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Classification of Dantzig-Wolfe reformulations for binary mixed integer programming problems

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  • Jans, Raf

Abstract

In this note, we provide a classification of Dantzig-Wolfe reformulations for Binary Mixed Integer Programming Problems. We specifically focus on modeling the binary conditions in the convexification approach to the Dantzig-Wolfe decomposition. For a general Binary Mixed Integer Programming problem, an extreme point of the overall problem does not necessarily correspond to an extreme point of the subproblem. Therefore, the binary conditions cannot in general be imposed on the new master problem variables but must be imposed on the original binary variables. In some cases, however, it is possible to impose the binary restrictions directly on the new master problem variables. The issue of imposing binary conditions on the original variables versus the master problem variables has not been discussed systematically for MIP problems in general in the literature and most of the research has been focused on the pure binary case. The classification indicates in which cases you can, and cannot, impose binary conditions on the new master problem variables.

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  • Jans, Raf, 2010. "Classification of Dantzig-Wolfe reformulations for binary mixed integer programming problems," European Journal of Operational Research, Elsevier, vol. 204(2), pages 251-254, July.
    • Handle: RePEc:eee:ejores:v:204:y:2010:i:2:p:251-254
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    2. Christian Tilk & Katharina Olkis & Stefan Irnich, 2020. "The Last-mile Vehicle Routing Problem with Delivery Options," Working Papers 2017, Gutenberg School of Management and Economics, Johannes Gutenberg-Universität Mainz.
    3. Wakui, Tetsuya & Hashiguchi, Moe & Yokoyama, Ryohei, 2020. "A near-optimal solution method for coordinated operation planning problem of power- and heat-interchange networks using column generation-based decomposition," Energy, Elsevier, vol. 197(C).
    4. Christian Tilk & Katharina Olkis & Stefan Irnich, 2021. "The last-mile vehicle routing problem with delivery options," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 43(4), pages 877-904, December.
    5. Christian Tilk & Michael Drexl & Stefan Irnich, 2018. "Nested Branch-and-Price-and-Cut for Vehicle Routing Problems with Multiple Resource Interdependencies," Working Papers 1801, Gutenberg School of Management and Economics, Johannes Gutenberg-Universität Mainz.
    6. Tilk, Christian & Drexl, Michael & Irnich, Stefan, 2019. "Nested branch-and-price-and-cut for vehicle routing problems with multiple resource interdependencies," European Journal of Operational Research, Elsevier, vol. 276(2), pages 549-565.
    7. Christian Tilk & Nicola Bianchessi & Michael Drexl & Stefan Irnich & Frank Meisel, 2015. "Branch-and-Price for the Active-Passive Vehicle-Routing Problem," Working Papers 1513, Gutenberg School of Management and Economics, Johannes Gutenberg-Universität Mainz.
    8. Christian Tilk & Nicola Bianchessi & Michael Drexl & Stefan Irnich & Frank Meisel, 2018. "Branch-and-Price-and-Cut for the Active-Passive Vehicle-Routing Problem," Transportation Science, INFORMS, vol. 52(2), pages 300-319, March.

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