Abstract
This paper presents a history of a difficult facility layout problem that falls into the category of the Koopmans–Beckmann variant of the quadratic assignment problem (QAP), wherein 30 facilities are to be assigned to 30 locations. The problem arose in 1972 as part of the design of a German University Hospital, Klinikum Regensburg. This problem, known as the Krarup 30a upon its inclusion in the QAP library of QAP instances, has remained an important example of one of the most difficult to solve. In 1999, two approaches provided multiple optimum solutions. The first was Thomas Stützle's analysis of fitness–distance correlation that resulted in the discovery of 256 global optima. The second was a new branch-and-bound enumeration that confirmed 133 of the 256 global optima found and proved that Stützle's 256 solutions were indeed optimum solutions. We report here on the steps taken to provide in-time heuristic solutions and the methods used to finally prove the optimum.
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Adams, W. P. and Johnson, T (1994) Improved linear programming-based lower bounds for the quadratic assignment problem, Quadratic Assignment and Related Problems, DIMACS Series on Discrete Mathematics and Theoretical Computer Science, 16, 43-76.
Anstreicher, K. M. and Brixius, N. W. (2000) Solving quadratic assignment problems using convex quadratic programming relaxations, University of Iowa Report, currently available on the Web at http://www.biz.uiowa.edu/faculty/anstreicher/qapqp2.ps
Assad, A. A. and Xu, W (1985) On lower bounds for a class of quadratic 0, 1 programs. Operations Research Letters, 4, 175-180.
Bazaraa, M. S. and Kirca, O. (1983) A branch-and-bound-based heuristic for solving the quadratic assignment problem. Naval Research Quarterly, 30, 287-304.
Boese, K. D. (1996) Models for iterative global optimization. Ph.D. Thesis, University of California, Computer Science Department, Los Angeles, USA.
Burkard, R. E. and Stratmann, K. H (1978) Numerical investigations on quadratic assignment problems. Naval Research Logistical Quarterly, 25, 129-148.
Burkard, R. E. and Bönniger, T. (1983) A heuristic for quadratic Boolean programs with applications to quadratic assignment problems. European Journal of Operational Research, 13, 374-386.
Burkard, R. E. and Rendl, F. (1984) A thermodynamically motivated simulation procedure of combinatorial optimization problems. European Journal of Operational Research, 17, 169-174.
Burkard, R. E., Karisch, S. E. and Rendl, F. (1991) QAPLIB—A quadratic assignment problem library. European Journal of Operational Research, 55, 115-119. (This can be found on the web at http://www.opt.math.tu-gratz.ac.at/qaplib).
Burkard, R. E. (1991) Locations with spatial interactions: The quadratic assignment problem. P. B. Mirchandani and R. L. Francis, (eds). Discrete Location Theory, John Wiley & Sons, 387-437.
Carraresi, P. and Malucelli, F. (1992) A new lower bound for the quadratic assignment problem. Operations Research, 40(Suppl. 1), S22-S27.
Clausen, J. and Perregaard, M. (1997) Solving large quadratic assignment problems in parallel. Computational Optimization and Applications, 8, 111-128.
Connolly, D. T. (1990) An improved annealing scheme for the QAP. European Journal of Operational Research, 46, 93-100.
Cung, V. D. (1997) A scatter search based approach for the quadratic assignment problem. Proceedings of ICEC' 97, pp. 165-170.
Fleurent, C. and Ferland, J. A. (1994) Genetic hybrids for the quadratic assignment problems, Quadratic Assignment and Related Problems. DIMACS Series on Discrete Mathematics and Theoretical Computer Science, 16, 173-187.
Gilmore, P. C. (1962) Optimal and suboptimal algorithms for the quadratic assignment problem. Journal of the Society of Industrial and Applied Mathematics, 10, 305-313.
Goux, J-P., Kulkarni, S., Linderoth, J. and Yoder, M. (2000) An enabling framework for master-worker computing applications on the computational grid, available from http://www.mcs.anl.gov/metaneos/papers/mw2.ps.
Grant, T. L. (1989) An evaluation and analysis of the resolvent sequence method for solving the quadratic assignment problem. Master's Thesis, University of Pennsylvania.
Hahn, P. M. (1968) Minimization of cost in assignment of codes to data transmission. Ph.D. Dissertation, University of Pennsylvania.
Hahn, P. M. and Grant, T. L. (1998) Lower bounds for the quadratic assignment problem based upon a dual formulation. Operations Research, 46, 912-922.
