Toll pricing framework under logit‐based stochastic user equilibrium constraints

This paper addresses the toll pricing framework for the first‐best pricing with logit‐based stochastic user equilibrium (SUE) constraints. The first‐best pricing is usually known as marginal‐cost toll, which can be obtained by solving a traffic assignment problem based on the marginal cost functions. The marginal‐cost toll, however, has rarely been implemented in practice, because it requires every specific link on the network to be charged. Thus, it is necessary to search for a substitute of the marginal cost pricing scheme, which can reduce the toll locations but still minimize the total travel time. The toll pricing framework is the set of all the substitute toll patterns of the marginal cost pricing. Assuming the users' route choice behavior following the logit‐based SUE principle, this paper has first derived a mathematical expression for the toll pricing framework. Then, by proposing an origin‐based variational inequality model for the logit‐based SUE problem, another toll pricing framework is built, which avoids path enumeration/storage. Finally, the numerical test shows that many alternative pricing patterns can inherently reduce the charging locations and total toll collected, while achieving the same equilibrium link flow pattern. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

A portfolio theory of route choice

Although many individual route choice models have been proposed to incorporate travel time variability as a decision factor, they are typically still deterministic in the sense that the optimal strategy requires choosing one particular route that maximizes utility. In contrast, this study introduces an individual route choice model where choosing a portfolio of routes instead of a single route is the best strategy for a rational traveler who cares about both journey time and lateness when facing stochastic network conditions. The proposed model is compared with UE and SUE models and the difference in both behavioral foundation and model characteristics is highlighted. A numerical example is introduced to demonstrate how such model can be used in traffic assignment problem. The model is then tested with GPS data collected in metropolitan Minneapolis–St. Paul, Minnesota. Our data suggest there is no single dominant route (defined here as a route with the shortest travel time for a 15 day period) in 18% of cases when links travel times are correlated. This paper demonstrates that choosing a portfolio of routes could be the rational choice of a traveler who wants to optimize route decisions under variability.     

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.     

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.     

Formulating the within-day dynamic stochastic traffic assignment problem from a Bayesian perspective

This study proposes a formulation of the within-day dynamic stochastic traffic assignment problem. Considering the stochastic nature of route choice behavior, we treat the solution to the assignment problem as the conditional joint distribution of route traffic, given that the network is in dynamic stochastic user equilibrium. We acquire the conditional joint probability distribution using Bayes’ theorem. A Metropolis–Hastings sampling scheme is developed to estimate the characteristics (e.g., mean and variance) of the route traffic. The proposed formulation has no special requirements for the traffic flow models and user behavior models, and so is easily implemented.     

Simultaneous determination of the equilibrium market penetration and compliance rate of advanced traveler information systems

We consider a specific advanced traveler information systems (ATIS) whose objective is to reduce drivers’ travel time uncertainty with recurrent network congestion through provision of traffic information. Since the provided information is still partial or imperfect, drivers equipped with an ATIS cannot always find the shortest travel time route and thus may not always comply with the advice provided by ATIS. Thus, there are three classes of drivers on a specific day: drivers without ATIS, drivers with ATIS but without compliance with ATIS advice, drivers with ATIS and in compliance with ATIS advice. All three classes of drivers make route choice in a stochastic manner, but with different degree of uncertainty of travel time on the network. In this paper we investigate the interactions among the three classes of drivers in an ATIS environment using a multiple behavior stochastic user equilibrium model. By assuming that the market penetration of ATIS is an increasing function of the actual private gain (time saving minus the cost associated with system use) derived from ATIS service, and the ATIS compliance rate of equipped drivers is given as the probability of the actual travel time of complied drivers being less than that of non-complied drivers, we determine the equilibrium market penetration and compliance rate of ATIS and the resulting equilibrium network flow pattern using an iterative solution procedure.     

A reliability‐based traffic assignment model for multi‐modal transport network under demand uncertainty

In densely populated and congested urban areas, the travel times in congested multi‐modal transport networks are generally varied and stochastic in practice. These stochastic travel times may be raised from day‐to‐day demand fluctuations and would affect travelers' route and mode choice behaviors according to their different expectations of on‐time arrival. In view of these, this paper presents a reliability‐based user equilibrium traffic assignment model for congested multi‐modal transport networks under demand uncertainty. The stochastic bus frequency due to the unstable travel time of bus route is explicitly considered. By the proposed model, travelers' route and mode choice behaviors are intensively explored. In addition, a stochastic state‐augmented multi‐modal transport network is adopted in this paper to effectively model probable transfers and non‐linear fare structures. A numerical example is given to illustrate the merits of the proposed model. Copyright © 2012 John Wiley & Sons, Ltd.     

