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.     

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.     

Reliable route guidance: A case study from Chicago

Reliable route guidance can be obtained by solving the reliable a priori shortest path problem, which finds paths that maximize the probability of arriving on time. The goal of this paper is to demonstrate the benefits and applicability of such route guidance using a case study. An adaptive discretization scheme is first proposed to improve the efficiency in computing convolution, a time-consuming step used in the reliable routing algorithm to obtain path travel time distributions. Methods to construct link travel time distributions from real data in the case study are then discussed. Particularly, the travel time distributions on arterial streets are estimated from linear regression models calibrated from expressway data. Numerical experiments demonstrate that optimal paths are substantially affected by the reliability requirement in rush hours, and that reliable route guidance could generate up to 5-15% of travel time savings. The study also verifies that existing algorithms can solve large-scale problems within a reasonable amount of time.     

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.     

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.     

A predictive dynamic traffic assignment model in congested capacity-constrained road networks

In this paper, a predictive dynamic traffic assignment model in congested capacity-constrained road networks is formulated. A traffic simulator is developed to incrementally load the traffic demand onto the network, and updates the traffic conditions dynamically. A time-dependent shortest path algorithm is also given to determine the paths with minimum actual travel time from an origin to all the destinations. The traffic simulator and time-dependent shortest path algorithm are employed in a method of successive averages to solve the dynamic equilibrium solution of the problem. A numerical example is given to illustrate the effectiveness of the proposed method.     

The dynamic shortest path problem with time-dependent stochastic disruptions

The dynamic shortest path problem with time-dependent stochastic disruptions consists of finding a route with a minimum expected travel time from an origin to a destination using both historical and real-time information. The problem is formulated as a discrete time finite horizon Markov decision process and it is solved by a hybrid Approximate Dynamic Programming (ADP) algorithm with a clustering approach using a deterministic lookahead policy and value function approximation. The algorithm is tested on a number of network configurations which represent different network sizes and disruption levels. Computational results reveal that the proposed hybrid ADP algorithm provides high quality solutions with a reduced computational effort.     

An improved Dial's algorithm for logit-based traffic assignment within a directed acyclic network

Abstract

Dial's algorithm is one of the most effective and popular procedures for a logit-type stochastic traffic assignment, as it does not require path enumeration over a network. However, a fundamental problem associated with the algorithm is its simple definition of ‘efficient paths’, which sometimes produces unrealistic flow patterns. In this paper, an improved algorithm based on the route extension coefficient is proposed in order to circumvent this problem, in which ‘efficient paths’ simultaneously consider link travel cost and minimum travel cost. Path enumeration is still not required and a similar computing efficiency with the original algorithm is guaranteed. A limitation of the algorithm is that it can only be applied to a directed acyclic network because a topological sorting algorithm is used to decide the order of the sequential calculation. A numerical example based on the Beijing subway network illustrates the effectiveness of the proposed algorithm. It is found that it is able to exclude most unrealistic paths, but include all reasonable paths when compared with path enumeration and the original Dial's algorithm.     

A two-stage stochastic optimization model for the transfer activity choice in metro networks

This research focuses on finding the best transfer schemes in metro networks. Using sample-based time-invariant link travel times to capture the uncertainty of a realistic network, a two-stage stochastic integer programming model with the minimized expected travel time and penalty value incurred by transfer activities is formulated. The first stage aims to find a sequence of potential transfer nodes (stations) that can compose a feasible path from origins to destinations in the transfer activity network, and the second stage provides the least time paths passing by the generated transfer stations in the first stage for evaluating the given transfer schemes and then outputs the best routing information. To solve our proposed model, an efficient hybrid algorithm, in which the label correcting algorithm is embedded into a branch and bound searching framework, is presented to find the optimal solutions of the considered problem. Finally, the numerical experiments are implemented in different scales of metro networks. The computational results demonstrate the effectiveness and performance of the proposed approaches even for the large-scale Beijing metro network.     

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.     

Modelling rain effects on risk‐taking behaviours of multi‐user classes in road networks with uncertainty

This paper proposes a new travel time reliability‐based traffic assignment model to investigate the rain effects on risk‐taking behaviours of different road users in networks with day‐to‐day demand fluctuations and variations in travel time. A generalized link travel time function is used to capture the rain effects on vehicle travel times and road conditions. This function is further incorporated into daily demand variations to investigate those travel time variations arising from demand uncertainty and rain condition. In view of these rain effects, road users' perception errors on travel times and risk‐taking behaviours on path choices are incorporated in the proposed model with the use of a logit‐based stochastic user equilibrium framework. This new model is formulated as a variational inequality problem in terms of path flows. A numerical example is used to illustrate the application of the proposed model for assessment of the rain effects on road networks with uncertainty.     

