita, D. S. (2012b). Three-dimensional container loading models with cargo stability and load bearing constraints. Computers and Operations Research, 39(1), 74–85.
Kenmochi, M., Imamichi, T., Nonobe, K., Yagiura, M., & Nagamochi, H. (2009). Exact algorithms for the two-dimensional strip packing problem with and without rotations. European Journal of Operational Research, 198, 73–83.
Kocjan, W., & Holmström, K. (2009). Computing stable loads for pallets. European Journal of Operational Research, 207(2), 980–985.
Lesh, N., Marks, J., & Mahon, A. M. (2004). Exhaustive approaches to 2D rectangular perfect packings. Information Processing Letters, 90, 7–14.
Lodi, A., Martello, S., & Vigo, D. (2004). Models and bounds for two-dimensional level packing problems. Journal of Combinatorial Optimization, 8, 363–379.
Martello, S., Monaci, M., & Vigo, D. (2003). An exact approach to the strip-packing problem. INFORMS Journal on Computing, 15(3), 310–319.
Ntene, N., & van Vuuren, J. H. (2009). A survey and comparison of guillotine heuristics for the 2D oriented offline strip packing problem. Discrete Optimization, 6, 174–188.
Queiroz, T. A., & Miyazawa, F. K. (2013). Two-dimensional strip packing problem with load balancing, load bearing and multi-drop constraints. International Journal of Production Economics, 145, 511–530.
Queiroz, T. A., Miyazawa, F. K., Wakabayashi, Y., & Xavier, E. C. (2012). Algorithms for 3D guillotine cutting problems: Unbounded knapsack, cutting stock and strip packing. Computers and Operations Research, 39, 200–212.
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.
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.
Wäscher, G., Haussner, H., & Schumann, H. (2007). An improved typology of cutting and packing problems. European Journal of Operational Research, 183(3), 1109–1130.
Zhang, D. F., Chen, S. D., & Liu, Y. J. (2007). An improved heuristic recursive strategy based on genetic algorithm for the strip rectangular packing problem. Acta Automatica Sinica, 33, 911–916.
Acknowledgments
The authors would like to thank the anonymous reviewers for their comments and suggestions. This study has been supported by CNPq and FAPEG.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
de Queiroz, T.A., Miyazawa, F.K. Order and static stability into the strip packing problem. Ann Oper Res 223, 137–154 (2014). https://doi.org/10.1007/s10479-014-1634-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-014-1634-2