Abstract
Two efficient neighborhood reduction schemes are proposed for the solution of the p-median problem on the plane. Their integration into a local search significantly reduces the run time with an insignificant deterioration in the quality of the solution. For completeness this fast local search is also embedded into one of the most powerful metaheuristic algorithms, which is a combination of a genetic algorithm and a new improvement algorithm, recently developed for this continuous location problem. Excellent results for instances with up to 1060 demand points with various values of p are reported. Eight new best known solutions for ten instances of a large problem with 3038 demand points and up to 500 facilities are also found.
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References
Berman, O., Drezner, Z., & Krass, D. (2011). Big segment small segment global optimization algorithm on networks. Networks, 58, 1–11.
Bongartz, I., Calamai, P. H., & Conn, A. R. (1994). A projection method for \(\ell _p\) norm location-allocation problems. Mathematical Programming, 66, 238–312.
Brimberg, J., & Drezner, Z. (2013). A new heuristic for solving the p-median problem in the plane. Computers & Operations Research, 40, 427–437.
Brimberg, J., Drezner, Z., Mladenović, N., & Salhi, S. (2014). A new local search for continuous location problems. European Journal of Operational Research, 232, 256–265.
Brimberg, J., Hansen, P., & Mladenović, N. (2006). Decomposition strategies for large-scale continuous location-allocation problems. IMA Journal of Management Mathematics, 17, 307–316.
Brimberg, J., Hansen, P., Mladenović, N., & Salhi, S. (2008). A survey of solution methods for the continuous location-allocation problem. International Journal of Operations Research, 5, 1–12.
Brimberg, J., Hansen, P., Mladenović, N., & Taillard, E. (2000). Improvements and comparison of heuristics for solving the uncapacitated multisource Weber problem. Operations Research, 48, 444–460.
Chen, P. C., Hansen, P., Jaumard, B., & Tuy, H. (1998). A fast algorithm for the greedy interchange for large-scale clustering and median location problems by D.-C. programming. Operations Research, 46, 548–562.
Chen, R. (1983). Solution of minisum and minimax location-allocation problems with euclidean distances. Naval Research Logistics Quarterly, 30, 449–459.
Christofides, N., & Beasley, J. E. (1982). A tree search algorithm for the p-median problem. European Journal of Operational Research, 10, 196–204.
Church, R. L. (2003). COBRA: A new formulation of the classic p-median location problem. Annals of Operations Research, 122, 103–120.
Church, R. L. (2008). BEAMR: An exact and approximate model for the p-median problem. Computers & Operations Research, 35, 417–426.
Cooper, L. (1963). Location-allocation problems. Operations Research, 11, 331–343.
Cooper, L. (1964). Heuristic methods for location-allocation problems. SIAM Review, 6, 37–53.
Drezner, Z. (1984). The planar two-center and two-median problems. Transportation Science, 18, 351–361.
Drezner, Z., Brimberg, J., Salhi, S., & Mladenović, N. (2015a). New heuristic algorithms for solving the planar \(p\)-median problem. Computers and Operations Research, 62, 296–304.
Drezner, Z., Brimberg, J., Salhi, S., & Mladenović, N. (2015b). New local searches for solving the multi-source Weber problem. Annals of Operations Research. doi:10.1007/s10479-015-1797-5.
Drezner, Z., Mehrez, A., & Wesolowsky, G. O. (1991). The facility location problem with limited distances. Transportation Science, 25, 183–187.
Drezner, Z., & Suzuki, A. (2004). The big triangle small triangle method for the solution of non-convex facility location problems. Operations Research, 52, 128–135.
Eilon, S., Watson-Gandy, C. D. T., & Christofides, N. (1971). Distribution management. New York: Hafner.
García, S., Labbé, M., & Marín, A. (2011). Solving large p-median problems with a radius formulation. INFORMS Journal on Computing, 23, 546–556.
Golden, B. L., Magnanti, T. L., & Nguyen, H. Q. (1977). Implementing vehicle routing algorithms. Networks, 7, 113–148.
Hansen, P., & Mladenović, N. (1997). Variable neighborhood search for the \(p\)-median. Location Science, 5, 207–226.
Hansen, P., Peeters, D., & Thisse, J.-F. (1981). On the location of an obnoxious facility. Sistemi Urbani, 3, 299–317.
Hillsman, E. (1979). A system for location-allocation analysis. Ph.D. thesis, University of Iowa, Iowa City.
Krau, S. (1997). Extensions du problème de Weber. Ph.D. thesis, École Polytechnique de Montréal.
Megiddo, N., & Supowit, K. J. (1984). On the complexity of some common geometric location problems. SIAM Journal on Computing, 13, 182–196.
Mladenović, N., Dražić, M., Kovačevic-Vujčić, V., & Čangalović, M. (2008). General variable neighborhood search for the continuous optimization. European Journal of Operational Research, 191(3), 753–770.
Mladenović, N., & Hansen, P. (1997). Variable neighborhood search. Computers & Operations Research, 24, 1097–1100.
Murtagh, B. A., & Niwattisyawong, S. R. (1982). An efficient method for the multi-depot location-allocation problem. Journal of the Operational Research Society, 33, 629–634.
Plastria, F. (1992). GBSSS, the generalized big square small square method for planar single facility location. European Journal of Operational Research, 62, 163–174.
Reinelt, G. (1991). TSLIB a traveling salesman library. ORSA Journal on Computing, 3, 376–384.
Rosing, K., & ReVelle, C. (1997). Heuristic concentration: A two stage solution construction. European Journal of Operational Research, 97, 75–86.
Rosing, K. E. (1992). An optimal method for solving the (generalized) multi-Weber problem. European Journal of Operational Research, 58, 414–426.
Rosing, K. E., & Harris, B. (1992). Algorithmic and technical improvements: Optimal solutions to the (generalized) multi-Weber problem. Papers in Regional Science, 71, 331–352.
Salhi, S., & Sari, M. (1997). A multi-level composite heuristic for the multi-depot vehicle fleet mix problem. European Journal of Operational Research, 103, 95–112.
Schöbel, A., & Scholz, D. (2010). The big cube small cube solution method for multidimensional facility location problems. Computers and Operations Research, 37, 115–122.
Sorensen, P. A., & Church, R. L. (1995). A comparison of strategies for data storage reduction in location-allocation problems. Geographical Systems, 3, 221–242.
Taillard, É. (2003). Heuristic methods for large centroid clustering problems. Journal of Heuristics, 9, 51–73.
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Drezner, Z., Salhi, S. Incorporating neighborhood reduction for the solution of the planar p-median problem. Ann Oper Res 258, 639–654 (2017). https://doi.org/10.1007/s10479-015-1961-y
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DOI: https://doi.org/10.1007/s10479-015-1961-y