Effect of providing traffic information estimated by a stochastic network equilibrium model with stochastic demand

To estimate travel times through road networks, in this study, we assume a stochastic demand and formulate a stochastic network equilibrium model whose travel times, flows, and demands are stochastic. This model enables us to examine network reliability under stochastic circumstances and to evaluate the effect of providing traffic information on travel times. For traffic information, we focus on travel time information and propose methods to evaluate the effect of providing that information. To examine the feasibility and validity of the proposed model and methods, we apply them to a simple network and the real road network of Kanazawa, Japan. The results indicate that providing ambulance drivers in Kanazawa with travel time information leads to an average reduction in travel time of approximately three minutes.     

First-best marginal cost toll for a traffic network with stochastic demand

First-best marginal cost pricing (MCP) in traffic networks has been extensively studied with the assumption of deterministic travel demand. However, this assumption may not be realistic as a transportation network is exposed to various uncertainties. This paper investigates MCP in a traffic network under stochastic travel demand. Cases of both fixed and elastic demand are considered. In the fixed demand case, travel demand is represented as a random variable, whereas in the elastic demand case, a pre-specified random variable is introduced into the demand function. The paper also considers a set of assumptions of traveler behavior. In the first case, it is assumed that the traveler considers only the mean travel time in the route choice decision (risk-neutral behavior), and in the second, both the mean and the variance of travel time are introduced into the route choice model (risk-averse behavior). A closed-form formulation of the true marginal cost toll for the stochastic network (SN-MCP) is derived from the variational inequality conditions of the system optimum and user equilibrium assignments. The key finding is that the calculation of the SN-MCP model cannot be made by simply substituting related terms in the original MCP model by their expected values. The paper provides a general function of SN-MCP and derives the closed-form SN-MCP formulation for specific cases with lognormal and normal stochastic travel demand. Four numerical examples are explored to compare network performance under the SN-MCP and other toll regimes.     

Consistent formulation of network equilibrium with stochastic flows

Traffic flows in real-life transportation systems vary on a daily basis. According to traffic flow theory, such variability should induce a similar variability in travel times, but this “internal consistency” is generally not captured by existing network equilibrium models. We present an internally-consistent network equilibrium approach, which considers two potential sources of flow variability: (i) daily variation in route choice and (ii) daily variation in origin–destination demand. We particularly aspire to a flexible formulation that permits alternative statistical assumptions, which allows the best fit to be made to observed variability data in particular applications. Joint probability distributions of route—and therefore link—flows are derived under several assumptions concerning stochastic driver behavior. A stochastic network equilibrium model with stochastic demands and route choices is formulated as a fixed point problem. We explore limiting cases which allow an equivalent convex optimization problem to be defined, and finally apply this method to a real-life network of Kanazawa City, Japan.     

The time-dependent vehicle routing problem with soft time windows and stochastic travel times

This paper studies a vehicle routing problem with time-dependent and stochastic travel times. In our problem setting, customers have soft time windows. A mathematical model is used in which both efficiency for service as well as reliability for customers are taken into account. Depending on whether service times are included or not, we consider two versions of this problem. Two metaheuristics are built: a Tabu Search and an Adaptive Large Neighborhood Search. We carry out our experiments for well-known problem instances and perform comprehensive analyses on the numerical results in terms of the computational time and the solution quality. Experiments confirm that the proposed procedure is effective to obtain very good solutions to be performed in real-life environment.     

A probabilistic model for vehicle scheduling based on stochastic trip times

Vehicle scheduling plays a profound role in public transit planning. Traditional approaches for the Vehicle Scheduling Problem (VSP) are based on a set of predetermined trips in a given timetable. Each trip contains a departure point/time and an arrival point/time whilst the trip time (i.e. the time duration of a trip) is fixed. Based on fixed durations, the resulting schedule is hard to comply with in practice due to the variability of traffic and driving conditions. To enhance the robustness of the schedule to be compiled, the VSP based on stochastic trip times instead of fixed ones is studied. The trip times follow the probability distributions obtained from the data captured by Automatic Vehicle Locating (AVL) systems. A network flow model featuring the stochastic trips is devised to better represent this problem, meanwhile the compatibility of any pair of trips is redefined based on trip time distributions instead of fixed values as traditionally done. A novel probabilistic model of the VSP is proposed with the objectives of minimizing the total cost and maximizing the on-time performance. Experiments show that the probabilistic model may lead to more robust schedules without increasing fleet size.     

