Reliable vehicle routing problem in stochastic networks with correlated travel times

  • Mojtaba Rajabi-Bahaabadi
  • Afshin Shariat-MohaymanyEmail author
  • Mohsen Babaei
  • Daniele Vigo
Original Paper


This study is motivated by the fact that travel times on road networks are correlated. However, all existing studies on the vehicle routing problem share a common simplifying assumption that arc travel times are independently distributed. In this paper, we address a variant of the vehicle routing problem with soft time windows in which travel times are treated as correlated random variables. To this end, correlations among arc travel times are modeled by a variance–covariance matrix. We use a mathematical model in which penalties are incurred for early and late arrival at each customer (violation of time window constraints). A Max–Min ant colony system is hybridized with a tabu search algorithm to solve the model. Results show that ignorance of correlations among arc travel times can significantly lead to inefficient solutions to the vehicle routing problem in stochastic networks. We also conduct an exploratory analysis of real travel time data. The results of the analysis demonstrate that travel times are significantly correlated and the shifted log-normal distribution is an appropriate candidate for modeling travel time uncertainty.


Vehicle routing problem Time windows Correlated travel times Hybrid algorithm 



  1. Adulyasak Y, Jaillet P (2016) Models and algorithms for stochastic and robust vehicle routing with deadlines. Transp Sci 50(2):608–626Google Scholar
  2. Alexiou D, Katsavounis S (2015) A multi-objective transportation routing problem. Oper Res 15(2):199–211Google Scholar
  3. Ando N, Taniguchi E (2006) Travel time reliability in vehicle routing and scheduling with time windows. Netw Spat Econ 6(3):293–311Google Scholar
  4. Bernard M, Hackney JK, Axhausen KW (2006) Correlation of link travel speeds. Paper presented at the Swiss transport research conferenceGoogle Scholar
  5. Binart S, Dejax P, Gendreau M, Semet F (2016) A 2-stage method for a field service routing problem with stochastic travel and service times. Comput Oper Res 65(1):64–75Google Scholar
  6. Braaten S, Gjønnes O, Hvattum LM, Tirado G (2017) Heuristics for the robust vehicle routing problem with time windows. Expert Syst Appl 77:136–147Google Scholar
  7. Brown DB, Sim M (2009) Satisficing measures for analysis of risky positions. Manag Sci 55(1):71–84Google Scholar
  8. Chan K, Lam W, Tam M (2009) Real-time estimation of arterial travel times with spatial travel time covariance relationships. Transp Res Rec J Transp Res Board 2121:102–109Google Scholar
  9. Chen L, Hà MH, Langevin A, Gendreau M (2014) Optimizing road network daily maintenance operations with stochastic service and travel times. Transp Res Part E Logist Transp Rev 64:88–102Google Scholar
  10. Chen M, Yu G, Chen P, Wang Y (2017) A copula-based approach for estimating the travel time reliability of urban arterial. Transp Res Part C Emerg Technol 82:1–23Google Scholar
  11. Cheng T, Haworth J, Wang J (2012) Spatio-temporal autocorrelation of road network data. J Geogr Syst 14(4):389–413Google Scholar
  12. Chu JC, Yan S, Huang H-J (2017) A multi-trip split-delivery vehicle routing problem with time windows for inventory replenishment under stochastic travel times. Netw Spat Econ 17(1):41–68Google Scholar
  13. Cordeau JF, Laporte G, Mercier A (2001) A unified tabu search heuristic for vehicle routing problems with time windows. J Oper Res Soc 52(8):928–936Google Scholar
  14. Dorigo M, Bonabeau E, Theraulaz G (2000) Ant algorithms and stigmergy. Future Gener Comput Syst 16(8):851–871Google Scholar
  15. Ehmke JF, Campbell AM, Urban TL (2015) Ensuring service levels in routing problems with time windows and stochastic travel times. Eur J Oper Res 240(2):539–550Google Scholar
  16. Fenton L (1960) The sum of log-normal probability distributions in scatter transmission systems. IRE Trans Commun Syst 8(1):57–67Google Scholar
  17. Garcia-Najera A, Bullinaria JA (2011) An improved multi-objective evolutionary algorithm for the vehicle routing problem with time windows. Comput Oper Res 38(1):287–300Google Scholar
  18. Gendreau M, Jabali O, Rei W (2016) 50th Anniversary invited article—future research directions in stochastic vehicle routing. Transp Sci 50(4):1163–1173Google Scholar
  19. Glover F (1986) Future paths for integer programming and links to artificial intelligence. Comput Oper Res 13(5):533–549Google Scholar
