Accounting for travel time variability in the optimal pricing of cars and buses

A number of studies have shown that in addition to travel time and cost as the common influences on mode, route and departure time choices, travel time variability plays an increasingly important role, especially in the presence of traffic congestion on roads and crowding on public transport. The dominant focus of modelling and implementation of optimal pricing that incorporates trip time variability has been in the context of road pricing for cars. The main objective of this paper is to introduce a non-trivial extension to the existing literature on optimal pricing in a multimodal setting, building in the role of travel time variability as a source of disutility for car and bus users. We estimate the effect of variability in travel time and bus headway on optimal prices (i.e., tolls for cars and fares for buses) and optimal bus capacity (i.e., frequencies and size) accounting for crowding on buses, under a social welfare maximisation framework. Travel time variability is included by adopting the well-known mean–variance model, using an empirical relationship between the mean and standard deviation of travel times. We illustrate our model with an application to a highly congested corridor with cars, buses and walking as travel alternatives in Sydney, Australia. There are three main findings that have immediate policy implications: (i) including travel time variability results in higher optimal car tolls and substantial increases in toll revenue, while optimal bus fares remain almost unchanged; (ii) when bus headways are variable, the inclusion of travel time variability as a source of disutility for users yields higher optimal bus frequencies; and (iii) including both travel time variability and crowding discomfort leads to higher optimal bus sizes.     

Pricing scheme design of ridesharing program in morning commute problem

This paper examines the dynamic user equilibrium of the morning commute problem in the presence of ridesharing program. Commuters simultaneously choose departure time from home and commute mode among three roles: solo driver, ridesharing driver, and ridesharing rider. Considering the congestion evolution over time, we propose a time-varying compensation scheme to maintain a positive ridesharing ridership at user equilibrium. To match the demand and the supply of ridesharing service over time, the compensation scheme should be set according to the inconvenience cost functions and the out-of-pocket cost functions. When the price charged per time unit is higher than the inconvenience cost per time unit perceived by the ridesharing drivers, the ridesharing participants will travel at the center of peak hours and solo drivers will commute at the two tails. Within the feasible region with positive ridership, the ridesharing program can reduce the congestion and all the commuters will be better off. To support system optimum (SO), we derive a time-varying toll combined with a flat ridesharing price from eliminating queuing delay. Under SO toll, the ridesharing program can attract more participants and have an enlarged feasible region. This reveals that the commuters are more tolerant to the inconvenience caused by sharing a ride at SO because of the lower travel time. Compared with no-toll equilibrium, both overall congestion and individual travel cost are further reduced at SO.     

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.     

On the degradation of performance for traffic networks with oblivious users

We consider the problem of characterizing user equilibria and optimal solutions for routing in a given network. We extend the known models by considering users oblivious to congestion in the following sense: While in the typical user equilibrium setting the users follow a strategy that minimizes their individual cost by taking into account the (dynamic) congestion due to the current routing pattern, an oblivious user ignores congestion altogether; instead, he or she decides his routing on the basis of cheapest routes on a network without any flow whatsoever. These cheapest routes can be, for example, the shortest paths in the network without any flow. This model tries to capture the fact that a certain percentage of travelers base their route simply on the distances they observe on a map, without thinking (or knowing, or caring) about the delays experienced on this route due to their fellow travelers. In this work we study the effect of such users using as the measure of network performance its price of anarchy, i.e., the ratio of the total latency experienced by the users (oblivious or not) at equilibrium over the social optimum.     

Existence of self-financing and Pareto-improving congestion pricing: Impact of value of time distribution

This paper considers a static congestion pricing model in which travelers select a mode from either, driving on highway or taking public transit, to minimize a combination of travel time, operating cost and toll. The focus is to examine how travelers’ value of time (VOT), which is continuously distributed in a population, affects the existence of a pricing-refunding scheme that is both self-financing (i.e. requiring no external subsidy) and Pareto-improving (i.e. reducing system travel time while making nobody worse off). A condition that insures the existence of a self-financing and Pareto-improving (SFPI) toll scheme is derived. Our derivation reveals that the toll authority can select a proper SFPI scheme to distribute the benefits from congestion pricing through a credit-based pricing scheme. Under mild assumptions, we prove that an SFPI toll always exists for concave VOT functions, of which the linear function corresponding to the uniform distribution is a special case. Existence conditions are also established for a class of rational functions. These results can be used to analyze more realistic VOT distributions such as log-normal distribution. A useful implication of our analysis is that the existence of an SFPI scheme is not guaranteed for general functional forms. Thus, external subsidies may be required to ensure Pareto-improving, even if policy-makers are willing to return all toll revenues to road users.     