A computational model for the probit‐based dynamic stochastic user optimal traffic assignment problem

This paper focuses on computational model development for the probit‐based dynamic stochastic user optimal (P‐DSUO) traffic assignment problem. We first examine a general fixed‐point formulation for the P‐DSUO traffic assignment problem, and subsequently propose a computational model that can find an approximated solution of the interest problem. The computational model includes four components: a strategy to determine a set of the prevailing routes between each origin–destination pair, a method to estimate the covariance of perceived travel time for any two prevailing routes, a cell transmission model‐based traffic performance model to calculate the actual route travel time used by the probit‐based dynamic stochastic network loading procedure, and an iterative solution algorithm solving the customized fixed‐point model. The Ishikawa algorithm is proposed to solve the computational model. A comparison study is carried out to investigate the efficiency and accuracy of the proposed algorithm with the method of successive averages. Two numerical examples are used to assess the computational model and the algorithm proposed. Results show that Ishikawa algorithm has better accuracy for smaller network despite requiring longer computational time. Nevertheless, it could not converge for larger network. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Accounting for stochastic variables in discrete choice models

The estimation of discrete choice models requires measuring the attributes describing the alternatives within each individual’s choice set. Even though some attributes are intrinsically stochastic (e.g. travel times) or are subject to non-negligible measurement errors (e.g. waiting times), they are usually assumed fixed and deterministic. Indeed, even an accurate measurement can be biased as it might differ from the original (experienced) value perceived by the individual.Experimental evidence suggests that discrepancies between the values measured by the modeller and experienced by the individuals can lead to incorrect parameter estimates. On the other hand, there is an important trade-off between data quality and collection costs. This paper explores the inclusion of stochastic variables in discrete choice models through an econometric analysis that allows identifying the most suitable specifications. Various model specifications were experimentally tested using synthetic data; comparisons included tests for unbiased parameter estimation and computation of marginal rates of substitution. Model specifications were also tested using a real case databank featuring two travel time measurements, associated with different levels of accuracy.Results show that in most cases an error components model can effectively deal with stochastic variables. A random coefficients model can only effectively deal with stochastic variables when their randomness is directly proportional to the value of the attribute. Another interesting result is the presence of confounding effects that are very difficult, if not impossible, to isolate when more flexible models are used to capture stochastic variations. Due the presence of confounding effects when estimating flexible models, the estimated parameters should be carefully analysed to avoid misinterpretations. Also, as in previous misspecification tests reported in the literature, the Multinomial Logit model proves to be quite robust for estimating marginal rates of substitution, especially when models are estimated with large samples.     

Robust transit network design with stochastic demand considering development density

This paper analyzes the influence of urban development density on transit network design with stochastic demand by considering two types of services, rapid transit services, such as rail, and flexible services, such as dial-a-ride shuttles. Rapid transit services operate on fixed routes and dedicated lanes, and with fixed schedules, whereas dial-a-ride services can make use of the existing road network, hence are much more economical to implement. It is obvious that the urban development densities to financially sustain these two service types are different. This study integrates these two service networks into one multi-modal network and then determines the optimal combination of these two service types under user equilibrium (UE) flows for a given urban density. Then we investigate the minimum or critical urban density required to financially sustain the rapid transit line(s). The approach of robust optimization is used to address the stochastic demands as captured in a polyhedral uncertainty set, which is then reformulated by its dual problem and incorporated accordingly. The UE principle is represented by a set of variational inequality (VI) constraints. Eventually, the whole problem is linearized and formulated as a mixed-integer linear program. A cutting constraint algorithm is adopted to address the computational difficulty arising from the VI constraints. The paper studies the implications of three different population distribution patterns, two CBD locations, and produces the resultant sequences of adding more rapid transit services as the population density increases.     

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.     

A scenario-based planning for the pickup and delivery problem with time windows,scheduled lines and stochastic demands

The Pickup and Delivery Problem with Time Windows, Scheduled Lines and Stochastic Demands (PDPTW-SLSD) concerns scheduling a set of vehicles to serve a set of requests, whose expected demands are known in distribution when planning, but are only revealed with certainty upon the vehicles’ arrival. In addition, a part of the transportation plan can be carried out on limited-capacity scheduled public transportation line services. This paper proposes a scenario-based sample average approximation approach for the PDPTW-SLSD. An adaptive large neighborhood search heuristic embedded into sample average approximation method is used to generate good-quality solutions. Computational results on instances with up to 40 requests (i.e., 80 locations) reveal that the integrated transportation networks can lead to operational cost savings of up to 16% compared with classical pickup and delivery systems.     

