The Capacitated Vehicle Routing Problem with Evidential Demands: A Belief-Constrained Programming Approach

  • Nathalie Helal
  • Frédéric Pichon
  • Daniel Porumbel
  • David Mercier
  • Éric Lefèvre
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9861)


This paper studies a vehicle routing problem, where vehicles have a limited capacity and customer demands are uncertain and represented by belief functions. More specifically, this problem is formalized using a belief function based extension of the chance-constrained programming approach, which is a classical modeling of stochastic mathematical programs. In addition, it is shown how the optimal solution cost is influenced by some important parameters involved in the model. Finally, some instances of this difficult problem are solved using a simulated annealing metaheuristic, demonstrating the feasibility of the approach.


Vehicle routing problem Stochastic programming Chance-constrained programming Belief functions 


  1. 1.
    Bodin, L.D., Golden, B.L., Assad, A.A., Ball, M.O.: Routing and scheduling of vehicles and crews: the state of the art. Comput. Oper. Res. 10(2), 63–212 (1983)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Charnes, A., Cooper, W.W.: Chance-constrained programming. Manag. Sci. 6(1), 73–79 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cordeau, J.-F., Laporte, G., Savelsbergh, M.W.P., Vigo, D.: Vehicle routing (Chap. 6). In: Barnhart, C., Laporte, G. (eds.) Transportation, Handbooks in Operations Research and Management Science, vol. 14, pp. 367–428. Elsevier, Amsterdam (2007)Google Scholar
  4. 4.
    Ferson, S., Tucker, W.T.: Sensitivity in risk analyses with uncertain numbers. Technical report, Sandia National Laboratories (2006)Google Scholar
  5. 5.
    Harmanani, H., Azar, D., Helal, N., Keirouz, W.: A simulated annealing algorithm for the capacitated vehicle routing problem. In: 26th International Conference on Computers and their Applications, New Orleans, USA (2011)Google Scholar
  6. 6.
    Kirby, M.J.L.: The current state of chance-constrained programming. In: Kuhn, H.W. (ed.) Proceedings of the Princeton Symposium on Mathematical Programming, pp. 93–111. Princeton University Press (1970)Google Scholar
  7. 7.
    Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimisation by simulated annealing. Science 220(4598), 671–680 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    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, 415–423 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Masri, H., Abdelaziz, F.B.: Belief linear programming. Int. J. Approx. Reason. 51, 973–983 (2010)CrossRefzbMATHGoogle Scholar
  10. 10.
    Mourelatos, Z.P., Zhou, J.: A design optimization method using evidence theory. J. Mech. Design 128, 901–908 (2006)CrossRefGoogle Scholar
  11. 11.
    Nassreddine, G., Abdallah, F., Denoeux, T.: State estimation using interval analysis and belief function theory: application to dynamic vehicle localization. IEEE Trans. Syst. Man Cybern. B 40(5), 1205–1218 (2010)CrossRefGoogle Scholar
  12. 12.
    Pichon, F., Dubois, D., Denoeux, T.: Relevance and truthfulness in information correction and fusion. Int. J. Approx. Reason. 53(2), 159–175 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Vehicle Routing Data Sets. Accessed 20 Mar 2016
  14. 14.
    Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)zbMATHGoogle Scholar
  15. 15.
    Srivastava, R.K., Deb, K., Tulshyan, R.: An evolutionary algorithm based approach to design optimization using evidence theory. J. Mech. Design 135(8), 081003-1–081003-12 (2013)CrossRefGoogle Scholar
  16. 16.
    Sungur, I., Ordónez, F., Dessouky, M.: A robust optimization approach for the capacitated vehicle routing problem with demand uncertainty. IIE Trans. 40, 509–523 (2008)CrossRefGoogle Scholar
  17. 17.
    Yager, R.R.: Arithmetic and other operations on Dempster-Shafer structures. Int. J. Man Mach. Stud. 25(4), 357–366 (1986)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Nathalie Helal
    • 1
  • Frédéric Pichon
    • 1
  • Daniel Porumbel
    • 2
  • David Mercier
    • 1
  • Éric Lefèvre
    • 1
  1. 1.University of Artois, EA 3926, Laboratoire de Génie Informatique et d’Automatique de l’Artois (LGI2A)BéthuneFrance
  2. 2.Conservatoire National des Arts et Métiers, EA 4629, CedricParisFrance

Personalised recommendations