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Robust Branch-and-Cut-and-Price for the Capacitated Vehicle Routing Problem

  • Conference paper
Integer Programming and Combinatorial Optimization (IPCO 2004)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 3064))

Abstract

The best exact algorithms for the Capacitated Vehicle Routing Problem (CVRP) have been based on either branch-and-cut or Lagrangean relaxation/column generation. This paper presents an algorithm that combines both approaches: it works over the intersection of two polytopes, one associated with a traditional Lagrangean relaxation over q-routes, the other defined by bound, degree and capacity constraints. This is equivalent to a linear program with exponentially many variables and constraints that can lead to lower bounds that are superior to those given by previous methods. The resulting branch-and-cut-and-price algorithm can solve to optimality all instances from the literature with up to 135 vertices. This doubles the size of the instances that can be consistently solved.

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References

  1. Achuthan, N., Caccetta, L., Hill, S.: Capacited vehicle routing problem: Some new cutting planes. Pacific J. of Operational Research 15, 109–123 (1998)

    MATH  MathSciNet  Google Scholar 

  2. Achuthan, N., Caccetta, L., Hill, S.: An improved branch-and-cut algorithm for the capacitated vehicle routing problem. Transportation Science 37, 153–169 (2003)

    Article  Google Scholar 

  3. Araque, J., Hall, L., Magnanti, T.: Capacitated trees, capacitated routing and associated polyhedra. Technical Report SOR-90-12, Princeton University (1990)

    Google Scholar 

  4. Araque, J., Kudva, G., Morin, T., Pekny, J.: A branch-and-cut algorithm for the vehicle routing problem. Annals of Operations Research 50, 37–59 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  5. Augerat, P.: Approche polyèdrale du problème de tournées de véhicles. PhD thesis, Institut National Polytechnique de Grenoble (1995)

    Google Scholar 

  6. Augerat, P., Belenguer, J., Benavent, E., Corberán, A., Naddef, D., Rinaldi, G.: Computational results with a branch and cut code for the capacitated vehicle routing problem. Technical Report 949-M, Université Joseph Fourier, Grenoble, France (1995)

    Google Scholar 

  7. Agarwal, Y., Mathur, K., Salkin, H.: A set-partitioning based exact algorithm for the vehicle routing problem. Networks 19, 731–739 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  8. Balinski, M., Quandt, R.: On an integer program for a delivery problem. Operations Research 12, 300–304 (1964)

    Article  Google Scholar 

  9. Blasum, U., Hochstättler, W.: Application of the branch and cut method to the vehicle routing problem. Technical Report ZPR2000-386, Zentrum fur Angewandte Informatik Köln (2000)

    Google Scholar 

  10. Christofides, N., Eilon, S.: An algorithm for the vehicle-dispatching problem. Operational Research Quarterly 20, 309–318 (1969)

    Article  Google Scholar 

  11. Christofides, N., Mingozzi, A., Toth, P.: Exact algorithms for the vehicle routing problem, based on spanning tree and shortest path relaxations. Mathematical Programming 20, 255–282 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  12. Cornuéjols, G., Harche, F.: Polyhedral study of the capacitated vehicle routing problem. Mathematical Programming 60, 21–52 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dantzig, G., Ramser, R.: The truck dispatching problem. Management Science 6, 80–91 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  14. Fisher, M.: Optimal solution of vehicle routing problem using minimum k-trees. Operations Research 42, 626–642 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  15. Fukasawa, R., Poggi de Aragão, M., Porto, O., Uchoa, E.: Robust branch-andcut- and-price for the capacitated minimum spanning tree problem. In: Proc. of the International Network Optimization Conference, Evry, France, pp. 231–236 (2003)

    Google Scholar 

  16. Fukasawa, R., Reis, M., Poggi de Aragão, M., Uchoa, E.: Robust branch-and-cutand- price for the capacitated vehicle routing problem. Technical Report RPEP Vol.3 no.8, Universidade Federal Fluminense, Engenharia de Produção, Niterói, Brazil (2003)

    Google Scholar 

  17. Hadjiconstantinou, E., Christofides, N., Mingozzi, A.: A new exact algorithm from the vehicle routing problem based on q-paths and k-shortest paths relaxations. In: Laporte, G., Gendreau, M. (eds.) Freight Transportation. Annals of Operations Research, vol. 61, pp. 21–44. Baltzer Science Publishers (1995)

    Google Scholar 

  18. Irnich, S., Villeneuve, D.: The shortest path problem with resource constraints and k-cycle elimination for k ≥ 3 (2003) (unpublished manuscript)

    Google Scholar 

  19. Laporte, G., Norbert, Y.: A branch and bound algorithm for the capacitated vehicle routing problem. Operations Research Spektrum 5, 77–85 (1983)

