Mathematical Programming

, Volume 106, Issue 3, pp 491–511 | Cite as

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

  • Ricardo Fukasawa
  • Humberto Longo
  • Jens Lysgaard
  • Marcus Poggi de Aragão
  • Marcelo Reis
  • Eduardo Uchoa
  • Renato F. Werneck


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 more than doubles the size of the instances that can be consistently solved.

Mathematics Subject Classification (2000)

20E28 20G40 20C20 


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  1. 1.
    Achuthan, N., Caccetta, L., Hill, S.: Capacited vehicle routing problem: Some new cutting planes. Asia-Pacific J. Oper. Res. 15, 109–123 (1998)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Achuthan, N., Caccetta, L., Hill, S.: An improved branch-and-cut algorithm for the capacitated vehicle routing problem. Transportation Sci. 37, 153–169 (2003)CrossRefGoogle Scholar
  3. 3.
    Agarwal, Y., Mathur, K., Salkin, H.: A set-partitioning based exact algorithm for the vehicle routing problem. Networks 19, 731–739 (1989)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Araque, J., Hall, L., Magnanti, T.: Capacitated trees, capacitated routing and polyhedra. Technical Report SOR-90-12. Princeton University, 1990Google Scholar
  5. 5.
    Araque, J., Kudva, G., Morin, T., Pekny, J.: A branch-and-cut algorithm for the vehicle routing problem. Ann. Oper. Res. 50, 37–59 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Augerat, P.: Approche polyèdrale du problème de tournées de véhicles. PhD thesis, Institut National Polytechnique de Grenoble, 1995Google Scholar
  7. 7.
    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, 1995Google Scholar
  8. 8.
    Balinski, M., Quandt, R.: On an integer program for a delivery problem. Oper. Res. 12, 300–304 (1964)Google Scholar
  9. 9.
    Barnhart, C., Hane, C., Vance, P.: Using branch-and-price-and-cut to solve origin-destination integer multicommodity flow problems. Oper. Res. 40, 318–326 (2000)CrossRefGoogle Scholar
  10. 10.
    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, 2000Google Scholar
  11. 11.
    Christofides, N., Eilon, S.: An algorithm for the vehicle-dispatching problem. Oper. Res. Q. 20, 309–318 (1969)MathSciNetGoogle Scholar
  12. 12.
    Christofides, N., Mingozzi, A., Toth, P.: Exact algorithms for the vehicle routing problem, based on spanning tree and shortest path relaxations. Math. Prog. 20, 255–282 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Cornuéjols, G., Harche, F.: Polyhedral study of the capacitated vehicle routing problem. Math. Prog. 60, 21–52 (1993)zbMATHCrossRefGoogle Scholar
  14. 14.
    Dantzig, G., Ramser, R.: The truck dispatching problem. Management Science 6, 80–91 (1959)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Desrochers, M., Desrosiers, J., Solomon, M.: A new optimization algorithm for the vehicle routing problem with time windows. Oper. Res. 40, 342–354 (1992)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Felici, G., Gentile, C., Rinaldi, G.: Solving large MIP models in supply chain management by branch & cut. Technical report, Istituto di Analisi dei Sistemi ed Informatica del CNR, Italy, 2000Google Scholar
  17. 17.
    Fisher, M.: Optimal solution of vehicle routing problem using minimum k-trees. Oper. Res. 42, 626–642 (1994)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Fukasawa, R., Poggi de Aragão, M., Porto, O., Uchoa, E.: Robust branch-and-cut-and-price for the capacitated minimum spanning tree problem. In: Proc. of the International Network Optimization Conference. Evry, France, 2003, pp. 231–236Google Scholar
  19. 19.
    Fukasawa, R., Reis, M., Poggi de Aragão, M., Uchoa, E.: Robust branch-and-cut-and-price for the capacitated vehicle routing problem. Technical Report RPEP Vol.3 no.8, Universidade Federal Fluminense, Engenharia de Produção, Niterói, Brazil, 2003Google Scholar
  20. 20.
    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., Michel Gendreau (eds.), Freight Transportation, 61 in Annals of Operations Research. Baltzer Science Publishers, 1995, pp. 21–44Google Scholar
  21. 21.
    Irnich, S., Villeneuve, D.: The shortest path problem with resource constraints and k-cycle elimination for k ≥ 3, 2003. To appear in JNFORMS Journal on computing. Available at
  22. 22.
    Kim, D., Barnhart, C., Ware, K., Reinhardt, G.: Multimodal express package delivery: A service network design application. Transportation Science 33, 391–407 (1999)zbMATHGoogle Scholar
  23. 23.
