IDEAS home Printed from https://ideas.repec.org/a/inm/orijoc/v31y2019i3p477-492.html
   My bibliography  Save this article

An Exact Algorithm for the Quadratic Multiknapsack Problem with an Application to Event Seating

Author

Listed:
  • David Bergman

    (Department of Operations and Information Management, University of Connecticut, Storrs, Connecticut 06268)

Abstract

Knapsack problems play a pivotal role in the operations research literature, with various generalizations proposed and studied over the last century. Of recent interest is the quadratic multiknapsack problem (QMKP). Despite a plethora of heuristics, no exact methods for the QMKP have been published in the literature. This paper presents an exact branch-and-price algorithm for the QMKP. Experimental results indicate that the proposed algorithm is far superior, both in terms of solution times and objective function bounds, to state-of-the-art optimization technology solving a standard encoding of the problem. In addition to the algorithmic contribution, this paper studies the optimization problem of seating attendees at events, an operational challenge faced by event organizers. An optimization model for table event seating is shown to be closely related to the QMKP, and computational testing indicates that the proposed algorithm is particularly well suited for this application.

Suggested Citation

  • David Bergman, 2019. "An Exact Algorithm for the Quadratic Multiknapsack Problem with an Application to Event Seating," INFORMS Journal on Computing, INFORMS, vol. 31(3), pages 477-492, July.
    • Handle: RePEc:inm:orijoc:v:31:y:2019:i:3:p:477-492
    DOI: 10.1287/ijoc.2018.0840
    as

