Advertisement

Exact algorithms for the chance-constrained vehicle routing problem

Full Length Paper Series B

Abstract

We study the chance-constrained vehicle routing problem (CCVRP), a version of the vehicle routing problem (VRP) with stochastic demands, where a limit is imposed on the probability that each vehicle’s capacity is exceeded. A distinguishing feature of our proposed methodologies is that they allow correlation between random demands, whereas nearly all existing exact methods for the VRP with stochastic demands require independent demands. We first study an edge-based formulation for the CCVRP, in particular addressing the challenge of how to determine a lower bound on the number of vehicles required to serve a subset of customers. We then investigate the use of a branch-and-cut-and-price (BCP) algorithm. While BCP algorithms have been considered the state of the art in solving the deterministic VRP, few attempts have been made to extend this framework to the VRP with stochastic demands. In contrast to the deterministic VRP, we find that the pricing problem for the CCVRP problem is strongly \(\mathcal {NP}\)-hard, even when the routes being priced are allowed to have cycles. We therefore propose a further relaxation of the routes that enables pricing via dynamic programming. We also demonstrate how our proposed methodologies can be adapted to solve a distributionally robust CCVRP problem. Numerical results indicate that the proposed methods can solve instances of CCVRP having up to 55 vertices.

Keywords

Stochastic vehicle routing Chance constraint Branch-and-cut-and-price 

Mathematics Subject Classification

90C10 90C15 90B06 

References

  1. 1.
    Baldacci, R., Mingozzi, A.: A unified exact method for solving different classes of vehicle routing problems. Math. Program. 120(2), 347–380 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Baldacci, R., Mingozzi, A., Roberti, R.: New route relaxation and pricing strategies for the vehicle routing problem. Oper. Res. 59(5), 1269–1283 (2011)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Beraldi, P., Bruni, M.E., Laganà, D., Musmanno, R.: The mixed capacitated general routing problem under uncertainty. Eur. J. Oper. Res. 240, 382–392 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bertsimas, D.J.: A vehicle routing problem with stochastic demand. Oper. Res. 40(3), 574–585 (1992)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Calafiore, G.C., El Ghaoui, L.: On distributionally robust chance-constrained linear programs. J. Optim. Theory Appl. 130(1), 1–22 (2006)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chen, W., Sim, M., Sun, J., Teo, C.P.: From CVaR to uncertainty set: implications in joint chance-constrained optimization. Oper. Res. 58(2), 470–485 (2010)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cheng, J., Delage, E., Lisser, A.: Distributionally robust stochastic knapsack problem. SIAM J. Optim. 24(3), 1485–1506 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Christiansen, C.H., Lysgaard, J., Wøhlk, S.: A branch-and-price algorithm for the capacitated arc routing problem with stochastic demands. Oper. Res. Lett. 37(6), 392–398 (2009)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Christofides, N., Mingozzi, A., Toth, P.: Exact algorithms for the vehicle routing problem, based on spanning tree and shortest path relaxations. Math. Program. 20(1), 255–282 (1981)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Clarke, G., Wright, J.W.: Scheduling of vehicles from a central depot to a number of delivery points. Oper. Res. 12(4), 568–581 (1964)CrossRefGoogle Scholar
  11. 11.
    Contardo, C., Martinelli, R.: A new exact algorithm for the multi-depot vehicle routing problem under capacity and route length constraints. Discr. Optim. 12, 129–146 (2014)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Dantzig, G.B., Ramser, J.H.: The truck dispatching problem. Manag. Sci. 6(1), 80–91 (1959)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Delage, E., Ye, Y.: Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. 58(3), 595–612 (2010)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Dinh, T., Fukasawa, R., Luedtke, J.: Exact algorithms for the chance-constrained vehicle routing problem. In: Louveaux, Q., Skutella, M. (eds.) Integer Programming and Combintatorial Optimization. Springer, Berlin (2016)Google Scholar
  15. 15.
    Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201–213 (2002)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Dror, M., Laporte, G., Louveaux, F.V.: Vehicle routing with stochastic demands and restricted failures. Z. Oper. Res. 37(3), 273–283 (1993)MathSciNetMATHGoogle Scholar
