A new day-to-day dynamic network vulnerability analysis approach with Weibit-based route adjustment process

Highlights

• Develop a new day-to-day dynamic network vulnerability analysis approach.

• Consider day-to-day network performance fluctuation based on a new day-to-day model.

• Use the Weibit model to capture travelers’ subjective perception error uncertainty.

• Use the mean-excess travel time measure to capture objective travel time uncertainty.

• Identify the critical bridges in the Winnipeg network.

Abstract

The disruption of critical components in a transportation network can bring about severe network performance degradation and requires a relatively long period to recover, which would lead to commuters’ day-to-day route choice adjustment. Under disruptions, there would be greater travel time variability (objective uncertainty) and travelers’ perception error uncertainty (subjective uncertainty) in the transportation network. However, no vulnerability analysis method in the literature can consider the day-to-day network performance fluctuation under uncertainties. In this paper, we develop a new day-to-day dynamic network vulnerability analysis approach that allows the consideration of day-to-day network performance fluctuation based on a new day-to-day dynamic model considering both objective travel time uncertainty and subjective perception error uncertainty. Compared to most existing day-to-day models that either adopt User Equilibrium (UE) or Logit-based route choice criterion, the new day-to-day model has two advantages: (1) the Weibit model is used to capture travelers’ subjective perception error uncertainty, which does not have the perfect information assumption in the UE model, or the identically distributed perception error assumption in the Logit model; and (2) the mean-excess travel time (METT) concept is used to capture the objective travel time uncertainty, which handles the inconsideration of travel time variability in most day-to-day models while remaining computational tractability. Based on the proposed day-to-day dynamic model, we develop a new component importance metric for network vulnerability analysis. This new metric characterizes the post-disruption day-dependent consequences to alleviate the limitation of only assessing the final static equilibrium consequence as in the existing studies of vulnerability analysis. Numerical examples are provided to demonstrate the features of the proposed day-to-day dynamic model and the new component importance metric, as well as their applicability in identifying the critical bridges in the Winnipeg network. The proposed approach provides a new decision support tool for planners and managers in assessing the consequences of disruptions, identifying the critical components, and determining the recovery schedules after disruptions.

Introduction

• Transportation network vulnerability analysis

Transportation system is one of the important lifelines that are vital to people’s health, safety, comfort, and economic activities. In light of the recent man-made and natural disasters frequently occurred in the world, disruptions to the transportation system can seriously damage the economic productivity of the society and cause inconvenience to peoples’ daily life. These man-made and natural disasters have emphasized the multi-faceted importance and vulnerability of the transportation networks to the society, and the need for government agencies and communities to prepare strategies to identify and strengthen the weakness of transportation networks to withstand losses caused by these disruptive events.

Vulnerability analysis is a powerful tool in identifying the critical components in transportation networks. Among the abundant studies on transportation network vulnerability analysis, the two widely cited definitions of vulnerability come from Berdica, 2002, D'Este and Taylor, 2003. Berdica (2002) defined vulnerability as the susceptibility of a system to threats and incidents that result in operational degradation, while D'Este and Taylor (2003) used the notion of accessibility to define that a network link is vulnerable if the loss of the link significantly diminishes the accessibility of the network. According to Taylor (2017), the main task of vulnerability analysis is to determine the network components (e.g., nodes and links), whose failures or degradations could significantly affect travelers’ behaviors and network performance. Therefore, vulnerability analysis is also known as the identification of the weakest/critical/important/vulnerable components. This identification task has wide applications in both the pre-disaster planning and post-disaster management (e.g., targeted protection or retrofitting, strategic location of rapid response and repair stations, evacuation route planning, and evacuation network monitoring) to ensure that the critical components of the network are adequately monitored and protected.