    Download full text from publisher

    File URL: https://doi.org/10.1287/ijoc.2018.0840
    Download Restriction: no
    File URL: https://libkey.io/10.1287/ijoc.2018.0840?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. Kurt M. Bretthauer & Bala Shetty & Siddhartha Syam, 1995. "A Branch and Bound Algorithm for Integer Quadratic Knapsack Problems," INFORMS Journal on Computing, INFORMS, vol. 7(1), pages 109-116, February.
    2. Tjalling C. Koopmans & Martin J. Beckmann, 1955. "Assignment Problems and the Location of Economic Activities," Cowles Foundation Discussion Papers 4, Cowles Foundation for Research in Economics, Yale University.
    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. Alberto Caprara & David Pisinger & Paolo Toth, 1999. "Exact Solution of the Quadratic Knapsack Problem," INFORMS Journal on Computing, INFORMS, vol. 11(2), pages 125-137, May.
    5. W. David Pisinger & Anders Bo Rasmussen & Rune Sandvik, 2007. "Solution of Large Quadratic Knapsack Problems Through Aggressive Reduction," INFORMS Journal on Computing, INFORMS, vol. 19(2), pages 280-290, May.
    6. Pisinger, David, 1999. "An exact algorithm for large multiple knapsack problems," European Journal of Operational Research, Elsevier, vol. 114(3), pages 528-541, May.
    7. Franklin Djeumou Fomeni & Adam N. Letchford, 2014. "A Dynamic Programming Heuristic for the Quadratic Knapsack Problem," INFORMS Journal on Computing, INFORMS, vol. 26(1), pages 173-182, February.
    8. Samir Elhedhli & Jean-Louis Goffin, 2005. "Efficient Production-Distribution System Design," Management Science, INFORMS, vol. 51(7), pages 1151-1164, July.
    9. Eugene L. Lawler, 1963. "The Quadratic Assignment Problem," Management Science, INFORMS, vol. 9(4), pages 586-599, July.
    10. Pessoa, Artur Alves & Hahn, Peter M. & Guignard, Monique & Zhu, Yi-Rong, 2010. "Algorithms for the generalized quadratic assignment problem combining Lagrangean decomposition and the Reformulation-Linearization Technique," European Journal of Operational Research, Elsevier, vol. 206(1), pages 54-63, October.
    11. Ming S. Hung & John C. Fisk, 1978. "An algorithm for 0‐1 multiple‐knapsack problems," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 25(3), pages 571-579, September.
    12. Fred Glover, 1975. "Improved Linear Integer Programming Formulations of Nonlinear Integer Problems," Management Science, INFORMS, vol. 22(4), pages 455-460, December.
    13. Frédéric Roupin, 2004. "From Linear to Semidefinite Programming: An Algorithm to Obtain Semidefinite Relaxations for Bivalent Quadratic Problems," Journal of Combinatorial Optimization, Springer, vol. 8(4), pages 469-493, December.
    14. Bretthauer, Kurt M. & Shetty, Bala, 2002. "The nonlinear knapsack problem - algorithms and applications," European Journal of Operational Research, Elsevier, vol. 138(3), pages 459-472, May.
    15. Yuning Chen & Jin-Kao Hao, 2015. "Iterated responsive threshold search for the quadratic multiple knapsack problem," Annals of Operations Research, Springer, vol. 226(1), pages 101-131, March.
    16. Cordeau, Jean-Francois & Gaudioso, Manlio & Laporte, Gilbert & Moccia, Luigi, 2007. "The service allocation problem at the Gioia Tauro Maritime Terminal," European Journal of Operational Research, Elsevier, vol. 176(2), pages 1167-1184, January.
    17. M. Dawande & J. Kalagnanam & P. Keskinocak & F.S. Salman & R. Ravi, 2000. "Approximation Algorithms for the Multiple Knapsack Problem with Assignment Restrictions," Journal of Combinatorial Optimization, Springer, vol. 4(2), pages 171-186, June.
    18. Billionnet, Alain & Calmels, Frederic, 1996. "Linear programming for the 0-1 quadratic knapsack problem," European Journal of Operational Research, Elsevier, vol. 92(2), pages 310-325, July.
    19. Yuji Nakagawa & Ross J. W. James & César Rego & Chanaka Edirisinghe, 2014. "Entropy-Based Optimization of Nonlinear Separable Discrete Decision Models," Management Science, INFORMS, vol. 60(3), pages 695-707, March.
    20. Jean-François Cordeau & Manlio Gaudioso & Gilbert Laporte & Luigi Moccia, 2006. "A Memetic Heuristic for the Generalized Quadratic Assignment Problem," INFORMS Journal on Computing, INFORMS, vol. 18(4), pages 433-443, November.
    21. Billionnet, Alain & Soutif, Eric, 2004. "An exact method based on Lagrangian decomposition for the 0-1 quadratic knapsack problem," European Journal of Operational Research, Elsevier, vol. 157(3), pages 565-575, September.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Philippe Olivier & Andrea Lodi & Gilles Pesant, 2021. "The Quadratic Multiknapsack Problem with Conflicts and Balance Constraints," INFORMS Journal on Computing, INFORMS, vol. 33(3), pages 949-962, July.
    2. Wu, Zhengtian & Jiang, Baoping & Karimi, Hamid Reza, 2020. "A logarithmic descent direction algorithm for the quadratic knapsack problem," Applied Mathematics and Computation, Elsevier, vol. 369(C).
    3. Nihal Berktaş & Hande Yaman, 2021. "A Branch-and-Bound Algorithm for Team Formation on Social Networks," INFORMS Journal on Computing, INFORMS, vol. 33(3), pages 1162-1176, July.
    4. Fleszar, Krzysztof, 2022. "A branch-and-bound algorithm for the quadratic multiple knapsack problem," European Journal of Operational Research, Elsevier, vol. 298(1), pages 89-98.
    5. Saharnaz Mehrani & Carlos Cardonha & David Bergman, 2022. "Models and Algorithms for the Bin-Packing Problem with Minimum Color Fragmentation," INFORMS Journal on Computing, INFORMS, vol. 34(2), pages 1070-1085, March.

