Assessing the effects of stochastic perception error under travel time variability

Abstract

Perceived mean-excess travel time is a new risk-averse route choice criterion recently proposed to simultaneously consider both stochastic perception error and travel time variability when making route choice decisions under uncertainty. The stochastic perception error is conditionally dependent on the actual travel time distribution, which is different from the deterministic perception error used in the traditional logit model. In this paper, we investigate the effects of stochastic perception error at three levels: (1) individual perceived travel time distribution and its connection to the classification by types of travelers and trip purposes, (2) route choice decisions (in terms of equilibrium flows and perceived mean-excess travel times), and (3) network performance measure (in terms of the total travel time distribution and its statistics). In all three levels, a curve fitting method is adopted to estimate the whole distribution of interest. Numerical examples are also provided to illustrate and visualize the above analyses. The graphical illustrations allow for intuitive interpretation of the effects of stochastic perception error at different levels. The analysis results could enhance the understanding of route choice behaviors under both (subjective) stochastic perception error and (objective) travel time uncertainty. Some suggestions are also provided for behavior data collection and behavioral modeling.

Appendices

Appendix 1

This appendix provides the equivalent variational inequality (VI) formulation of the SMETE conditions. First, a feasible route flow pattern f should satisfy the following basic conditions:

$$ \sum\limits_{{p \in P^{w} }} {f_{p}^{w} } = q^{w} ,\;\;\forall w \in W, $$

(12)

$$ v_{a} = \sum\limits_{w \in W} {\sum\limits_{{p \in P^{w} }} {f_{p}^{w} \delta_{pa}^{w} } } ,\;\;\forall a \in A, $$

(13)

$$ f_{p}^{w} \ge 0,\;\;\forall p \in P^{w} ,\;w \in W, $$

(14)

where v _a is the flow on link a; $\delta_{pa}^{w}$=1 if route p connecting O–D pair w uses link a, and 0, otherwise. Eq. (12) is the travel demand conservation constraint; Eq. (13) is a definitional constraint that sums up all route flows that pass through a given link; and Eq. (14) is a non-negativity constraint on the route flows. In the following, we use Ω to denote the constraint set that consists of Eqs. (12)–(14).

The SMETE conditions can be equivalently formulated as the following variational inequality (VI) problem, which is to find a route flow pattern ${\mathbf{f}}^{*} \in \Upomega$, such that

$$ {\tilde{\varvec{\eta }}}\left( {{\mathbf{f}}^{*} } \right)^{T}\left( {{\mathbf{f}} - {\mathbf{f}}^{*} } \right) \ge 0,\;\;\forall {\mathbf{f}} \in \Upomega $$

(15)

For the existence of the equilibrium route flow pattern and the equivalence between the SMETE conditions and the solution to the above VI problem, interested readers may refer to Chen and Zhou (2009) and Chen et al. (2011).

Appendix 2

This appendix provides a detailed derivation of the probability distribution statistics for the lognormal distributed travel demand uncertainty. Chen and Zhou (2009) assumed the uncertainty only comes from the random free-flow travel time. As is well-known, day-to-day demand fluctuation and link capacity degradation are two main uncertainty sources in the transportation system. Also, most of the recent literature considered either demand uncertainty (e.g., Chen and Zhou 2010; Chen et al. 2011; Shao et al. 2006a, b) or capacity uncertainty (e.g., Lo et al. 2006; Siu and Lo 2006), or both (e.g., Lam et al. 2008; Shao et al. 2008; Siu and Lo 2008) in the traffic equilibrium models under uncertainty. In this study, we consider the day-to-day demand fluctuation as a representative uncertainty source. Following Zhou and Chen (2008), the lognormal distribution is adopted to characterize the demand uncertainty. It is a nonnegative, asymmetrical distribution and has already been adopted as a more realistic approximation of the fluctuated travel demand to investigate the uncertainty propagation in the four-step travel demand forecasting procedure (Zhao and Kockelman 2002).

The mean and variance of the lognormal distributed travel demand between a generic O–D pair w are denoted as q ^w and ε ^w, respectively. Consider the commonly used BPR (Bureau of Public Roads) function:

$$ T_{a} = t_{a}^{0} \left[ {1 + \theta \left( {\frac{{V_{a} }}{{C_{a} }}} \right)^{\beta } } \right], $$

(16)

where T _a and V _a are the random link travel time and flow, respectively; $t_{a}^{0}$ and C _a are, respectively, the deterministic link free-flow travel time and capacity; θ and β are BPR parameters. Then, the probability distribution or statistics of flow and travel time variables can be derived as shown in Table 3. One can see the actual and perceived travel time distributions are explicitly derived from the travel demand and SPE distributions. This approach is different from the empirical studies on the perception of travel time unreliability using stated preference surveys (e.g., Tseng et al. 2009).

Table 3 Statistics of flow and travel time variables

Using the Cornish and Fisher (1937)’s asymptotic expansion, we can estimate the PTTB at a user-specified confidence level α as

$$ \tilde{\xi }_{p}^{w} \left( \alpha \right) \approx Cum\left[ {\left( {\tilde{C}_{p}^{w} } \right)^{1} } \right] + \psi_{p}^{w} \left( \alpha \right) \cdot \sqrt {Cum\left[ {\left( {\tilde{C}_{p}^{w} } \right)^{2} } \right]} , $$

(17)

where $\psi_{p}^{w} \left( \alpha \right)$ is related to the skewness and kurtosis of the perceived TTD. It can be calculated as follows:

$$ \begin{aligned} \psi_{p}^{w} \left( \alpha \right) = & \Upphi^{ - 1} \left( \alpha \right) + \left( {1/6} \right)\left[ {\left( {\Upphi^{ - 1} \left( \alpha \right)} \right)^{2} - 1} \right] \cdot S\left[ {\tilde{C}_{p}^{w} } \right]\\ & + \left( {1/24} \right)\left[ {\left( {\Upphi^{ - 1} \left( \alpha \right)} \right)^{3} - 3\Upphi^{ - 1} \left( \alpha \right)} \right] \cdot K\left[ {\tilde{C}_{p}^{w} } \right]\\ & - \left( {1/36} \right)\left[ {2\left( {\Upphi^{ - 1} \left( \alpha \right)} \right)^{3} - 5\Upphi^{ - 1} \left( \alpha \right)} \right] \cdot \left( {S\left[ {\tilde{C}_{p}^{w} } \right]} \right)^{2} , \end{aligned} $$

(18)

where $\Upphi^{ - 1} \left( \cdot \right)$ is the inverse of the standard normal cumulative distribution function (CDF); $S\left[ {\tilde{C}_{p}^{w} } \right]$ and $K\left[ {\tilde{C}_{p}^{w} } \right]$ are the skewness and kurtosis of $\tilde{C}_{p}^{w}$ for quantifying the asymmetry and peakedness of the probability distribution of $\tilde{C}_{p}^{w}$, respectively.