Travel time resilience of roadway networks under disaster
Introduction
More than 4 million miles of U.S. public roads serve approximately 90% of passenger transport in the country (BTS, 2013). Natural and human-caused hazards threaten this roadway network and create the possibility for significant economic loss due to damage. Damage caused by Hurricane Irene to the Vermont transportation network amounted to $65 million (Lunderville, 2012). The collapse of the I-35 W Bridge over the Mississippi River interrupted more than 140,000 daily vehicular trips causing more than $0.4 million increase in daily passenger trip costs due to traffic rerouting (Zhu et al., 2010). The repair and reconstruction costs of transportation infrastructure systems after Hurricane Katrina were estimated to have exceeded $32 billion (Sundeen and Reed, 2006).
Transportation infrastructure systems are also attractive targets for malicious acts. Recent examples include bombings of passenger rail systems in London (2005), Madrid (2004), and Mumbai (2006). Since the 1990s, more than 25% of terrorist attacks have either targeted surface transportation systems or used them to provide access to other targets (Murray-Tuite, 2008). In addition to resulting physical damage, these events have long-term socio-economic and psychological impacts. Furthermore, they affect traveler decisions. Gordon et al. (2007) identified a 6% reduction in passenger trips and a sizable shift from public transit services to private automobiles during a two-year period following the 9/11 attack.
To prevent significant loss from disaster events, whether caused by a malicious act, an accident or technology failure, or nature, the transportation system must be resilient, and thus able to cope with disaster impact. This notion of resilience was initially conceptualized in relation to stability of ecological systems under a disruptive event (Holling, 1973). Holling included in the paper’s resilience definition the system’s ability to bounce back to a state of equilibrium post-event. This definition of resilience as including the system’s ability both to resist and adapt to disruptions has been applied in many settings since then. These settings include transportation and civil infrastructure systems. Quantitative approaches that adopt this definition of resilience in the context of transportation applications include: (Nair et al., 2010, Cox et al., 2011, Chen and Miller-Hooks, 2012, Faturechi et al., 2014).
Faturechi and Miller-Hooks (2014) provide a synthesis of approximately 200 works in the literature on resilience and other related measures, including coping capacity, robustness, flexibility and recovery, in the context of transportation. In addition to works that focus on resilience estimation, there are works that determine optimal pre-event mitigation or preparedness strategies (Losada et al., 2012), post-event response actions (Chen and Miller-Hooks, 2012, Vugrin et al., 2010) or both (Miller-Hooks et al., 2012) with the goal of maximizing resilience. A single paradigm for understanding and optimizing resilience and related measures that builds on the existence or nonexistence of possible actions that can be taken pre- or post-disaster is provided in (Faturechi and Miller-Hooks, 2014).
All prior works related to the maximization of resilience consider only applications in which resilience enhancing actions are chosen with the aim of achieving a system optimal solution. Such solutions inherently assume that the users of the system will follow these system optimal directives. For example, traffic might be centrally directed to use predetermined routes seeking a system optimum implementation. This is appropriate in many applications, such as in freight networks where the goods to be moved are not cognizant. Several relevant works involving network design under supply or demand uncertainty explicitly recognize the ability of people to make their own decisions regarding their path choice, often with the goal of selfishly maximizing their own utility functions. This is discussed in detail in (Nagurney and Qiang, 2012). These works generally involve a bilevel program structure, where design decisions, such as capacity expansion of a network link, are taken at the upper level, while the response of travelers to the supply offerings is assessed at the lower level. Supply uncertainty typically arises from day-to-day incidents, like traffic accidents, that may cause degradation in network performance. The impact of demand uncertainty is typically measured through variations in travel speeds and, thus, travel times. Chen et al. (2011) provide an extensive review of this literature.
A few works in the literature employ a similar bilevel structure in addressing network design or enhancement problems in the context of disaster mitigation. Specifically, these works consider retrofit (Fan and Liu, 2010) and expansion (Lo and Tung, 2003, Dimitriou and Stathopoulous, 2008) actions with the aim of reducing the impact of potential disaster events on network performance. Link capacities are only known with certainty post-disaster. These works build in the capacity uncertainty within the lower-level problem, where the system users take decisions only after the disaster scenario is realized. They propose inexact solution techniques in which the complicating complementarity constraints are relaxed or other heuristics. Through such mitigative actions, system reliability (the probability of continued functionality post-event) or robustness (a post-event functionality measure) are improved. Both are measures of inherent coping capacity and omit from the evaluation the system’s ability to adapt. Faturechi and Miller-Hooks (2014) provide a framework for understanding these and other measures in the context of resilience.
In the earlier works where uncertainty in supply (e.g. link capacities) was considered in the lower level, a User Equilibrium (UE) is determined for each potential event scenario given upper-level decisions, and upper-level decisions are taken deterministically. Achieving a UE assumes fully-adaptive behavior by system users. The users are presumed to have perfect information about the state of the roadway network in choosing their paths from their historical travel experiences. Despite a rich literature on travel behavior, modeling such behavior under disruption has received little attention. This lack of attention to post-disaster travel behavior is mentioned in (Zhu et al., 2010). In this context, this assumption of perfect information by all users might be valid only long after the event’s initial occurrence at which time system users have enough information to adapt their travel behavior to the new situation, and a new UE is established. However, shortly after the event occurrence, such an assumption is likely erroneous.