A multi‐class mean‐excess traffic equilibrium model with elastic demand

Recent empirical studies have revealed that travel time variability plays an important role in travelers' route choice decisions. To simultaneously account for both reliability and unreliability aspects of travel time variability, the concept of mean‐excess travel time (METT) was recently proposed as a new risk‐averse route choice criterion. In this paper, we extend the mean‐excess traffic equilibrium model to include heterogeneous risk‐aversion attitudes and elastic demand. Specifically, this model explicitly considers (1) multiple user classes with different risk‐aversions toward travel time variability when making route choice decisions under uncertainty and (2) the elasticity of travel demand as a function of METT when making travel choice decisions under uncertainty. This model is thus capable of modeling travelers' heterogeneous risk‐averse behaviors with both travel choice and route choice considerations. The proposed model is formulated as a variational inequality problem and solved via a route‐based algorithm using the modified alternating direction method. Numerical analyses are also provided to illustrate the features of the proposed model and the applicability of the solution algorithm. Copyright © 2012 John Wiley & Sons, Ltd.     

Investigating the learning effects of route guidance and traffic advisories on route choice behavior

This paper describes a study to investigate the effects of route guidance and traffic advisories on driver's route choice behavior. The study is a two-factor experiment with repeated measures on one factor where the between-subjects factor is the type of traveler information provided and the repeated, within-subjects factor is trips made between a specified origin and destination. Participants were recruited and randomly assigned to one of four groups: group 1 having only a basic map of the network; group 2 having access only to route guidance, group 3 having access to traffic advisory information, and group 4 having access to both route guidance and traffic advisory information. Each participant completed 15 trips between a specified origin-destination pair on a hypothetical network. The results of this study indicate that there may be significant short-term advantages to providing in-vehicle routing and navigation information to unfamiliar drivers. However, the results also indicate that the format and amount of information provided may not be significant as the benefits to having route guidance diminish when drivers become more familiar with the travel network.     

On the flexibility of using marginal distribution choice models in traffic equilibrium

Traffic equilibrium models are fundamental to the analysis of transportation systems. The stochastic user equilibrium (SUE) model which relaxes the perfect information assumption of the deterministic user equilibrium is one such model. The aim of this paper is to develop a new user equilibrium model, namely the MDM-SUE model, that uses the marginal distribution model (MDM) as the underlying route choice model. In this choice model, the marginal distributions of the path utilities are specified but the joint distribution is not. By focusing on the joint distribution that maximizes expected utility, we show that MDM-SUE exists and is unique under mild assumptions on the marginal distributions. We develop a convex optimization formulation for the MDM-SUE. For specific choices of marginal distributions, the MDM-SUE model recreates the optimization formulation of logit SUE and weibit SUE. Moreover, the model is flexible since it can capture perception variance scaling at the route level and allows for modeling different user preferences by allowing for skewed distributions and heavy tailed distributions. The model can also be generalized to incorporate bounded support distributions and discrete distributions which allows to distinguish between used and unused routes within the SUE framework. We adapt the method of successive averages to develop an efficient approach to compute MDM-SUE traffic flows. In our numerical experiments, we test the ability of MDM-SUE to relax the assumption that the error terms are independently and identically distributed random variables as in the logit models and study the additional modeling flexibility that MDM-SUE provides on small-sized networks as well as on the large network of the city of Winnipeg. The results indicate that the model provides both modeling flexibility and computational tractability in traffic equilibrium.     