Hahn, P. M. Grant, T. L. and Hall, N. (1998) A branch-and-bound algorithm for the quadratic assignment problem based on the Hungarian method. European Journal of Operational Research, 108, 629-640.
Hahn, P. M., Hightower, W., Johnson, T., Guignard-Spielberg, M. and Roucairol, C. (2001) Tree elaboration strategies in branch and bound algorithms for solving the quadratic assignment problem. Yugoslav Journal of Operations Research, 11, 41-60. Also University of Pennsylvania Systems Engineering Department. Working Report (currently available on the web at http://www.seas.upenn.edu/ ~ hahn).
House, R. W., Nelson, L. D. and Rado, T. (1965) Computer studies of a certain class of linear integer programs. A. Lavi and T. P. Vogl, (eds). Recent Advances in Optimization Techniques, John Wiley & Sons, 241-280.
Johnson, T. A. (1992) New linear programming-based solution procedures for the quadratic assignment problem. Ph.D. Dissertation, Clemson University, Clemson, SC.
Jones, T. and Forrest, S. (1995) Fitness distance correlation as a measure of problem difficulty for genetic algorithms. L. J. Eshelman, ed. Proceedings of the 6th International Conference on Genetic Algorithms, Morgan Kaufman, San Francisco, pp. 184-190.
Kaku, B. K. and Thompson, G. L. (1986) An exact algorithm for the general assignment problem. European Journal of Operational Research, 23, 382-390.
Krarup, J. (1972) Quadratic Assignment, DATA, 3/72, 12-15.
Krarup, J. and Pruzan, P. M. (1978) Computer-aided layout design. Mathematical Programming Study, 9, 75-94.
Koopmans, T. C. and Beckmann, M. J. (1957) Assignment Problems and the location of economic activities. Econometrica, 25, 53-76.
Lawler, E. L. (1963) The quadratic assignment problem. Management Science, 9, 586-599.
Li, Y., Pardalos, P. M. and Resende, M. G. (1994) A greedy randomized adaptive search procedure for the quadratic assignment problem. DIMACS Series on Discrete Mathematics and Theoretical Computer Science, 16, 237-261.
Mautor, T. and Roucairol, C. (1994) A new exact algorithm for the solution of quadratic assignment problems. Discrete Applied Mathematics, 55, 281-293.
Merz, P. and Freisleben, B. (1997) A genetic local search approach to the Quadratic Assignment Problem. Proceedings of the Seventh International Conference on Genetic Algorithms (ICGA'97), pp. 465-472.
Nugent, C. E., Vollman, T. E. and Ruml, J. (1968) An experimental comparison of techniques for the assignment of facilities to locations. Operations Research, 16, 150-173.
Pardalos, P. M. and Crouse, J. V. (1989) A parallel algorithm for the quadratic assignment problem. Proceedings of the 1989 Supercomputing Conference, ACM Press, pp. 351-360.
Pierce, J. F. and Crowston, W. B. (1971) Tree-search algorithms for quadratic assignment problems. Naval Research Logistics Quarterly, 18, 1-36.
Roucairol, C. (1979) A reduction method for the quadratic assignment problem. Operations Research Verfahren-Methods of Operations Research, 32, 185-187.
Skorin-Kapov, J. (1990) Tabu search applied to the quadratic assignment problem. ORSA Journal on Computing, 2, 33-45.
Stützle, T. (1997) MAX-MIN ant system for quadratic assignment problems. Research Report AIDA-97-04, Department of Computer Science, Darmstadt University of Technology, Germany.
Stützle, T. (1999) Iterated local search for the quadratic assignment problem. The European Journal of Operational Research, Research Report AIDA-99-03, Department of Computer Science, Darmstadt University of Technology, Germany (submitted).
Stützle, T. and Dorigo, M. (1999) ACO algorithms for the quadratic assignment problem. D. Corne, M. Dorigo and F. Glover, eds. New Ideas in Optimization, McGraw-Hill, London, UK, 33-50.
Taillard, E. D. (1991) Robust tabu search for the quadratic assignment problem. Parallel Computing, 17, 443-455.
Taillard, E. D. (1995) Comparison of iterative searches for the quadratic assignment problem. Location Science, 3, 87-105.
Tate, D. M. and Smith, A. E. (1995) A genetic approach to the quadratic assignment problem. Computers and Operations Research, 22, 73-83.
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Hahn, P.M., Krarup, J. A hospital facility layout problem finally solved. Journal of Intelligent Manufacturing 12, 487–496 (2001). https://doi.org/10.1023/A:1012252420779
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DOI: https://doi.org/10.1023/A:1012252420779