  17. 17.
    El Ghaoui, L., Oks, M., Oustry, F.: Worst-case value-at-risk and robust portfolio optimization: a conic programming approach. Oper. Res. 51(4), 543–556 (2003)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Erdoğan, E., Iyengar, G.: Ambiguous chance constrained problems and robust optimization. Math. Program. 107(1), 37–61 (2005)MathSciNetMATHGoogle Scholar
  19. 19.
    Fukasawa, R., Longo, H., Lysgaard, J., de Aragão, M.P., Reis, M., Uchoa, E., Werneck, R.F.: Robust branch-and-cut-and-price for the capacitated vehicle routing problem. Math. Program. 106(3), 491–511 (2006)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Gauvin, C., Desaulniers, G., Gendreau, M.: A branch-cut-and-price algorithm for the vehicle routing problem with stochastic demands. Comput. Oper. Res. 50, 141–153 (2014)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Gounaris, C.E., Wiesemann, W., Floudas, C.A.: The robust capacitated vehicle routing problem under demand uncertainty. Oper. Res. 61(3), 677–693 (2013)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Hanasusanto, G.A., Roitch, V., Kuhn, D., Wiesemann, W.: A distributionally robust perspective on uncertainty quantification and chance constrained programming. Math. Program. 151(1), 35–62 (2015)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Irnich, S., Desaulniers, G.: Shortest path problems with resource constraints. In: Desaulniers, G., Desrosiers, J., Solomon, M. (eds.) Column Gener. Springer, Berlin (2005)Google Scholar
  24. 24.
    Jiang, R., Guan, Y.: Data-driven chance constrained stochastic program. Math. Program. 58, 291–327 (2016)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Laporte, G., Louveaux, F., Mercure, H.: Models and exact solutions for a class of stochastic location-routing problems. Eur. J. Oper. Res. 39(1), 71–78 (1989)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Laporte, G., Louveaux, F., Mercure, H.: The vehicle routing problem with stochastic travel times. Trans. Sci. 26(3), 161–170 (1992)CrossRefMATHGoogle Scholar
  27. 27.
    Laporte, G., Louveaux, F.V., Van Hamme, L.: An integer L-shaped algorithm for the capacitated vehicle routing problem with stochastic demands. Oper. Res. 50(3), 415–423 (2002)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Luedtke, J., Ahmed, S.: A sample approximation approach for optimization with probabilistic constraints. SIAM J. Optim. 19, 674–699 (2008)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Pecin, D., Pessoa, A., Poggi, M., Uchoa, E.: Improved branch-cut-and-price for capacitated vehicle routing. In: Integer Programming and Combinatorial Optimization. Lecture Notes in Computer Science, vol. 8494, pp. 393–403. Springer (2014)Google Scholar
  30. 30.
    Pessoa, A.A., Pugliese, L.D.P., Guerriero, F., Poss, M.: Robust constrained shortest path problems under budgeted uncertainty. http://www.optimization-online.org/DB_FILE/2014/10/4601.pdf (2014)
  31. 31.
    Secomandi, N., Margot, F.: Reoptimization approaches for the vehicle-routing problem with stochastic demands. Oper. Res. 57(1), 214–230 (2009)CrossRefMATHGoogle Scholar
  32. 32.
    Song, Y., Luedtke, J.R., Küçükyavuz, S.: Chance-constrained binary packing problems. INFORMS J. Comput. 26, 735–747 (2014)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Stewart, W.R., Golden, B.L.: Stochastic vehicle routing: a comprehensive approach. Eur. J. Oper. Res. 14(4), 371–385 (1983)CrossRefMATHGoogle Scholar
  34. 34.
    Sturm, J.F.: Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones. Optim. Methods Softw. 11(1–4), 625–653 (1999)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Yang, W., Xu, H.: Distributionally robust chance constraints for non-linear uncertainties. Math. Program. 155(1), 231–265 (2014)MathSciNetMATHGoogle Scholar
  36. 36.
    Yang, W.H., Mathur, K., Ballou, R.H.: Stochastic vehicle routing problem with restocking. Trans. Sci. 34(1), 99–112 (2000)CrossRefMATHGoogle Scholar
  37. 37.
    Zymler, S., Kuhn, D., Rustem, B.: Distributionally robust joint chance constraints with second-order moment information. Math. Program. 137, 167–198 (2011)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2017

Authors and Affiliations

  1. 1.Department of Industrial and Systems Engineering and Wisconsin Institute for DiscoveryUniversity of Wisconsin-MadisonMadisonUSA
  2. 2.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada

Personalised recommendations