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Reducing the 0-1 knapsack problem with a single continuous variable to the standard 0-1 knapsack problem

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  • Büther, Marcel
  • Briskorn, Dirk

Abstract

The 0-1 knapsack problem with a single continuous variable (KPC) is a natural extension of the binary knapsack problem (KP), where the capacity is not any longer fixed but can be extended which is expressed by a continuous variable. This variable might be unbounded or restricted by a lower or upper bound, respectively. This paper concerns techniques in order to reduce several variants of KPC to KP which enables us to employ approaches for KP. We propose both, an equivalent reformulation and a heuristic one bringing along less computational effort. We show that the heuristic reformulation can be customized in order to provide solutions having an objective value arbitrarily close to the one of the original problem.

Suggested Citation

  • Büther, Marcel & Briskorn, Dirk, 2007. "Reducing the 0-1 knapsack problem with a single continuous variable to the standard 0-1 knapsack problem," Manuskripte aus den Instituten für Betriebswirtschaftslehre der Universität Kiel 629, Christian-Albrechts-Universität zu Kiel, Institut für Betriebswirtschaftslehre.
    • Handle: RePEc:zbw:cauman:629
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    1. WOLSEY, Laurence A., 2003. "Strong formulations for mixed integer programs: valid inequalities and extended formulations," LIDAM Reprints CORE 1627, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Martello, Silvano & Pisinger, David & Toth, Paolo, 2000. "New trends in exact algorithms for the 0-1 knapsack problem," European Journal of Operational Research, Elsevier, vol. 123(2), pages 325-332, June.
    3. Michael F. Gorman & Sanjay Ahire, 2006. "A Major Appliance Manufacturer Rethinks Its Inventory Policies for Service Vehicles," Interfaces, INFORMS, vol. 36(5), pages 407-419, October.
    4. R M Nauss, 2004. "The elastic generalized assignment problem," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 55(12), pages 1333-1341, December.
    5. Silvano Martello & David Pisinger & Paolo Toth, 1999. "Dynamic Programming and Strong Bounds for the 0-1 Knapsack Problem," Management Science, INFORMS, vol. 45(3), pages 414-424, March.
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    Cited by:

    1. M Büther, 2010. "Reducing the elastic generalized assignment problem to the standard generalized assignment problem," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 61(11), pages 1582-1595, November.
    2. Büther, Marcel, 2008. "Beam search for the elastic generalized assignment problem," Manuskripte aus den Instituten für Betriebswirtschaftslehre der Universität Kiel 634, Christian-Albrechts-Universität zu Kiel, Institut für Betriebswirtschaftslehre.
    3. Büther, Marcel, 2007. "Reducing the elastic generalized assignment problem to the standard generalized assignment problem," Manuskripte aus den Instituten für Betriebswirtschaftslehre der Universität Kiel 632, Christian-Albrechts-Universität zu Kiel, Institut für Betriebswirtschaftslehre.
    4. Briskorn, Dirk & Knust, Sigrid, 2008. "On Circular 2-Factorizations of the Complete Tripartite Graph," Manuskripte aus den Instituten für Betriebswirtschaftslehre der Universität Kiel 636, Christian-Albrechts-Universität zu Kiel, Institut für Betriebswirtschaftslehre.

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