IDEAS home Printed from https://ideas.repec.org/a/wly/navres/v43y1996i5p673-689.html
   My bibliography  Save this article

Solving the generalized knapsack problem with variable coefficients

Author

Listed:
  • Kaj Holmberg
  • Kurt Jörnsten

Abstract

In this article we present methods based on Lagrangian duality and decomposition techniques for the generalized knapsack problem with variable coefficients. The Lagrangian dual is solved with subgradient optimization or interval bisection. We also describe a heuristic that yields primal feasible solutions. Combining the Lagrangian relaxation with a primal (Benders) subproblem yields the subproblem phase in cross decomposition. By using averages in this procedure, we get the new mean‐value cross‐decomposition method. Finally, we describe how to insert this into a globally convergent generalized Benders decomposition framework, in the case that there is a duality gap. Encouraging computational results for the optimal generating unit commitment problem are presented. © 1996 John Wiley & Sons, Inc.

Suggested Citation

  • Kaj Holmberg & Kurt Jörnsten, 1996. "Solving the generalized knapsack problem with variable coefficients," Naval Research Logistics (NRL), John Wiley & Sons, vol. 43(5), pages 673-689, August.
    • Handle: RePEc:wly:navres:v:43:y:1996:i:5:p:673-689
    DOI: 10.1002/(SICI)1520-6750(199608)43:53.0.CO;2-2
    as

