Abstract
The Capacitated Facility Location Problem (CFLP) is to locate a set of facilities with capacity constraints, to satisfy at the minimum cost the order-demands of a set of clients. A multi-source version of the problem is considered in which each client can be served by more than one facility.
In this paper we present a reformulation of the CFLP based on Mixed Dicut Inequalities, a family of minimum knapsack inequalities of a mixed type, containing both binary and continuous (flow) variables. By aggregating flow variables, any Mixed Dicut Inequality turns into a binary minimum knapsack inequality with a single continuous variable. We will refer to the convex hull of the feasible solutions of this minimum knapsack problem as the Mixed Dicut polytope.
We observe that the Mixed Dicut polytope is a rich source of valid inequalities for the CFLP: basic families of valid CFLP inequalities, like Variable Upper Bounds, Cover, Flow Cover and Effective Capacity Inequalities, are valid for the Mixed Dicut polytope. Furthermore we observe that new families of valid inequalities for the CFLP can be derived by the lifting procedures studied for the minimum knapsack problem with a single continuous variable.
To deal with large-scale instances, we have developed a Branch-and-Cut-and-Price algorithm, where the separation algorithm consists of the complete enumeration of the facets of the Mixed Dicut polytope for a set of candidate Mixed Dicut Inequalities. We observe that our procedure returns inequalities that dominate most of the known classes of inequalities presented in the literature. We report on computational experience with instances up to 1000 facilities and 1000 clients to validate the approach.
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References
Aardal, K.: On the solution of one and two-level capacitated facility location problems by the cutting plane approach. Ph.D. thesis, Université Catholique de Louvain, Louvain-la-Neuve, Belgium (1992)
Aardal, K.: Capacitated facility location: separation algorithms and computational experience. Math. Program. 81, 149–175 (1998)
Aardal, K., Pochet, Y., Wolsey, L.A.: Capacitated facility location: valid inequalities and facets. Math. Operat. Res. 20, 562–582 (1995)
Anbil, R., Barahona, F.: The volume algorithm: producing primal solutions with a subgradient method. Math. Program. Ser. A 87, 385–399 (2000)
Avella, P., Boccia, M., Sforza, A., Vasiliev, I.: An effective heuristic for large-scale capacitated plant location problems. Technical Report, available online at http://www.ing.unisannio.it/boccia/ (2004)
Avella, P., Sassano, A., Vasil’ev, I.: Computational study of large-scale p-median problems. Math. Program. 109(1), 89–114 (2007)
Barahona, F., Chudak, F.A.: Near-optimal solution to large scale facility location problems. Internal report, IBM research division, T.J. Watson Research Center, RC 21606 (1999)
Barber, C.B., Dobkin, D.P., Huhdanpaa, H.T.: The quickhull algorithm for convex hulls. ACM Trans. Math. Softw. 22(4), 469–483 (1996); http://www.qhull.org
Beasley, J.E.: OR-Library: distributing test problems by electronic mail. J. Operat. Res. Soc. 41(11), 1069–1072 (1990)
Beasley, J.E.: Lagrangean heuristics for location problems. Eur. J. Operat. Res. 65, 383–399 (1993)
Bertsimas, D., Tsitsiklis, J.N.: Introduction to Linear Optimization. Athena Scientific, Belmont (1997)
Chudak, F.A., Williamson, D.P.: Improved approximation algorithms for capacitated facility location problems. In: 7th International IPCO Conference Proceedings, pp. 99–113 (1999)
Cornuejols, G., Sridharan, R., Thizy, J.M.: A comparison of heuristics and relaxations for the capacitated plant location problem. Eur. J. Oper. Res. 50, 280–297 (1991)
Klose, A., Görtz, S.: A branch-and-price algorithm for the capacitated facility location problem. Eur. J. Operat. Res. (2006, forthcoming). Available online at http://www.sciencedirect.com
Korupolu, M.R., Plaxton, C.G., Rajaraman, R.: Analysis of a local search heuristic for facility location problems. DIMACS Technical Report, pp. 98–30 (1998)
Leung, J.M.Y., Magnanti, T.L.: Valid inequalities and facets of the capacitated plant location problem. Math. Program. 44, 271–291 (1989)
Marchand, H., Wolsey, L.A.: The 0-1 knapsack problem with a single continuous variable. Math. Program. 85, 15–33 (1999)
Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley, New York (1988)
Ortega, F., Wolsey, L.: A branch and cut algorithm for the single commodity uncapacitated fixed charge network flow problem. Networks 41, 143–158 (2003)
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Avella, P., Boccia, M. A cutting plane algorithm for the capacitated facility location problem. Comput Optim Appl 43, 39–65 (2009). https://doi.org/10.1007/s10589-007-9125-x
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DOI: https://doi.org/10.1007/s10589-007-9125-x