Abstract
In this paper, we propose a novel nonexpected route travel time (NERTT) model, which belong to the rankdependent expected utility model. The NERTT consists of two parts, which are the route travel time distribution and the distortion function. With the strictly increasing and strictly concave distortion function, we can prove that the route travel time in the proposed model is riskaverse, which is the main focus of this paper. We show two different reduction methods from the NERTT model to the travel time budget model and meanexcess travel time model. One method is based on the properly selected distortion functions and the other one is based on a general distortion function. Besides, the behavioral inconsistency of the expected utility model in the route choice can be overcome with the proposed model. The NERTT model can also be generalized to the nonexpected disutility (NED) model, and some relationship between the NED model and the route choice model based on the cumulative prospect theory can be shown. This indicates that the proposed model has some generality. Finally, we develop a nonexpected riskaverse user equilibrium model and formulate it as a variational inequality (VI) problem. A heuristic gradient projection algorithm with column generation is used to solve the VI. The proposed model and algorithm are tested on some hypothetical traffic networks and on some largescale traffic networks.
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Notes
In the main body of the paper, we focus on the modeling of the riskaverse route travel time, while the CPTbased route choice model does not belong to the riskaverse analysis merely ( Xu et al. 2011a, b). Therefore, we postpone the analysis of the NED and its relationship to the CPTbased route choice model in the appendix, which can demonstrate the generality of the proposed model.
In this paper, we assume the probability distribution of the route travel time is continuous.
Note here that the utility can be in the form of negative route travel time.
The Gauss error function is defined as \( erf(x)=\frac{2}{\sqrt{\pi}}{\int}_0^x \exp \left({t}^2\right) dt \).
In this simple case, the value of parameter α in the Wang function is 2.
In this paper, we use this simple example to demonstrate the rightness of the proposed propositions. In practice, one can estimate the route travel time distribution (e.g., Westgate et al. 2016).
The convergence rate is 10^{−7}, which is achieved in 15 iterations, and we only show the three iterations for a better presentation.
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Acknowledgements
The authors thank the anonymous reviewers for their insightful comments and suggestions that significantly improved the quality of earlier versions of the paper. This study is supported by the National Basic Research Program of China (973 ProgramNo. 2012CB725405). The first author’s work is also supported by the Fundamental Research Funds for the Central Universities (Grant No. CXZZ13_0116) and by the Scientific Research Foundations of Graduate School of Southeast University (Grant No. YBJJ1344). The China Scholarship Council (CSC) provided a scholarship that supports the first author to visit Rensselaer Polytechnic Institute from 2014 to 2016. The second author is partially supported by the National Science Foundation (NSF) grant CMMI1055555 and The Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning via a grant to the Shanghai Maritime University. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the NSF or NSFC.
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Appendix
Appendix
1.1 Generalization to the disutility function
In Section 3, we discussed the NERTT model and showed the relationship of the NERTT and TTB/METT. In this appendix, we show that the NERTT can be generalized to embed a disutility function, i.e., the NED model. Note here that we can also generalize the NERTT by embed the utility function, because utility is the negative disutility, and the following discussions can be conducted in a similar way. In fact, in Appendix A.2, the relationship between the NED and the CPTbased route choice model is shown by embedding the utility function. In fact, different disutility functions can be adopted in the NED model and the relationship between the NED model and the CPTbased route choice model can be derived in a special case, which further demonstrates the generality of the NED model.

Definition A.1 (Nonexpected disutility). The nonexpected disutility is the expectation of the random disutility under an appropriate distortion of the original distribution, i.e., \( {\rho}_g\left( d(T)\right)={\int}_0^{\infty } g\left( D\left( d(t)\right)\right) dt \), where d(t) is the disutility function.
The property of the NED model can also be shown in the similar way as shown in Proposition 1. Mirchandani and Soroush (1987) introduced two different disutility functions and we summarize them as following.^{Footnote 8}

Definition A.2 (Linear disutility function, LDF). For the linear disutility function d _{ LDF }, d _{ LDF }(x) = β _{0} + β _{1} x, β _{1} > 0.