    Download full text from publisher

    File URL: https://doi.org/10.1002/(SICI)1520-6750(199608)43:53.0.CO;2-2
    Download Restriction: no
    File URL: https://libkey.io/10.1002/(SICI)1520-6750(199608)43:53.0.CO;2-2?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. T. L. Magnanti & R. T. Wong, 1981. "Accelerating Benders Decomposition: Algorithmic Enhancement and Model Selection Criteria," Operations Research, INFORMS, vol. 29(3), pages 464-484, June.
    2. VAN ROY, Tony J., 1983. "Cross decomposition for mixed integer programming," LIDAM Reprints CORE 496, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Holmberg, Kaj, 1992. "Linear mean value cross decomposition: A generalization of the Kornai-Liptak method," European Journal of Operational Research, Elsevier, vol. 62(1), pages 55-73, October.
    4. Robert Wilson, 1966. "Programming Variable Factors," Management Science, INFORMS, vol. 13(1), pages 144-151, September.
    5. Bazaraa, Mokhtar S. & Goode, Jamie J., 1979. "A survey of various tactics for generating Lagrangian multipliers in the context of Lagrangian duality," European Journal of Operational Research, Elsevier, vol. 3(4), pages 322-338, July.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ogbe, Emmanuel & Li, Xiang, 2017. "A new cross decomposition method for stochastic mixed-integer linear programming," European Journal of Operational Research, Elsevier, vol. 256(2), pages 487-499.
    2. Altay, Nezih & Robinson Jr., Powell E. & Bretthauer, Kurt M., 2008. "Exact and heuristic solution approaches for the mixed integer setup knapsack problem," European Journal of Operational Research, Elsevier, vol. 190(3), pages 598-609, November.
    3. Walter Rei & Jean-François Cordeau & Michel Gendreau & Patrick Soriano, 2009. "Accelerating Benders Decomposition by Local Branching," INFORMS Journal on Computing, INFORMS, vol. 21(2), pages 333-345, May.
    4. Holmberg, Kaj, 1997. "Mean value cross decomposition applied to integer programming problems," European Journal of Operational Research, Elsevier, vol. 97(1), pages 124-138, February.
    5. Munoz, F.D. & Hobbs, B.F. & Watson, J.-P., 2016. "New bounding and decomposition approaches for MILP investment problems: Multi-area transmission and generation planning under policy constraints," European Journal of Operational Research, Elsevier, vol. 248(3), pages 888-898.
    6. Peiling Wu & Joseph C. Hartman & George R. Wilson, 2005. "An Integrated Model and Solution Approach for Fleet Sizing with Heterogeneous Assets," Transportation Science, INFORMS, vol. 39(1), pages 87-103, February.
    7. Ragheb Rahmaniani & Shabbir Ahmed & Teodor Gabriel Crainic & Michel Gendreau & Walter Rei, 2020. "The Benders Dual Decomposition Method," Operations Research, INFORMS, vol. 68(3), pages 878-895, May.
    8. Georgios Saharidis & Marianthi Ierapetritou, 2013. "Speed-up Benders decomposition using maximum density cut (MDC) generation," Annals of Operations Research, Springer, vol. 210(1), pages 101-123, November.
    9. Hanif Sherali & Brian Lunday, 2013. "On generating maximal nondominated Benders cuts," Annals of Operations Research, Springer, vol. 210(1), pages 57-72, November.
    10. Elisangela Martins de Sá & Ivan Contreras & Jean-François Cordeau & Ricardo Saraiva de Camargo & Gilberto de Miranda, 2015. "The Hub Line Location Problem," Transportation Science, INFORMS, vol. 49(3), pages 500-518, August.
    11. Brech, Claus-Henning & Ernst, Andreas & Kolisch, Rainer, 2019. "Scheduling medical residents’ training at university hospitals," European Journal of Operational Research, Elsevier, vol. 274(1), pages 253-266.
    12. Lim, Gino J. & Bard, Jonathan F., 2016. "Benders decomposition and an IP-based heuristic for selecting IMRT treatment beam anglesAuthor-Name: Lin, Sifeng," European Journal of Operational Research, Elsevier, vol. 251(3), pages 715-726.
    13. Halit Üster & Panitan Kewcharoenwong, 2011. "Strategic Design and Analysis of a Relay Network in Truckload Transportation," Transportation Science, INFORMS, vol. 45(4), pages 505-523, November.
    14. Hanif D. Sherali & Ki-Hwan Bae & Mohamed Haouari, 2013. "An Integrated Approach for Airline Flight Selection and Timing, Fleet Assignment, and Aircraft Routing," Transportation Science, INFORMS, vol. 47(4), pages 455-476, November.
    15. Yossiri Adulyasak & Jean-François Cordeau & Raf Jans, 2015. "Benders Decomposition for Production Routing Under Demand Uncertainty," Operations Research, INFORMS, vol. 63(4), pages 851-867, August.
    16. Mazzola, Joseph B. & Neebe, Alan W., 1999. "Lagrangian-relaxation-based solution procedures for a multiproduct capacitated facility location problem with choice of facility type," European Journal of Operational Research, Elsevier, vol. 115(2), pages 285-299, June.
    17. Jean-François Cordeau & Federico Pasin & Marius Solomon, 2006. "An integrated model for logistics network design," Annals of Operations Research, Springer, vol. 144(1), pages 59-82, April.
    18. Pourbabai, B. & Ashayeri, J. & Van Wassenhove, L.N., 1992. "Strategic marketing, production, and distribution planning of an integrated manufacturing system," Other publications TiSEM 16c2bacb-2c2b-427e-b429-c, Tilburg University, School of Economics and Management.
    19. Aliakbari Sani, Sajad & Bahn, Olivier & Delage, Erick, 2022. "Affine decision rule approximation to address demand response uncertainty in smart Grids’ capacity planning," European Journal of Operational Research, Elsevier, vol. 303(1), pages 438-455.
    20. Imai, Akio & Nishimura, Etsuko & Current, John, 2007. "A Lagrangian relaxation-based heuristic for the vehicle routing with full container load," European Journal of Operational Research, Elsevier, vol. 176(1), pages 87-105, January.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:wly:navres:v:43:y:1996:i:5:p:673-689. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Wiley Content Delivery (email available below). General contact details of provider: https://doi.org/10.1002/(ISSN)1520-6750 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.