The influence of (toll-related) travel costs in residential location decisions of households: A stated choice approach

In this paper, we investigate the impact of travel costs, in particular toll costs, on the residential location choice of households, using a stated choice survey. Within the stated choice experiment, car drivers that frequently face traffic congestion, traded-off several trip-related (including toll costs) and house/location-related factors in their decision where to locate. If we look at the influence of different variables, toll and fuel costs seem to be important. Respondents are more sensitive to travel costs (i.e. toll and fuel costs) than to equally high (monthly) housing costs. Travel time appears to play a less important role, as indicated by a low value of time (VOT). In addition, location-related factors, such as the type of location and the number of bedrooms, turn out to be important factors as well. It can be concluded that respondents generally speaking prefer to pay higher housing costs and accept longer travel times to avoid (high) travel costs. Finally, if we look at the difference in preferences in relation to toll and fuel cost, we can conclude that toll costs are valued more negatively than fuel costs, although the differences are small.     

Optimal joint distance and time toll for cordon-based congestion pricing

This paper addresses the optimal toll design problem for the cordon-based congestion pricing scheme, where both a time-toll and a nonlinear distance-toll (i.e., joint distance and time toll) are levied for each network user’s trip in a pricing cordon. The users’ route choice behaviour is assumed to follow the Logit-based stochastic user equilibrium (SUE). We first propose a link-based convex programming model for the Logit-based SUE problem with a joint distance and time toll pattern. A mathematical program with equilibrium constraints (MPEC) is developed to formulate the optimal joint distance and time toll design problem. The developed MPEC model is equivalently transformed into a semi-infinite programming (SIP) model. A global optimization method named Incremental Constraint Method (ICM) is designed for solving the SIP model. Finally, two numerical examples are used to assess the proposed methodology.     

Time-dependent area-based pricing for multimodal systems with heterogeneous users in an agent-based environment

In this paper, we investigate an area-based pricing scheme for congested multimodal urban networks with the consideration of user heterogeneity. We propose a time-dependent pricing scheme where the tolls are iteratively adjusted through a Proportional–Integral type feedback controller, based on the level of vehicular traffic congestion and traveler’s behavioral adaptation to the cost of pricing. The level of congestion is described at the network level by a Macroscopic Fundamental Diagram, which has been recently applied to develop network-level traffic management strategies. Within this dynamic congestion pricing scheme, we differentiate two groups of users with respect to their value-of-time (which related to income levels). We then integrate incentives, such as improving public transport services or return part of the toll to some users, to motivate mode shift and increase the efficiency of pricing and to attain equitable savings for all users. A case study of a medium size network is carried out using an agent-based simulator. The developed pricing scheme demonstrates high efficiency in congestion reduction. Comparing to pricing schemes that utilize similar control mechanisms in literature which do not treat the adaptivity of users, the proposed pricing scheme shows higher flexibility in toll adjustment and a smooth behavioral stabilization in long-term operation. Significant differences in behavioral responses are found between the two user groups, highlighting the importance of equity treatment in the design of congestion pricing schemes. By integrating incentive programs for public transport using the collected toll revenue, more efficient pricing strategies can be developed where savings in travel time outweigh the cost of pricing, achieving substantial welfare gain.     

Minimizing CO2e emissions by setting a road toll

The main purpose of this paper is to develop a bi-level pricing model to minimize the CO₂e emissions and the total travel time in a small road network. In the lower level of the model, it is assumed that users of the road network find a dynamic user equilibrium which minimizes the total costs of those in the system. For the higher level of the model, different road toll strategies are applied in order to minimize the CO₂e emissions. The model has been applied to an illustrative example. It shows the effects on traffic flows, revenues, total time and CO₂e emissions for different numbers of servers collecting tolls and different pricing strategies over a morning peak traffic period. The results show that the CO₂e emissions produced can be significantly affected by the number of servers and the type of toll strategy employed. The model is also used to find the best toll strategy when there is a constraint on the revenue that is required to be raised from the toll and how this affects the emissions produced. Further runs compare strategies to minimize the CO₂e emissions with those that minimize total travel time in the road system. In the illustrative example, the results for minimizing CO₂e emissions are shown to be similar to the results obtained from minimizing the total travel time.     

Optimal locations and travel time display for variable message signs

This paper first develops a network equilibrium model with the travel time information displayed via variable message signs (VMS). Specifically, the equilibrium considers the impact of the displayed travel time information on travelers’ route choices under the recurrent congestion, with the endogenous utilization rates of displayed information by travelers. The existence of the equilibrium is proved and an iterative solution procedure is provided. Then, we conduct the sensitivity analyses of the network equilibrium and further propose a paradox, i.e., providing travel time information via VMS to travelers may degrade the network performance under some poor designs. Therefore, we investigate the problem of designing the VMS locations and travel time display within a given budget, and formulate it as a mixed integer nonlinear program, solved by an active-set algorithm. Lastly, numerical examples are presented to offer insights on the equilibrium results and optimal designs of VMS.     

