A bi-objective user equilibrium model of travel time reliability in a road network

Travel time, travel time reliability and monetary cost have been empirically identified as the most important criteria influencing route choice behaviour. We concentrate on travel time and travel time reliability and review two prominent user equilibrium models incorporating these two factors. We discuss some shortcomings of these models and propose alternative bi-objective user equilibrium models that overcome the shortcomings. Finally, based on the observation that both models use standard deviation of travel time within their measure of travel time reliability, we propose a general travel time reliability bi-objective user equilibrium model. We prove that this model encompasses those discussed previously and hence forms a general framework for the study of reliability related user equilibrium. We demonstrate and validate our concepts on a small three-link example.     

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.     

Pareto-improving transportation network design and ownership regimes

Private provision of public roads signifies co-existence of free, public-tolled and private-tolled roads. This paper investigates the Pareto-improving transportation network design problem under various ownership regimes by allowing joint choice of road pricing and capacity enhancement on free links. The problem of interest is formulated as a bi-objective mathematical programming model that considers the travel cost of road users in each origin-destination pair and the investment return of the whole network. The non-dominated Pareto-improving solutions of toll and/or capacity enhancement schemes are sought for achieving a win-win situation. A sufficient condition is provided for the existence of the non-dominated Pareto-improving schemes and then the properties of those schemes are analyzed. It is found that, under some mild assumptions, the optimal capacity enhancement is uniquely determined by the link flow under any non-dominated Pareto-improving scheme. As a result, the joint road pricing and capacity enhancement problem reduces to a bi-objective second-best road pricing problem. A revenue distribution mechanism with return rate guarantee is proposed to implement the non-dominated Pareto-improving schemes.     

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.     

Pareto-improving congestion pricing and revenue refunding with multiple user classes

This study investigates Pareto-improving congestion pricing and revenue refunding schemes in general transportation networks, which make every road user better off as compared with the situation without congestion pricing. We consider user heterogeneity in value of time (VOT) by adopting a multiclass user model with fixed origin–destination (OD) demands. We first prove that an OD and class-based Pareto-improving refunding scheme exists if and only if the total system monetary travel disutility is reduced. In view of the practical difficulty in identifying individual user’s VOT, we further investigate class-anonymous refunding schemes that give the same amount of refund to all user classes traveling between the same OD pair regardless of their VOTs. We establish a sufficient condition for the existence of such OD-specific but class-anonymous Pareto-improving refunding schemes, which needs information only on the average toll paid and average travel time for trips between each OD pair.     

On some traffic equilibrium theory paradoxes

We consider the asymmetric equilibrium problem with fixed demands in a transportation network where the travel cost on each link may depend on the flow on this as well as other links of the network and we study how the travellers' cost is affected by changes in the travel demand or addition of new routes. Assuming that the travel cost functions are strongly monotone, we derive formulas which express, under certain conditions, how a change in travel demand associated with a particular origin-destination (O / D) pair will affect the travelers' cost for any O / D pair. We then use these formulas to show that an increase in the travel demand associated with a particular O / D pair (all other remaining fixed) always results in an increase in the travelers' cost on that O / D pair, however, the travelers' cost on other O / D pairs may decrease. We then derive formulas yielding, under certain conditions, the change in travelers' cost on every O / D pair induced by the addition of a new path. These can be used to determine, whether Braess' paradox occurs in the network. We then show that when a new path is added, the travelers' cost associated with the particular O / D pair joined by this path will decrease (hence Braess' paradox does not occur) if a test matrix is positive semidefinite.     

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.     

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.     

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.     

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.     

Validation of a schedule‐based capacity restraint transit assignment model for a large‐scale network

This paper describes the application of a capacity restraint trip assignment algorithm to a real, large‐scale transit network and the validation of the results. Unlike the conventional frequency‐based approach, the network formulation of the proposed model is dynamic and schedule‐based. Transit vehicles are assumed to operate to a set of pre‐determined schedules. Passengers are assumed to select paths based on a generalized cost function including in‐vehicle and out‐of‐vehicle time and line change penalty. The time‐varying passenger demand is loaded onto the network by a time increment simulation method, which ensures that the capacity restraint of each vehicle during passenger boarding is strictly observed. The optimal‐path and path‐loading algorithms are applied iteratively by the method of successive averages until the network converges to the predictive dynamic user equilibrium. The Hong Kong Mass Transit Railway network is used to validate the model results. The potential applications of the model are also discussed.     