  20. Guimarans D, Dominguez O, Juan AA, Martinez E (2016) A multi-start simheuristic for the stochastic two-dimensional vehicle routing problem. Paper presented at the winter simulation conference, WashingtonGoogle Scholar
  21. Guimarans D, Dominguez O, Panadero J, Juan AA (2018) A simheuristic approach for the two-dimensional vehicle routing problem with stochastic travel times. Simul Model Pract Theory 89:1–14Google Scholar
  22. Hsu C-I, Hung S-F, Li H-C (2007) Vehicle routing problem with time-windows for perishable food delivery. J Food Eng 80(2):465–475Google Scholar
  23. Isukapati I, List G, Williams B, Karr A (2013) Synthesizing route travel time distributions from segment travel time distributions. Transp Res Rec J Transp Res Board 2396:71–81Google Scholar
  24. Jaillet P, Qi J, Sim M (2016) Routing optimization under uncertainty. Oper Res 64(1):186–200Google Scholar
  25. Jenelius E, Koutsopoulos HN (2013) Travel time estimation for urban road networks using low frequency probe vehicle data. Transp Res Part B Methodol 53:64–81Google Scholar
  26. Kenyon AS, Morton DP (2003) Stochastic vehicle routing with random travel times. Transp Sci 37(1):69–82Google Scholar
  27. Lam WHK, Shao H, Sumalee A (2008) Modeling impacts of adverse weather conditions on a road network with uncertainties in demand and supply. Transp Res Part B Methodol 42(10):890–910Google Scholar
  28. Lambert V, Laporte G, Louveaux F (1993) Designing collection routes through bank branches. Comput Oper Res 20(7):783–791Google Scholar
  29. Laporte G, Louveaux F, Mercure H (1992) The vehicle routing problem with stochastic travel times. Transp Sci 26(3):161–170Google Scholar
  30. Lau HC, Sim M, Teo KM (2003) Vehicle routing problem with time windows and a limited number of vehicles. Eur J Oper Res 148(3):559–569Google Scholar
  31. Lecluyse C, Van Woensel T, Peremans H (2009) Vehicle routing with stochastic time-dependent travel times. 4OR Q J Oper Res 7(4):363–377Google Scholar
  32. Li X, Tian P, Leung SCH (2010) Vehicle routing problems with time windows and stochastic travel and service times: models and algorithm. Int J Prod Econ 125(1):137–145Google Scholar
  33. List GF et al (2014) Guide to establishing monitoring programs for travel time reliability, SHRP 2 Report S2-L02-RR-2. Transportation Research Board, WashingtonGoogle Scholar
  34. Ma Z, Koutsopoulos HN, Ferreira L, Mesbah M (2017) Estimation of trip travel time distribution using a generalized Markov chain approach. Transp Res Part C Emerg Technol 74:1–21Google Scholar
  35. Miranda DM, Conceição SV (2016) The vehicle routing problem with hard time windows and stochastic travel and service time. Expert Syst Appl 64:104–116Google Scholar
  36. Mohamadi A, Yaghoubi S, Pishvaee MS (2016) Fuzzy multi-objective stochastic programming model for disaster relief logistics considering telecommunication infrastructures: a case study. Oper Res. Google Scholar
  37. Nannen V, Eiben AE (2007) Efficient relevance estimation and value calibration of evolutionary algorithm parameters. Paper presented at the congress on evolutionary computation, SingaporeGoogle Scholar
  38. Nguyen VA, Jiang J, Ng KM, Teo KM (2016) Satisficing measure approach for vehicle routing problem with time windows under uncertainty. Eur J Oper Res 248(2):404–414Google Scholar
  39. Park D, Rilett LR (1999) Forecasting freeway link travel times with a multilayer feedforward neural network. Comput Aided Civ Infrastruct Eng 14(5):357–367Google Scholar
  40. Rajabi-Bahaabadi M, Shariat-Mohaymany A, Babaei M, Ahn CW (2015) Multi-objective path finding in stochastic time-dependent road networks using non-dominated sorting genetic algorithm. Expert Syst Appl 42(12):5056–5064Google Scholar
  41. Ramezani M, Geroliminis N (2012) On the estimation of arterial route travel time distribution with Markov chains. Transp Res Part B Methodol 46(10):1576–1590Google Scholar
  42. Reyes-Rubiano LS, Faulin J, Calvet L, Juan AA (2017) A simheuristic approach for freight transportation in smart cities. Paper presented at the winter simulation conference Las VegasGoogle Scholar
  43. Režnar T, Martinovič J, Slaninová K, Grakova E, Vondrák V (2017) Probabilistic time-dependent vehicle routing problem. CEJOR 25(3):545–560Google Scholar
  44. Russell RA, Urban TL (2008) Vehicle routing with soft time windows and erlang travel times. J Oper Res Soc 59(9):1220–1228Google Scholar
  45. Shao Z-J, Gao S-P, Wang S-S (2009) A hybrid particle swarm optimization algorithm for vehicle routing problem with stochastic travel time. In: Cao B-Y, Zhang C-Y, Li T-F (eds) Fuzzy information and engineering, vol 1. Springer, Berlin, Heidelberg, pp 566–574. Google Scholar
