## Abstract

Travel time reliability has been recognized as an important factor in cost–benefit analysis in a transportation network. To estimate the benefit and cost of travel time reliability, several network models have been proposed to estimate the monetary values both of travel time and of travel time reliability based on the driver’s route choice behavior. In this study, we propose a network model that addresses monetary values that are specific to each origin–destination pair in a road network. The proposed model is formulated as a utility maximization problem with constraints in a manner similar to studies that address the value of travel time only. The proposed model has the same equilibrium conditions as a multiclass user equilibrium traffic assignment problem with elastic demand in which the risk-averse driver’s route choice behavior is explicitly considered. By accounting for data availability, we propose a parameter estimation method based on the maximum likelihood method with two alternative approaches; namely link flow-based and route flow-based approaches. However, the two proposed methods require link flow data only. Numerical experiments are carried out to demonstrate the performance of the two proposed methods together with some insightful findings.

### Similar content being viewed by others

## References

Bates, J., Polak, J., Jones, P., Cook, A.: The valuation of reliability for personal travel. Transp. Res. Part E Log. Transp. Rev.

**37**, 191–229 (2001)Becker, G.S.: A theory of the allocation of time. Econ. J.

**75**(299), 493–517 (1965)Bras, R.L., Georgakakos, K.P.: Real time nonlinear filtering techniques in streamflow forecasting: a statistical linearization approach. In: Proceedings of the 3rd International Symposium on Stochastic Hydraulics, pp. 95–105 (1980)

Brownstone, D., Small, K.A.: Valuing time and reliability: assessing the evidence from road pricing demonstrations. Transp. Res. Part A Policy Pract.

**39**(4), 279–293 (2005)Carrion, C., Levinson, D.: Value of travel time reliability: a review of current evidence. Transportation Research Part A: Policy and Practice.

**46**(4), 720–741 (2012)Castillo, E., Calviño, A., Nogal, M., Lo, H.K.: On the probabilistic and physical consistency of traffic random variables and models. Comput. Aided Civ. Infrastruct. Eng.

**29**, 496–517 (2014)Chen, A., Zhou, Z.: The α-reliable mean-excess traffic equilibrium model with stochastic travel times. Transp. Res. Part B Methodol.

**44**(4), 493–513 (2010)Chen, X.M., Xiong, C., He, X., Zhu, Z., Zhang, L.: Time-of-day vehicle mileage fees for congestion mitigation and revenue generation: a simulation-based optimization method and its real-world application. Transp. Res. Part C Emerg. Technol.

**63**, 71–95 (2016)Clark, S., Watling, D.: Modelling network travel time reliability under stochastic demand. Transp. Res. Part B Methodol.

**39**(2), 119–140 (2005)Cowell, F.A.: Measuring Inequality, 3rd edn. Oxford University Press, Oxford (2011)

Dafermos, S.: The traffic assignment problem for multi-class user transportation networks. Transp. Sci.

**6**(1), 73–87 (1972)DeSerpa, A.C.: A theory of the economics of time. Econ. J.

**81**(324), 828–846 (1971)Engelson, L., Fosgerau, M.: Additive measures of travel time variability. Transp. Res. Part B Methodol.

**45**(10), 1560–1571 (2011)Fischer, A.: A special Newton-type optimization method. Optimization

**24**(3–4), 269–284 (1992)Fosgerau, M., Karlström, A.: The value of reliability. Transp. Res. Part B Methodol.

**44**(1), 38–49 (2010)Fukushima, M., Luo, Z.Q., Pang, J.S.: A globally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints. Comput. Optim. Appl.

**10**(1), 5–34 (1998)Gan, H., Bai, Y.: The effect of travel time variability on route choice decision: a generalized linear mixed model based analysis. Transportation

**41**(2), 339–350 (2014)Gartner, N.H.: Optimal traffic assignment with elastic demands. Transp. Sci.

**14**(2), 174–191 (1980)Gibrat, R.: Les Inégalités économiques. Librairie du Recueil Sirey, Paris (1931)

Hensher, D.A.: The sensitivity of the valuation of travel time savings to the specification of unobserved effects. Transp. Res. Part E Log. Transp. Rev.

