Improved Branch-Cut-and-Price for Capacitated Vehicle Routing

  • Diego Pecin
  • Artur Pessoa
  • Marcus Poggi
  • Eduardo Uchoa
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8494)


The best performing exact algorithms for the Capacitated Vehicle Routing Problem are based on the combination of cut and column generation. Some authors could obtain reduced duality gaps by also using a restricted number of cuts over the Master LP variables, stopping separation before the pricing becomes prohibitively hard. This work introduces a technique for greatly decreasing the impact on the pricing of the Subset Row Cuts, thus allowing much more such cuts to be added. The newly proposed Branch-Cut-and-Price algorithm also incorporates and combines for the first time (often in an improved way) several elements found in previous works, like route enumeration and strong branching. All the instances used for benchmarking exact algorithms, with up to 199 customers, were solved to optimality. Moreover, some larger instances with up to 360 customers, only considered before by heuristic methods, were solved too.


Exact Algorithm Column Generation Dual Variable Vehicle Capacity Label Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baldacci, R., Christofides, N., Mingozzi, A.: An exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts. Mathematical Programming 115(2), 351–385 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Baldacci, R., Mingozzi, A., Roberti, R.: New route relaxation and pricing strategies for the vehicle routing problem. Operations Research 59(5), 1269–1283 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Contardo, C.: A new exact algorithm for the multi-depot vehicle routing problem under capacity and route length constraints. Technical report, Archipel-UQAM 5078, Université du Québec à Montréal, Canada (2012)Google Scholar
  5. 5.
    Fukasawa, R., Longo, H., Lysgaard, J., Poggi de Aragão, M., Reis, M., Uchoa, E., Werneck, R.F.: Robust branch-and-cut-and-price for the capacitated vehicle routing problem. Mathematical Programming 106(3), 491–511 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Golden, B., Wasil, E., Kelly, J., Chao, I.: The impact of metaheuristics on solving the vehicle routing problem: Algorithms, problem sets, and computational results. In: Fleet Management and Logistics, pp. 33–56. Springer (1998)Google Scholar
  7. 7.
    Irnich, S., Desaulniers, G., Desrosiers, J., Hadjar, A.: Path-reduced costs for eliminating arcs in routing and scheduling. INFORMS Journal on Computing 22(2), 297–313 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Jepsen, M., Petersen, B., Spoorendonk, S., Pisinger, D.: Subset-row inequalities applied to the vehicle-routing problem with time windows. Oper. Research 56, 497–511 (2008)CrossRefzbMATHGoogle Scholar
  9. 9.
    Lysgaard, J., Letchford, A., Eglese, R.: A new branch-and-cut algorithm for the capacitated vehicle routing problem. Mathematical Programming 100, 423–445 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Martinelli, R., Pecin, D., Poggi, M.: Efficient elementary and restricted non-elementary route pricing for routing problems (2012) (submitted)Google Scholar
  11. 11.
    Pessoa, A., Poggi de Aragão, M., Uchoa, E.: Robust branch-cut-and-price algorithms for vehicle routing problems. In: Golden, B., Raghavan, S., Wasil, E. (eds.) The Vehicle Routing Problem: Latest Advances and New Challenges, pp. 297–325. Springer (2008)Google Scholar
  12. 12.
    Pessoa, A., Sadykov, R., Uchoa, E., Vanderbeck, F.: In-out separation and column generation stabilization by dual price smoothing. In: Bonifaci, V., Demetrescu, C., Marchetti-Spaccamela, A. (eds.) SEA 2013. LNCS, vol. 7933, pp. 354–365. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  13. 13.
    Pessoa, A., Uchoa, E., Poggi de Aragão, M., Rodrigues, R.: Exact algorithm over an arc-time-indexed formulation for parallel machine scheduling problems. Mathematical Programming Computation 2(3-4), 259–290 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    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, pp. 56–61 (2003)Google Scholar
  15. 15.
    Righini, G., Salani, M.: Symmetry helps: bounded bi-directional dynamic programming for the elementary shortest path problem with resource constraints. Discrete Optimization 3(3), 255–273 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Røpke, S.: Branching decisions in branch-and-cut-and-price algorithms for vehicle routing problems. Presentation in Column Generation 2012 (2012)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Diego Pecin
    • 1
  • Artur Pessoa
    • 2
  • Marcus Poggi
    • 1
  • Eduardo Uchoa
    • 2
  1. 1.Departamento de InformáticaPontifícia Universidade Católica do Rio de JaneiroBrazil
  2. 2.Departamento de Engenharia de ProduçãoUniversidade Federal FluminenseBrazil

Personalised recommendations