Abstract
In this paper we present a heuristic method to generate constrained two-dimensional guillotine cutting patterns. This problem appears in different industrial processes of cutting rectangular plates to produce ordered items, such as in the glass, furniture and circuit board business. The method uses a state space relaxation of a dynamic programming formulation of the problem and a state space ascent procedure of subgradient optimization type. We propose the combination of this existing approach with an and/or-graph search and an inner heuristic that turns infeasible solutions provided in each step of the ascent procedure into feasible solutions. Results for benchmark and randomly generated instances indicate that the method’s performance is competitive compared to other methods proposed in the literature. One of its advantages is that it often produces a relatively tight upper bound to the optimal value. Moreover, in most cases for which an optimal solution is obtained, it also provides a certificate of optimality.
Similar content being viewed by others
References
Alvarez-Valdés, R., Parajón, A., & Tamarit, J. (2002). A tabu search algorithm for large-scale guillotine (un)constrained two-dimensional cutting problems. Computers and Operations Research, 29, 925–947.
Arenales, M., Morabito, R., & Yanasse, H. (Eds.) (1999). Cutting and packing problems. Pesquisa Operacional, 19(2), 107–299.
Beasley, J. E. (1985). Algorithms for unconstrained two-dimensional guilhotine cutting. Journal of the Operational Research Society, 36(4), 297–306.
Bischoff, E., & Waescher, G. (Eds.) (1995). Cutting and packing. European Journal of Operational Research, 84, 3.
Burke, E., Kendall, G., & Whitwell, G. (2004). A new placement heuristic for the orthogonal stock-cutting problem. Operations Research, 52(4), 655–671.
Christofides, N., & Hadjiconstantinou, E. (1995). An exact algorithm for orthogonal 2-d cutting problems using guillotine cuts. European Journal of Operational Research, 83, 21–38.
Christofides, N., Mingozzi, A., & Toth, P. (1981). State-space relaxation procedure for the computation of bounds to routing problems. Networks, 11, 145–164.
Christofides, N., & Whitlock, C. (1977). An algorithm for two-dimensional cutting problems. Operations Research, 25(1), 30–44.
Cui, Y. (2008). Heuristic and exact algorithms for generating homogeneous constrained three-staged cutting patterns. Computers and Operations Research, 35, 212–225.
Cung, V., Hifi, M., & Le Cun, B. (2000). Constrained two-dimensional guillotine cutting stock problems: A best-first branch-and-bound algorithm. International Transactions in Operational Research, 7, 185–201.
Dowsland, K., & Dowsland, W. (1992). Packing problems. European Journal of Operational Research, 56, 2–14.
Dowsland, K., Herbert, E., Kendall, G., & Burke, E. (2006). Using tree search bounds to enhance a genetic algorithm approach to two rectangle packing problems. European Journal of Operational Research, 168, 390–402.
Dyckhoff, H., & Finke, U. (1992). Cutting and packing in production and distribution: typology and bibliography. Heidelberg: Springer.
Dyckhoff, H., Scheithauer, G., & Terno, J. (1997). Cutting and packing. In M. Amico, F. Maffioli, & S. Martello (Eds.), Annoted bibliographies in combinatorial optimization (pp. 393–414). New York: Wiley.
Dyckhoff, H., & Waescher, G. (Eds.) (1990). Cutting and packing. European Journal of Operational Research, 44, 2.
ESICUP—Euro special interest group on cutting and packing. Available in: http://www.apdio.pt/esicup/ (accessed in 2008).
Fayard, D., Hifi, M., & Zissimopoulos, V. (1998). An efficient approach for large-scale two-dimensional guillotine cutting stock problems. Journal of the Operational Research Society, 49, 1270–1277.
Fontes, D., Hadjiconstantinou, E., & Christofides, N. (2006). Lower bounds from state space relaxations for concave cost network flow problems. Journal of Global Optimization, 34, 97–125.
Gilmore, P., & Gomory, R. (1966). The theory and computation of knapsack functions. Operations Research, 14, 1045–1074.
Hadjiconstantinou, E., & Christofides, N. (1995). A new exact algorithm for the vehicle routing problem based on q-paths and k-shortest paths relaxations. Annals of Operations Research, 61, 21–43.
