Efficient solution algorithm for finding spatially dependent reliable shortest path in road networks

Travel times are generally stochastic and spatially correlated in congested road networks. However, very few existing route guidance systems (RGS) can provide reliable guidance services to aid travellers planning their trips with taking account explicitly travel time reliability constraint. This study aims to develop such a RGS with particular consideration of travellers' concern on travel time reliability in congested road networks with uncertainty. In this study, the spatially dependent reliable shortest path problem (SD‐RSPP) is formulated as a multi‐criteria shortest path‐finding problem in road networks with correlated link travel times. Three effective dominance conditions are established for links with different levels of travel time correlations. An efficient algorithm is proposed to solve SD‐RSPP by adaptively using three established dominance conditions. The complexities of road networks in reality are also explicitly considered. To demonstrate the applicability of proposed algorithm, a comprehensive case study is carried out in Hong Kong. The results of case study show that the proposed solution algorithm is robust to take account of travellers' multiple routing criteria. Computational results demonstrate that the proposed solution algorithm can determine the reliable shortest path on real‐time basis for large‐scale road networks. Copyright © 2016 John Wiley & Sons, Ltd.     

Travel time estimation from sparse floating car data with consistent path inference: A fixed point approach

Estimation of urban network link travel times from sparse floating car data (FCD) usually needs pre-processing, mainly map-matching and path inference for finding the most likely vehicle paths that are consistent with reported locations. Path inference requires a priori assumptions about link travel times; using unrealistic initial link travel times can bias the travel time estimation and subsequent identification of shortest paths. Thus, the combination of path inference and travel time estimation is a joint problem. This paper investigates the sensitivity of estimated travel times, and proposes a fixed point formulation of the simultaneous path inference and travel time estimation problem. The methodology is applied in a case study to estimate travel times from taxi FCD in Stockholm, Sweden. The results show that standard fixed point iterations converge quickly to a solution where input and output travel times are consistent. The solution is robust under different initial travel times assumptions and data sizes. Validation against actual path travel time measurements from the Google API and an instrumented vehicle deployed for this purpose shows that the fixed point algorithm improves shortest path finding. The results highlight the importance of the joint solution of the path inference and travel time estimation problem, in particular for accurate path finding and route optimization.     

Finding the k reliable shortest paths under travel time uncertainty

This paper investigates the problem of finding the K reliable shortest paths (KRSP) in stochastic networks under travel time uncertainty. The KRSP problem extends the classical K loopless shortest paths problem to the stochastic networks by explicitly considering travel time reliability. In this study, a deviation path approach is established for finding K α-reliable paths in stochastic networks. A deviation path algorithm is proposed to exactly solve the KRSP problem in large-scale networks. The A^* technique is introduced to further improve the KRSP finding performance. A case study using real traffic information is performed to validate the proposed algorithm. The results indicate that the proposed algorithm can determine KRSP under various travel time reliability values within reasonable computational times. The introduced A^* technique can significantly improve KRSP finding performance.     

A copula-based approach for estimating the travel time reliability of urban arterial

Estimating the travel time reliability (TTR) of urban arterial is critical for real-time and reliable route guidance and provides theoretical bases and technical support for sophisticated traffic management and control. The state-of-art procedures for arterial TTR estimation usually assume that path travel time follows a certain distribution, with less consideration about segment correlations. However, the conventional approach is usually unrealistic because an important feature of urban arterial is the dependent structure of travel times on continuous segments. In this study, a copula-based approach that incorporates the stochastic characteristics of segments travel time is proposed to model arterial travel time distribution (TTD), which serves as a basis for TTR quantification. First, segments correlation is empirically analyzed and different types of copula models are examined. Then, fitting marginal distributions for segment TTD is conducted by parametric and non-parametric regression analysis, respectively. Based on the estimated parameters of the models, the best-fitting copula is determined in terms of the goodness-of-fit tests. Last, the model is examined at two study sites with AVI data and NGSIM trajectory data, respectively. The results of path TTD estimation demonstrate the advantage of the proposed copula-based approach, compared with the convolution model without capturing segments correlation and the empirical distribution fitting methods. Furthermore, when considering the segments correlation effect, it was found that the estimated path TTR is more accurate than that by the convolution model.     

