Abstract.
We present a new branch-and-cut algorithm for the capacitated vehicle routing problem (CVRP). The algorithm uses a variety of cutting planes, including capacity, framed capacity, generalized capacity, strengthened comb, multistar, partial multistar, extended hypotour inequalities, and classical Gomory mixed-integer cuts.
For each of these classes of inequalities we describe our separation algorithms in detail. Also we describe the other important ingredients of our branch-and-cut algorithm, such as the branching rules, the node selection strategy, and the cut pool management. Computational results, for a large number of instances, show that the new algorithm is competitive. In particular, we solve three instances (B-n50-k8, B-n66-k9 and B-n78-k10) of Augerat to optimality for the first time.
Similar content being viewed by others
References
Achuthan, N.R., Caccetta, L., Hill, S.P.: ‘‘Capacitated vehicle routing problem: some new cutting planes’’. Asia-Pac. J. Oper. Res. 15, 109–123 (1998)
Agarwal, Y., Mathur, K., Salkin, H.M.: ‘‘A set-partitioning-based exact algorithm for the vehicle routing problem’’. Networks 19, 731–749 (1989)
Araque, J.R.: ‘‘Lots of combs of different sizes for vehicle routing’’. Discussion paper. Center for Operations Research and Econometrics. Catholic University of Louvain, Belgium, 1990
Araque, J.R., Hall, L.A., Magnanti, T.L.: ‘‘Capacitated trees, capacitated routing and associated polyhedra’’. Discussion paper. Center for Operations Research and Econometrics, Catholic University of Louvain, Belgium, 1990
Araque, J.R., Kudva, G., Morin, T.L., Pekny, J.F.: ‘‘A branch-and-cut algorithm for the vehicle routing problem’’. Ann. Oper. Res. 50, 37–59 (1994)
Augerat, P.: Approche Polyèdrale du Problème de Tournées de Véhicules. PhD thesis, Institut National Polytechnique de Grenoble, 1995
Augerat, P., Belenguer, J.M., Benavent, E., Corberán, A., Naddef, D., Rinaldi, G.:‘‘Computational results with a branch-and-cut code for the capacitated vehicle routing problem’’. Research report RR949-M. ARTEMIS-IMAG, France, 1995
Augerat, P., Belenguer, J.M., Benavent, E., Corberán, A., Naddef, D.: ‘‘Separating capacity constraints in the CVRP using tabu search’’. Eur. J. Opl. Res. 106, 546–557 (1998)
Balas, E., Ceria, S., Cornuéjols, G., Natraj, N.: ‘‘Gomory cuts revisited’’. Oper. Res. Lett. 19, 1–10 (1996)
Ball, M.O., Magnanti, T.L., Monma, C.L., Nemhauser, G.L.: (eds.) Network Routing. Handbooks on Operations Research and Management Science, 8. Amsterdam: Elsevier, 1995
Blasum, U.: ‘‘Anwendung des Branch & Cut Verfahrens auf das kapazitierte Vehicle-Routing Problem’’. PhD thesis, Universität zu Köln, 1999
Blasum, U., Hochstättler, W.: ‘‘Application of the branch-and-cut method to the vehicle routing problem’’. Technical report. Universität zu Köln, 2002
Campos,V., Corberán, A., Mota, E.: ‘‘Polyhedral results for a vehicle routing problem’’. Eur. J. Opl. Res. 52, 75–85 (1991)
Carpaneto, G., Martello, S., Toth, P.:‘‘Algorithms and codes for the assignment problem’’. Ann. Oper. Res. 13, 193–223 (1988)
Christofides, N., Eilon, S.: ‘‘An algorithm for the vehicle dispatching problem’’. Oper. Res. Quarterly 20, 309–318 (1969)
Christofides, N., Mingozzi, A., Toth, P.:‘‘Exact algorithms for the vehicle routing problem based on the spanning tree and shortest path relaxations’’. Math. Program. 20, 255–282 (1981)
Cornuéjols, G., Harche, F.: ‘‘Polyhedral study of the capacitated vehicle routing problem’’. Math. Program. 60, 21–52, (1993)
Dantzig, G.B., Ramser, R.H.: ‘‘The truck dispatching problem’’. Manage. Sci. 6, 80–91 (1959)