Hifi, M. (1997a). An improvement of Viswanathan and Bagchi’s exact algorithm for constrained two-dimensional cutting stock. Computers and Operations Research, 24(8), 727–736.
Hifi, M. (1997b). The DH/KD algorithm: a hybrid approach for unconstrained cutting problems. European Journal of Operational Research, 97, 41–52.
Hifi, M. (Ed.) (2002). Special issue: Cutting and packing problems. Studia Informatica Universalis, 2(1), 1–161.
Hifi, M. (2004). Dynamic programming and hill-climbing techniques for constrained two-dimensional cutting stock problems. Journal of Combinatorial Optimization, 8, 65–84.
Lodi, A., Martello, S., & Monaci, M. (2002). Two-dimensional packing problems: a survey. European Journal of Operational Research, 141, 241–252.
Lucena, A. (2004). Non delayed relax-and-cut algorithms (Working Paper). Universidade Federal do Rio de Janeiro, Brazil.
Martello, S. (Ed.) (1994a) Special issue: Knapsack, packing and cutting, Part I: One dimensional knapsack problems. INFOR, 32, 3.
Martello, S. (Ed.) (1994b) Special issue: Knapsack, packing and cutting, Part II: Multidimensional knapsack and cutting stock problems. INFOR, 32, 4.
Mukhacheva, E. A. (Ed.). (1997). Decision making under conditions of uncertainty: cutting–packing problems. The International Scientific Collection, Ufa, Russia.
Morabito, R., & Arenales, M. (1994). An and/or-graph approach to the container loading problem. International Transactions in Operational Research, 1(1), 59–73.
Morabito, R., & Arenales, M. (1996). Staged and constrained two-dimensional guilhotine cutting problems: An and/or-graph approach. European Journal of Operational Research, 94, 548–560.
Morabito, R., Arenales, M., & Arcaro, V. (1992). An and/or-graph approach for two-dimensional cutting problems. European Journal of Operational Research, 58(2), 263–271.
Mornar, V., & Khoshnevis, B. (1997). A cutting stock procedure for printed circuit board production. Computers and Industrial Engineering, 32(1), 57–66.
Oliveira, J. F., & Ferreira, J. S. (1990). An improved version of Wang’s algorithm for two-dimensional cutting problems. European Journal of Operational Research, 44, 256–266.
Oliveira, J. F., & Waescher, G. (Eds.) (2007). Special issue on cutting and packing. European Journal of Operational Research, 183.
Parada, V., Alvarenga, A. G., & Diego, J. (1995). Exact solutions for constrained two-dimensional cutting problems. European Journal of Operational Research, 84, 633–644.
Parada, V., Sepulveda, M., Solar, M., & Gomes, A. (1998). Solution for the constrained guillotine cutting problem by simulated annealing. Computers and Operations Research, 25, 37–47.
Parada, V., Palma, R., Sales, D., & Gomes, A. (2000). A comparative numerical analysis for the guillotine two-dimensional cutting problem. Annals of Operations Research, 96, 245–254.
Sweeney, P., & Paternoster, E. (1992). Cutting and packing problems: a categorized, application-oriented research bibliography. Journal of the Operational Research Society, 43, 691–706.
Vasko, F. J. (1989). A computational improvement to Wang’s two-dimensional cutting stock algorithm. Computers and Industrial Engineering, 16(1), 109–115.
Viswanathan, K. V., & Bagchi, A. (1993). Best-first search methods for constrained two-dimensional cutting stock problems. Operations Research, 41(4), 768–776.
Waescher, G., Haussner, H., & Schumann, H. (2007). An improved typology of cutting and packing problems. European Journal of Operational Research, 183, 1109–1130.
Wang, P. Y. (1983). Two algorithms for constrained two-dimensional cutting stock problems. Operations Research, 31, 573–586.
Wang, P. Y., & Waescher, G. (Eds.) (2002). Cutting and packing. European Journal of Operational Research, 141(2), 239–469.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Morabito, R., Pureza, V. A heuristic approach based on dynamic programming and and/or-graph search for the constrained two-dimensional guillotine cutting problem. Ann Oper Res 179, 297–315 (2010). https://doi.org/10.1007/s10479-008-0457-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-008-0457-4