Study on mean-standard deviation shortest path problem in stochastic and time-dependent networks: A stochastic dominance based approach

This paper studies a mean-standard deviation shortest path model, also called travel time budget (TTB) model. A route’s TTB is defined as this route’s mean travel time plus a travel time margin, which is the route travel time’s standard deviation multiplied with a factor. The TTB model violates the Bellman’s Principle of Optimality (BPO), making it difficult to solve it in any large stochastic and time-dependent network. Moreover, it is found that if path travel time distributions are skewed, the conventional TTB model cannot reflect travelers’ heterogeneous risk-taking behavior in route choice. This paper proposes to use the upper or lower semi-standard deviation to replace the standard deviation in the conventional TTB model (the new models are called derived TTB models), because these derived TTB models can well capture such heterogeneous risk-taking behavior when the path travel time distributions are skewed. More importantly, this paper shows that the optimal solutions of these two derived TTB models must be non-dominated paths under some specific stochastic dominance (SD) rules. These finding opens the door to solve these derived TTB models efficiently in large stochastic and time-dependent networks. Numerical examples are presented to illustrate these findings.     

Determining structural route components from GPS traces

Analysis of GPS traces shows that people often do not use the least cost path through the transportation network while making trips. This leads to the question which structural path characteristics can be used to construct realistic route choice sets for use in traffic simulation models. In this paper, we investigate the hypothesis that, for utilitarian trips, the route between origin and destination consists of a small number of concatenated least cost paths. The hypothesis is verified by analyzing routes extracted from large sets of recorded GPS traces which constitute revealed preference information. Trips have been extracted from the traces and for each trip the path in the transportation network is determined by map matching. This is followed by a path decomposition phase for which the algorithm constitutes the first contribution of this paper. There are multiple ways to split a given path in a directed graph into a minimal number of subpaths of minimal cost. By calculating two specific path splittings, it is possible to identify subsets of the vertices (splitVertexSuites) that can be used to generate every possible minimum path splitting by taking one vertex from each such subset. As a second contribution, we show how the extracted information is used in microscopic travel simulation. The distribution for the size of the minimum decomposition, extracted from the GPS traces, can be used in constrained enumeration methods for route choice set generation. The sets of vertices that can act as boundary vertices separating consecutive route parts contain way points (landmarks) having a particular meaning to their user. The paper explains the theoretical aspects of route splitting as well as the process to extract splitVertexSuites from big data. It reports statistical distributions extracted from sets of GPS traces for both multimodal person movements and unimodal car trips.     

Proactive route guidance to avoid congestion

We propose a proactive route guidance approach that integrates a system perspective: minimizing congestion, and a user perspective: minimizing travel inconvenience. The approach assigns paths to users so as to minimize congestion while not increasing their travel inconvenience too much. A maximum level of travel inconvenience is ensured and a certain level of fairness is maintained by limiting the set of considered paths for each Origin-Destination pair to those whose relative difference with respect to the shortest (least-duration) path, called travel inconvenience, is below a given threshold. The approach hierarchically minimizes the maximum arc utilization and the weighted average experienced travel inconvenience. Minimizing the maximum arc utilization in the network, i.e., the ratio of the number of vehicles entering an arc per time unit and the maximum number of vehicles per time unit at which vehicles can enter the arc and experience no slowdown due to congestion effects, is a system-oriented objective, while minimizing the weighted average experienced travel inconvenience, i.e., the average travel inconvenience over all eligible paths weighted by the number of vehicles per time unit that traverse the path, is a user-oriented objective. By design, to ensure computational efficiency, the approach only solves linear programming models. In a computational study using benchmark instances reflecting a road infrastructure encountered in many cities, we analyze, for different levels of maximum travel inconvenience and, the minimum maximum arc utilization and the weighted average experienced travel inconvenience. We find that accepting relatively small levels of maximum travel inconvenience can result in a significant reduction, or avoiding, of congestion.     

