Abstract
Assigning and scheduling vehicle routes in a stochastic time-dependent environment is a crucial management problem. The assumption that in a real-life environment everything goes according to an a priori determined static schedule is unrealistic. Our methodology builds on earlier work in which the traffic congestion is captured in an analytical way using queueing theory. The congestion is then applied to the VRP problem. In this paper, we introduce the variability in traffic flows into the model. This allows for an evaluation of the routes based on the uncertainty involved. Different experiments show that the risk taking behavior of the planner can be taken into account during optimization. As more weight is given to the variability component, the resulting optimal route will take a slightly longer travel time, but will be more reliable. We propose a powerful objective function that is easily implemented and that captures the trade-off between the average travel time and its variance. The evaluation of the solution is done in terms of the 95th-percentile of the travel time distribution (assumed to be lognormal), which reflects well the quality of the solution in this stochastic time-dependent environment.
Similar content being viewed by others
References
Augerat P, Belenguer JM, Benavent E, Corber A, Naddef D (1998) Separating capacity constraints in the CVRP using tabu search. Eur J Oper Res 106: 546–557
Beaulieu NC, Xie Q (2004) An optimal lognormal approximation to lognormal sum distributions. IEEE Trans Vehicular Technol 53(2): 479–489
Berry DS, Belmont DM (1951) Distribution of vehicle speeds and travel times. In: Proceedings of 2nd Berkeley symposium on mathematical and statistical probabability, pp 589–602
Best MJ, Grauer RR (1991) Sensitivity analysis for mean-variance portfolio problems. Manage Sci 37(8): 980–989
Chen C, van Zwet E, Varaiya P, Skabardonis A (2003) Travel time reliability as a measure of service. Technical report, Transporation Research Board
DfT (2000) Transport 2010: the 10 year plan. Technical report, DETR, July
DfT (2003) Delivering better transport—progress report. Technical report
Donati AV, Montemanni R, Casagrande N, Rizzoli AE, Gambardella LM (2003) Time dependent vehicle routing problem with a multi ant colony system. Technical Report IDSIA −02-03, International IDSIA, 2003
Finkel AM (1990) A simple formula for calculating the mass density of a lognormally-distributed characteristic: Applications to risk analysis. Risk Anal 10(2): 291–301
Fu L, Rilett LR (1998) Expected shortest path in dynamic and stochastic traffic networks. Transp Res Record 32(7): 499–516
Gao S, Chabini I (2002) The best routing policy problem in stochastic time-dependent networks. Transp Res Record 1783: 188–196
Gao S, Chabini I (2006) Optimal routing policy problem in stochastic time-dependent networks. Transp Res B 40: 93–122
Gendreau M, Hertz A, Laporte G (1994) A tabu search heuristic for the vehicle routing problem. Manage Sci 40(10): 1276–1290
Gendreau M, Laporte G, Séguin R (1996) A tabu search heuristic for the vehicle routing problem with stochastic demands and customers. Oper Res 44(3): 469–477
Gendreau M, Laporte G, Séguin R (1996b) Stochastic vehicle routing. Eur J Oper Res 88(1): 3–12
Glover F (1986) Future paths for integer programming and links to artificial intelligence. Comp Oper Res 13(5): 533–549
Grauer RR, Hakansson NH (1993) On the use of mean–variance and quadratic approximations in implementing dynamic investment strategies: a comparison of returns and investment policies. Manage Sci 39(7): 856–871
Haghani A, Jung S (2005) A dynamic vehicle routing problem with time-dependent travel times. Comp Oper Res 32: 2959–2986
Hall RW (1986) The fastest path through a network with random time-dependent travel times. Transp Sci 20(3): 182–188
He RR, Kornhauser AL, Ran B (2005) Essentially best routes in dynamic and stochastic transportation network. Int J Vehicle Inf Commun Syst 1(1–2): 1–14
Heidemann D (1996) A queueing theory approach to speed-flow-density relationships. In: Proceedings of the 13th international symposium on transportation and traffic theory, Lyon, France, 1996. Transporation and traffic theory
Hertz A, Laporte G, Mittaz M (2000) A tabu search heuristic for the capacitated arc routing problem. Oper Res 48(1): 129–135
Ichoua S, Gendreau M, Potvin J-Y (2003) Vehicle dispatching with time-dependent travel times. Eur J Oper Res 144: 379–396
Kharoufeh JP, Gautam N (2004) Deriving link travel-time distributions via stochastic speed processes. Transp Sci 38(1): 97–106
Kwon J, Coifman B, Bickel P (2000) Day-to-day travel time trends and travel-time prediction from loop-detector data. Transp Res Record 1717: 120–129
Laporte G (1992) The vehicle routing problem: An overview of exact and approximate algorithms. Eur J Oper Res 59(3): 345–358
Laporte G, Louveaux F, Mercure H (1992) The vehicle routing problem with stochastic travel times. Transp Sci 26(3): 161–170
Malandraki C, Daskin MS (1992) Time dependent vehicle routing problems: Formulations, properties and heuristic algorithms. Transp Sci 26(3): 185–200
Mulvey JM, Vanderbei RJ, Zenios SA (1995) Robust optimization of large-scale systems. Oper Res 43(2): 264–281
Osman IH (1991) Metastrategy Simulated Annealing and Tabu Search Algorithms for Combinatorial Optimization Problems. PhD Thesis, Imperial College London, The Management School
Osman IH (1993) Vehicle routing and scheduling: Applications, algorithms and developments. In: Proceeding of the international conference on industrial logistics, Rennes
Pirlot M (1996) General local search methods. Eur J Oper Res 92: 493–511
Rockwell Software Inc. (2005) Arena user’s guide. Rockwell Software Inc., USA
Taniguchi E, Thompson RG, Yamada T, Van Duin R (2001) City logistics: network modelling and intelligent transport systems. Pergamon, New York
Van Woensel T, Vandaele N (2006) Empirical validation of a queueing approach to uninterrupted traffic flows. A Quart J Oper Res 4(1): 59–72
Van Woensel T, Creten R, Vandaele N (2001) Managing the environmental externalities of traffic logistics: the issue of emissions. POMS J Spec Issue Environ Manage Oper 10(2)
Van Woensel T, Kerbache L, Peremans H, Vandaele N (2008) Vehicle routing with dynamic travel times: A queueing approach. EJOR 186(3): 990–1007
Vandaele N, Van Woensel T, Verbruggen A (2000) A queueing based traffic flow model. Transp Res D 5(2): 121–135
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lecluyse, C., Van Woensel, T. & Peremans, H. Vehicle routing with stochastic time-dependent travel times. 4OR-Q J Oper Res 7, 363–377 (2009). https://doi.org/10.1007/s10288-009-0097-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10288-009-0097-9