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Cumulative Vehicle Routing Problem: A Column Generation Approach

  • Daya Ram Gaur
  • Rishi Ranjan Singh
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8959)

Abstract

Cumulative vehicle routing problems are a simplified model of fuel consumption in vehicle routing problems. Here we study computationally, an approach for constructing approximate solutions to cumulative vehicle routing problems based on rounding solutions to a linear program. The linear program is based on the set cover formulation, and is solved using column generation. The pricing sub-problem is solved using dynamic programming. Simulation results show that the simple scalable strategy computes solutions with cost close to the lower bound given by the linear programming relaxation.

Keywords

Routing Transportation Linear programming Traveling salesman Column generation 

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References

  1. 1.
    Altinkemer, K., Gavish, B.: Technical Note: Heuristics for Delivery Problems with Constant Error Guarantees. Transportation Science 24(4), 294–297 (1990)CrossRefzbMATHGoogle Scholar
  2. 2.
    Augerat, P., Belenguer, J.M., Benavent, E., Corberan, A., Naddef, D., Rinaldi, G.: Computational results with a branch and cut code for the capacitated vehicle routing problem, Tech. Rep. 1 RR949-M, ARTEMIS-IMAG, Grenoble France (1995)Google Scholar
  3. 3.
    Balas, E.: New classes of efficiently solvable generalized traveling salesman problems. Annals of Operations Research (56), 529–558 (1999)Google Scholar
  4. 4.
    Balinski, M.L., Quandt, R.E.: On an integer program for a delivery problem. Operations Research 12, 300–304 (1964)CrossRefGoogle Scholar
  5. 5.
    Bianco, L., Mingozzi, A., Ricciardelli, S.: The traveling salesman problem with cumulative costs. Networks 23(2), 81–91 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Blum, A., Chalasani, P., Coppersmith, D., Pulleyblank, W.R., Raghavan, P., Sudan, M.: The minimum latency problem. In: STOC, pp. 163–171 (1994)Google Scholar
  7. 7.
    Bramel, J., Simchi-Levi, D.: Probabilistic Analysis and Practical Algorithms for the Vehicle Routing Problem with Time Windows. Operations Research 44, 501–509 (1996)CrossRefzbMATHGoogle Scholar
  8. 8.
    Chabrier, A.: Vehicle Routing Problem with elementary shortest path based column generation. Computers & Operations Research 33(10), 2972–2990 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Christofides, N., Eilon, S.: An algorithm for the vehicle-dispatching problem. In: OR, pp. 309–318 (1969)Google Scholar
  10. 10.
    Cullen, F.H., Jarvis, J.J., Ratliff, H.D.: Set partitioning based heuristics for interactive routing. Networks 11(2), 125–143 (1981)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Gaur, D.R., Mudgal, A., Singh, R.R.: Routing vehicles to minimize fuel consumption. Operations Research Letters 41(6), 576–580 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Fakcharoenphol, J., Harrelson, C., Rao, S.: The k-traveling repairman problem. In: Proceedings of the fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 655–664. Society for Industrial and Applied Mathematics (January 2003)Google Scholar
  13. 13.
    Feillet, D., Dejax, P., Gendreau, M., Gueguen, C.: An exact algorithm for the elementary shortest path problem with resource constraints: Application to some vehicle routing problems. Networks 44(3), 216–229 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Hassin, R.: Approximation schemes for the restricted shortest path problem. Mathematics of Operations Research 17(1), 36–42 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Irnich, S., Desaulniers, G.: Shortest path problems with resource constraints. In: Desaulniers, G., Desrosiers, J., Solomon, M.M. (eds.) Column generation, GERAD 25th Anniversary Series, ch. 2, pp. 33–65. Springer (2005)Google Scholar
  16. 16.
    Kara, İ., Kara, B.Y., Yetiş, M.K.: Cumulative Vehicle Routing Problems. In: Caric, T., Gold, H. (eds.) Vehicle Routing Problem, pp. 85–98. I-Tech Education and Publishing KG, Vienna (2008)Google Scholar
  17. 17.
    Kara, İ., Kara, B.Y., Yetis, M.K.: Energy Minimizing Vehicle Routing Problem. In: Dress, A., Xu, Y., Zhu, B. (eds.) COCOA. LNCS, vol. 4616, pp. 62–71. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  18. 18.
    Lysgaard, J., Wohlk, S.: A branch-and-cut-and-price algorithm for the cumulative capacitated vehicle routing problem. European Journal of Operational Research (August 2013)Google Scholar
  19. 19.
    Raghavan, P., Thompson, C.D.: Randomized Rounding: A Technique for Provably Good Algorithms and Algorithmic Proofs. Combinatorica 7(4), 365–374 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Rinaldi, G., Yarrow, L.: Optimizing a 48-city traveling salesman problem: A case study in combinatorial problem solving, New York University, Graduate School of Business Administration (1985)Google Scholar
  21. 21.
    Rousseau, L.M., Gendreau, M., Pesant, G., Focacci, F.: Solving VRPTWs with constraint programming based column generation. Annals of Operations Research 130(1-4), 199–216 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Sahin, B., Yilmaz, H., Ust, Y., Guneri, A.F., Gulsun, B.: An approach for analyzing transportation costs and a case study. European Journal of Operational Research 193, 1–11 (2009)CrossRefzbMATHGoogle Scholar
  23. 23.
    Toth, P., Vigo, D.: The Vehicle Routing Problem. SIAM (2001)Google Scholar
  24. 24.
    Vazirani, V.V.: Approximation Algorithms. Springer, New York (2001)Google Scholar
  25. 25.
    Williamson, D.P., Shmoys, D.B.: The Design of Approximation Algorithms, 1st edn. Cambridge University Press, New York (2011)CrossRefzbMATHGoogle Scholar
  26. 26.
    Xiao, Y., Zhao, Q., Kaku, I., Xu, Y.: Development of a fuel consumption optimization model for the capacitated vehicle routing problem. Computers & Operations Research 39(7), 1419–1431 (2012)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Daya Ram Gaur
    • 1
  • Rishi Ranjan Singh
    • 2
  1. 1.Department of Mathematics and Computer ScienceUniversity of LethbridgeLethbridgeCanada
  2. 2.Department of Computer Science and EngineeringIndian Institute of Technology RoparRupnagarIndia

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