# A COMPARATIVE STUDY OF SOLVING QUADRATIC ASSIGNMENT PROBLEMS USING SOME STANDARD MINLP SOLVERS

### Toni Lastusilta, Tapio Westerlund

#### Abstract

The Quadratic Assignment Problem (QAP) has important application areas, for example, facility layout (Dickey and Hopkins, 1972) and electronic component placement (Rabak and Sichman, 2003). The NP-hard problem already becomes difficult and time consuming to solve satisfactorily for small applications. It is therefore of interest to investigate how well standard MINLP methods can provide good solutions within a reasonable time, even though global optimality can not be guaranteed. In this study we focus on solving a subset of 50 problems in the QAP library (Burkard et al., 1997). We use a standard Mixed-Integer Non-Linear Programming (MINLP) formulation modelled in the General Algebraic Modeling System (GAMS) (Rosenthal, 2010). The solution quality and solution time is evaluated for the solvers AlphaECP, Bonmin, DICOPT and SBB. We compare the solvers when a 1 hour time limit per problem is used, where the solvers are started from 3 random start points, i.e. initial variable levels. Furthermore, we investigate how well the most promising solver DICOPT performs when started from 50 random start points for 22 problems for which the global optimal solution is known.

#### References

- Bos, J., 1993. Zoning in Forest Management: a Quadratic Assignment Problem Solved by Simulated Annealing. Journal of Environmental Management, 37, pp. 127- 145
- Burkard, R. E., Karisch, S. E., Rendl, F., 1997. QAPLIB - A Quadratic Assignment Problem Library. Journal of Global Optimization 10,pp. 391-403.
- Çela, E., 1998.The Quadratic Assignment Problem: Theory and Algorithms. Kluwer Academic Publishers.
- Dickey, J. W., Hopkins J. W., 1972. Campus building arrangement using TOPAZ. Transportation Research, 6, pp.59-68.
- Elshafei, A. N., 1977. Hospital Layout as a Quadratic Assignment Problem. Operational Research Quarterly, Pergamon Press Vol. 28 (1ii), pp.167-179.
- Hahn, P., Anjos, M., 2002. QAPLIB - A Quadratic Assignment Problem Library Online. University of Pennsylvania, Engineering and Applied Science.
- Koopmans, T. C., Beckmann, M., 1957. Assignment Problems and the Location of Economic Activities. Econometrica, Vol. 25 (1), pp. 53-76.
- Miranda G., Luna, H. P. L., Mateus, G. R., Ferreira, R. P. M., 2005.A performance guarantee heuristic for electronic components placement problems including thermal effects. Computers & Operations Research, 32, pp. 2937-2957
- Rabak, C. S., Sichman, J. S., 2003. Using A-Teams to optimize automatic insertion of electronic components Advanced Engineering Informatics, 17, pp. 95-106.
- Rosenthal R. E., 2010, GAMS -A User's Guide, GAMS Development Corporation, <http://www.gams.com/dd/ docs/bigdocs/GAMSUsersGuide.pdf> accessed 25.3.2011

#### Paper Citation

#### in Harvard Style

Lastusilta T. and Westerlund T. (2011). ** A COMPARATIVE STUDY OF SOLVING QUADRATIC ASSIGNMENT PROBLEMS USING SOME STANDARD MINLP SOLVERS** . In *Proceedings of 1st International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,* ISBN 978-989-8425-78-2, pages 409-412. DOI: 10.5220/0003598604090412

#### in Bibtex Style

@conference{simultech11,

author={Toni Lastusilta and Tapio Westerlund},

title={ A COMPARATIVE STUDY OF SOLVING QUADRATIC ASSIGNMENT PROBLEMS USING SOME STANDARD MINLP SOLVERS},

booktitle={Proceedings of 1st International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,},

year={2011},

pages={409-412},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0003598604090412},

isbn={978-989-8425-78-2},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of 1st International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,

TI - A COMPARATIVE STUDY OF SOLVING QUADRATIC ASSIGNMENT PROBLEMS USING SOME STANDARD MINLP SOLVERS

SN - 978-989-8425-78-2

AU - Lastusilta T.

AU - Westerlund T.

PY - 2011

SP - 409

EP - 412

DO - 10.5220/0003598604090412