Skip to main content
Book cover

Conference on Algorithms and Discrete Applied Mathematics

CALDAM 2015: Algorithms and Discrete Applied Mathematics pp 262–274Cite as

Cumulative Vehicle Routing Problem: A Column Generation Approach

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNTCS,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

This is a preview of subscription content, access via your institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-14974-5_25
  • Chapter length: 13 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   54.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-14974-5
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   69.99
Price excludes VAT (USA)
Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Altinkemer, K., Gavish, B.: Technical Note: Heuristics for Delivery Problems with Constant Error Guarantees. Transportation Science 24(4), 294–297 (1990)

    CrossRef  MATH  Google Scholar 

  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. Balas, E.: New classes of efficiently solvable generalized traveling salesman problems. Annals of Operations Research (56), 529–558 (1999)

    Google Scholar 

  4. Balinski, M.L., Quandt, R.E.: On an integer program for a delivery problem. Operations Research 12, 300–304 (1964)

    CrossRef  Google Scholar 

  5. Bianco, L., Mingozzi, A., Ricciardelli, S.: The traveling salesman problem with cumulative costs. Networks 23(2), 81–91 (1993)

    CrossRef  MATH  MathSciNet  Google Scholar 

  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. Bramel, J., Simchi-Levi, D.: Probabilistic Analysis and Practical Algorithms for the Vehicle Routing Problem with Time Windows. Operations Research 44, 501–509 (1996)

    CrossRef  MATH  Google Scholar 

  8. Chabrier, A.: Vehicle Routing Problem with elementary shortest path based column generation. Computers & Operations Research 33(10), 2972–2990 (2006)

    CrossRef  MATH  MathSciNet  Google Scholar 

  9. Christofides, N., Eilon, S.: An algorithm for the vehicle-dispatching problem. In: OR, pp. 309–318 (1969)

    Google Scholar 

  10. Cullen, F.H., Jarvis, J.J., Ratliff, H.D.: Set partitioning based heuristics for interactive routing. Networks 11(2), 125–143 (1981)

    CrossRef  MathSciNet  Google Scholar 

  11. Gaur, D.R., Mudgal, A., Singh, R.R.: Routing vehicles to minimize fuel consumption. Operations Research Letters 41(6), 576–580 (2013)

    CrossRef  MATH  MathSciNet  Google Scholar 

  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. 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)

    CrossRef  MATH  MathSciNet  Google Scholar 

  14. Hassin, R.: Approximation schemes for the restricted shortest path problem. Mathematics of Operations Research 17(1), 36–42 (1992)

    CrossRef  MATH  MathSciNet  Google Scholar 

  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. 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. 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)

    CrossRef  Google Scholar 

  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. Raghavan, P., Thompson, C.D.: Randomized Rounding: A Technique for Provably Good Algorithms and Algorithmic Proofs. Combinatorica 7(4), 365–374 (1987)

    CrossRef  MATH  MathSciNet  Google Scholar 

  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. 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)

    CrossRef  MATH  MathSciNet  Google Scholar 

  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)

    CrossRef  MATH  Google Scholar 

  23. Toth, P., Vigo, D.: The Vehicle Routing Problem. SIAM (2001)

    Google Scholar 

  24. Vazirani, V.V.: Approximation Algorithms. Springer, New York (2001)

    Google Scholar 

  25. Williamson, D.P., Shmoys, D.B.: The Design of Approximation Algorithms, 1st edn. Cambridge University Press, New York (2011)

    CrossRef  MATH  Google Scholar 

  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)

    CrossRef  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive, Lethbridge, Alberta, Canada

    Daya Ram Gaur

  2. Department of Computer Science and Engineering, Indian Institute of Technology Ropar, Nangal Road, Rupnagar, Punjab, 140001, India

    Rishi Ranjan Singh

Authors
  1. Daya Ram Gaur
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rishi Ranjan Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science and Engineering, I.I.T. Kanpur, 208016, India

    Sumit Ganguly

  2. School of Computing Science, Simon Fraser University, V5A 1S6, Burnaby, BC, Canada

    Ramesh Krishnamurti

Rights and permissions

Reprints and Permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Gaur, D.R., Singh, R.R. (2015). Cumulative Vehicle Routing Problem: A Column Generation Approach. In: Ganguly, S., Krishnamurti, R. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2015. Lecture Notes in Computer Science, vol 8959. Springer, Cham. https://doi.org/10.1007/978-3-319-14974-5_25

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-14974-5_25

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-14973-8

  • Online ISBN: 978-3-319-14974-5

  • eBook Packages: Computer ScienceComputer Science (R0)