A GRASP with path-relinking for the p-median problem

Given n customers and a set F of m potential facilities, the p-median problem consists in finding a subset of F with p facilities such that the cost of serving all customers is minimized. This is a well-known NPcomplete problem with important applications in location science and classification (clustering). We present here a GRASP (Greedy Randomized Adaptive Search Procedure) with path-relinking to find near-optimal solutions to this problem. Empirical results on instances from the literature suggest that this is a very robust algorithm, performing at least as well as other methods, and often better in terms of both running time and solution quality.

[1]  Belén Melián-Batista,et al.  The Parallel Variable Neighborhood Search for the p-Median Problem , 2002, J. Heuristics.

[2]  Éric D. Taillard,et al.  Heuristic Methods for Large Centroid Clustering Problems , 2003, J. Heuristics.

[3]  Roberto D. Galvão,et al.  A Dual-Bounded Algorithm for the p-Median Problem , 1980, Oper. Res..

[4]  L. A. Lorena,et al.  Stabilizing column generation using Lagrangean/surrogate relaxation: an application to p-median location problems , 2003 .

[5]  Robert E. Markland,et al.  Theory and Application of an Optimizing Procedure for Lock Box Location Analysis , 1981 .

[6]  Alfred A. Kuehn,et al.  A Heuristic Program for Locating Warehouses , 1963 .

[7]  M. J. Hodgson.  Toward More Realistic Allocation in Location—Allocation Models: An Interaction Approach , 1978 .

[8]  Kenneth E. Rosing,et al.  An Empirical Investigation of the Effectiveness of a Vertex Substitution Heuristic , 1997 .

[9]  Pierre Hansen,et al.  Stabilized column generation , 1998, Discret. Math..

[10]  Panos M. Pardalos,et al.  GRASP With Path Relinking For The Three-Index Assignment Problem , 2000 .

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

[12]  George L. Nemhauser,et al.  Note--On "Location of Bank Accounts to Optimize Float: An Analytic Study of Exact and Approximate Algorithms" , 1979 .

[13]  Luiz Antonio Nogueira Lorena,et al.  Lagrangean/Surrogate Heuristics for p-Median Problems , 2000 .

[14]  G. Nemhauser,et al.  Exceptional Paper—Location of Bank Accounts to Optimize Float: An Analytic Study of Exact and Approximate Algorithms , 1977 .

[15]  R. A. Whitaker,et al.  A Fast Algorithm For The Greedy Interchange For Large-Scale Clustering And Median Location Problems , 1983 .

[16]  K. E. Rosing,et al.  The p-Median and its Linear Programming Relaxation: An Approach to Large Problems , 1979 .

[17]  Mauricio G. C. Resende,et al.  Grasp: An Annotated Bibliography , 2002 .

[18]  Fred Glover,et al.  Tabu Search and Adaptive Memory Programming — Advances, Applications and Challenges , 1997 .

[19]  Mauricio G. C. Resende,et al.  On the implementation of a swap-based local search procedure for the p -median problem ∗ , 2002 .

[20]  Pierre Hansen,et al.  Variable Neighborhood Decomposition Search , 1998, J. Heuristics.

[21]  J. Current,et al.  Heuristic concentration and Tabu search: A head to head comparison , 1998 .

[22]  Takuji Nishimura,et al.  Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator , 1998, TOMC.

[23]  Celso C. Ribeiro,et al.  A Hybrid GRASP with Perturbations for the Steiner Problem in Graphs , 2002, INFORMS J. Comput..

[24]  J. Current,et al.  An efficient tabu search procedure for the p-Median Problem , 1997 .

[25]  C. Revelle,et al.  Heuristic concentration: Two stage solution construction , 1997 .

[26]  M. Resende,et al.  A probabilistic heuristic for a computationally difficult set covering problem , 1989 .

[27]  F. E. Maranzana,et al.  On the Location of Supply Points to Minimize Transport Costs , 1964 .

[28]  P. Hansen,et al.  Variable neighborhood search for the p-median , 1997 .

[29]  C. Revelle,et al.  A Lagrangean heuristic for the maximal covering location problem , 1996 .

[30]  Mikkel Thorup.  Quick k-Median, k-Center, and Facility Location for Sparse Graphs , 2001, ICALP.

[31]  J. Beasley.  A note on solving large p-median problems , 1985 .

[32]  Mauricio G. C. Resende,et al.  Greedy Randomized Adaptive Search Procedures , 1995, J. Glob. Optim..

[33]  Rafael Martí,et al.  GRASP and Path Relinking for 2-Layer Straight Line Crossing Minimization , 1999, INFORMS J. Comput..

[34]  F. Glover,et al.  Fundamentals of Scatter Search and Path Relinking , 2000 .

[35]  Gerhard Reinelt,et al.  TSPLIB - A Traveling Salesman Problem Library , 1991, INFORMS J. Comput..

[36]  Francesco E. Maranzana,et al.  On the Location of Supply Points to Minimize Transportation Costs , 1963, IBM Syst. J..