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Stabilized Column Generation for the Temporal Knapsack Problem using Dual- Optimal Inequalities

Author

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  • Timo Gschwind

    (Johannes Gutenberg-Universität Mainz, Germany)

  • Stefan Irnich

    (Johannes Gutenberg-Universität Mainz, Germany)

Abstract

We present two new methods to stabilize column-generation algorithms for the Temporal Knapsack Problem (TKP). Caprara et al. [Caprara A, Furini F, and Malaguti E (2013) Uncommon Dantzig-Wolfe Reformulation for the Temporal Knapsack Problem. INFORMS J. on Comp. 25(3):560–571] 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.

Suggested Citation

  • Timo Gschwind & Stefan Irnich, 2014. "Stabilized Column Generation for the Temporal Knapsack Problem using Dual- Optimal Inequalities," Working Papers 1413, Gutenberg School of Management and Economics, Johannes Gutenberg-Universität Mainz, revised 13 Nov 2014.
    • Handle: RePEc:jgu:wpaper:1413
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    File URL: https://download.uni-mainz.de/RePEc/pdf/Discussion_Paper_1413.pdf
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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. 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.
    3. 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.
    4. 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.
    5. 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.
    6. Marco E. Lübbecke & Jacques Desrosiers, 2005. "Selected Topics in Column Generation," Operations Research, INFORMS, vol. 53(6), pages 1007-1023, December.
    7. Timo Gschwind & Stefan Irnich, 2014. "Dual Inequalities for Stabilized Column Generation Revisited," Working Papers 1407, Gutenberg School of Management and Economics, Johannes Gutenberg-Universität Mainz, revised 23 Jul 2014.
    8. 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. Timo Gschwind & Stefan Irnich, 2016. "Dual Inequalities for Stabilized Column Generation Revisited," INFORMS Journal on Computing, INFORMS, vol. 28(1), pages 175-194, February.

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    Keywords

    Column generation; dual inequalities; stabilization;
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