Reliable vehicle routing problem in stochastic networks with correlated travel times

Abstract

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.

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References

  1. Adulyasak Y, Jaillet P (2016) Models and algorithms for stochastic and robust vehicle routing with deadlines. Transp Sci 50(2):608–626

    Article  Google Scholar 

  2. Alexiou D, Katsavounis S (2015) A multi-objective transportation routing problem. Oper Res 15(2):199–211

    Google Scholar 

  3. Ando N, Taniguchi E (2006) Travel time reliability in vehicle routing and scheduling with time windows. Netw Spat Econ 6(3):293–311

    Article  Google Scholar 

  4. Bernard M, Hackney JK, Axhausen KW (2006) Correlation of link travel speeds. Paper presented at the Swiss transport research conference

  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):64–75

    Article  Google Scholar 

  6. Braaten S, Gjønnes O, Hvattum LM, Tirado G (2017) Heuristics for the robust vehicle routing problem with time windows. Expert Syst Appl 77:136–147

    Article  Google Scholar 

  7. Brown DB, Sim M (2009) Satisficing measures for analysis of risky positions. Manag Sci 55(1):71–84

    Article  Google 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 2121:102–109

    Article  Google 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 64:88–102

    Article  Google 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 82:1–23

    Article  Google Scholar 

  11. Cheng T, Haworth J, Wang J (2012) Spatio-temporal autocorrelation of road network data. J Geogr Syst 14(4):389–413

    Article  Google 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):41–68

    Article  Google 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):928–936

    Article  Google Scholar 

  14. Dorigo M, Bonabeau E, Theraulaz G (2000) Ant algorithms and stigmergy. Future Gener Comput Syst 16(8):851–871

    Article  Google 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):539–550

    Article  Google Scholar 

  16. Fenton L (1960) The sum of log-normal probability distributions in scatter transmission systems. IRE Trans Commun Syst 8(1):57–67

    Article  Google 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):287–300

    Article  Google Scholar 

  18. Gendreau M, Jabali O, Rei W (2016) 50th Anniversary invited article—future research directions in stochastic vehicle routing. Transp Sci 50(4):1163–1173

    Article  Google Scholar 

  19. Glover F (1986) Future paths for integer programming and links to artificial intelligence. Comput Oper Res 13(5):533–549

    Article  Google 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, Washington

  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 89:1–14

    Article  Google 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):465–475

    Article  Google 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 2396:71–81

    Article  Google Scholar 

  24. Jaillet P, Qi J, Sim M (2016) Routing optimization under uncertainty. Oper Res 64(1):186–200

    Article  Google 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 53:64–81

    Article  Google Scholar 

  26. Kenyon AS, Morton DP (2003) Stochastic vehicle routing with random travel times. Transp Sci 37(1):69–82

    Article  Google 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):890–910

    Article  Google Scholar 

  28. Lambert V, Laporte G, Louveaux F (1993) Designing collection routes through bank branches. Comput Oper Res 20(7):783–791

    Article  Google Scholar 

  29. Laporte G, Louveaux F, Mercure H (1992) The vehicle routing problem with stochastic travel times. Transp Sci 26(3):161–170

    Article  Google 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):559–569

    Article  Google 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):363–377

    Article  Google 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):137–145

    Article  Google 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, Washington

    Google 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 74:1–21

    Article  Google Scholar 

  35. Miranda DM, Conceição SV (2016) The vehicle routing problem with hard time windows and stochastic travel and service time. Expert Syst Appl 64:104–116

    Article  Google 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. https://doi.org/10.1007/s12351-016-0285-2

    Article  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, Singapore

  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):404–414

    Article  Google 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):357–367

    Article  Google 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):5056–5064

    Article  Google 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):1576–1590

    Article  Google 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 Vegas

  43. Režnar T, Martinovič J, Slaninová K, Grakova E, Vondrák V (2017) Probabilistic time-dependent vehicle routing problem. CEJOR 25(3):545–560

    Article  Google Scholar 

  44. Russell RA, Urban TL (2008) Vehicle routing with soft time windows and erlang travel times. J Oper Res Soc 59(9):1220–1228

    Article  Google 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. https://doi.org/10.1007/978-3-540-88914-4_70

    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, Barcelona

  47. Solomon MM (1984) Vehicle routing and scheduling with time window constraints: models and algorithms. University of Pennsylvania, Philadelphia

    Google Scholar 

  48. Solomon MM (1987) Algorithms for the vehicle routing and scheduling problems with time window constraints. Oper Res 35(2):254–265

    Article  Google Scholar 

  49. Stützle T, Hoos HH (2000) MAX–MIN ant system. Future Gener Comput Syst 16(8):889–914

    Article  Google 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 1964:81–90

    Article  Google Scholar 

  51. Taguchi G, Chowdhury S, Wu Y (2005) Taguchi’s quality engineering handbook. Wiley, Hoboken

    Google 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):170–186

    Article  Google 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):214–224

    Article  Google 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):789–799

    Article  Google 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):1–6

    Article  Google 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 2238:86–96

    Article  Google Scholar 

  57. Unal R, De B (1991) Design for cost and quality: the robust design approach. J Parametr 11(1):73–93

    Article  Google 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. https://doi.org/10.1287/trsc.2017.0748

    Article  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 72:207–220

    Article  Google 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, Washington

    Google 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):322–333

    Article  Google 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, Washington

  63. Yang H, Zhao L, Ye D, Ma J (2017) Disturbance management for vehicle routing with time window changes. Oper Res. https://doi.org/10.1007/s12351-017-0363-0

    Article  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):2277–2290

    Article  Google 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):471–496

    Article  Google 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, Qingdao

  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 2334:1–9

    Article  Google 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):33–49

    Article  Google Scholar 

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Correspondence to Afshin Shariat-Mohaymany.

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Rajabi-Bahaabadi, M., Shariat-Mohaymany, A., Babaei, M. et al. Reliable vehicle routing problem in stochastic networks with correlated travel times. Oper Res Int J 21, 299–330 (2021). https://doi.org/10.1007/s12351-019-00452-w

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Keywords

  • Vehicle routing problem
  • Time windows
  • Correlated travel times
  • Hybrid algorithm