Modeling stochastic perception error in the mean-excess traffic equilibrium model

In this paper, we extend the α-reliable mean-excess traffic equilibrium (METE) model of Chen and Zhou (Transportation Research Part B 44(4), 2010, 493-513) by explicitly modeling the stochastic perception errors within the travelers’ route choice decision processes. In the METE model, each traveler not only considers a travel time budget for ensuring on-time arrival at a confidence level α, but also accounts for the impact of encountering worse travel times in the (1 − α) quantile of the distribution tail. Furthermore, due to the imperfect knowledge of the travel time variability particularly in congested networks without advanced traveler information systems, the travelers’ route choice decisions are based on the perceived travel time distribution rather than the actual travel time distribution. In order to compute the perceived mean-excess travel time, an approximation method based on moment analysis is developed. It involves using the conditional moment generation function to derive the perceived link travel time, the Cornish-Fisher Asymptotic Expansion to estimate the perceived travel time budget, and the Acerbi and Tasche Approximation to estimate the perceived mean-excess travel time. The proposed stochastic mean-excess traffic equilibrium (SMETE) model is formulated as a variational inequality (VI) problem, and solved by a route-based solution algorithm with the use of the modified alternating direction method. Numerical examples are also provided to illustrate the application of the proposed SMETE model and solution method.     

Application of the support vector machine and heuristic k-shortest path algorithm to determine the most eco-friendly path with a travel time constraint

This study aims to determine an eco-friendly path that results in minimum CO₂ emissions while satisfying a specified budget for travel time. First, an aggregated CO₂ emission model for light-duty cars is developed in a link-based level using a support vector machine. Second, a heuristic k-shortest path algorithm is proposed to solve the constrained shortest path problem. Finally, the CO₂ emission model and the proposed eco-routing model are validated in a real-world network. Specifically, the benefit of the trade-off between CO₂ emission reduction and the travel time budget is discussed by carrying out sensitivity analysis on a network-wide scale. A greater spare time budget may enable the eco-routing to search for the most eco-friendly path with higher probability. Compared to the original routes selected by travelers, the eco-friendly routes can save an average of 11% of CO₂ emissions for the trip OD pairs with a straight distance between 6 km and 9 km when the travel time budget is set to 10% above the least travel time. The CO₂ emission can also be reduced to some degree for other OD pairs by using eco-routing. Furthermore, the impact of market penetration of eco-routing users is quantified on the potential benefit for the environment and travel-time saving.     

International experience with speed limits during and prior to the energy crisis of 1973–4

This paper presents an artificial neural network (ANN) based method for estimating route travel times between individual locations in an urban traffic network. Fast and accurate estimation of route travel times is required by the vehicle routing and scheduling process involved in many fleet vehicle operation systems such as dial‐a‐ride paratransit, school bus, and private delivery services. The methodology developed in this paper assumes that route travel times are time‐dependent and stochastic and their means and standard deviations need to be estimated. Three feed‐forward neural networks are developed to model the travel time behaviour during different time periods of the day‐the AM peak, the PM peak, and the off‐peak. These models are subsequently trained and tested using data simulated on the road network for the City of Edmonton, Alberta. A comparison of the ANN model with a traditional distance‐based model and a shortest path algorithm is then presented. The practical implication of the ANN method is subsequently demonstrated within a dial‐a‐ride paratransit vehicle routing and scheduling problem. The computational results show that the ANN‐based route travel time estimation model is appropriate, with respect to accuracy and speed, for use in real applications.     

Time-dependent Hyperstar algorithm for robust vehicle navigation

The vehicle navigation problem studied in Bell (2009) is revisited and a time-dependent reverse Hyperstar algorithm is presented. This minimises the expected time of arrival at the destination, and all intermediate nodes, where expectation is based on a pessimistic (or risk-averse) view of unknown link delays. This may also be regarded as a hyperpath version of the Chabini and Lan (2002) algorithm, which itself is a time-dependent A^* algorithm. Links are assigned undelayed travel times and maximum delays, both of which are potentially functions of the time of arrival at the respective link. Probabilities for link use are sought that minimise the driver’s maximum exposure to delay on the approach to each node, leading to the determination of a pessimistic expected time of arrival at the destination and all intermediate nodes. Since the context considered is vehicle navigation, the probability of link use measures link attractiveness, so a link with a zero probability of use is unattractive while a link with a probability of use equal to one will have no attractive alternatives. A solution algorithm is presented and proven to solve the problem provided the node potentials are feasible and a FIFO condition applies to undelayed link travel times. The paper concludes with a numerical example.     