Transportation network vulnerability analysis has received a great deal of attention in the past decade. Different perspectives, measures, and evaluation methods have been proposed to study the transportation network vulnerability (see, e.g., Chen et al., 2007a, Chen et al., 2007b, Murray and Grubesic, 2007, Nagurney and Qiang, 2010, Luathep et al., 2011, Chen et al., 2012, Ho et al., 2013, El-Rashidy and Grant-Muller, 2014, Gedik et al., 2014, Jenelius and Mattsson, 2015, Reggiani et al., 2015, Oliveira et al., 2016, Wang et al., 2016a, Wang et al., 2016b, Bell et al., 2017, Calatayud et al., 2017, Bababeik et al., 2018, Haghighi et al., 2018, Garcia-Palomares et al., 2018, Xu et al., 2018b, Gu et al., 2020). As reviewed by Mattsson and Jenelius, 2015, Xu et al., 2018b, most existing methods of transportation network vulnerability analysis are based on enumerating/generating/simulating disruption scenarios and then evaluating the network performance based on some performance measures (e.g., origin–destination (O-D) connectivity, total system cost, efficiency, and accessibility). The most widely used model in vulnerability analysis is the classical (static) user equilibrium (UE) traffic assignment model. To the best knowledge of the authors, very few studies used within-day or day-to-day dynamic traffic assignment models in the network vulnerability analysis. Jenelius, 2007, Cats and Jenelius, 2014 considered the dynamics and information effects in the vulnerability analysis of road networks and transit networks, respectively. The “dynamics” considered in their studies belong to the within-day or short-term dynamics, which may be more applicable to the disruptions whose impact period and the subsequent recovery process are within a few hours. However, many disruptions in the transportation network are usually associated with severe infrastructure damages and a relatively long-term reconstruction process (from a few weeks to a few months). Such disruptions and reconstructions will cause the travelers’ day-to-day route or mode choice adjustment afterwards. For example, on May 23, 2016, a truck accident damaged a four-lane segment in the Middle-Ring elevated expressway of Shanghai and also blocked its corresponding surface roads. The reconstruction lasted 14 days. This expressway segment accommodated daily traffic of more than 61,000 vehicles, whose route choices had to be adjusted after this truck accident (Tian and Chen, 2019). Empirical studies have also revealed that the traffic flow conditions of many elevated expressways and surface roads fluctuated dramatically during these 14 days and also afterwards (e.g., Li, 2018, Wang, 2019). Hence, it is important and necessary to explicitly model the day-to-day route choice adjustment in the transportation network vulnerability analysis.

Other than the characteristics of the considered disruption mentioned above, the widely-used static UE traffic assignment has the following drawbacks in the transportation network vulnerability analysis: (1) The existence or attainability of this theoretically ideal UE state has been questioned by many studies (e.g., Iida et al., 1992, Friesz et al., 1994). Instead, the network flows evolve from one realizable disequilibrium state to another; and (2) Managers may activate some recovery strategies far earlier than the long-term “final equilibrium” state after the disruption, which means that it is more meaningful to understand the evolution process rather than just the “final equilibrium” state. Ignoring the post-disaster network flow evolution process may underestimate the negative consequences, and potentially lead to bias or cost-ineffectiveness in the recovery strategy decision-making as well as the inability of differentiating the impact of various recovery schedules.

• Day-to-day dynamic traffic assignment

As mentioned above, the disruption of a network component is likely to cause fluctuations of the transportation network states (including both supply and demand) and change commuters’ route choice decisions from one day to another. To evaluate the system-wide performance after the disruption, an appropriate day-to-day dynamic traffic assignment model is needed to predict the network flow evolution.

Day-to-day dynamic traffic assignment models have been extensively studied in the literature. They can be classified into different categories according to different dimensions, such as discrete-time models versus continuous-time models, link-based models versus path-based models, and deterministic models versus stochastic models. For a more comprehensive review on day-to-day models, readers may refer to Cantarella and Watling, 2016, Zhou et al., 2017.

In this paper, we classify the current day-to-day models according to two dimensions: route choice criterion and uncertainty consideration (i.e., subjective and objective uncertainty). Herein the subjective uncertainty is referred to as the travelers’ perception error due to their imperfect knowledge about the network states. The objective uncertainty is referred to as the travel time variability that widely exists in transportation systems. In reality, both subjective uncertainty and objective uncertainty affect the travelers’ route choice decisions, especially after significant disruptions. Table 1 provides a selective summary of the existing models.

From Table 1, we can see that the existing studies mainly adopted either deterministic or Logit-based stochastic UE (SUE) route choice models. Also, most of them usually assumed that the link travel times on each day are deterministic without uncertainty. On the one hand, the unrealistic perfect information assumption of the deterministic UE model and the independently and identically distributed (IID) perception error assumption in the Logit-based SUE model, have been widely recognized and criticized by both researchers and practitioners. On the other hand, these models ignore the travel time variability that widely exists in daily travel. Many factors can contribute to the travel time variability, e.g., adverse weather, traffic incidents and accidents, work zone, special events, infrastructure disruption, etc. Empirical studies (e.g., Abdel-Aty et al., 1995) have revealed that travel time variability plays an important role in travelers’ route choice decisions, and it is either the most or second most important factor for most commuters. Very few studies have considered both objective and subjective uncertainties simultaneously in the day-to-day dynamic models.