Optimal speed detector density for the network with travel time information

In the expressway network, detectors are installed on the links for detecting the travel time information while the predicted travel time can be provided by the route guidance system (RGS). The speed detector density can be determined to influence flow distributions in such a way that the precision of the travel time information and the social cost of the speed detectors are optimized, provided that each driver chooses the minimum perceived travel time path in response to the predicted travel time information. In this paper, a bilevel programming model is proposed for the network with travel time information provided by the RGS. The lower-level problem is a probit-based traffic assignment model, while the upper-level problem is to determine the speed detector density that minimizes the measured travel time error variance as well as the social cost of the speed detectors. The sensitivity analysis based algorithm is proposed for the bilevel programming problem. Numerical examples are provided to illustrate the applications of the proposed model and of the solution algorithm.     

Application of Lagrangian relaxation approach to α-reliable path finding in stochastic networks with correlated link travel times

This study investigates the important problem of determining a reliable path in a stochastic network with correlated link travel times. First, the distribution of path travel time is quantified by using trip records from GPS probe vehicles. Second, the spatial correlation of link travel time is explicitly considered by using a correlation coefficient matrix, which is incorporated into the α-reliable path problem by Cholesky decomposition. Third, the Lagrangian relaxation based framework is used to handle the α-reliable path problem, by which the intractable problem with a non-linear and non-additive structure can be decomposed into several easy-to-solve problems. Finally, the path-finding performance of this approach is tested on a real-world network. The results show that 15 iterations of calculation can yield a small relative gap between upper and lower bounds of the optimal solution and the average running time is about 5 s for most OD settings. The applicability of α-reliable path finding is validated by a case study.     

Forecasting journey time distribution with consideration to abnormal traffic conditions

Travel time is an important index for managers to evaluate the performance of transportation systems and an intuitive measure for travelers to choose routes and departure times. An important part of the literature focuses on predicting instantaneous travel time under recurrent traffic conditions to disseminate traffic information. However, accurate travel time prediction is important for assessing the effects of abnormal traffic conditions and helping travelers make reliable travel decisions under such conditions. This study proposes an online travel time prediction model with emphasis on capturing the effects of anomalies. The model divides a path into short links. A Functional Principal Component Analysis (FPCA) framework is adopted to forecast link travel times based on historical data and real-time measurements. Furthermore, a probabilistic nested delay operator is used to calculate path travel time distributions. To ensure that the algorithm is fast enough for online applications, parallel computation architecture is introduced to overcome the computational burden of the FPCA. Finally, a rolling horizon structure is applied to online travel time prediction. Empirical results for Guangzhou Airport Expressway indicate that the proposed method can capture an abrupt change in traffic state and provide a promising and reliable travel time prediction at both the link and path levels. In the case where the original FPCA is modified for parallelization, accuracy and computational effort are evaluated and compared with those of the sequential algorithm. The proposed algorithm is found to require only a piece rather than a large set of traffic incident records.     

Short‐term travel time forecasts for transport information system in hong kong

In this paper, a case study is carried out in Hong Kong for demonstration of the Transport Information System (TIS) prototype. A traffic flow simulator (TFS) is presented to forecast the short‐term travel times that can be served as a predicted travel time database for the TIS in Hong Kong. In the TFS, a stochastic deviation coefficient is incorporated to simulate the minute‐by‐minute fluctuation of traffic flows within the peak hour period. The purposes of the case study are: 1) to show the applicability of the TFS for larger‐scale road network; and 2) to illustrate the short‐term forecasting of path travel times in practice. The results of the case study show that the TFS can be applied to real network effectively. The predicted travel times are compared with the observed travel times on the selected paths for an OD pair. The results show that the observed path travel times fall in the 90% confidence interval of the predicted path travel times.     

Designing heterogeneous sensor networks for estimating and predicting path travel time dynamics: An information-theoretic modeling approach

With a particular emphasis on the end-to-end travel time prediction problem, this paper proposes an information-theoretic sensor location model that aims to minimize total travel time uncertainties from a set of point, point-to-point and probe sensors in a traffic network. Based on a Kalman filtering structure, the proposed measurement and uncertainty quantification models explicitly take into account several important sources of errors in the travel time estimation/prediction process, such as the uncertainty associated with prior travel time estimates, measurement errors and sampling errors. By considering only critical paths and limited time intervals, this paper selects a path travel time uncertainty criterion to construct a joint sensor location and travel time estimation/prediction framework with a unified modeling of both recurring and non-recurring traffic conditions. An analytical determinant maximization model and heuristic beam-search algorithm are used to find an effective lower bound and solve the combinatorial sensor selection problem. A number of illustrative examples and one case study are used to demonstrate the effectiveness of the proposed methodology.     