The subject of this lower-level problem is the period arising shortly after the occurrence of a disaster event in which short-term, contingency plans can be implemented. According to a user behavior survey of De Palma and Rochat (1999), users have high flexibility in their route choice shortly after the occurrence of an event. That is, user behavior is characterized as being semi-adaptive given limited information on network conditions, including information on damage and completion of repair (Iida et al., 2000). Thus, the lower-level problem is formulated as a Partial UE (PUE) traffic assignment problem. This concept of a PUE was introduced in Sumalee and Watling (2008) for modeling user behavior in post-disaster circumstances. Recently, He and Liu (2012) proposed a prediction–correction traffic assignment model to capture day-to-day travel behavior post-disaster that may also have some relevance. Their model incorporates gradually improving information about network conditions over time.
This paper incorporates user behavior in the measurement and maximization of travel time resilience for roadway networks given under a set of possible disaster scenarios. The problem of quantifying and optimizing travel time resilience (i.e. the Travel Time Resilience Problem (TTRP)) is formulated as a bilevel, three-stage stochastic program with lower-level equilibrium (PUE) constraints. Both upper- and lower-level problems involve capacity uncertainty. The upper-level includes a three-stage decision making process in which both pre- and post-event resilience enhancing actions may be taken. The decision process is informed as information is revealed at each stage, and is compatible with the Disaster Management Life-Cycle (DMLC) (Waugh, 2000): (1) pre-event expansion and retrofit as mitigation options to enhance the coping capacity of the road network, (2) pre-event preparedness where resources are acquired and prepositioned shortly in advance of a predicted event occurrence to facilitate response actions, and (3) post-event short-term response actions taken post-disaster to restore network capacity, minimize the extent of damage, and/or protect the remaining facilities. A multi-hazard perspective is taken, whereby actions that may be effective in one scenario may be ineffective in another. An exact Progressive Hedging Algorithm (adapted from Rockafellar and Wets (1991)) is presented for solution of a single-level equivalent to this bilevel, three-stage stochastic program.
Whether addressing day-to-day incident-induced traffic congestion or disaster events, including pre-event or both pre- and post-event actions for enhancing system performance, or employing a UE or PUE, these problems involving uncertainty in available system capacity in which user response to network supply decisions is captured can be mathematically modeled as Stochastic Mathematical Programs with Equilibrium Constraints (SMPECs). Thus, they are a type of Stochastic Network Design Problem (SNDP). A general discussion on the properties of SMPECs and potential solution techniques can be found in (Patriksson, 2008). In addition to its contributions to resilience measurement, this paper extends the study of SNDPs (SMPECs) generally. Key to its contributions are its consideration of supply uncertainty in both upper- and lower-level problems, incorporation of a three-stage stochastic program in the upper-level to capture key relevant DMLC stages, use of a PUE rather than a UE in traffic assignment as is appropriate for the disaster-context, and application of cutting-edge linearization methodologies for dealing with complementarity constraints and nonlinear, nonseparable travel time functions, as well as nonlinear design decision terms.
The contributions of this work are derived from: (a) development of a SMPEC-based framework for conceptualizing a measure of resilience compatible with the stages of the DMLC, (b) using a Partial UE to model post-disaster behavior of roadway network users, (c) transformation of the proposed SMPEC into a Stochastic Mixed Integer Program (SMIP) through the application of cutting-edge piecewise linearization techniques, and (d) adaptation of an exact solution technique based on the Progressive Hedging Algorithm to solve the transformed problem. It must be noted that a number of assumptions were required in developing the proposed model and solution schemes. First, only supply-oriented uncertainties were taken into account. That is, travel demand is assumed to be inelastic to the disaster impact. Second, a PUE adequately captures driver route-choice behavior post-disaster.
The next section introduces the problem formulation. This is followed by description of the solution method in Section 3 and application on an illustrative example in Section 4.
Section snippets
Problem formulation
At the upper level of the proposed TTRP, mitigation, preparedness and response actions are chosen with information from the lower level about the resulting total travel time for all O–D pairs that can be expected given upper level choices. The upper level acts as the leader, determining the optimal supply decisions. The lower level acts as the follower, wherein affected system users selfishly determine their paths with knowledge of the upper-level decisions. The optimal solution to the bilevel
Solving the TTRP
The bilevel TTRP (2), (3), (4), (5), (6), (7), (8), (9), (10), (11), (12) is solved by first reducing it to a single-level problem as is often done in the literature. To accomplish this, the lower-level problem is eliminated and corresponding Karush–Kuhn–Tucker (KKT) conditions are embedded within the upper-level problem. Larsson and Patriksson (1995) showed, in a similar context involving a bilevel program with a related UE formulation in the lower level, that use of the KKT conditions in
Illustrative example
The model and solution method are illustrated on a test network with 6 nodes and 16 links representing a highway system as illustrated in Fig. 5. The network was first introduced in (Suwansirikul et al., 1987) and has been used for similar purposes in many works (e.g. Li et al., 2012). Links 2, 5, 6 and 9 represent bridges. Four OD pairs are considered, and the corresponding travel demand is reported in Table 2. The network data, including the values of link travel time function parameters from
Conclusions
This paper proposes a novel stochastic network design formulation for maximizing travel time resilience for roadway networks. In particular, it targets freeway networks. The problem explicitly addresses the first three stages of the decision processes of the disaster management life cycle, specifically pre-event mitigation and preparedness, and post-event response. Decisions are taken at each stage based upon the evolution of uncertainty over the stages. The problem is formulated as a bilevel,
Acknowledgments
This work was funded by the National Science Foundation. This support is gratefully acknowledged, but implies no endorsement of the findings.
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