Edmonds, J.: ‘‘Maximum matching and a polyhedron with 0–1 vertices’’. J. Res. Nat. Bur. Standards. 69B, 125–130 (1965)
Fisher, M.L.: ‘‘Optimal solution of vehicle routing problems using minimum K-trees’’. Oper. Res. 42, 626–642 (1994)
Gomory, R.E.: ‘‘An algorithm for the mixed-integer problem’’. Report RM-2597, Rand Corporation, 1960 (unpublished)
Gouveia, L.: ‘‘A result on projection for the vehicle routing problem’’. Eur. J. Opl. Res. 85, 610–624 (1995)
Grötschel, M., Lovász, L., Schrijver, A.J.: Geometric Algorithms in Combinatorial Optimization. Springer, 1988
Grötschel, M., Padberg, M.W.: ‘‘On the symmetric travelling salesman problem I: inequalities’’. Math. Program. 16, 265–280 (1979)
Grötschel, M., Padberg, M.W.: ‘‘On the symmetric travelling salesman problem II: lifting theorems and facets’’. Math. Program. 16, 281–302 (1979)
Hadjiconstantinou, E., Christofides, N., Mingozzi, A.: ‘‘A new exact algorithm for the vehicle routing problem based on q-paths and k-shortest paths relaxations’’. Ann. Oper. Res. 61, 21–43 (1996)
Laporte, G.: ‘‘The vehicle routing problem: an overview of exact and approximate algorithms’’. Eur. J. Opl. Res. 59, 345–358, (1992)
Laporte, G.: ‘‘Vehicle routing’’. In: Dell’Amico, Maffioli, Martello (eds.) Annotated Bibliographies in Combinatorial Optimization. New York, Wiley, 1997
Laporte, G., Nobert, Y.: ‘‘Comb inequalities for the vehicle routing problem’’. Methods of Oper. Res. 51, 271–276 (1984)
Letchford, A.N., Eglese, R.W., Lysgaard, J.: ‘‘Multistars, partial multistars and the capacitated vehicle routing problem’’. Math. Program. 94, 21–40 (2002)
Martello, S., Toth, P.: Knapsack Problems: Algorithms and Computer Implementations. Chichester: Wiley, 1990
Miller, D.L.: ‘‘A matching-based exact algorithm for capacitated vehicle routing problems’’. ORSA J. Comp. 7, 1–9 (1995)
Naddef, D., Rinaldi, G.: ‘‘Branch-and-cut algorithms for the capacitated VRP’’. In: P.Toth, D.Vigo (eds.), The Vehicle Routing Problem. SIAM Monographs on Discr. Math. Appl. 9, 2002
Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. New York: Wiley, 1988
Padberg, M.W., Rao, M.R.: ‘‘Odd minimum cut-sets and b-matchings’’. Math. Oper. Res. 7, 67–80 (1982)
Padberg, M.W., Rinaldi, G.: ‘‘Facet identification for the symmetric traveling salesman polytope’’. Math. Program. 47, 219–257 (1990)
Padberg, M.W., Rinaldi, G.: ‘‘A branch-and-cut algorithm for the resolution of large-scale symmetric travelling salesman problems’’. SIAM Rev. 33, 60–100 (1991)
Pereira, F.B., Tavares, J., Machado, P., Costa, E.: ‘‘GVR: a new genetic representation for the vehicle routing problem’’. In: M.O’Neill et al.(eds.), Proceedings of AICS 2002. Berlin: Springer-Verlag, 2002, pp. 95–102
Ralphs, T.K.: ‘‘Parallel branch and cut for capacitated vehicle routing’’. To appear in Parallel Computing
Ralphs, T.K., Kopman, L., Pulleyblank, W.R., Trotter, L.E.: ‘‘On the capacitated vehicle routing problem’’. Math. Program. 94, 343–359 (2003)
Reinelt, G.: ‘‘TSPLIB: A travelling salesman problem library’’. ORSA J. Comp. 3, 376–384 (1991) URL: http://www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95/
Toth, P., Vigo, D.: (eds.), The Vehicle Routing Problem. SIAM Monographs on Discr. Math. Appl. 9, (2002)
Xu, J., Kelly, J.P.: ‘‘A network flow-based tabu search heuristic for the vehicle routing problem’’. Transportation Sci. 30, 379–393 (1996)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lysgaard, J., Letchford, A. & Eglese, R. A new branch-and-cut algorithm for the capacitated vehicle routing problem. Math. Program., Ser. A 100, 423–445 (2004). https://doi.org/10.1007/s10107-003-0481-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-003-0481-8