Definition A.3 (Exponential disutility function, EDF). For the exponential disutility function d _{ EDF }, d _{ EDF } = β _{1} exp (β _{2} x + β _{0}), β _{1} > 0, and β _{2} > 0.
Here we can find the analytical form of the NED with LDF and EDF under the normally distributed route travel time, and the analytical form of NED with LDF under the lognormally distributed route travel time. However, we cannot find the analytical form of the NED with EDF under the lognormally distributed route travel time and other NEDs with other probability distributions, which can be obtained with the MonteCarlo simulation. Therefore, the following three propositions can be reached.

Proposition A.1. If the route travel time follows the normal distribution, the NED with LDF is β _{1}(μ + ασ) + β _{0}.
Proof
If we choose the LDF, we can obtain the following Eq. (11),
Substitute Eq. (11) into the Wang function, we can get \( \begin{array}{l} g\left[{D}_{u(T)}(t)\right]=\Phi \left[{\Phi}^{1}\left(\Phi \left[\frac{\mu \frac{t{\beta}_0}{\beta_1}}{\sigma}\right]\right)+\alpha \right]=\Phi \left[\frac{\mu \frac{t{\beta}_0}{\beta_1}}{\sigma}+\alpha \right]=\Phi \left[\frac{\mu \frac{t{\beta}_0}{\beta_1}+\alpha \sigma}{\sigma}\right]\\ {}=1\Phi \left[\frac{\frac{t{\beta}_0}{\beta_1}\left(\mu +\alpha \sigma \right)}{\sigma}\right]={D}_Y(t)\end{array} \).
Therefore, \( \frac{Y{\beta}_0}{\beta_1}\sim N\left(\mu +\alpha \sigma, {\sigma}^2\right) \), and \( Y\sim N\left({\beta}_1\left(\mu +\alpha \sigma \right)+{\beta}_0,{\beta}_1^2{\sigma}^2\right) \). Thus,\( {E}_g\left( u(T)\right)={\int}_0^{+\infty }{D}_Y(t) dt= E(Y)={\beta}_1\left(\mu +\alpha \sigma \right)+{\beta}_0 \), which complete the proof.

Proposition A.2. If the route travel time follows the normal distribution, the NED with EDF is \( {\beta}_1 \exp \left({\beta}_2\left(\mu +\alpha \sigma \right)+{\beta}_0+\frac{\beta_2^2{\sigma}^2}{2}\right). \)
Proof
If we choose the EDF, we can obtain the following Eq. (12),
Substitute Eq. (12) into the Wang function, we can get \( \begin{array}{l} g\left[{D}_{u(T)}(t)\right]=\Phi \left[{\Phi}^{1}\left(\Phi \left[\frac{\mu \frac{ \ln \left(\frac{t}{\beta_1}\right){\beta}_0}{\beta_2}}{\sigma}\right]\right)+\alpha \right]=\Phi \left[\frac{\mu \frac{ \ln \left(\frac{t}{\beta_1}\right){\beta}_0}{\beta_2}}{\sigma}+\alpha \right]\\ {}=\Phi \left[\frac{\mu \frac{ \ln \left(\frac{t}{\beta_1}\right){\beta}_0}{\beta_2}+\alpha \sigma}{\sigma}\right]=1\Phi \left[\frac{\frac{ \ln \left(\frac{t}{\beta_1}\right){\beta}_0}{\beta_2}\left(\mu +\alpha \sigma \right)}{\sigma}\right]={D}_Y(t)\end{array} \).
Therefore, \( \frac{ \ln \left(\frac{Y}{\beta_1}\right){\beta}_0}{\beta_2}\sim N\left(\mu +\alpha \sigma, {\sigma}^2\right) \), and \( \ln \left(\frac{Y}{\beta_1}\right)\sim N\left({\beta}_2\left(\mu +\alpha \sigma \right)+{\beta}_0,{\beta}_2^2{\sigma}^2\right) \). Thus, \( {E}_g\left( u(T)\right)={\int}_0^{+\infty }{D}_Y(t) dt= E(Y)={\beta}_1 E\left(\frac{Y}{\beta_1}\right)={\beta}_1 \exp \left({\beta}_2\left(\mu +\alpha \sigma \right)+{\beta}_0+\frac{\beta_2^2{\sigma}^2}{2}\right) \), which complete the proof.

