Journal of Advanced Transportation

Volume 2018, Article ID 5294185, 13 pages

https://doi.org/10.1155/2018/5294185

## A Reliability-Based Network Equilibrium Model with Adaptive Risk-Averse Travelers

^{1}College of Architecture and Civil Engineering, Beijing University of Technology, Beijing 100124, China^{2}Beijing Key Laboratory of Transportation Engineering, Beijing 100124, China

Correspondence should be addressed to Lei Zhao; moc.361@5021oahziel

Received 28 October 2018; Accepted 28 November 2018; Published 27 December 2018

Academic Editor: Eneko Osaba

Copyright © 2018 Lei Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, route free-flow travel time is taken as the lower bound of route travel time to examine its impacts on budget time and reliability for degradable transportation networks. A truncated probability density distribution with respect to route travel time is proposed and the corresponding travel time budget (TTB) model is derived. The budget time and reliability are compared between TTB models with and without truncated travel time distribution. Under truncated travel time distribution, the risk-averse levels of travelers are adaptive, which are affected by the characteristics of the used routes besides the confidence level of travelers. Then, a TTB-based stochastic user equilibrium (SUE) is developed to model travelers’ route choice behavior. Moreover, its equivalent variational inequality (VI) problem is formulated and a route-based algorithm is used to solve the proposed model. Numerical results indicate that route travel time boundary produces a great influence on decision cost and route choice behavior of travelers.

#### 1. Introduction

Most studies indicate that travel time reliability (TTR) is an essential travel decision cost of travelers [1–3]. In particular, Abdel-Aty et al. (1977) pointed out that TTR was one of the major elements influencing commuters' route choice considerations [4]. However, the classical user equilibrium (UE) principle assumes that travelers are risk-neutral and their decision behavior only depends on the mean travel time in deterministic transportation networks. Obviously, it leaves the travel time uncertainty out of consideration and neglects the risk preferences of travelers (i.e., risk aversion and risk seeking). Meanwhile, empirical studies demonstrated that majority of travelers are risk-neutral and they would like to take an additional payment to avoid the congestion and travel risk [5, 6].

In transportation systems, uncertainty is unavoidable, which mainly derives from roadway capacity variation and travel demand fluctuation. Both supply and demand variations bring about travel time uncertainty. In order to model travelers’ route choice behavior under uncertainty, a variety of traffic equilibrium models incorporating TTR are developed [7–11]. Typically, Lo et al. (2006) proposed the within budget time reliability (WBTR) model [12]. The notion of travel time budget (TTB) in WBTR model is given to represent the decision cost of travelers. The TTB refers to the sum of the mean travel time and a safety margin to arrive on time. The standard deviation of route travel time is used to measure the risk of route choice. However, the WBTR model disregards the effects of unreliable side of route travel time on route choice. By incorporating simultaneously the reliable and unreliable sides, the mean-excess traffic equilibrium (METE) model was developed by Chen and Zhou (2010) [13]. In the METE model, travelers are assumed to use such a route to make TTR ensured and unreliable impacts can be minimized. Subsequently, Chen et al. (2011) further introduced the perception errors of travelers to the METE model and proposed the stochastic mean-excess traffic equilibrium (SMETE) model [14]. In this case, the perceived travel time distribution is derived to model travelers' route choice behavior rather than the actual one. Different from the previous studies, Watling (2006) considered travelers' schedule delay and proposed a late arrive penalized user equilibrium (LAPUE) model [15]. Recently, the multiobjective equilibrium problems have been intensively investigated [16–18]. In this regard, Wang et al. (2014) provided a biobjective user equilibrium (BUE) condition, which is the state that no traveler can improve either his/her expected travel time or risk or both without worsening the other by unilaterally changing routes [17].

It can be found that most of aforementioned traffic equilibrium models with the TTR are based on the degradable transportation networks. As mentioned above, travel demand fluctuation would also lead to travel time uncertainty. Shao et al. (2006) developed the UE model with demand uncertainty [19]. Then, the model was further extended by considering travelers’ perception errors and heterogeneity, and the reliability stochastic user equilibrium (RSUE) model was developed [20]. Lam et al. (2008) assumed the uncertainties mainly due to adverse weather condition and proposed a traffic equilibrium model with doubly uncertainties in supply and demand sides [21]. Zhou and Chen (2008) compared and analyzed the equilibrium results of three UE models with stochastic travel demand, using a simple network to illustrate the differences, and examined how these models address the travel risk [22]. Siu and Lo (2008) formulated a multiclass equilibrium model, in which the demand composes of infrequent travelers and commuters [23]. Sun and Gao (2012) assumed the probability distributions of travel demand and link capacities are unknown and developed the robust traffic equilibrium model [24]. In addition, considered uncertainties in demand and supply are also extended to the network design problems (NDPs) to develop the robust optimization models [25, 26].

