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Stabilized column generation for the temporal knapsack problem using dual-optimal inequalities

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  • Timo Gschwind

    (Johannes Gutenberg University Mainz)

  • Stefan Irnich

    (Johannes Gutenberg University Mainz)

Abstract

We present two new methods to stabilize column-generation algorithms for the temporal knapsack problem (TKP). Caprara et al. (INFORMS J Comp 25(3):560–571, 2013] were the first to suggest the use of branch-and-price algorithms for Dantzig–Wolfe reformulations of the TKP. Herein, the respective pricing problems are smaller-sized TKP that can be solved with a general-purpose MIP solver or by dynamic programming. Our stabilization methods are tailored to the TKP as they use (deep) dual-optimal inequalities, that is, inequalities known to be fulfilled by all (at least some) optimal dual solutions to the linear relaxation. Extensive computational tests reveal that both new stabilization techniques are helpful. Several previously unsolved instances are now solved to proven optimality.

Suggested Citation

  • Timo Gschwind & Stefan Irnich, 2017. "Stabilized column generation for the temporal knapsack problem using dual-optimal inequalities," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 39(2), pages 541-556, March.
    • Handle: RePEc:spr:orspec:v:39:y:2017:i:2:d:10.1007_s00291-016-0463-x
    DOI: 10.1007/s00291-016-0463-x
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    References listed on IDEAS

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    1. Alberto Caprara & Fabio Furini & Enrico Malaguti, 2013. "Uncommon Dantzig-Wolfe Reformulation for the Temporal Knapsack Problem," INFORMS Journal on Computing, INFORMS, vol. 25(3), pages 560-571, August.
    2. Hatem Ben Amor & Jacques Desrosiers & José Manuel Valério de Carvalho, 2006. "Dual-Optimal Inequalities for Stabilized Column Generation," Operations Research, INFORMS, vol. 54(3), pages 454-463, June.
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    4. Desrosiers, Jacques & Gauthier, Jean Bertrand & Lübbecke, Marco E., 2014. "Row-reduced column generation for degenerate master problems," European Journal of Operational Research, Elsevier, vol. 236(2), pages 453-460.
    5. Timo Gschwind & Stefan Irnich, 2016. "Dual Inequalities for Stabilized Column Generation Revisited," INFORMS Journal on Computing, INFORMS, vol. 28(1), pages 175-194, February.
    6. P. C. Gilmore & R. E. Gomory, 1961. "A Linear Programming Approach to the Cutting-Stock Problem," Operations Research, INFORMS, vol. 9(6), pages 849-859, December.
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    8. Marco E. Lübbecke & Jacques Desrosiers, 2005. "Selected Topics in Column Generation," Operations Research, INFORMS, vol. 53(6), pages 1007-1023, December.
    9. François Vanderbeck, 2005. "Implementing Mixed Integer Column Generation," Springer Books, in: Guy Desaulniers & Jacques Desrosiers & Marius M. Solomon (ed.), Column Generation, chapter 0, pages 331-358, Springer.
    10. Alberto Caprara & Enrico Malaguti & Paolo Toth, 2011. "A Freight Service Design Problem for a Railway Corridor," Transportation Science, INFORMS, vol. 45(2), pages 147-162, May.
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    Cited by:

    1. 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.
    2. Clautiaux, F. & Detienne, B. & Guillot, G., 2021. "An iterative dynamic programming approach for the temporal knapsack problem," European Journal of Operational Research, Elsevier, vol. 293(2), pages 442-456.
    3. Martinovic, J. & Strasdat, N. & Valério de Carvalho, J. & Furini, F., 2023. "A combinatorial flow-based formulation for temporal bin packing problems," European Journal of Operational Research, Elsevier, vol. 307(2), pages 554-574.

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