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Improved state space relaxation for constrained two-dimensional guillotine cutting problems

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  • Velasco, André Soares
  • Uchoa, Eduardo

Abstract

Christofides and Hadjiconstantinou (1995) introduced a dynamic programming state space relaxation for obtaining upper bounds for the Constrained Two-dimensional Guillotine Cutting Problem. The quality of those bounds depend on the chosen item weights, they are adjusted using a subgradient-like algorithm. This paper proposes Algorithm X, a new weight adjusting algorithm based on integer programming that provably obtains the optimal weights. In order to obtain even better upper bounds, that algorithm is generalized into Algorithm X2 for obtaining optimal two-dimensional item weights. We also present a full hybrid method, called Algorithm X2D, that computes those strong upper bounds but also provides feasible solutions obtained by: (1) exploring the suboptimal solutions hidden in the dynamic programming matrices; (2) performing a number of iterations of a GRASP based primal heuristic; and (3) executing X2H, an adaptation of Algorithm X2 to transform it into a primal heuristic. Extensive experiments with instances from the literature and on newly proposed instances, for both variants with and without item rotation, show that X2D can consistently deliver high-quality solutions and sharp upper bounds. In many cases the provided solutions are certified to be optimal.

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  • Velasco, André Soares & Uchoa, Eduardo, 2019. "Improved state space relaxation for constrained two-dimensional guillotine cutting problems," European Journal of Operational Research, Elsevier, vol. 272(1), pages 106-120.
  • Handle: RePEc:eee:ejores:v:272:y:2019:i:1:p:106-120
    DOI: 10.1016/j.ejor.2018.06.016
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    References listed on IDEAS

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    3. Fabio Furini & Enrico Malaguti & Dimitri Thomopulos, 2016. "Modeling Two-Dimensional Guillotine Cutting Problems via Integer Programming," INFORMS Journal on Computing, INFORMS, vol. 28(4), pages 736-751, November.
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    11. Krzysztof Fleszar, 2016. "An Exact Algorithm for the Two-Dimensional Stage-Unrestricted Guillotine Cutting/Packing Decision Problem," INFORMS Journal on Computing, INFORMS, vol. 28(4), pages 703-720, November.
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    Cited by:

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    3. Oliviana Xavier Nascimento & Thiago Alves Queiroz & Leonardo Junqueira, 2022. "A MIP-CP based approach for two- and three-dimensional cutting problems with staged guillotine cuts," Annals of Operations Research, Springer, vol. 316(2), pages 805-835, September.

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