In the above models, route travel time are commonly assumed to follow continuous normal distribution or lognormal distribution. However, these distributions imply that the lower boundary of route travel time approaches negative infinity or zero. To the best of our knowledge, route travel time is bounded and there is at least a lower boundary, namely, free-flow travel time. Yan et al. (2015) studied the effects of link speed limits on travelers’ route choice and the network performance [27]. This study indicated that the mean and variance of total travel time can be reduced by imposing speed limits, but the total TTB of a network increases. Xu et al. (2017) showed clearly that the minimum travel time equals the link length divided by the speed limit value after imposing a speed limit scheme [28]. They indicated that a speed limit scheme on uncertain road networks would affect the network flow reallocation. In the study, the minimum travel time is derived from speed limits. Nevertheless, the route travel time without a speed limit should also have a lower bound, and the travel time uncertain profile would also change.

This paper aims to examine the impacts of route travel time boundary on travel cost and route choice of travelers. To this end, we firstly propose the probability density function (PDF) with the route free-flow travel time as a truncation of original distribution. Subsequently, the TTB model with truncated travel time distribution is derived and compared with the original TTB model in terms of budget time and TTR. Then, the TTB-based SUE models are developed to model the impacts of route travel time boundary on travelers’ route choice.

The remainder of this paper is organized as follows. Section 2 presents the effect of route travel time boundary on PDF. Section 3 develops the TTB model with truncated travel time distribution under uncertainty. Section 4 gives the TTB-based SUE model. Then, we use numerical examples to illustrate the effect of route travel time boundary on network equilibrium in Section 5. Finally, Section 6 summarizes some concluding remarks.

#### 2. Effect of Route Travel Time Boundary on Uncertainty

In this section, the route free-flow time is considered as the lower bound of route travel time, and the effects of route travel time boundary on uncertainty are examined by proposing a truncated PDF of route travel time. The relationships and differences between TTB models with and without route travel time boundary are derived.

Let us consider a stochastic network , which is composed of nodes and directed links with capacity variations. Let be the free-flow time, the traffic flow, and the capacity of link , respectively. Then, link travel time can be determined by the bureau of public roads (BPR) link performance function, i.e.,where and are, respectively, the deterministic parameters of the BPR link performance function. Assume that link capacity follows an uniform distribution, , where is the design capacity associated with link and denotes the capacity degradable coefficient, . Further, we suppose that link capacity distributions are independent, and thus route travel time is normally distributed , based on the central limit theorem. The mean and standard deviation of route travel time are, respectively,where and are, respectively, the mean and standard deviation of link travel time. is a binary variable, which equals 1 when link is on route , and 0 otherwise. is a set of all routes between OD pair , and is a set of all OD pairs in the transportation network.

Combining the derivation of the mean and standard deviation of link travel time in [12], (2) and (3) can be rewritten as follows.

##### 2.1. Truncated Route Travel Time Distribution

This paper assumes that the route free-flow time is a determinate constant associated with specific route in the transportation network. Travel time uncertainty is only caused by the link capacity degradation. As derived above, route travel time is normally distributed under such case. A normal distribution implies that the lower bound or minimum route travel time is close to negative infinity. However, the minimum route travel time must be a positive constant, namely, the route free-flow travel time. In other words, the free-flow travel time is the lower bound of route travel time, and route travel time cannot be less than the free-flow travel time. More specifically, the probability of route travel time being less than the free-flow travel time should be zero. Hence, we develop the PDF with the free-flow travel time as a truncation of original distribution without the lower bound, i.e.,where and are the PDF and CDF of the original normal distribution associated with route travel time . is the route free-flow travel time and . and are the mean and standard deviation of route travel time, respectively.

For the ease of elaboration, a single-route network with the mean of 20 and the standard deviation of 5 is used. Assume the free-flow travel time is 15. The PDFs of route travel time with and without lower bound are shown in Figure 1. The x-axis shows the change in route travel time. The blue solid line is the PDF curve without the lower bound. The green dotted line represents the PDF curve with the free-flow travel time as a truncation of the original distribution. From this figure, one can see that the original distribution is truncated and becomes taller due to the appearance of fixed denominator. Moreover, the probability equals zero when route travel time with the lower bound is less than free-flow travel time. Obviously, the new PDF still fulfills the conservation property of a valid PDF according to (6). In addition, rather than truncated continuous distributions, other modeling ways can also be used to model the impacts of the lower bound of route travel time. Xu et al. (2018) elaborated the reality and feasibility of truncated travel time distribution and its mathematical tractability [28].