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

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.

[1]  George A. Marcoulides,et al.  Modern methods for business research , 1998 .

[2]  Fred W. Glover,et al.  Future paths for integer programming and links to artificial intelligence , 1986, Comput. Oper. Res..

[3]  Franz Rendl,et al.  The Quadratic Assignment Problem , 2002 .

[4]  Alfonsas Misevicius,et al.  Ruin and Recreate Principle Based Approach for the Quadratic Assignment Problem , 2003, GECCO.

[5]  Said Salhi,et al.  Heuristic Search Methods , 1998 .

[6]  T. Koopmans,et al.  Assignment Problems and the Location of Economic Activities , 1957 .

[7]  Zvi Drezner,et al.  The multiple server location problem , 2007, J. Oper. Res. Soc..

[8]  Zvi Drezner,et al.  Facility location - applications and theory , 2001 .

[9]  Polly Bart,et al.  Heuristic Methods for Estimating the Generalized Vertex Median of a Weighted Graph , 1968, Oper. Res..

[10]  É. Taillard.  COMPARISON OF ITERATIVE SEARCHES FOR THE QUADRATIC ASSIGNMENT PROBLEM. , 1995 .

[11]  Alfonsas Misevicius,et al.  A Tabu Search Algorithm for the Quadratic Assignment Problem , 2005, Comput. Optim. Appl..

[12]  Mark S. Daskin,et al.  3 Discrete Network Location Models , 2001 .

[13]  John E. Beasley,et al.  OR-Library: Distributing Test Problems by Electronic Mail , 1990 .

[14]  C. Watson-Gandy.  Heuristic procedures for the m-partial cover problem on a plane , 1982 .

[15]  Mark S. Daskin,et al.  Network and Discrete Location: Models, Algorithms and Applications , 1995 .

[16]  Alfonsas Misevicius,et al.  An improved hybrid genetic algorithm: new results for the quadratic assignment problem , 2004, Knowl. Based Syst..

[17]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[18]  Alfonsas Misevicius.  Genetic algorithm hybridized with ruin and recreate procedure: application to the quadratic assignment problem , 2003, Knowl. Based Syst..

[19]  Z. Drezner.  Note-On a Modified One-Center Model , 1981 .

[20]  Zvi Drezner,et al.  Recent Advances for the Quadratic Assignment Problem with Special Emphasis on Instances that are Difficult for Meta-Heuristic Methods , 2005, Ann. Oper. Res..

[21]  É. Taillard,et al.  Adaptive memories for the Quadratic Assignment Problems , 1997 .