The analysis of regulation on private highway investment under a build-operate-transfer scheme

The build-operate-transfer (BOT) approach has become an attractive instrument for public facility provision, especially for a project that faces difficulty with public finance. This study analyzes the regulation alternatives on private highway investment under a BOT scheme and their impacts on traffic flows, travel costs, toll, capacity, and social welfare (total user-benefit in the traffic system including congestion). For comparison, five cases are analyzed: (1) No BOT with maximizing welfare, (2) No BOT with breaking even on finance, (3) BOT without regulation, (4) BOT with a minimum flow constraint (the total users will not be less than those in Case 1), and (5) BOT with a maximum travel cost constraint (the travel cost for users on a non-tolled road will not exceed the maximum tolerance). After each case is modeled and simulated on some functional forms, we find that the case of BOT with regulations performs between the cases of maximizing welfare and that of maximizing profit. From the perspective of the government, regulation has less power in a project with low elastic demand. Furthermore, even when the regulation is strict, a high cost-efficient firm with BOT could result in a higher level of social welfare than that without a BOT scheme.     

Road pricing for congestion control with unknown demand and cost functions

It is widely recognized that precise estimation of road tolls for various pricing schemes requires a few pieces of information such as origin–destination demand functions, link travel time functions and users’ valuations of travel time savings, which are, however, not all readily available in practice. To circumvent this difficulty, we develop a convergent trial-and-error implementation method for a particular pricing scheme for effective congestion control when both the link travel time functions and demand functions are unknown. The congestion control problem of interest is also known as the traffic restraint and road pricing problem, which aims at finding a set of effective link toll patterns to reduce link flows to below a desirable target level. For the generalized traffic equilibrium problem formulated as variational inequalities, we propose an iterative two-stage approach with a self-adaptive step size to update the link toll pattern based on the observed link flows and given flow restraint levels. Link travel time and demand functions and users’ value of time are not needed. The convergence of the iterative toll adjustment algorithm is established theoretically and demonstrated on a set of numerical examples.     

Tour-based mode choice of joint household travel patterns on weekend and weekday

Joint household travel, with or without joint participation in an activity, constitutes a fundamental aspect in modelling activity-based travel behaviour. This paper examines joint household travel arrangements and mode choices using a utility maximising approach. An individual tour-based mode choice model is formulated contingent on the choice of joint tour patterns where joint household activities and shared ride arrangements are recognised as part of the joint household decision-making that influences the travel modes of each household member. Two models, one for weekend and one for weekday, are estimated using empirical data from the Sydney Household Travel Survey. The results show that weekend travel is characterised by a high joint household activity participation rate while weekday travel is distinguished by more intra-household shared ride arrangements. The arrangements of joint household travel are highly associated with travel purpose, social and mobility constraints and household resources. On weekends, public transport is mainly used by captive users (i.e., no-car households and students) and its share is about half of that on weekdays. Also, the value of travel time savings (VOTs) are found to be higher on weekends than on weekdays, running entirely counter to the common belief that weekend VOTs are lower than weekday VOTs. This paper highlights the importance of studying joint household travel and using different transport management measures for alleviating traffic congestion on weekdays and weekends.     

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.     

Viability of new road infrastructure with heterogeneous users

This paper explores the importance of heterogeneity in value of time and route choice when assessing the viability of new road infrastructure to alleviate congestion problems. The model incorporates strategic interaction between road operators in a cost-benefit framework and several competitive regimes are considered. It is then employed to establish the financial and socio-economic viability of a congestion pricing demonstration entering Madrid city centre, where road users have to choose between a free but highly congested road and a priced free-flowing road (semi-private regime). A logit estimation is undertaken with information from a questionnaire among road users in the Eastern Madrid area to obtain users’ value of time and of congestion.The tolls obtained generate a traffic reallocation towards the new roadway such that revenues suffice to render the infrastructure socio-economically viable. The private and the low toll regimes generate similar welfare gains that are close to the first best. Yet the former supposes large losses to users. The low toll and the semi-private regimes do not raise such distributional concerns. However, the low toll regime requires a sufficiently high traffic growth rate to make it financially viable; this does not happen for the other competitive regimes.     

