Finding the most reliable path with and without link travel time correlation: A Lagrangian substitution based approach

Tao Xing, Xuesong Zhou
2011 Transportation Research Part B: Methodological  
Path travel time reliability is an essential measure of the quality of service for transportation systems and an important attribute in travelers' route and departure time scheduling. This paper investigates a fundamental problem of finding the most reliable path under different spatial correlation assumptions, where the path travel time variability is represented by its standard deviation. To handle the nonlinear and nonadditive cost functions introduced by the quadratic forms of the standard
more » ... eviation term, a Lagrangian substitution approach is adopted to estimate the lower bound of the most reliable path solution through solving a sequence of standard shortest path problems. A subgradient algorithm is used to iteratively improve the solution quality by reducing the optimality gap. To characterize the link travel time correlation structure associated with the end-to-end trip time reliability measure, this research develops a sampling-based method to dynamically construct a proxy objective function in terms of travel time observations from multiple days. The proposed algorithms are evaluated under a large-scale Bay Area, California network with real-world measurements. 2 1. Introduction 1. Motivation As traffic systems can be viewed as stochastic processes with non-deterministic demand and capacity distributions, travel time reliability has been widely recognized as an important element of a traveler's route and departure time scheduling. In recent years, operating agencies have begun to shift their focus more toward monitoring and improving the reliability of transportation systems through probe-based data collection, integrated corridor management and advanced traveler information provision. With a growing trend of incorporating trip time variability into traffic network analysis models, finding reliable path alternatives motivates substantial algorithmic development efforts. For many common route finding criteria, such as physical distance and travel time, the (generalized) path cost functions are linear and additive across different links, so the resulting optimization problem can be directly solved by the standard label correcting or label setting shortest path algorithms. In an early study by Sen et al. (2001) , the path travel time reliability is modeled as a linear combination of travel time mean and variance, and the resulting 0-1 quadratic integer program is solved by as a sequence of parametric subproblems. However, most end-to-end trip reliability measures, as discussed below, lead to nonlinear and nonadditive cost functions, which considerably increase the complexity and impose challenges for the path search procedures. A wide range of definitions and formulations have been proposed to measure travel time reliability, including (1) 90 th -or 95 th -percentile travel time, buffer and planning time index, (2) on-time arrival probability, (3) travel time variation expressed in terms of standard deviation or coefficient of variation. The first two definitions (1 and 2) are built on the probability distribution function of travel time. For example, in a report by Cambridge Systematics (2005), the 95 thpercentile travel time is defined as the travel time within which 95 th -percentile trips are completed, and the buffer and planning time indexes can be further calculated from the 90 th -or 95 th -percentile travel time. The on-time arrival probability measure, on the other hand, considers the percentage of trips that are completed within a reasonable buffered travel time (e.g. average travel time plus 20% buffer). In a study by Fan et al. (2005a), a path finding algorithm was proposed to minimize the probability of arriving at the destination later than a specified arrival time. Recently, Nie and Wu (2009a) developed solution algorithms with firstorder stochastic dominance rules for the routing problem with on-time arrival reliability. The third type of models, which characterizes the travel time reliability measure in terms of standard deviation, has been calibrated in various empirical studies (e.g. Small, 1982; Noland et al., 1998; Noland and Polak, 2002) , and the corresponding utility function is also incorporated in dynamic traffic assignment models (e.g. Zhou et al., 2008) . It is important to recognize that, within a Kalman filtering framework, which is the building block of real-time traffic state estimation and prediction systems (e.g. Ashok and Ben-Akiva, 1993; Zhou and Mahmassani, 2007) , the variance of travel time estimates, and accordingly its standard deviation, can be analytically derived and calculated through a recursive estimation error propagation formula. In comparison, the first two types of reliability measures must be assessed by relatively complicated numerical probabilistic methods. To allow further extensions in real-time traffic prediction and route guidance systems, we consider the most reliable path problem with a linear disutility function of mean trip travel time and its standard deviation:   min mean var   . In particular, we want to address two fundamental challenges introduced by this special functional form. First, the standard deviation of path travel time is not a linear summation of the standard deviation of related link travel times. Second, the square root transformation associated with the standard deviation term is, in fact, a concave function, so it is difficult to directly apply many convex programming techniques in this application. Literature review Several previous pioneering research efforts have been devoted to addressing computational issues caused by nonlinear, nonadditive or concave objective functions in the shortest path problem. The early work by Henig (1986) presented efficient approximate methods on the shortest path problem with two criteria, which are assumed to be quasiconcave or quasiconvex utility functions. Scott and Bernstein (1997) developed an iterative solution method for the shortest path problem where the value of time function is nonlinear and non-decreasing. In their algorithm, the search space is decomposed to a series of resourceconstrained shortest path subproblems, which can be solved by the Lagrangian relaxation (LR) technique
doi:10.1016/j.trb.2011.06.004 fatcat:v2b6cnbdc5c2jd6djpcumsb2p4