    Kohl, N., Desrosiers, J., Madsen, O., Solomon, M., Soumis, F.: 2-Path cuts for the vehicle routing problem with time windows. Transportation Science 33, 101–116 (1999)zbMATHGoogle Scholar
  24. 24.
    Laporte, G., Norbert, Y.: A branch and bound algorithm for the capacitated vehicle routing problem. Oper. Res. Spektrum 5, 77–85 (1983)zbMATHCrossRefGoogle Scholar
  25. 25.
    Letchford, A., Eglese, R., Lysgaard, J.: Multistars, partial multistars and the capacitated vehicle routing problem. Math. Prog. 94, 21–40 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Letchford, A., Reinelt, G., Theis, D.: A faster exact separation algorithm for blossom inequalities. In: Proceedings of the X IPCO, volume 3064 of Lecture Notes in Computer Science. New York, 2004, pp. 196–205Google Scholar
  27. 27.
    Letchford, A.N., Salazar, J.J.: Projection results for vehicle routing. Math. Prog. 2004, To appearGoogle Scholar
  28. 28.
    Longo, H., Poggi de Aragão, M., Uchoa, E.: Solving capacitated arc routing problems using a transformation to the CVRP. Computers & Operations Research, 2004, To appearGoogle Scholar
  29. 29.
    Lysgaard, J.: CVRPSEP: A package of separation routines for the capacitated vehicle routing problem, 2003. Available at
  30. 30.
    Lysgaard, J., Letchford, A., Eglese, R.: A new branch-and-cut algorithm for the capacitated vehicle routing problem. Math. Prog. 100 (2), 423–445 (2004)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Martinhon, C., Lucena, A., Maculan, N.: Stronger k-tree relaxations for the vehicle routing problem. European J. Oper. Res. 158 (1), 56–71 (2004)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Miller, D.: A matching based exact algorithm for capacitated vehicle routing problems. ORSA J. Comput. 7, 1–9 (1995)zbMATHGoogle Scholar
  33. 33.
    Naddef, D., Rinaldi, G.: Branch-and-cut algorithms for the capacitated VRP. In: Toth, P., Vigo, D. (eds.) The Vehicle Routing Problem, chapter 3. SIAM, 2002, pp. 53–84Google Scholar
  34. 34.
    Padberg, M., Rao, M.: Odd minimum cut-sets and b-matchings. Math. Oper. Res. 7 (1), 67–80 (1982)MathSciNetGoogle Scholar
  35. 35.
    Pigatti, A.: Modelos e algoritmos para o problema de alocação generalizada e aplicações. Master's thesis, Pontifí cia Universidade Católica do Rio de Janeiro, Brazil, July 2003Google Scholar
  36. 36.
    Poggi de Aragão, M., Uchoa, E.: Integer program reformulation for robust branch-and-cut-and-price. In: Annals of Mathematical Programming in Rio. Búzios, Brazil, 2003, pp. 56–61Google Scholar
  37. 37.
    Ralphs, T.: Parallel branch and cut for capacitated vehicle routing. Parallel Computing 29, 607–629 (2003)CrossRefGoogle Scholar
  38. 38.
    Ralphs, T., Kopman, L., Pulleyblank, W., Trotter, L. Jr.: On the capacitated vehicle routing problem. Math. Prog. 94, 343–359 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Toth, P., Vigo, D.: Models, relaxations and exact approaches for the capacitated vehicle routing problem. Discrete Applied Mathematics 123, 487–512 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Toth, P., Vigo, D.:The Vehicle Routing Problem. Monographs on Discrete Mathematics and Applications. SIAM, 2002Google Scholar
  41. 41.
    Van den Akker, J., Hurkens, C., Savelsbergh, M.: Time-indexed formulation for machine scheduling problems: column generation. INFORMS J. on Computing 12, 111–124 (2000)zbMATHCrossRefGoogle Scholar
  42. 42.
    Vanderbeck, F.: Lot-sizing with start-up times. Management Science 44, 1409–1425 (1998)zbMATHCrossRefGoogle Scholar
  43. 43.
    Wenger, K.M.: Generic Cut Generation Methods for Routing Problems. PhD thesis, Institute of Computer Science, University of Heidelberg, 2003Google Scholar
  44. 44.
    Werneck, R.F., Setubal, J.C.: Finding minimum congestion spanning trees. ACM J. of Experimental Algorithmics, 5, 2000Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Ricardo Fukasawa
    • 1
  • Humberto Longo
    • 2
  • Jens Lysgaard
    • 3
  • Marcus Poggi de Aragão
    • 4
  • Marcelo Reis
    • 4
  • Eduardo Uchoa
    • 5
  • Renato F. Werneck
    • 6
  1. 1.School of Industrial and Systems EngineeringGeorgiaTechUSA
  2. 2.Instituto de InformáticaUniversidade Federal de GoiásBrazil
  3. 3.Department of Accounting, Finance and LogisticsAarhus School of BusinessDenmark
  4. 4.Departamento de InformáticaPUC Rio de JaneiroBrazil
  5. 5.Departamento de Engenharia de ProduçãoUniversidade Federal FluminenseBrazil
  6. 6.Department of Computer SciencePrinceton UniversityUSA

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