New technology and the modeling of risk-taking behavior in congested road networks

Intelligent transport systems provide various means to improve traffic congestion in road networks. Evaluation of the benefits of these improvements requires consideration of commuters’ response to reliability and/or uncertainty of travel time under various circumstances. Various disruptions cause recurrent or non-recurrent congestion on road networks, which make road travel times intrinsically fluctuating and unpredictable. Confronted with such uncertain traffic conditions, commuters are known to develop some simple decision-making process to adjust their travel choices. This paper represents the decision-making process involved in departure-time and route choices as risk-taking behavior under uncertainty. An expected travel disutility function associated with commuters’ departure-time and route choices is formulated with taking into account the travel delay (due the recurrent congestion), the uncertainty of travel times (due to incident-induced congestion) and the consequent early or late arrival penalty. Commuters are assumed to make decision on the departure-time and route choices on the basis of the minimal expected travel disutility. Thus the network will achieve a simultaneous route and departure-time user equilibrium, in which no commuter can decrease his or her expected disutility by unilaterally changing the route or departure-time. The equilibrium is further formulated as an equivalent nonlinear complementarity problem and is then converted into an unconstrained minimization problem with the use of a gap function suggested recently. Two algorithms based on the Nelder–Mead multidimensional simplex method and the heuristic route/time-swapping approach, are adapted to solve the problem. Finally, numerical example is given to illustrate the application of the proposed model and algorithms.     

How much is too much for tolled road users: Toll saturation and the implications for car commuting value of travel time savings?

The current practice of forecasting the demand for new tolled roads typically assumes that car users are prepared to pay a higher toll for a shorter journey, and they will keep doing so as long as the toll cost is not higher than their current value of travel time savings. Practice ignores the possibility that there could be a point when motorists stop driving on toll roads due to a toll budget constraint. The unconstrained toll budget assumption may be valid in networks where the addition of a new toll road does not result in a binding budget constraint that car users may have for using toll roads (although it could also be invoked for existing tolled routes through a reduction in use of a tolled route). In a road network like Sydney which offers a growing number of (linked) tolled roads, the binding budget constraint may be invoked, and hence including additional toll links might in turn reduce the car users’ willingness to pay for toll roads to save the same amount of travel time. When this occurs, car users are said to reach a toll saturation point (or threshold) and begin to consider avoiding one or more toll roads. Whilst toll saturation has important implications for demand forecasting and planning of toll roads, this type of behaviour has not been explored in the literature. We investigate the influence that increasing toll outlays has on preferences of car commuters to use one or more tolled roads as the number of tolled roads increases. The Sydney metropolitan area offers a unique laboratory to test this phenomenon, with nine tolled roads currently in place and another five in planning. The evidence supports the hypothesis that the value of travel time savings decreases as a consequence of toll saturation.     

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.     

The Stockholm congestion – charging trial 2006: Overview of effects

The Stockholm congestion charging trial in 2006 demonstrated the effects of a full-scale time-differentiated urban road toll scheme. Improvements in travel times were large enough to be perceived by the general public. This was pivotal to the radical change of public attitudes that occurred during the trial and that resulted in a positive outcome of a subsequent referendum on a proposal for making the system permanent. This paper summarises the effects of the trial and analyses to what extent targets were met. Effects on congestion reduction were larger than anticipated, which also resulted in favourable economic and environmental effects. The trial showed that a single-cordon toll could affect traffic within a large area, i.e., not just close to the zone limits.     

An advanced traveler general information system for Fresno,California

This study estimates the effects of an advanced traveler general information system (ATGIS), which includes fuel consumption and health-related emissions cost information on transportation network users’ travel choice behavior for recurrent congestion conditions. The effects are estimated using four different formulations based on four different behavioral assumptions. Incorporating stochastic features in link cost estimation rather than in route choice, we provide a novel modeling approach that enables us to use transportation planning models of major metropolitan areas without a need for major computationally-expensive changes in the existing models. We examined the effects of an ATGIS on the Fresno, CA, road network and found several interesting results. First, the ATGIS impact is closely related to pre-system (prior to the implementation of an ATGIS) perceived fuel and emissions costs. Total travel time in the city can be reduced by 17% (no pre-system perceived costs) to 1% (accurate pre-system perceived costs), and even increased by 1% (higher-than-actual pre-system perceived costs). Second, the addition of emissions costs, although negligible relative to fuel and time costs, can effectively reduce total system-wide travel time by up to 1% and fuel consumption by up to 0.6% during peak hours. Third, the ATGIS can reduce annual social costs by as much as $1053 million (high gas price, no pre-system perception) to $48 million (medium gas price, accurate pre-system perception), which are comparable to social cost savings by a congestion pricing (CP) scheme in the study area.