**37**(2–3), 129–142 (2001)Hjorth, K., Börjesson, M., Engelson, L., Fosgerau, M.: Estimating exponential scheduling preferences. Transp. Res. Part B Methodol.

**81**(1), 230–251 (2015)Jackson, W.B., Jucker, J.V.: An empirical study of travel time variability and travel choice behavior. Transp. Sci.

**16**(4), 460–475 (1981)Kalecki, M.: On the Gibrat distribution. Econometrica

**13**(2), 161–170 (1945)Kato, T., Uchida, K.: A study on benefit estimation that considers the values of travel time and travel time reliability in road networks. Transportm. A Transp. Sci.

**14**(1–2), 89–109 (2018)Kroes, K., Koster, P., Peer, S.: A practical method to estimate the benefits of improved road network reliability: an application to departing air passengers. Transportation

**45**(5), 1433–1448 (2018)Lam, T.C., Small, K.A.: The value of time and reliability: measurement from a value pricing experiment. Transp. Res. Part E Log. Transp. Rev.

**37**(2–3), 231–251 (2001)Lam, W.H.K., Shao, H., Sumalee, A.: Modeling impacts of adverse weather conditions on a road network with uncertainties in demand and supply. Transp. Res. Part B Methodol.

**42**, 890–910 (2008)Li, Z., Hensher, D.A., Rose, J.M.: Willingness to pay for travel time reliability in passenger transport: a review and some new empirical evidence. Transp. Res. Part E Log. Transp. Rev.

**46**(3), 384–403 (2010)Li, H., Tu, H.Z., Zhang, X.N.: Travel time variations over time and routes: endogenous congestion with degradable capacities. Transportm. B Transp. Dyn.

**5**(1), 60–81 (2017)Lo, H.K., Chen, A.: Traffic equilibrium problem with route-specific costs: formulation and algorithms. Transp. Res. Part B Methodol.

**34**, 493–513 (2000)Lo, H.K., Tung, Y.K.: Network with degradable links: capacity analysis and design. Transp. Res. Part B Methodol.

**37**(4), 345–363 (2003)Lo, H.K., Luo, X.W., Siu, B.W.Y.: Degradable transport network: travel time budget of travelers with heterogeneous risk aversion. Transp. Res. Part B Methodol.

**40**, 792–806 (2006)Marcotte, P., Zhu, D.: Equilibria with infinitely many differentiated classes of customers. Complementarity and variational problems. In: Proceedings of the 13th International Conference on Complementarity Problems, Jong-Shi Pang and Michael Ferris, eds., SIAM, Philadelphia, pp. 234–258 (2000)

Markowitz, H.: Portfolio Selection: Efficient Diversification of Investments. Wiley, New York (1959)

Nagurney, A., Dong, J.: A multiclass, multicriteria traffic network equilibrium model with elastic demand. Transp. Res. Part B Methodol.

**36**(5), 445–469 (2002)Nakayama, S., Watling, D.: Consistent formulation of network equilibrium with stochastic flows. Transp. Res. Part B Methodol.

**66**, 50–69 (2014)Nakayama, S., Connors, R., Watling, D. (2009): Estimation of parameters of network equilibrium models: a maximum likelihood method and statistical properties of network flow. In: Transportation and Traffic Theory 2009: Golden Jubilee, , pp. 39–56. Springer, New York (2009)

Nguyen, S., Dupuis, C.: An efficient method for computing traffic equilibria in networks with asymmetric transportation costs. Transp. Sci.

**18**, 185–202 (1984)Nie, Y.M.: Multi-class percentile user equilibrium with flow-dependent stochasticity. Transp. Res. Part B Methodol.