Probe vehicle based real-time traffic monitoring on urban roadways

Travel time estimation and prediction on urban arterials is an important component of Active Traffic and Demand Management Systems (ATDMS). This paper aims in using the information of GPS probes to augment less dynamic but available information describing arterial travel times. The direction followed in this paper chooses a cooperative approach in travel time estimation using static information describing arterial geometry and signal timing, semi-dynamic information of historical travel time distributions per time of day, and utilizes GPS probe information to augment and improve the latter. First, arterial travel times are classified by identifying different travel time states, then link travel time distributions are approximated using mixtures of normal distributions. If prior travel time data is available, travel time distributions can be estimated empirically. Otherwise, travel time distribution can be estimated based on signal timing and arterial geometry. Real-time GPS travel time data is then used to identify the current traffic condition based on Bayes Theorem. Moreover, these GPS data can also be used to update the parameters of the travel time distributions using a Bayesian update. The iterative update process makes the posterior distributions more and more accurate. Finally, two comprehensive case studies using the NGSIM Peachtree Street dataset, and GPS data of Washington Avenue in Minneapolis, were conducted. The first case study estimated prior travel time distributions based on signal timing and arterial geometry under different traffic conditions. Travel time data were classified and corresponding distributions were updated. In addition, results from the Bayesian update and EM algorithm were compared. The second case study first tested the methodologies based on real GPS data and showed the importance of sample size. In addition, a methodology was proposed to distinguish new traffic conditions in the second case study.     

Distributions of travel time variability on urban roads

Reliability is an important factor in route, mode and also departure time choice analysis and is a key performance indicator for transport systems. However, the current metrics used to measure travel time variability may be not sufficient to fully represent reliability. Better understanding of the distributions of travel times is needed for the development of improved metrics for reliability. A comprehensive data analysis involving the assessment of longitudinal travel time data for two urban arterial road corridors in Adelaide, Australia, demonstrates that the observed distributions are more complex than previously assumed. The data sets demonstrate strong positive skew, very long upper tails, and sometimes bimodality. This paper proposes the use of alternative statistical distributions for travel time variability, with the Burr Type XII distribution emerging as an appropriate model for both links and routes. This statistical distribution has some attractive properties that make it suitable for explicit definition of many travel time reliability metrics. Copyright © 2011 John Wiley & Sons, Ltd.     

Performance measure for reliable travel time of emergency vehicles

Travel time is very critical for emergency response and emergency vehicle (EV) operations. Compared to ordinary vehicles (OVs), EVs are permitted to break conventional road rules to reach the destination within shorter time. However, very few previous studies address the travel time performance of EVs. This study obtained nearly 4-year EV travel time data in Northern Virginia (NOVA) region using 76,000 preemption records at signalized intersections. First, the special characteristics of EV travel time are explored in mean, median, standard deviation and also the distribution, which display largely different characteristics from that of OVs in previous studies. Second, a utility-based model is proposed to quantify the travel time performance of EVs. Third, this paper further investigates two important components of the utility model: benchmark travel time and standardized travel time. The mode of the distribution is chosen as benchmark travel time, and its nonlinear decreasing relationship with the link length is revealed. At the same time, the distribution of standardized travel time is fitted with different candidate distributions and Inv. Gaussian distribution is proved to be the most suitable one. Finally, to validate the proposed model, we implement the model in case studies to estimate link and route travel time performance. The results of route comparisons also show that the proposed model can support EV route choice and eventually improve EV service and operations.     

