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A new exact discrete linear reformulation of the quadratic assignment problem

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  • Nyberg, Axel
  • Westerlund, Tapio

Abstract

The quadratic assignment problem (QAP) is a challenging combinatorial problem. The problem is NP-hard and in addition, it is considered practically intractable to solve large QAP instances, to proven optimality, within reasonable time limits. In this paper we present an attractive mixed integer linear programming (MILP) formulation of the QAP. We first introduce a useful non-linear formulation of the problem and then a method of how to reformulate it to a new exact, compact discrete linear model. This reformulation is efficient for QAP instances with few unique elements in the flow or distance matrices. Finally, we present optimal results, obtained with the discrete linear reformulation, for some previously unsolved instances (with the size n=32 and 64), from the quadratic assignment problem library, QAPLIB.

Suggested Citation

  • Nyberg, Axel & Westerlund, Tapio, 2012. "A new exact discrete linear reformulation of the quadratic assignment problem," European Journal of Operational Research, Elsevier, vol. 220(2), pages 314-319.
    • Handle: RePEc:eee:ejores:v:220:y:2012:i:2:p:314-319
    DOI: 10.1016/j.ejor.2012.02.010
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    References listed on IDEAS

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    1. Adams, Warren P. & Guignard, Monique & Hahn, Peter M. & Hightower, William L., 2007. "A level-2 reformulation-linearization technique bound for the quadratic assignment problem," European Journal of Operational Research, Elsevier, vol. 180(3), pages 983-996, August.
    2. Loiola, Eliane Maria & de Abreu, Nair Maria Maia & Boaventura-Netto, Paulo Oswaldo & Hahn, Peter & Querido, Tania, 2007. "A survey for the quadratic assignment problem," European Journal of Operational Research, Elsevier, vol. 176(2), pages 657-690, January.
    3. de Klerk, E. & Sotirov, R., 2007. "Exploiting Group Symmetry in Semidefinite Programming Relaxations of the Quadratic Assignment Problem," Discussion Paper 2007-44, Tilburg University, Center for Economic Research.
    4. Zvi Drezner & Peter Hahn & Éeric Taillard, 2005. "Recent Advances for the Quadratic Assignment Problem with Special Emphasis on Instances that are Difficult for Meta-Heuristic Methods," Annals of Operations Research, Springer, vol. 139(1), pages 65-94, October.
    5. Kaufman, L. & Broeckx, F., 1978. "An algorithm for the quadratic assignment problem using Bender's decomposition," European Journal of Operational Research, Elsevier, vol. 2(3), pages 207-211, May.
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    Cited by:

    1. Matteo Fischetti & Michele Monaci & Domenico Salvagnin, 2012. "Three Ideas for the Quadratic Assignment Problem," Operations Research, INFORMS, vol. 60(4), pages 954-964, August.
    2. Zvi Drezner & Alfonsas Misevičius & Gintaras Palubeckis, 2015. "Exact algorithms for the solution of the grey pattern quadratic assignment problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 82(1), pages 85-105, August.
    3. Feizollahi, Mohammad Javad & Feyzollahi, Hadi, 2015. "Robust quadratic assignment problem with budgeted uncertain flows," Operations Research Perspectives, Elsevier, vol. 2(C), pages 114-123.
    4. 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.
    5. E. de Klerk & R. Sotirov & U. Truetsch, 2015. "A New Semidefinite Programming Relaxation for the Quadratic Assignment Problem and Its Computational Perspectives," INFORMS Journal on Computing, INFORMS, vol. 27(2), pages 378-391, May.

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