**45**(10), 1641–1659 (2011)Prakash, A.A., Seshadri, R., Srinivasan, K.K.: A consistent reliability-based user-equilibrium problem with risk-averse users and endogenous travel time correlations: formulation and solution algorithm. Transp. Res. Part B

**114**, 171–198 (2018)Shao, H., Lam, W.H.K., Tam, M.L.: A reliability-based stochastic traffic assignment model for network with multiple user classes under uncertainty in demand. Netw. Spat. Econ.

**6**, 173–204 (2006)Shao, H., Lam, W.H.K., Tam, M.L., Yuan, X.-M.: Modelling rain effects on risk-taking behaviours of multi-user classes in road networks with uncertainty. J. Adv. Transp.

**42**(3), 265–290 (2008)Shao, H., Lam, W.H.K., Sumalee, A., Chen, A., Hazelton, M.L.: Estimation of mean and covariance of peak hour origin-destination demands from day-to-day traffic counts. Transp. Res. Part B Methodol.

**68**, 52–75 (2014)Shao, H., Lam, W.H.K., Sumalee, A., Hazelton, M.L.: Estimation of mean and covariance of stochastic multi-class OD demands from classified traffic counts. Transp. Res. Procedia

**7**, 192–211 (2015)Sikka, N., Hanley, P.: What do commuters think travel time reliability is worth? Calculating economic value of reducing the frequency and extent of unexpected delays. Transportation

**40**(5), 903–919 (2013)Siu, B.W.Y., Lo, H.K.: Doubly uncertain transportation network: degradable capacity and stochastic demand. Eur. J. Oper. Res.

**191**(1), 164–179 (2008)Siu, B.W.Y., Lo, H.K.: Punctuality-based route and departure time choice. Transportm. A Transp. Sci.

**10**(7), 585–621 (2014)Small, K.A.: The scheduling of consumer activities: work trips. Am. Econ. Rev.

**72**(3), 467–479 (1982)Srivastava, M.S., von Rosen, D.: Regression models with unknown singular covariance matrix. Linear Algebra Appl.

**354**(1–3), 255–273 (2002)Steck, S., Kolarova, V., Bahamonde-Birke, F., Trommer, S., Lenz, B.: How autonomous driving may affect the value of travel time savings for commuting. Transp. Res. Rec. J. Transp. Res. Board (2018). https://doi.org/10.1177/0361198118757980

Sutton, J.: Gibrat’s legacy. J. Econ. Lit.

**35**(1), 40–59 (1997)Uchida, K.: A study on impact of stochastic traffic capacity on travel times of a road network. Doboku Gakkai Ronbunshuu D

**66**(4), 431–441 (2010).**(in Japanese)**Uchida, K.: Estimating the value of travel time and of travel time reliability in road networks. Transp. Res. Part B Methodol.

**66**, 129–147 (2014)Uchida, K.: Travel time reliability estimation model using observed link flows in a road network. Comput. Aided Civ. Infrastruct. Eng.

**30**(6), 449–463 (2015)Uchida, K., Munehiro, K.: Impact of stochastic traffic capacity on travel time in road network. In: The 89th Annual Meeting of the Transportation Research Board (2010)

Uchida, K., Kato, T.: A simplified network model for travel time reliability analysis in a road network. J. Adv. Transp. (2017). https://doi.org/10.1155/2017/4941535

Varian, H.R.: Microeconomic Analysis, 2nd edn. W.W. Norton & Co Inc, New York (1992)

Vickrey, W.S.: Congestion theory and transport investment. Am. Econ. Rev.

**59**(2), 251–260 (1969)Wang, J.Y.T., Ehrgott, M., Chen, A.: A bi-objective user equilibrium model of travel time reliability in a road network. Transp. Res. Part B Methodol.

