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Finding a cluster of points and the grey pattern quadratic assignment problem

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Abstract

In this paper we propose a model which aims at selecting a tight cluster from a set of points. The same formulation applies also to the grey pattern problem where the objective is to find a set of black dots in a rectangular grid with a given density so that the dots are spread as evenly as possible. A branch and bound algorithm and five heuristic approaches are proposed. Computational results demonstrate the efficiency of these approaches. Seven grey pattern problems are solved to optimality and for eight additional grey pattern problems the best known solution is improved. The cluster problem on a network is solved for 40 problems with the number of points ranging between 100 and 900 and the size of the cluster ranging between 5 and 200. Twenty one problems were solved optimally and the remaining 19 problems were heuristically solved in a very short computer time with excellent results.

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References

  • Beasley JE (1990) OR-library—distributing test problems by electronic mail. J Oper Res Soc 41:1069–1072. Also available at http://mscmga.ms.ic.ac.uk/jeb/orlib/pmedinfo.html

    Google Scholar 

  • Berman O, Drezner Z (2005) The multiple server location problem. J Oper Res Soc (in press)

  • Current J, Daskin M, Schilling D (2002) Discrete network location models. Ch. 3 In: Drezner Z, Hamacher HW (eds) Location analysis: applications and theory, pp 81–118

  • Daskin M (1995) Network and discrete location: models, algorithms and applications. Wiley, New York

    Google Scholar 

  • Drezner Z (1981) On a modified one-center model. Manage Sci 27:848–851

    Article  Google Scholar 

  • Drezner Z, Hahn PM, Taillard ED (2005) Recent advances for the quadratic assignment problem with special emphasis on instances that are difficult for meta-heuristic methods. Ann Oper Res 139:65–94

    Google Scholar 

  • Drezner Z, Marcoulides GA (2005) On the range of tabu tenure in solving quadratic assignment problems. Under review

  • Glover F (1986) Future paths for integer programming and links to artificial intelligence. Comput Oper Res 13:533–549

    Article  Google Scholar 

  • Glover F, Laguna M (1997) Tabu search. Kluwer, Boston

    Google Scholar 

  • Kirkpatrick S, Gelat CD, Vecchi MP (1983) Optimization by simulated annealing. Science 220:671–680

    Article  Google Scholar 

  • Koopmans TC, Beckmann MJ (1957) Assignment problems and the location of economics activities. Econometrica 25:53–76

    Article  Google Scholar 

  • Misevicius A (2003a) Ruin and recreate principle based approach for the quadratic assignment problem. Lecture notes in computer science, vol 2723. In: Cant-Paz E, Foster JA, Deb K, et al (eds) Genetic and Evolutionary Computation—GECCO 2003 (Chicago, USA), Proceedings, Part I, Springer, Berlin Heidelberg New York pp. 598–609

  • Misevicius A (2003b) Genetic algorithm hybridized with ruin and recreate procedure: application to the quadratic assignment problem. Knowl Based Syst 16:261–268

    Article  Google Scholar 

  • Misevicius A (2004) An improved hybrid genetic algorithm: new results for the quadratic assignment problem. Knowl Based Syst 17:65–73

    Article  Google Scholar 

  • Misevicius A (2005) A tabu search algorithm for the quadratic assignment problem. Working paper, Kaunas University of Technology, Kaunas, Lithuania Comput Optim Appl 30:95–111

  • Rendl F (2002) The quadratic assignment problem. In: Drezner Z, Hamacher H (eds) Facility location: applications and theory. Springer, Berlin Heidelberg New York

    Google Scholar 

  • Salhi S (1998) Heuristic search methods. In: Marcoulides G (ed) Modern methods for business research. Lawrence Erlbaum Associates, Mahwah, NJ

    Google Scholar 

  • Taillard ED (1995) Comparison of iterative searches for the quadratic assignment problem. Location Science 3:87–105

    Article  Google Scholar 

  • Taillard ED, Gambardella LM (1997) Adaptive memories for the quadratic assignment problem. Tech. report IDSIA-8797. Lugano, Switzerland

    Google Scholar 

  • Teitz MB, Bart P (1968) Heuristic methods for estimating the generalized vertex median of a weighted graph. Oper Res 16:955-961

    Google Scholar 

  • Watson-Gandy CDT (1982) Heuristic procedures for the M-partial cover problem on a plane. Eur J Oper Res 11:149-157

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. College of Business and Economics, California State University-Fullerton, Fullerton, CA, 92834, USA

    Zvi Drezner

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  1. Zvi Drezner
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Correspondence to Zvi Drezner.

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Drezner, Z. Finding a cluster of points and the grey pattern quadratic assignment problem. OR Spectrum 28, 417–436 (2006). https://doi.org/10.1007/s00291-005-0010-7

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