  46. Smit SK, Eiben AE (2010) Beating the world champion evolutionary algorithm via REVAC tuning. Paper presented at the congress on evolutionary computation, BarcelonaGoogle Scholar
  47. Solomon MM (1984) Vehicle routing and scheduling with time window constraints: models and algorithms. University of Pennsylvania, PhiladelphiaGoogle Scholar
  48. Solomon MM (1987) Algorithms for the vehicle routing and scheduling problems with time window constraints. Oper Res 35(2):254–265Google Scholar
  49. Stützle T, Hoos HH (2000) MAX–MIN ant system. Future Gener Comput Syst 16(8):889–914Google Scholar
  50. Sumalee A, Watling D, Nakayama S (2006) Reliable network design problem: case with uncertain demand and total travel time reliability. Transp Res Rec J Transp Res Board 1964:81–90Google Scholar
  51. Taguchi G, Chowdhury S, Wu Y (2005) Taguchi’s quality engineering handbook. Wiley, HobokenGoogle Scholar
  52. Taillard É, Badeau P, Gendreau M, Guertin F, Potvin J-Y (1997) A tabu search heuristic for the vehicle routing problem with soft time windows. Transp Sci 31(2):170–186Google Scholar
  53. Taş D, Dellaert N, van Woensel T, de Kok T (2013) Vehicle routing problem with stochastic travel times including soft time windows and service costs. Comput Oper Res 40(1):214–224Google Scholar
  54. Taş D, Gendreau M, Dellaert N, van Woensel T, de Kok T (2014) Vehicle routing with soft time windows and stochastic travel times: a column generation and branch-and-price solution approach. Eur J Oper Res 236(3):789–799Google Scholar
  55. Tavakkoli-Moghaddam R, Alinaghian M, Salamat-Bakhsh A, Norouzi N (2012) A hybrid meta-heuristic algorithm for the vehicle routing problem with stochastic travel times considering the driver’s satisfaction. J Ind Eng Int 8(1):1–6Google Scholar
  56. Thompson R, Taniguchi E, Yamada T (2011) Estimating the benefits of considering travel time variability in urban distribution. Transp Res Rec J Transp Res Board 2238:86–96Google Scholar
  57. Unal R, De B (1991) Design for cost and quality: the robust design approach. J Parametr 11(1):73–93Google Scholar
  58. Vareias AD, Repoussis PP, Tarantilis CD (2017) Assessing customer service reliability in route planning with self-imposed time windows and stochastic travel times. Transp Sci. Google Scholar
  59. Wang Z, Lin W-H (2017) Incorporating travel time uncertainty into the design of service regions for delivery/pickup problems with time windows. Expert Syst Appl 72:207–220Google Scholar
  60. Wang Y, Araghi BN, Malinovskiy Y, Corey J, Cheng T (2014) Error assessment for emerging traffic data collection devices, Report WA-RD 810.1. University of Washington, WashingtonGoogle Scholar
  61. Westgate BS, Woodard DB, Matteson DS, Henderson SG (2016) Large-network travel time distribution estimation for ambulances. Eur J Oper Res 252(1):322–333Google Scholar
  62. Yang Y, Qin Y, Li X, Tian Y, Jia L (2015) Correlation patterns of highway segment travel times. Paper presented at the 94th annual meeting of the transportation research board, WashingtonGoogle Scholar
  63. Yang H, Zhao L, Ye D, Ma J (2017) Disturbance management for vehicle routing with time window changes. Oper Res. Google Scholar
  64. Zhang T, Chaovalitwongse WA, Zhang Y (2012) Scatter search for the stochastic travel-time vehicle routing problem with simultaneous pick-ups and deliveries. Comput Oper Res 39(10):2277–2290Google Scholar
  65. Zhang J, Lam WHK, Chen BY (2013) A stochastic vehicle routing problem with travel time uncertainty: trade-off between cost and customer service. Netw Spat Econ 13(4):471–496Google Scholar
  66. Zhao T, Nie Y, Wu X, Zhang Y (2014) Empirical analysis of the dependence structure in traffic data using copula function. Paper presented at the service operations and logistics, and informatics (SOLI), IEEE international conference on, QingdaoGoogle Scholar
  67. Zockaie A, Nie Y, Wu X, Mahmassani H (2013) Impacts of correlations on reliable shortest path finding. Transp Res Rec J Transp Res Board 2334:1–9Google Scholar
  68. Zou Y, Zhu X, Zhang Y, Zeng X (2014) A space–time diurnal method for short-term freeway travel time prediction. Transp Res Part C Emerg Technol 43(Part 1):33–49Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Mojtaba Rajabi-Bahaabadi
    • 1
  • Afshin Shariat-Mohaymany
    • 1
    Email author
  • Mohsen Babaei
    • 2
  • Daniele Vigo
    • 3
  1. 1.School of Civil EngineeringIran University of Science and TechnologyTehranIran
  2. 2.Department of Civil Engineering, Faculty of EngineeringBu-Ali Sina UniversityHamadanIran
  3. 3.Department of Electrical, Electronic and Information EngineeringUniversity of BolognaBolognaItaly

Personalised recommendations