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A survey of dual-feasible and superadditive functions

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  • François Clautiaux
  • Cláudio Alves
  • José Valério de Carvalho

Abstract

Dual-feasible functions are valuable tools that can be used to compute both lower bounds for different combinatorial problems and valid inequalities for integer programs. Several families of functions have been used in the literature. Some of them were defined explicitly, and others not. One of the main objectives of this paper is to survey these functions, and to state results concerning their quality. We clearly identify dominant subsets of functions, i.e. those which may lead to better bounds or stronger cuts. We also describe different frameworks that can be used to create dual-feasible functions. With these frameworks, one can get a dominant function based on other ones. Two new families of dual-feasible functions obtained by applying these methods are proposed in this paper. We also performed a computational comparison on the relative strength of the functions presented in this paper for deriving lower bounds for the bin-packing problem and valid cutting planes for the pattern minimization problem. Extensive experiments on instances generated using methods described in the literature are reported. In many cases, the lower bounds are improved, and the linear relaxations are strengthened. Copyright Springer Science+Business Media, LLC 2010

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  • François Clautiaux & Cláudio Alves & José Valério de Carvalho, 2010. "A survey of dual-feasible and superadditive functions," Annals of Operations Research, Springer, vol. 179(1), pages 317-342, September.
  • Handle: RePEc:spr:annopr:v:179:y:2010:i:1:p:317-342:10.1007/s10479-008-0453-8
    DOI: 10.1007/s10479-008-0453-8
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    2. Jean-François Côté & Mohamed Haouari & Manuel Iori, 2021. "Combinatorial Benders Decomposition for the Two-Dimensional Bin Packing Problem," INFORMS Journal on Computing, INFORMS, vol. 33(3), pages 963-978, July.
    3. Iori, Manuel & de Lima, Vinícius L. & Martello, Silvano & Miyazawa, Flávio K. & Monaci, Michele, 2021. "Exact solution techniques for two-dimensional cutting and packing," European Journal of Operational Research, Elsevier, vol. 289(2), pages 399-415.
    4. François Clautiaux & Cláudio Alves & José Valério de Carvalho & Jürgen Rietz, 2011. "New Stabilization Procedures for the Cutting Stock Problem," INFORMS Journal on Computing, INFORMS, vol. 23(4), pages 530-545, November.
    5. Alves, Cláudio & de Carvalho, José Valério & Clautiaux, François & Rietz, Jürgen, 2014. "Multidimensional dual-feasible functions and fast lower bounds for the vector packing problem," European Journal of Operational Research, Elsevier, vol. 233(1), pages 43-63.
    6. Jean-François Côté & Michel Gendreau & Jean-Yves Potvin, 2020. "The Vehicle Routing Problem with Stochastic Two-Dimensional Items," Transportation Science, INFORMS, vol. 54(2), pages 453-469, March.
    7. Arbib, Claudio & Marinelli, Fabrizio, 2017. "Maximum lateness minimization in one-dimensional bin packing," Omega, Elsevier, vol. 68(C), pages 76-84.
    8. Mauro Dell'Amico & José Carlos Díaz Díaz & Manuel Iori, 2012. "The Bin Packing Problem with Precedence Constraints," Operations Research, INFORMS, vol. 60(6), pages 1491-1504, December.
    9. Leggieri, Valeria & Haouari, Mohamed, 2017. "Lifted polynomial size formulations for the homogeneous and heterogeneous vehicle routing problems," European Journal of Operational Research, Elsevier, vol. 263(3), pages 755-767.
    10. Liao, Chung-Shou & Hsu, Chia-Hong, 2013. "New lower bounds for the three-dimensional orthogonal bin packing problem," European Journal of Operational Research, Elsevier, vol. 225(2), pages 244-252.
    11. Pereira, Jordi, 2016. "Procedures for the bin packing problem with precedence constraints," European Journal of Operational Research, Elsevier, vol. 250(3), pages 794-806.
    12. Sebastian Kraul & Markus Seizinger & Jens O. Brunner, 2023. "Machine Learning–Supported Prediction of Dual Variables for the Cutting Stock Problem with an Application in Stabilized Column Generation," INFORMS Journal on Computing, INFORMS, vol. 35(3), pages 692-709, May.
    13. Delorme, Maxence & Iori, Manuel & Martello, Silvano, 2016. "Bin packing and cutting stock problems: Mathematical models and exact algorithms," European Journal of Operational Research, Elsevier, vol. 255(1), pages 1-20.
    14. Tao Wu & Kerem Akartunal? & Raf Jans & Zhe Liang, 2017. "Progressive Selection Method for the Coupled Lot-Sizing and Cutting-Stock Problem," INFORMS Journal on Computing, INFORMS, vol. 29(3), pages 523-543, August.

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