**66**, 4–15 (2014)Watling, D.: A second-order stochastic network equilibrium model, I: theoretical foundation. Transp. Sci.

**36**(2), 149–166 (2002)Watling, D.: User equilibrium traffic network assignment with stochastic travel times and late arrival penalty. Eur. J. Oper. Res.

**175**(3), 1539–1556 (2006)Wu, X., Nie, Y.: Modeling heterogeneous risk-taking behavior in route choice: a stochastic dominance approach. Transp. Res. Part A Policy Pract.

**45**(9), 896–915 (2011)Xu, X., Chen, A., Zhou, Z., Cheng, L.: A multi-class mean-excess traffic equilibrium model with elastic demand. J. Adv. Transp.

**48**(3), 203–222 (2014)Yin, Y., Lam, W.H.K., Ieda, H.: New technology and the modeling of risk-taking behavior in congested road networks. Transp. Res. Part C Emerg. Technol.

**12**(3–4), 171–192 (2004)Zhang, L., Xiong, C.F.: A novel agent-based modelling framework for travel time reliability analysis. Transportm. B Transp. Dyn.

**5**(1), 82–99 (2017)

## Acknowledgements

The work described in this paper was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. PolyU R5029-18).

## Author information

### Authors and Affiliations

### Contributions

TK: Literature search and review, Mathematical analysis, Edit equations, Manuscript writing and editing, Content planning. KU: Comments for the manuscript, Literature search and review, Content planning, Manuscript writing and editing WHKL: Comments for the manuscript, Literature search and review, Manuscript writing and editing. AS: Comments for the manuscript, Literature search and review, Manuscript writing and editing.

### Corresponding author

## Ethics declarations

### Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Appendix

### Appendix

By using an approximation shown in Markowitz (1959), the expected utility function shown by (28) can be approximated as

where \(b_{i}\) is a positive parameter on the utility function. We calculate the error of this approximation by numerical simulation in the case of \(q_{i}\) = 1000 and \(cv_{i}\) = 0.3. The result shows that the error of this approximation is at most 1.5%. We considered that it is enough small to ignore. Following Kato and Uchida (2018), \(b_{i}\) is calculated as follows.

We set the parameter \(\alpha_{i}\) in (28) as follows

where \(q_{i}^{ * }\) is the O–D flow under equilibrium conditions. By multiplying \(\ln \left( {1 + q_{i} } \right)\) and \(\alpha_{i}\) in (92), the following relationship is obtained at \(q_{i} = q_{i}^{ * }\).

(93) shows that (28) can approximate the expected utility function of \(E\left[ {u\left( {\mathbf{Q}} \right)} \right] = E\left[ {\sum\limits_{i} {b_{i} \cdot \ln \left( {1 + Q_{i} } \right)} } \right]\), because \(b_{i} \cdot E\left[ {\ln \left( {1 + Q_{i} } \right)} \right] \approx \alpha_{i} \cdot \ln \left( {1 + q_{i} } \right)\). Two partial derivatives of the left and right sides of (93) with respect to \(q_{i}\) are shown as follows.

The difference between (94) and (95) is negligible at \(q_{i} = q_{i}^{ * }\). This implies that the marginal value of the utility function shown by (28) is almost the same as that of the expected utility of \(E\left[ {u\left( {\mathbf{Q}} \right)} \right] = E\left[ {\sum\nolimits_{i} {b_{i} \cdot \ln \left( {1 + Q_{i} } \right)} } \right]\) in the vicinity of \(q_{i} = q_{i}^{ * }\).

## Rights and permissions

## About this article

### Cite this article

Kato, T., Uchida, K., Lam, W.H.K. *et al.* Estimation of the value of travel time and of travel time reliability for heterogeneous drivers in a road network.
*Transportation* **48**, 1639–1670 (2021). https://doi.org/10.1007/s11116-020-10107-x

Published:

Issue Date:

DOI: https://doi.org/10.1007/s11116-020-10107-x