    Article  MATH  Google Scholar 

  20. Letchford, A., Eglese, R., Lysgaard, J.: Multistars, partial multistars and the capacitated vehicle routing problem. Mathematical Programming 94, 21–40 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  21. Letchford, A., Reinelt, G., Theis, D.: A faster exact separation algorithm for blossom inequalities. In: Bienstock, D., Nemhauser, G.L. (eds.) IPCO 2004. LNCS, vol. 3064, pp. 196–205. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  22. Lysgaard, J.: CVRPSEP: A package of separation routines for the capacitated vehicle routing problem (2003), Available at http://www.asb.dk/~lys

  23. Lysgaard, J., Letchford, A., Eglese, R.: A new branch-and-cut algorithm for the capacitated vehicle routing problem. Mathematical Programming (2003) (to appear)

    Google Scholar 

  24. Martinhon, C., Lucena, A., Maculan, N.: A relax and cut algorithm for the vehicle routing problem. European J. of Operational Research (2003) (to appear)

    Google Scholar 

  25. Miller, D.: A matching based exact algorithm for capacitated vehicle routing problems. ORSA J. on Computing 7, 1–9 (1995)

    MATH  Google Scholar 

  26. Naddef, D., Rinaldi, G.: Branch-and-cut algorithms for the capacitated VRP. In: Toth, P., Vigo, D. (eds.) The Vehicle Routing Problem, pp. 53–84. SIAM, Philadelphia (2002)

    Chapter  Google Scholar 

  27. Pigatti, A.: Modelos e algoritmos para o problema de alocação generalizada e aplicações. Master’s thesis, Pontifícia Universidade Católica do Rio de Janeiro, Brazil (2003)

    Google Scholar 

  28. Poggi de Aragão, M., Uchoa, E.: Integer program reformulation for robust branchand- cut-and-price. In: Annals of Mathematical Programming in Rio, Búzios, Brazil, pp. 56–61 (2003)

    Google Scholar 

  29. Ralphs, T.: Parallel branch and cut for capacitated vehicle routing. Parallel Computing 29, 607–629 (2003)

    Article  Google Scholar 

  30. Ralphs, T., Kopman, L., Pulleyblank, W., Jr., L.T.: On the capacitated vehicle routing problem. Mathematical Programming 94 (2003) 343–359

    Google Scholar 

  31. Toth, P., Vigo, D.: Models, relaxations and exact approaches for the capacitated vehicle routing problem. Discrete Applied Mathematics 123, 487–512 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  32. Toth, P., Vigo, D.: The Vehicle Routing Problem. Monographs on Discrete Mathematics and Applications. SIAM, Philadelphia (2002)

    Book  MATH  Google Scholar 

  33. Wenger, K.: Generic Cut Generation Methods for Routing Problems. PhD thesis, Institute of Computer Science, University of Heidelberg (2003)

    Google Scholar 

  34. Werneck, R.F., Setubal, J.C.: Finding minimum congestion spanning trees. ACM J. of Experimental Algorithmics 5 (2000)

    Google Scholar 

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Author information

Authors and Affiliations

  1. School of Industrial and Systems Engineering, GeorgiaTech, USA

    Ricardo Fukasawa

  2. Department of Management Science and Logistics, Aarhus School of Business, Denmark

    Jens Lysgaard

  3. Departamento de Informática, PUC Rio de Janeiro, Brazil

    Marcus Poggi de Aragão & Marcelo Reis

  4. Departamento de Engenharia de Produção, Universidade Federal Fluminense, Brazil

    Eduardo Uchoa

  5. Department of Computer Science, Princeton University, USA

    Renato F. Werneck

Authors
  1. Ricardo Fukasawa
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  2. Jens Lysgaard
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  3. Marcus Poggi de Aragão
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  4. Marcelo Reis
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  5. Eduardo Uchoa
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  6. Renato F. Werneck
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Editor information

Editors and Affiliations

  1. Department of IEOR, Columbia University, 500 West 120th Street, 10027, New York, NY, USA

    Daniel Bienstock

  2. H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA

    George Nemhauser

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© 2004 Springer-Verlag Berlin Heidelberg

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Fukasawa, R., Lysgaard, J., Poggi de Aragão, M., Reis, M., Uchoa, E., Werneck, R.F. (2004). Robust Branch-and-Cut-and-Price for the Capacitated Vehicle Routing Problem. In: Bienstock, D., Nemhauser, G. (eds) Integer Programming and Combinatorial Optimization. IPCO 2004. Lecture Notes in Computer Science, vol 3064. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-25960-2_1

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  • DOI: https://doi.org/10.1007/978-3-540-25960-2_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-22113-5

  • Online ISBN: 978-3-540-25960-2

  • eBook Packages: Springer Book Archive

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