Table 1 also shows that some limited studies indeed have developed stochastic day-to-day models to incorporate different sources of stochasticity, including the objective uncertainty such as demand or capacity uncertainty, and the subjective uncertainty such as random route choice behavior. Compared to deterministic models that assume the travelers’ route choice mechanism per day is determined in advance, stochastic models consider uncertainty in travelers’ route choice decisions and can provide the probability distribution of flow states (Watling and Cantarella, 2015, Cantarella and Watling, 2016, Hazelton and Parry, 2016). Although stochastic models seem to be more general than deterministic models, the computational burden may hinder their applications in large-scale transportation networks. For example, Cantarella and Watling (2016) proposed a stochastic day-to-day model that considers Logit-based route choice and demand fluctuation, which was solved by Monte Carlo techniques due to the multinomial distribution of route flows. In addition, they still inherited the IID assumption of perception error due to the Logit-based route choice model. Note that the day-to-day dynamic network vulnerability analysis addressed in this study needs to run a day-to-day dynamic model for each generated/enumerated disruption scenario.

Hence, to support the day-to-day dynamic network vulnerability analysis, it is necessary to develop a new day-to-day dynamic model that can reasonably capture the effects of both subjective perception error uncertainty and objective travel time uncertainty on the adjustment of travelers’ route choice, while having computational tractability in large-scale networks.

Motivated by the above observations, this paper develops a new network vulnerability analysis method based on a day-to-day dynamic model under uncertainties. The proposed method includes the following two parts.

Part I- Modeling day-to-day dynamics under uncertainties: We explicitly model two types of uncertainties simultaneously in the day-to-day dynamic model: objective uncertainty due to travel time variability, and subjective uncertainty due to the travelers’ imperfect knowledge about the network states. For the objective uncertainty, we use the mean-excess travel time (METT) concept (Chen and Zhou, 2010) as the travel cost to capture travelers’ risk-averseness against travel time variability. For the subjective uncertainty, we provide a new exploration using an advanced discrete choice model (i.e., the Weibit model with the Weibull perception error distribution) to alleviate the drawbacks of the Logit model. With the Weibit model, we can capture the route-specific perception variance in the route adjustment process, rather than assuming a fixed and constant perception variance among different routes as in the Logit model. Then, the objective uncertainty and subjective uncertainty are integrated into the day-to-day modeling framework. Mathematical properties of the proposed model are also examined.

Part II- Day-to-day dynamic network vulnerability analysis: In evaluating the consequence of disruption scenarios, the existing (static) vulnerability analysis methods based on the static UE or Logit-SUE model only consider the consequence at the final equilibrium state, thus leading to the abovementioned drawbacks. To alleviate these drawbacks, we apply the proposed day-to-day dynamic model to develop a new metric of component importance for network-wide vulnerability analysis. Methodologically, the area circled by the network performance evolution curve derived by the proposed model and the pre-disruption normal network state is quantified. This new metric allows the consideration of both the performance dimension and time dimension, i.e., day-dependent consequence rather than the final equilibrium state. A case study in the realistic Winnipeg network is conducted to demonstrate the advantages of the proposed new metric in identifying the critical bridges.

In summary, the main contributions of this paper are twofold: (1) the development of a day-to-day dynamic model considering both subjective perception error uncertainty (via the Weibit model) and objective travel time uncertainty (via the METT concept) to alleviate the drawbacks of the current day-to-day models that mostly adopt UE or Logit-based route choice criterion as well as ignore the travel time variability, and (2) the development of a component importance metric for day-to-day dynamic network-wide vulnerability analysis by characterizing the post-disruption day-dependent consequence to alleviate the limitations of only considering the final static equilibrium consequence as in the existing studies of vulnerability analysis.

The remainder of this paper is organized as follows. Section 2 presents the day-to-day dynamic model under uncertainties. Section 3 provides a new importance measure for network vulnerability analysis. Section 4 uses numerical examples to demonstrate the features of the proposed model and the new importance measure for network vulnerability analysis. Finally, conclusions and future research directions are summarized in Section 5. The notations and their explanations in this paper are summarized in Table D.1 in the Appendix D.