Proposition A.3. If the route travel time follows the lognormal distribution, the NED with LDF is \( {\beta}_1 \exp \left(\mu +\alpha \sigma +\frac{\sigma^2}{2}\right)+{\beta}_0. \)
Proof
If we choose the EDF, we can obtain the following Eq. (13),
Substitute Eq. (13) into the Wang function, we can get \( \begin{array}{l} g\left[{D}_{u(T)}(t)\right]=\Phi \left[{\Phi}^{1}\left(\Phi \left[\frac{\mu  \ln \left(\frac{t{\beta}_0}{\beta_1}\right)}{\sigma}\right]\right)+\alpha \right]=\Phi \left[\frac{\mu  \ln \left(\frac{t{\beta}_0}{\beta_1}\right)}{\sigma}+\alpha \right]\\ {}=\Phi \left[\frac{\mu  \ln \left(\frac{t{\beta}_0}{\beta_1}\right)+\alpha \sigma}{\sigma}\right]=1\Phi \left[\frac{ \ln \left(\frac{t{\beta}_0}{\beta_1}\right)\left(\mu +\alpha \sigma \right)}{\sigma}\right]={D}_Y(t)\\ {}\end{array} \).
Therefore, \( \ln \left(\frac{Y{\beta}_0}{\beta_1}\right)\sim N\left(\mu +\alpha \sigma, {\sigma}^2\right) \). Thus, we can get \( {E}_g\left( u(T)\right)={\int}_0^{+\infty }{D}_Y(t) dt= E(Y)={\beta}_1 E\left(\frac{Y{\beta}_0}{\beta_1}\right)+{\beta}_0={\beta}_1 \exp \left(\mu +\alpha \sigma +\frac{\sigma^2}{2}\right)+{\beta}_0 \), which complete the proof.
1.2 Relationship between the NED and the CPTbased route choice model
In this subsection, we continue to show the relationship between the NED model and the CPTbased route choice model.

Proposition A.4. Given a reference point t _{0}, \( {\rho}_g\left( d(T)\right)={\int}_0^{\infty } g\left( D\left( d(t)\right)\right) dt \) can be written as \( {\rho}_g\left( d(T)\right)={\int}_0^{t_0} g\left( D\left( d(t)\right)\right) dt+{\int}_{t_0}^{+\infty } g\left( D\left( d(t)\right)\right) dt \). If the distortion function \( g(x)=\frac{x^{\gamma}}{{\left({x}^{\gamma}+{\left(1 x\right)}^{\gamma}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$\gamma $}\right.}} \) and the utility function \( d(t)=\left\{\begin{array}{cc}\hfill {\left({t}_0 t\right)}^{\beta_0}\hfill & \hfill t\le {t}_0\hfill \\ {}\hfill {\beta}_1{\left( t{t}_0\right)}^{\beta_2}\hfill & \hfill t>{t}_0\hfill \end{array}\right. \) are adopted, we can get the CPTbased route choice model in Xu et al. (2011a).
Proof of proposition A.4 can be found in Tversky and Kahneman (1992) and Connors and Sumalee (2009), which is omitted here. From proposition A.4, we can see that given the reference point, the NED model is decomposed into the sum of two NED models restricted on a smaller travel time interval. This helps to show the relationship between NED model and the CPTbased route choice model. If there is no reference point and the disutility function is d(t) = t, the difference between the (special) CPTbased route choice model and the NERTT model is the distortion function, which shows travelers’ decision weights for the probabilities.
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Ji, X., Ban, X.(., Li, M. et al. Nonexpected Route Choice Model under Risk on Stochastic Traffic Networks. Netw Spat Econ 17, 777–807 (2017). https://doi.org/10.1007/s1106701793443
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DOI: https://doi.org/10.1007/s1106701793443