Modeling heterogeneous risk-taking behavior in route choice: A stochastic dominance approach

This paper proposes a unified approach to modeling heterogonous risk-taking behavior in route choice based on the theory of stochastic dominance (SD). Specifically, the first-, second-, and third-order stochastic dominance (FSD, SSD, TSD) are respectively linked to insatiability, risk-aversion and ruin-aversion within the framework of utility maximization. The paths that may be selected by travelers of different risk-taking preferences can be obtained from the corresponding SD-admissible paths, which can be generated using general dynamic programming. This paper also analyzes the relationship between the SD-based approach and other route choice models that consider risk-taking behavior. These route choice models employ a variety of reliability indexes, which often make the problem of finding optimal paths intractable. We show that the optimal paths with respect to these reliability indexes often belong to one of the three SD-admissible path sets. This finding offers not only an interpretation of risk-taking behavior consistent with the SD theory for these route choice models, but also a unified and computationally viable solution approach through SD-admissible path sets, which are usually small and can be generated without having to enumerate all paths. A generic label-correcting algorithm is proposed to generate FSD-, SSD-, and TSD-admissible paths, and numerical experiments are conducted to test the algorithm and to verify the analytical results.     

Stochastic transit equilibrium

We present a transit equilibrium model in which boarding decisions are stochastic. The model incorporates congestion, reflected in higher waiting times at bus stops and increasing in-vehicle travel time. The stochastic behavior of passengers is introduced through a probability for passengers to choose boarding a specific bus of a certain service. The modeling approach generates a stochastic common-lines problem, in which every line has a chance to be chosen by each passenger. The formulation is a generalization of deterministic transit assignment models where passengers are assumed to travel according to shortest hyperpaths. We prove existence of equilibrium in the simplified case of parallel lines (stochastic common-lines problem) and provide a formulation for a more general network problem (stochastic transit equilibrium). The resulting waiting time and network load expressions are validated through simulation. An algorithm to solve the general stochastic transit equilibrium is proposed and applied to a sample network; the algorithm works well and generates consistent results when considering the stochastic nature of the decisions, which motivates the implementation of the methodology on a real-size network case as the next step of this research.     

Preferences for travel time under risk and ambiguity: Implications in path selection and network equilibrium

In this paper, we study the preferences for uncertain travel times in which probability distributions may not be fully characterized. In evaluating an uncertain travel time, we explicitly distinguish between risk, where the probability distribution is precisely known, and ambiguity, where it is not. In particular, we propose a new criterion called ambiguity-aware CARA travel time (ACT) for evaluating uncertain travel times under various attitudes of risk and ambiguity, which is a preference based on blending the Hurwicz criterion and Constant Absolute Risk Aversion (CARA). More importantly, we show that when the uncertain link travel times are independently distributed, finding the path that minimizes travel time under the ACT criterion is essentially a shortest path problem. We also study the implications on Network Equilibrium (NE) model where travelers on the traffic network are characterized by their knowledge of the network uncertainty as well as their risk and ambiguity attitudes under the ACT. We derive and analyze the existence and uniqueness of solutions under NE. Finally, we obtain the Price of Anarchy that characterizes the inefficiency of this new equilibrium. The computational study suggests that as uncertainty increases, the influence of selfishness on inefficiency diminishes.     

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.     

Joint travel problem in space–time multi-state supernetworks

Joint travel problem (JTP) is an extension of the classic shortest path problem and relevant to shared mobility. A pioneering endeavor via supernetwork framework has been put forward to model two-person JTP. However, it was only addressed in the static context and with the assumption of zero waiting disutility, which resulted in no or weak synchronization among the travelers. This paper proposes a space–time multi-state supernetwork framework to address JTP for conducting one joint activity in the time-dependent context. Space–time synchronization and various choice facets related to joint travel are captured systematically. Two-person JTP is first discussed in a uni-modal transport network, and further extended to incorporate multi-modal and multi-person respectively. Stage-wise recursive formulations are proposed to find the optimal joint paths. It is found that JTP is a variant of Steiner tree problem by reduction and the number of meeting/departing points has no impact on the run-time complexity in space–time multi-state supernetworks.