Modelling route choice behaviour in a tolled road network with a time surplus maximisation bi-objective user equilibrium model

In this paper, we propose a novel approach to model route choice behaviour in a tolled road network with a bi-objective approach, assuming that all users have two objectives: (1) minimise travel time; and (2) minimise toll cost. We assume further that users have different preferences in the sense that for any given path with a specific toll, there is a limit on the time that an individual would be willing to spend. Different users can have different preferences represented by this indifference curve between toll and time. Time surplus is defined as the maximum time minus the actual time. Given a set of paths, the one with the highest (or least negative) time surplus will be the preferred path for the individual. This will result in a bi-objective equilibrium solution satisfying the time surplus maximisation bi-objective user equilibrium (TSmaxBUE) condition. That is, for each O–D pair, all individuals are travelling on the path with the highest time surplus value among all the efficient paths between this O–D pair.We show that the TSmaxBUE condition is a proper generalisation of user equilibrium with generalised cost function, and that it is equivalent to bi-objective user equilibrium. We also present a multi-user class version of the TSmaxBUE condition and demonstrate our concepts with illustrative examples.     

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

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 non-linear and non-additive cost functions introduced by the quadratic forms of the standard deviation 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.     

Estimating freeway route travel time distributions with consideration to time‐of‐day,inclement weather,and traffic incidents

This paper develops an efficient probabilistic model for estimating route travel time variability, incorporating factors of time‐of‐day, inclement weather, and traffic incidents. Estimating the route travel time distribution from historical link travel time data is challenging owing to the interactions among upstream and downstream links. Upon creating conditional probability function for each link travel time, we applied Monte Carlo simulation to estimate the total travel time from origin to destination. A numerical example of three alternative routes in the City of Buffalo shows several implications. The study found that weather conditions, except for snow, incur minor impact on off‐peak and weekend travel time, whereas peak travel times suffer great variations under different weather conditions. On top of that, inclement weather exacerbates route travel time reliability, even when mean travel time increases moderately. The computation time of the proposed model is linearly correlated to the number of links in a route. Therefore, this model can be used to obtain all the origin to destination travel time distributions in an urban region. Further, this study also validates the well‐known near‐linear relation between the standard deviation of travel time per unit distance and the corresponding mean value under different weather conditions. Copyright © 2016 John Wiley & Sons, Ltd.     

Safety-based path finding in urban areas for older drivers and bicyclists

This paper presents a safety-based path finding methodology for older drivers and bicyclists in an urban area. The paths are estimated based on costs consisting of both safety and travel time. Safety is evaluated against potential risk of a crash involving an older driver (or a bicyclist) with other vehicles present on the road. To accomplish this, simple formulations are developed for safety indicators of streets and intersections, which are actually generic irrespective of the type of road user. Traffic attributes such as speed and density, driver attributes such as perception-reaction time and street attributes of length and tire-to-road friction coefficient are taken into account in building the safety indicators. Thus, the safety indicators do not necessarily require historical crash data which may or may not be available during path finding. Subsequently, a multi-objective shortest path algorithm is presented that identifies the best path (the non-inferior path) from amongst a set of selected safest paths with due considerations to travel time incurred on each. A simple application example of the proposed methodology is demonstrated on an existing street network system from the City of College Station, Texas. The contributions of this research are twofold – first, the safety indicators can be used by planners in determining high crash potential sites – streets and/or intersections – and second, the safety-based path finding methodology developed in this paper can be integrated with modern day route planning devices and tools in guiding older drivers and bicyclists within an Intelligent Transportation Systems framework.     

Development of realistic shortest path algorithm considering lane changes

Lane changes occur as many times as turning movements are needed while following a designated path. The cost of a route with many lane changes is likely to be more expensive than that with less lane changes, and unrealistic paths with impractical lane changes should be avoided for drivers' safety. In this regard, a new algorithm is developed in this study to find the realistic shortest path considering lane changing. The proposed algorithm is a modified link‐labeling Dijkstra algorithm considering the effective lane‐changing time that is a parametric function of the prevailing travel speed and traffic density. The parameters were estimated using microscopic traffic simulation data, and the numerical test demonstrated the performance of the proposed algorithm. It was found that the magnitude of the effect of the effective lane‐changing time on determining the realistic shortest path is nontrivial, and the proposed algorithm has capability to exclude links successfully where the required lane changes are practically impossible. Copyright © 2015 John Wiley & Sons, Ltd.