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Near-optimal solutions to large-scale facility location problems. (English) Zbl 1140.90442
Summary: We investigate the solution of large-scale instances of the capacitated and uncapacitated facility location problems. Let $n$ be the number of customers and $m$ the number of potential facility sites. For the uncapacitated case we solved instances of size $m\times n=3000\times 3000$; for the capacitated case the largest instances were $1000\times 1000$. We use heuristics that produce a feasible integer solution and use a Lagrangian relaxation to obtain a lower bound on the optimal value. In particular, we present new heuristics whose gap from optimality was generally below $1\%$. The heuristics combine the volume algorithm and randomized rounding. For the uncapacitated facility location problem, our computational experiments show that our heuristic compares favorably against DUALOC.

MSC:
90B85Continuous location
90C59Approximation methods and heuristics
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References:
[1] Aardal, K.: Capacitated facility locationseparation algorithms and computational experience. Math. programming 81, 149-175 (1998) · Zbl 0919.90096
[2] Ahn, S.; Cooper, C.; Cornuéjols, G.; Frieze, A. M.: Probabilistic analysis of a relaxation for the k-median problem. Math. oper. Res. 13, 1-31 (1988) · Zbl 0653.90049
[3] Balinski, M. L.: Integer programmingmethods, uses, computation. Management sci. 12, No. 3, 253-313 (1965) · Zbl 0129.12004
[4] Barahona, F.; Anbil, R.: The volume algorithmproducing primal solutions with a subgradient method. Math. programming 87, 385-399 (2000) · Zbl 0961.90058
[5] F.A. Chudak, Improved approximation algorithms for the uncapacitated facility location problem, Ph.D. Thesis, Cornell University, 1998. · Zbl 0910.90201
[6] Chudak, F. A.: Improved approximation algorithms for uncapacitated facility location. Proceedings of the 6th IPCO conference, 180-194 (1998) · Zbl 0910.90201
[7] F.A. Chudak, D.B. Shmoys, Improved approximation algorithms for the uncapacitated facility location problem, SIAM J. Comput. 33 (2003) 1, 1 -- 25. · Zbl 1044.90056
[8] Cornuéjols, G.; Nemhauser, G. L.; Wolsey, L. A.: The uncapacitated facility location problem. Discrete location theory, 119-171 (1990)
[9] Cornuéjols, G.; Sridharan, R.; Thizy, J. M.: A comparison of heuristics and relaxations for the capacitated plant location problem. European J. Oper. res. 50, 280-297 (1991) · Zbl 0734.90046
[10] Erlenkotter, D.: A dual-based procedure for uncapacitated facility location. Oper. res. 26, 992-1009 (1978) · Zbl 0422.90053
[11] D. Erlenkotter, Program DUALOC --- Version II, Distributed on request, 1991.
[12] A.V. Goldberg, An efficient implementation of a scaling minimum-cost flow algorithm, Technical Report STAN-CS-92-1439, Stanford University, 1992.
[13] Held, M.; Wolfe, P.; Crowder, H. P.: Validation of subgradient optimization. Math. programming 49, 62-88 (1991) · Zbl 0284.90057
[14] C. Lemaréchal, Nondifferential optimization, in: G.L. Nemhauser, A.H.G. Rinnoy Kan, M.J. Todd (Eds.), Optimization, Handbooks in Operations Research, North Holland, Amsterdam, 1989, pp. 529 -- 572.
[15] P. Mirchandani, R. Francis (Eds.), Discrete Location Theory, Wiley, New York, 1990. · Zbl 0718.00021
[16] Raghavan, P.; Thompson, C. D.: Randomized rounding. Combinatorica 7, 365-374 (1987) · Zbl 0651.90052
[17] D.B. Shmoys, É. Tardos, K. Aardal, Approximation algorithms for facility location problems, in: Proceedings of the 29th ACM Symposium on Theory of Computing, 1997, pp. 265 -- 274. · Zbl 0962.68008
[18] Wolfe, P.: A method of conjugate subgradients for minimizing nondifferentiable functions. Math. programming study 3, 145-173 (1975) · Zbl 0369.90093