Tollroads are only part of the overall trip: the error of our ways in past willingness to pay studies

With rare exception, actual tollroad traffic in many countries has failed to reproduce forecast traffic levels, regardless of whether the assessment is made after an initial year of operation or as long as 10 years after opening. Pundits have offered many reasons for this divergence, including optimism bias, strategic misrepresentation, the promise to equity investors of early returns on investment, errors in land use forecasts, and specific assumptions underlying the traffic assignment models used to develop traffic forecasts. One such assumption is the selection of a behaviourally meaningful value of travel time savings (VTTS) for use in a generalised cost or generalised time user benefit expression that is the main behavioural feature of the traffic assignment (route choice) model. Numerous empirical studies using stated choice experiments have designed choice sets of alternatives as if users choose a tolled route or a free route under the (implied) assumption that the tolled route is tolled for the entire trip. Reality is often very different, with a high incidence of use of a non-tolled road leading into and connecting out of a tolled link. In this paper we recognise this feature of route choice and redesign the stated choice experiment to account for it. Furthermore, this study is a follow up to a previous study undertaken before a new toll road was in place, and it benefits from real exposure to the new toll road. We find that the VTTS is noticeably reduced, and if the VTTS is a significant contributing influence on errors on traffic forecasts, then the lower estimates make sense behaviourally.     

The nonlinear equation system approach to solving dynamic user optimal simultaneous route and departure time choice problems

Dynamic user optimal simultaneous route and departure time choice (DUO-SRDTC) problems are usually formulated as variational inequality (VI) problems whose solution algorithms generally require continuous and monotone route travel cost functions to guarantee convergence. However, the monotonicity of the route travel cost functions cannot be ensured even if the route travel time functions are monotone. In contrast to traditional formulations, this paper formulates a DUO-SRDTC problem (that can have fixed or elastic demand) as a system of nonlinear equations. The system of nonlinear equations is a function of generalized origin-destination (OD) travel costs rather than route flows and includes a dynamic user optimal (DUO) route choice subproblem with perfectly elastic demand and a quadratic programming (QP) subproblem under certain assumptions. This study also proposes a solution method based on the backtracking inexact Broyden–Fletcher–Goldfarb–Shanno (BFGS) method, the extragradient algorithm, and the Frank-Wolfe algorithm. The BFGS method, the extragradient algorithm, and the Frank-Wolfe algorithm are used to solve the system of nonlinear equations, the DUO route choice subproblem, and the QP subproblem, respectively. The proposed formulation and solution method can avoid the requirement of monotonicity of the route travel cost functions to obtain a convergent solution and provide a new approach with which to solve DUO-SRDTC problems. Finally, numeric examples are used to demonstrate the performance of the proposed solution method.     

Marginal cost congestion pricing based on the network fundamental diagram

Congestion pricing schemes have been traditionally derived based on analytical representations of travel demand and traffic flows, such as in bottleneck models. A major limitation of these models, especially when applied to urban networks, is the inconsistency with traffic dynamics and related phenomena such as hysteresis and the capacity drop. In this study we propose a new method to derive time-varying tolling schemes using the concept of the Network Fundamental Diagram (NFD). The adopted method is based on marginal cost pricing, while it also enables to account realistically for the dynamics of large and heterogeneous traffic networks. We derive two alternative cordon tolls using network-aggregated traffic flow conditions: a step toll that neglects the spatial distribution of traffic by simply associating the marginal costs of any decrease in production within the NFD to the surplus of traffic; and a step toll that explicitly accounts for how network performance is also influenced by the spatial variance in a 3D-NFD. This pricing framework is implemented in the agent-based simulation model MATSim and applied to a case study of the city of Zurich. The tolling schemes are compared with a uniform toll, and they highlight how the inhomogeneous distribution of traffic may compromise the effectiveness of cordon tolls.     

A study of traveler behavior under traffic information-provided conditions in the Beijing area

ABSTRACT

The quality of traffic information has become one of the most important factors that can affect the distribution of urban and highway traffic flow by changing the travel route, transportation mode, and travel time of travelers and trips. Past research has revealed traveler behavior when traffic information is provided. This paper summarizes the related study achievements from a survey conducted in the Beijing area with a specially designed questionnaire considering traffic conditions and the provision of traffic information services. With the survey data, a Logit model is estimated, and the results indicate that travel time can be considered the most significant factor that affects highway travel mode choice between private vehicles and public transit, whereas trip purpose is the least significant factor for private vehicle usage for both urban and highway travel.     