    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. Gabriel Lopez Zenarosa & Oleg A. Prokopyev & Eduardo L. Pasiliao, 2021. "On exact solution approaches for bilevel quadratic 0–1 knapsack problem," Annals of Operations Research, Springer, vol. 298(1), pages 555-572, March.
    2. D. Quadri & E. Soutif & P. Tolla, 2009. "Exact solution method to solve large scale integer quadratic multidimensional knapsack problems," Journal of Combinatorial Optimization, Springer, vol. 17(2), pages 157-167, February.
    3. Caprara, Alberto, 2008. "Constrained 0-1 quadratic programming: Basic approaches and extensions," European Journal of Operational Research, Elsevier, vol. 187(3), pages 1494-1503, June.
    4. Dell’Amico, Mauro & Delorme, Maxence & Iori, Manuel & Martello, Silvano, 2019. "Mathematical models and decomposition methods for the multiple knapsack problem," European Journal of Operational Research, Elsevier, vol. 274(3), pages 886-899.
    5. Pessoa, Artur Alves & Hahn, Peter M. & Guignard, Monique & Zhu, Yi-Rong, 2010. "Algorithms for the generalized quadratic assignment problem combining Lagrangean decomposition and the Reformulation-Linearization Technique," European Journal of Operational Research, Elsevier, vol. 206(1), pages 54-63, October.
    6. Yuning Chen & Jin-Kao Hao, 2015. "Iterated responsive threshold search for the quadratic multiple knapsack problem," Annals of Operations Research, Springer, vol. 226(1), pages 101-131, March.
    7. Britta Schulze & Michael Stiglmayr & Luís Paquete & Carlos M. Fonseca & David Willems & Stefan Ruzika, 2020. "On the rectangular knapsack problem: approximation of a specific quadratic knapsack problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(1), pages 107-132, August.
    8. M. Gaudioso & L. Moccia & M. F. Monaco, 2010. "Repulsive Assignment Problem," Journal of Optimization Theory and Applications, Springer, vol. 144(2), pages 255-273, February.
    9. Nihal Berktaş & Hande Yaman, 2021. "A Branch-and-Bound Algorithm for Team Formation on Social Networks," INFORMS Journal on Computing, INFORMS, vol. 33(3), pages 1162-1176, July.
    10. Hoto, R.S.V. & Matioli, L.C. & Santos, P.S.M., 2020. "A penalty algorithm for solving convex separable knapsack problems," Applied Mathematics and Computation, Elsevier, vol. 387(C).
    11. Monique Guignard, 2020. "Strong RLT1 bounds from decomposable Lagrangean relaxation for some quadratic 0–1 optimization problems with linear constraints," Annals of Operations Research, Springer, vol. 286(1), pages 173-200, March.
    12. Fabio Furini & Emiliano Traversi, 2019. "Theoretical and computational study of several linearisation techniques for binary quadratic problems," Annals of Operations Research, Springer, vol. 279(1), pages 387-411, August.
    13. Jesus Cunha & Luidi Simonetti & Abilio Lucena, 2016. "Lagrangian heuristics for the Quadratic Knapsack Problem," Computational Optimization and Applications, Springer, vol. 63(1), pages 97-120, January.
    14. Alain Billionnet & Éric Soutif, 2004. "Using a Mixed Integer Programming Tool for Solving the 0–1 Quadratic Knapsack Problem," INFORMS Journal on Computing, INFORMS, vol. 16(2), pages 188-197, May.
    15. Z. Y. Wu & Y. J. Yang & F. S. Bai & M. Mammadov, 2011. "Global Optimality Conditions and Optimization Methods for Quadratic Knapsack Problems," Journal of Optimization Theory and Applications, Springer, vol. 151(2), pages 241-259, November.
    16. Bretthauer, Kurt M. & Shetty, Bala, 2002. "The nonlinear knapsack problem - algorithms and applications," European Journal of Operational Research, Elsevier, vol. 138(3), pages 459-472, May.
    17. Richard J. Forrester & Warren P. Adams & Paul T. Hadavas, 2010. "Concise RLT forms of binary programs: A computational study of the quadratic knapsack problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 57(1), pages 1-12, February.
    18. Silva, Allyson & Coelho, Leandro C. & Darvish, Maryam, 2021. "Quadratic assignment problem variants: A survey and an effective parallel memetic iterated tabu search," European Journal of Operational Research, Elsevier, vol. 292(3), pages 1066-1084.
    19. Alberto Caprara & David Pisinger & Paolo Toth, 1999. "Exact Solution of the Quadratic Knapsack Problem," INFORMS Journal on Computing, INFORMS, vol. 11(2), pages 125-137, May.
    20. Edirisinghe, Chanaka & Jeong, Jaehwan, 2019. "Indefinite multi-constrained separable quadratic optimization: Large-scale efficient solution," European Journal of Operational Research, Elsevier, vol. 278(1), pages 49-63.

    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:inm:orijoc:v:31:y:2019:i:3:p:477-492. 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: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    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.