Time-dependent green vehicle routing problem with stochastic vehicle speeds: An approximate dynamic programming algorithm

This paper addresses a Time Dependent Capacitated Vehicle Routing Problem with stochastic vehicle speeds and environmental concerns. The problem has been formulated as a Markovian Decision Process. As distinct from the traditional attempts on the problem, while estimating the amount of fuel consumption and emissions, the model takes time-dependency and stochasticity of the vehicle speeds into account. The Time Dependent Capacitated Vehicle Routing Problem is known to be NP-Hard for even deterministic settings. Incorporating uncertainty to the problem increases complexity, which renders classical optimization methods infeasible. Therefore, we propose an Approximate Dynamic Programming based heuristic as a decision aid tool for the problem. The proposed Markovian Decision Model and Approximate Dynamic Programming based heuristic are flexible in terms that more environmentally friendly solutions can be obtained by changing the objective function from cost minimization to emissions minimization. The added values of the proposed decision support tools have been shown through computational analyses on several instances. The computational analyses show that incorporating vehicle speed stochasticity into decision support models has potential to improve the performance of resulting routes in terms of travel duration, emissions and travel cost. In addition, the proposed heuristic provides promising results within relatively short computation times.     

A stochastic optimization model for transit network timetable design to mitigate the randomness of traveling time by adding slack time

Transit network timetabling aims at determining the departure time of each trip of all lines in order to facilitate passengers transferring either to or from a bus. In this paper, we consider a bus timetabling problem with stochastic travel times (BTP-STT). Slack time is added into timetable to mitigate the randomness in bus travel times. We then develop a stochastic integer programming model for the BTP-STT to minimize the total waiting time cost for three types of passengers (i.e., transferring passengers, boarding passengers and through passengers). The mathematical properties of the model are characterized. Due to its computational complexity, a genetic algorithm with local search (GALS) is designed to solve our proposed model (OPM). The numerical results based on a small bus network show that the timetable obtained from OPM reduces the total waiting time cost by an average of 9.5%, when it is tested in different scenarios. OPM is relatively effective if the ratio of the number of through passengers to the number of transferring passengers is not larger than a threshold (e.g., 10 in our case). In addition, we test different scale instances randomly generated in a practical setting to further verify the effectiveness of OPM and GALS. We also find that adding slack time into timetable greatly benefits transferring passengers by reducing the rate of transferring failure.     

Finding time‐robust fuel‐efficient paths for a call‐taxi in a stochastic city road network

Intra‐city commuting is being revolutionized by call‐taxi services in many developing countries such as India. A customer requests a taxi via phone, and it arrives at the right time and at the right location for the pick‐up. This mode of intra‐city travel has become one of the most reliable and convenient modes of transportation for customers traveling for business and non‐business purposes. The increased number of vehicles on city roads and raising fuel costs has prompted a new type of transportation logistics problem of finding a fuel‐efficient and quickest path for a call‐taxi through a city road network, where the travel times are stochastic. The stochastic travel time of the road network is induced by obstacles such as the traffic signals and intersections. The delay and additional fuel consumption at each of these obstacles are calculated that are later imputed to the total travel time and fuel consumption of a path. A Monte‐Carlo simulation‐based approach is proposed to identify unique fuel‐efficient paths between two locations in a city road network where each obstacle has a delay distribution. A multi‐criteria score is then assigned to each unique path based on the probability that the path is fuel efficient, the average travel time of the path and the coefficient of variation of the travel times of the path. Copyright © 2016 John Wiley & Sons, Ltd.     

Estimating metro passengers’ path choices by combining self-reported revealed preference and smart card data

With the help of automated fare collection systems in the metro network, more and more smart card (SC) data has been widely accumulated, which includes abundant information (i.e., Big Data). However, its inability to record passengers’ transfer information and factors affecting passengers’ travel behaviors (e.g., socio-demographics) limits further potential applications. In contrast, self-reported Revealed Preference (RP) data can be collected via questionnaire surveys to include those factors; however, its sample size is usually very small in comparison to SC data. The purpose of this study is to propose a new set of approaches of estimating metro passengers’ path choices by combining self-reported RP and SC data. These approaches have the following attractive features. The most important feature is to jointly estimate these two data sets based on a nested model structure with a balance parameter by accommodating different scales of the two data sets. The second feature is that a path choice model is built to incorporate stochastic travel time budget and latent individual risk-averse attitude toward travel time variations, where the former is derived from the latter and the latter is further represented based on a latent variable model with observed individual socio-demographics. The third feature is that an algorithm of combining the two types of data is developed by integrating an Expectation-Maximization algorithm and a nested logit model estimation method. The above-proposed approaches are examined based on data from Guangzhou Metro, China. The results show the superiority of combined data over single data source in terms of both estimation and forecasting performance.     