A marginal utility day-to-day traffic evolution model based on one-step strategic thinking

Most deterministic day-to-day traffic evolution models, either in continuous-time or discrete-time space, have been formulated based on a fundamental assumption on driver route choice rationality where a driver seeks to maximize her/his marginal benefit defined as the difference between the perceived route costs. The notion of rationality entails the exploration of the marginal decision rule from economic theory, which states that a rational individual evaluates his/her marginal utility, defined as the difference between the marginal benefit and the marginal cost, of each incremental decision. Seeking to analyze the marginal decision rule in the modeling of deterministic day-to-day traffic evolution, this paper proposes a modeling framework which introduces a term to capture the marginal cost to the driver induced by route switching. The proposed framework enables to capture both benefit and cost associated with route changes. The marginal cost is then formulated upon the assumption that drivers are able to predict other drivers’ responses to the current traffic conditions, which is adopted based on the notion of strategic thinking of rational players developed in behavior game theory. The marginal cost based on 1-step strategic thinking also describes the “shadow price” of shifting routes, which helps to explain the behavioral tendency of the driver perceiving the cost-sensitivity to link/route flows. After developing a formulation of the marginal utility day-to-day model, its theoretical properties are analyzed, including the invariance property, asymptotic stability, and relationship with the rational behavioral adjustment process.     

Network reliability‐based optimal toll design

Optimal toll design from a network reliability point of view is addressed in this paper. Improving network reliability is proposed as a policy objective of road pricing. A reliability‐based optimal toll design model, where on the upper level network performance including travel time reliability is optimized, while on the lower level a dynamic user‐equilibrium is achieved, is presented. Road authorities aim to optimize network travel time reliability by setting tolls in a network design problem. Travelers are influenced by these tolls and make route and trip decisions by considering travel times and tolls. Network performance reliability is analyzed for a degradable network with elastic and fluctuated travel demand, which integrates reliability and uncertainty, dynamic network equilibrium models, and Monte Carlo methods. The proposed model is applied to a small hypothesized network for which optimal tolls are derived. The network travel time reliability is indeed improved after implementing optimal tolling system. Trips may have a somewhat higher, but more reliable, travel time.     

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.     

Optimal tolls for multi-class traffic: Analytical formulations and policy implications

This paper puts together an analytical formulation to compute optimal tolls for multi-class traffic. The formulation is comprised of two major modules. The first one is an optimization component aimed at computing optimal tolls assuming a Stackelberg game in which the toll agency sets the tolls, and the equilibrating traffic plays the role of the followers. The optimization component is supported by a set of cost models that estimate the externalities as a function of a multivariate vector of traffic flows. These models were estimated using Taylor series expansions of the output obtained from traffic simulations of a hypothetical test case. Of importance to the paper is the total travel time function estimated using this approach that expresses total travel time as a multivariate function of the traffic volumes. The formulation presented in the paper is then applied to a variety of scenarios to gain insight into the optimality of current toll policies. The optimal tolls are computed for two different cases: independent tolls, and tolls proportional to passenger car equivalencies (PCE).The numerical results clearly show that setting tolls proportional to PCEs leads to lower values of welfare that are on average 15% lower than when using independent tolls, though, in some cases the total welfare could be up to 33% lower. This is a consequence of two factors. First, the case of independent tolls has more degrees of freedom than the case of tolls proportional to PCEs. Second, tolls proportional to PCEs do not account for externalities other than congestion, which is likely to lead to lower welfare values.The analytical formulations and numerical results indicate that, because the total travel time is a non-linear function of the traffic volumes, the marginal social costs and thus the optimal congestion tolls also depend on the traffic volumes for each vehicle class. As a result of this, for the relatively low volumes of truck traffic observed in real life, the optimal congestion tolls for trucks could indeed be either lower or about the same as for passenger cars. This stand in sharp contrast with what is implied in the use of PCEs, i.e., that the contribution to congestion are constant. This latter assumption leads to optimal truck congestion tolls that are always proportional to the PCE values.The comparison of the toll ratios (truck tolls divided by passenger car tolls) for both observed and optimal conditions suggests that the tolls for small trucks are about the right level, maybe a slightly lower than optimal. However, the analysis of the toll ratio for large trucks seems to indicate a significant overcharge. The estimates show that the average observed toll ratio for large trucks is even higher than the maximum optimal toll ratio found in the numerical experiments. This suggests that the tolls for large trucks are set on the basis of revenue generation principles while the passenger car tolls are being set based on a mild form of welfare maximization. This leads to a suboptimal cross-subsidization of passenger car traffic in detriment of an important sector of the economy.