# 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.

## Keywords

Quadratic assignment problem Mixed integer linear programming## Notes

### 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.

## 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. CrossRefGoogle 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. CrossRefGoogle 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/. CrossRefGoogle 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. CrossRefGoogle 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. Google Scholar - 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. CrossRefGoogle Scholar - Hahn, P., & Grant, T. (1998). Lower bounds for the quadratic assignment problem based upon a dual formulation.
*Operations Research*,*46*(6), 912–922. CrossRefGoogle 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. CrossRefGoogle 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. CrossRefGoogle Scholar - Lawler, E. L. (1963). The quadratic assignment problem.
*Management Science*,*9*(4), 586–599. CrossRefGoogle 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. CrossRefGoogle 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. CrossRefGoogle Scholar - Xia, Y., & Yuan, Y. X. (2006). A new linearization method for quadratic assignment problems.
*Optimization Methods and Software*,*21*(5), 805–818. CrossRefGoogle 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. CrossRefGoogle 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.