Estimation of the value of travel time and of travel time reliability for heterogeneous drivers in a road network

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.

Appendix

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

$$\sum\limits_{i} {E\left[ {b_{i} \cdot \ln \left( {1 + Q_{i} } \right)} \right]} = \sum\limits_{i} {b_{i} \cdot E\left[ {\ln \left( {1 + Q_{i} } \right)} \right]} \approx \sum\limits_{i} {b_{i} \cdot \left( {\ln \left( {1 + q_{i} } \right) - \frac{1}{2} \cdot \frac{{\left( {cv_{i} \cdot q_{i} } \right)^{2} }}{{\left( {1 + q_{i} } \right)^{2} }}} \right)} \,$$

(90)

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.

$$b_{i} = \frac{{\varphi_{i} + \pi_{i} + \theta_{i} }}{{\sum\nolimits_{k \in I} {\left( {\varphi_{k} + \pi_{k} + \theta_{k} } \right)} }} \, \forall i \in I$$

(91)

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

$$\begin{aligned} \alpha_{i} & = \frac{1}{{\ln \left( {1 + q_{i}^{ * } } \right)}} \cdot b_{i} \cdot \left( {\ln \left( {1 + q_{i}^{ * } } \right) - \frac{1}{2} \cdot \frac{{\left( {cv_{i} \cdot q_{i}^{ * } } \right)^{2} }}{{\left( {1 + q_{i}^{ * } } \right)^{2} }}} \right) \\ & = b_{i} \cdot \left( {1 - \frac{1}{2} \cdot \frac{{\left( {cv_{i} \cdot q_{i}^{ * } } \right)^{2} }}{{\left( {1 + q_{i}^{ * } } \right)^{2} \cdot \ln \left( {1 + q_{i}^{ * } } \right)}}} \right)\quad \forall i \in {\text{I}} \\ \end{aligned}$$

(92)

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 $q_{i} = q_{i}^{ * }$.

$$\begin{aligned} \alpha_{i} \cdot \ln \left( {1 + q_{i} } \right) & = b_{i} \cdot \left( {1 - \frac{1}{2} \cdot \frac{{\left( {cv_{i} \cdot q_{i}^{ * } } \right)^{2} }}{{\left( {1 + q_{i}^{ * } } \right)^{2} \cdot \ln \left( {1 + q_{i} } \right)}}} \right) \cdot \ln \left( {1 + q_{i} } \right) \\ & = b_{i} \cdot \left( {\ln \left( {1 + q_{i} } \right) - \frac{1}{2} \cdot \frac{{\left( {cv_{i} \cdot q_{i} } \right)^{2} }}{{\left( {1 + q_{i} } \right)^{2} }}} \right)\quad \forall i \in {\text{I}} \\ \end{aligned}$$

(93)

(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.

$$\frac{\partial }{{\partial q_{i} }} \cdot \left( {b_{i} \cdot \left( {1 - \frac{1}{2} \cdot \frac{{\left( {cv_{i} \cdot q_{i}^{ * } } \right)^{2} }}{{\left( {1 + q_{i}^{ * } } \right)^{2} \cdot \ln \left( {1 + q_{i}^{ * } } \right)}}} \right) \cdot \ln \left( {1 + q_{i} } \right)} \right) = b_{i} \cdot \left( {\frac{1}{{1 + q_{i} }} - \frac{{\left( {cv_{i} } \right)^{2} \cdot q_{i}^{ * } }}{{\left( {1 + q_{i}^{ * } } \right)^{2} \cdot \left( {1 + q_{i} } \right)}} \cdot \frac{{q_{i}^{ * } }}{{2 \cdot \ln \left( {1 + q_{i}^{ * } } \right)}}} \right)$$

(94)

$$\frac{\partial }{{\partial q_{i} }} \cdot \left( {b_{i} \cdot \left( {\ln \left( {1 + q_{i} } \right) - \frac{1}{2} \cdot \frac{{\left( {cv_{i} \cdot q_{i} } \right)^{2} }}{{\left( {1 + q_{i} } \right)^{2} }}} \right)} \right) = b_{i} \cdot \left( {\frac{1}{{1 + q_{i} }} - \frac{{\left( {cv_{i} } \right)^{2} \cdot q_{i} }}{{\left( {1 + q_{i} } \right)^{3} }}} \right)$$

(95)

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 $q_{i} = q_{i}^{ * }$.