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Solving the quadratic assignment problem by means of general purpose mixed integer linear programming solvers

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Abstract

The Quadratic Assignment Problem (QAP) can be solved by linearization, where one formulates the QAP as a mixed integer linear programming (MILP) problem. On the one hand, most of these linearizations are tight, but rarely exploited within a reasonable computing time because of their size. On the other hand, Kaufman and Broeckx formulation (Eur. J. Oper. Res. 2(3):204–211, 1978) is the smallest of these linearizations, but very weak. In this paper, we analyze how the Kaufman and Broeckx formulation can be tightened to obtain better QAP-MILP formulations. As shown in our numerical experiments, these tightened formulations remain small but computationally effective to solve the QAP by means of general purpose MILP solvers.

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References

  • Adams, W. P., Guignard, M., Hahn, P. M., & Hightower, W. L. (2007). A level-2 reformulation-linearization technique bound for the quadratic assignment problem. European Journal of Operational Research, 180(3), 983–996.

    Article  Google Scholar 

  • Adams, W. P., & Johnson, T. A. (1994). Improved linear programming-based lower bounds for the quadratic assignment problem. In DIMACS series in discrete mathematics and theoretical computer science (Vol. 16, pp. 43–75), Providence: American Mathematical Society.

    Google Scholar 

  • Baotic, M. (2004). Matlab interface for CPLEX, http://control.ee.ethz.ch/~hybrid/cplexint.php.

  • Burkard, R. E., Dell’Amico, M., & Martello, S. (2009). Assignment problems (pp. 203–304). Philadelphia: SIAM.

    Book  Google Scholar 

  • Burkard, R. E., Karisch, S. E., & Rendl, F. (1997). QAPLIB—a quadratic assignment problem library. Journal of Global Optimization, 10(4), 391–403. http://www.seas.upenn.edu/qaplib/.

    Article  Google Scholar 

  • Drezner, Z., Hahn, P. M., & Taillard, É. D. (2005). Recent advances for the quadratic assignment problem with special emphasis on instances that are difficult for meta-heuristic methods. Annals of Operations Research, 139(1), 65–94.

    Article  Google Scholar 

  • Eschermann, B., & Wunderlich, H. J. (1990). Optimized synthesis of self-testable finite state machines. In 20th international symposium on fault-tolerant computing (FTCS 20), Newcastle Upon Tyne, UK, June.

    Google Scholar 

  • Fischetti, M., Monaci, M., & Salvagnin, D. (2012). Three ideas for the quadratic assignment problem. Operations Research, in press.

  • Gilmore, P. C. (1962). Optimal and suboptimal algorithms for the quadratic assignment problem. Journal of the Society for Industrial and Applied Mathematics, 10(2), 305–313.

    Article  Google Scholar 

  • Hahn, P., & Grant, T. (1998). Lower bounds for the quadratic assignment problem based upon a dual formulation. Operations Research, 46(6), 912–922.

    Article  Google Scholar 

  • Kaufman, L., & Broeckx, F. (1978). An algorithm for the quadratic assignment problem using Benders’ decomposition. European Journal of Operational Research, 2(3), 204–211.

    Article  Google Scholar 

  • Koopmans, T. C., & Beckmann, M. J. (1957). Assignment problems and the location of economic activities. Econometrica. Journal of the Econometric Society, 25(1), 53–76.

    Article  Google Scholar 

  • Lawler, E. L. (1963). The quadratic assignment problem. Management Science, 9(4), 586–599.

    Article  Google Scholar 

  • Loiola, E. M., Abreu, N. M. M., Boaventura-Netto, P. O., Hahn, P., & Querido, T. (2007). A survey for the quadratic assignment problem. European Journal of Operational Research, 176(2), 657–690.

    Article  Google Scholar 

  • Peng, J., Mittelmann, H., & Li, X. (2010). A new relaxation framework for quadratic assignment problems based on matrix splitting. Mathematical Programming Computation, 2(1), 59–77.

    Article  Google Scholar 

  • Xia, Y., & Yuan, Y. X. (2006). A new linearization method for quadratic assignment problems. Optimization Methods and Software, 21(5), 805–818.

    Article  Google Scholar 

  • Zhang, H. Z., Beltran-Royo, C., & Constantino, M. (2010). Effective formulation reductions for the quadratic assignment problem. Computers and Operations Research, 37(11), 2007–2016.

    Article  Google Scholar 

  • Zhang, H. Z., Beltran-Royo, C., & Ma, L. (2010). Solving the quadratic assignment problem by means of general purpose mixed integer linear programming solvers. http://www.optimization-online.org/DB_HTML/2010/05/2622.html, pp. 1–16.

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Acknowledgements

Thanks are due to the support of the National Natural Science Foundation of China (Grant No. 70871081), and to the support from MAEC-AECID fellowship from the Spanish government. We wish to thank the Faculty of Sciences of the University of Lisbon for providing the software Cplex 11.2. We also thank the support of the grant S2009/esp-1594 from the ‘Comunidad de Madrid’ (Spain) and the grant MTM2009-14039-C06-03 from the Spanish government.

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Authors and Affiliations

  1. Statistics and Operations Research, Rey Juan Carlos University, Madrid, Spain

    Huizhen Zhang & Cesar Beltran-Royo

  2. School of Management, University of Shanghai for Science and Technology, Shanghai, China

    Huizhen Zhang & Liang Ma

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  1. Huizhen Zhang
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  2. Cesar Beltran-Royo
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  3. Liang Ma
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Correspondence to Huizhen Zhang.

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Zhang, H., Beltran-Royo, C. & Ma, L. Solving the quadratic assignment problem by means of general purpose mixed integer linear programming solvers. Ann Oper Res 207, 261–278 (2013). https://doi.org/10.1007/s10479-012-1079-4

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