Cluster Computing

, Volume 22, Supplement 6, pp 15459–15470 | Cite as

Solving the time-dependent multi-trip vehicle routing problem with time windows and an improved travel speed model by a hybrid solution algorithm

  • Yan Sun
  • Danzhu Wang
  • Maoxiang LangEmail author
  • Xuesong Zhou


In this study, we explore a time-dependent multi-trip vehicle routing problem (TDMTVRP) with an improved travel speed model. This problem is set in a scenario that (a) a set of customers with fixed demands and service time windows have to be served in a sequence of service trips which originate and terminate at a distribution centre, (b) the service trips will be assigned to a fleet of vehicles with fixed capacities and maximum allowable working durations each day, (c) each vehicle can perform more than one service trip, and (d) the link travel times varies with vehicle travel speeds which results from congestion effects during different time of day in urban areas. The aim of the TDMTVRP model is to find an optimal strategy to minimize the vehicle utilized times and their total scheduling time. A continuous piecewise linear function is first introduced to represent the variation and transition of vehicle travel speeds with the time of the day instead of the traditional staircase travel speed function. Then a hybrid solution algorithm is developed by using the nearest-neighbour heuristic to obtain an initial solution and Tabu search heuristic to search the optimal solution. Finally, an experimental case study is used to verify the feasibility of the proposed model and algorithm. The experimental results indicate that compared with the CVRPTW (capacitated vehicle routing problem with time windows) model, the TDMTVRP model proposed in this study can both decrease the vehicle utilized times dramatically and shorten the vehicle travel distances slightly in dealing with the vehicle routing problem.


Vehicle routing problem Time-dependent Multi-trip Time window Nearest-neighbour heuristic Tabu search 



This study was supported by the Shandong Provincial Natural Science Foundation of China under Grant No. ZR2017BG010, the Shandong Provincial Social Science Planning of China under Grant No. 16DGLJ06, and the Shandong Provincial Higher Educational Science and Technology Program of China under Grant No. J17KA189. The authors would also like to thank Dr. Cyril B. Lucas from the United Kingdom, author of the book Atomic and Molecular Beams: Production and Collimation, for helping us edit and polish the language of this paper.


  1. 1.
    Crevier, B., Cordeau, J.F., Laporte, G.: The multi-depot vehicle routing problem with inter-depot routes. Eur. J. Oper. Res. 176(2), 756–773 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Taillard, É.D., Laporte, G., Gendreau, M.: Vehicle routeing with multiple use of vehicles. J. Oper. Res. Soc. 47(8), 1065–1070 (1996)CrossRefGoogle Scholar
  3. 3.
    Ichoua, S., Gendreau, M., Potvin, J.Y.: Vehicle dispatching with time-dependent travel times. Eur. J. Oper. Res. 144(2), 379–396 (2003)CrossRefGoogle Scholar
  4. 4.
    Donati, A.V., Montemanni, R., Casagrande, N., et al.: Time dependent vehicle routing problem with a multi ant colony system. Eur. J. Oper. Res. 185(3), 1174–1191 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Salhi, S., Petch, R.J.: A GA based heuristic for the vehicle routing problem with multiple trips. J. Math. Model. Algorithms 6(4), 591–613 (2007)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Baldacci, R., Christofides, N., Mingozzi, A.: An exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts. Math. Program 115(2), 351–385 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Baldacci, R., Mingozzi, A., Roberti, R.: New route relaxation and pricing strategies for the vehicle routing problem. Oper. Res. 59(5), 1269–1283 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Liu, S., Huang, W., Ma, H.: An effective genetic algorithm for the fleet size and mix vehicle routing problems. Transp. Res. Part E 45(3), 434–445 (2009)CrossRefGoogle Scholar
  9. 9.
    Chen, X., Li, J.: Enhanced vehicle routing algorithm with time windows based on heuristic algorithm. J. Comput. Inf. Syst. 7(11), 3886–3892 (2011)Google Scholar
  10. 10.
    Tavakkoli-Moghaddam, R., Gazanfari, M., Alinaghian, M., et al.: A new mathematical model for a competitive vehicle routing problem with time windows solved by simulated annealing. J. Manuf. Syst. 30(2), 83–92 (2011)CrossRefGoogle Scholar
  11. 11.
    Garcia-Najera, A., Bullinaria, J.A.: An improved multi-objective evolutionary algorithm for the vehicle routing problem with time windows. Comput. Oper. Res. 38(1), 287–300 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gan, X., Wang, Y., Li, S., et al.: Vehicle routing problem with time windows and simultaneous delivery and pick-up service based on MCPSO. Math. Prob. Eng. (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Pop, P.C., Kara, I., Marc, A.H.: New mathematical models of the generalized vehicle routing problem and extensions. Appl. Math. Model. 36(1), 97–107 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Li, H., Lu, Y., Zhang, J., et al.: Solving the tractor and semi-trailer routing problem based on a heuristic approach. Math. Prob. Eng. (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Moon, I.K., Lee, J.H., Seong, J.: Vehicle routing problem with time windows considering overtime and outsourcing vehicles. Expert Syst. Appl. 39(18), 13202–13213 (2012)CrossRefGoogle Scholar
  16. 16.
    Zhang, J., Wang, W., Zhao, Y., et al.: Multiobjective quantum evolutionary algorithm for the vehicle routing problem with customer satisfaction. Math. Prob. Eng. (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lang, M., Wang, Y., Zhou, X.: A two-stage algorithm for a dynamic multi-trip vehicle scheduling problem. WASE Int. Conf. Inf. Eng. 2010, 188–191 (2010)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Management Science and EngineeringShandong University of Finance and EconomicsJinanChina
  2. 2.Transportation and Economics Research Institute, China Academy of Railway SciencesBeijingChina
  3. 3.School of Traffic and TransportationBeijing Jiaotong UniversityBeijingChina
  4. 4.School of Sustainable Engineering and the Built EnvironmentArizona State UniversityTempeUSA

Personalised recommendations