Scheduling of airport runway operations using stochastic branch and bound methods

In this paper we present a solution methodology based on the stochastic branch and bound algorithm to find optimal, or close to optimal, solutions to the stochastic airport runway scheduling problem. The objective of the scheduling problem is to find a sequence of aircraft operations on one or several runways that minimizes the total makespan, given uncertain aircraft availability at the runway. Enhancements to the general stochastic branch and bound algorithm are proposed and we give the specific details pertaining to runway scheduling. We show how the algorithm can be terminated early with solutions that are close to optimal, and investigate the impact of the uncertainty level. The computational experiment indicates that the sequences obtained using the stochastic branch and bound algorithm have, on average, 5–7% shorter makespans than sequences obtained using deterministic sequencing models. In addition, the proposed algorithm is able to solve instances with 14 aircraft using less than 1 min of computation time.     

Dynamic demand estimation and prediction for traffic urban networks adopting new data sources

Nowadays, new mobility information can be derived from advanced traffic surveillance systems that collect updated traffic measurements, both in fixed locations and over specific corridors or paths. Such recent technological developments point to challenging and promising opportunities that academics and practitioners have only partially explored so far.The paper looks at some of these opportunities within the Dynamic Demand Estimation problem (DDEP). At first, data heterogeneity, accounting for different sets of data providing a wide spatial coverage, has been investigated for the benefit of off-line demand estimation. In an attempt to mimic the current urban networks monitoring, examples of complex real case applications are being reported where route travel times and route choice probabilities from probe vehicles are exploited together with common link traffic measurements.Subsequently, on-line detection of non-recurrent conditions is being recorded, adopting a sequential approach based on an extension of the Kalman Filter theory called Local Ensemble Transformed Kalman Filter (LETKF).Both the off-line and the on-line investigations adopt a simulation approach capable of capturing the highly nonlinear dependence between the travel demand and the traffic measurements through the use of dynamic traffic assignment models. Consequently, the possibility of using collected traffic information is enhanced, thus overcoming most of the limitations of current DDEP approaches found in the literature.     

Multi-scenario optimization approach for assessing the impacts of advanced traffic information under realistic stochastic capacity distributions

In this study, to incorporate realistic discrete stochastic capacity distribution over a large number of sampling days or scenarios (say 30–100 days), we propose a multi-scenario based optimization model with different types of traveler knowledge in an advanced traveler information provision environment. The proposed method categorizes commuters into two classes: (1) those with access to perfect traffic information every day, and (2) those with knowledge of the expected traffic conditions (and related reliability measure) across a large number of different sampling days. Using a gap function framework or describing the mixed user equilibrium under different information availability over a long-term steady state, a nonlinear programming model is formulated to describe the route choice behavior of the perfect information (PI) and expected travel time (ETT) user classes under stochastic day-dependent travel time. Driven by a computationally efficient algorithm suitable for large-scale networks, the model was implemented in a standard optimization solver and an open-source simulation package and further applied to medium-scale networks to examine the effectiveness of dynamic traveler information under realistic stochastic capacity conditions.     

c-SPSA: Cluster-wise simultaneous perturbation stochastic approximation algorithm and its application to dynamic origin–destination matrix estimation

The simultaneous perturbation stochastic approximation (SPSA) algorithm has been used in the literature for the solution of the dynamic origin–destination (OD) estimation problem. Its main advantage is that it allows quite general formulations of the problem that can include a wide range of sensor measurements. While SPSA is relatively simple to implement, its performance depends on a set of parameters that need to be properly determined. As a result, especially in cases where the gradient of the objective function changes quickly, SPSA may not be as stable and even diverge. A modification of the SPSA algorithm, referred to as c-SPSA, is proposed which applies the simultaneous perturbation approximation of the gradient within a small number of carefully constructed “homogeneous” clusters one at a time, as opposed to all elements at once. The paper establishes the theoretical properties of the new algorithm with an upper bound for the bias of the gradient estimate and shows that it is lower than the corresponding SPSA bias. It also proposes a systematic approach, based on the k-means algorithm, to identify appropriate clusters. The performance of c-SPSA, with alternative implementation strategies, is evaluated in the context of estimating OD flows in an actual urban network. The results demonstrate the efficiency of the proposed c-SPSA algorithm in finding better OD estimates and achieve faster convergence and more robust performance compared to SPSA with fewer overall number of function evaluations.     

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.