Engineering condition assessment of cycling infrastructure: Cyclists’ perceptions of satisfaction and comfort

The UK National Cycle Network comprises 23,660 km of cycling and walking paths of which a significant percentage is dedicated off-road infrastructure. This represents a significant civil engineering infrastructure asset that currently contributes to the provision of a sustainable transport mode option nationwide. Commuting and recreational cyclists have observed the often hazardous conditions on these paths. There are various simple measures that could be taken to improve the maintenance of such off-road paths. Reliance on walk-over surveys (direct visual inspection) and path users notifying the local authority may not be tackling maintenance in a resource efficient manner. The proposed inspection method includes the use of an instrumented bicycle to examine cycle path condition through user perception of satisfaction and quality. A questionnaire was conducted to identify the attributes of off-road cycling infrastructure people find most important in relation to their personal satisfaction. An exploratory factor analysis was undertaken on perception study data to elucidate the determination of the variables associated with perceived user satisfaction. The study has shown that people find maintenance issues to be of high importance, especially surface issues. From exploratory factor analysis of results, satisfaction has been found to load with comfort and safety. Field testing was then conducted using subjective user opinions and objective vibration data. These results were then used to assist the creation of dedicated user perception based surface condition rating-scales.     

Auction-based highway reservation system an agent-based simulation study

Based on previous studies of a highway reservation system, this paper proposed an auction-based implementation, in which the users can bid for the right to use a route during a certain period of time. This paper models the auction system with MATSim using an agent-based simulation technique. The agents adopt their own bidding logic in the auction, and the price converges after around 130 iterations, when the number of users using the reserved highway and the total collected revenue become stable. When the overall demand changes, the collected revenue ranges from 5 to 11 dollars per user, and from 0.7 to 1.5 dollars per mile. The auction system can transfer more consumer surplus to the toll road operators, since it is a personalized tolling mechanism. The users are using the reservation system as insurance of a guaranteed congestion-free travel. The auction-based highway reservation shows great potential as a new traffic management system.     

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.     

Morning commute problem considering route choice, user heterogeneity and alternative system optima

This paper extends the bottleneck model to study congestion behavior of morning commute and its implications to transportation economics. The proposed model considers simultaneous route and departure time choices of heterogenous users who are distinguished by their valuation of travel time and punctual arrival. Moreover, two dynamic system optima are considered: one minimizes system cost in the unit of monetary value (i.e., the conventional system optimum, or SO) and the other minimizes system cost in the unit of travel time (i.e., the time-based SO, or TSO). Analytical solutions of no-toll equilibrium, SO and TSO are provided and the welfare effects of the corresponding dynamic congestion pricing options are examined, with and without route choice. The analyses suggest that TSO provides a Pareto-improving solution to the social inequity issue associated with SO. Although a TSO toll is generally discriminatory, anonymous TSO tolls do exist under certain circumstances. Unlike in the case with homogenous users, an SO toll generally alters users’ route choices by tolling the poorer users off the more desirable road, which worsens social inequity. Numerical examples are presented to verify analytical results.     

The existence, uniqueness and computation of an arc-based dynamic network user equilibrium formulation

In this paper, a dynamic user equilibrium traffic assignment model with simultaneous departure time/route choices and elastic demands is formulated as an arc-based nonlinear complementarity problem on congested traffic networks. The four objectives of this paper are (1) to develop an arc-based formulation which obviates the use of path-specific variables, (2) to establish existence of a dynamic user equilibrium solution to the model using Brouwer's fixed-point theorem, (3) to show that the vectors of total arc inflows and associated minimum unit travel costs are unique by imposing strict monotonicity conditions on the arc travel cost and demand functions along with a smoothness condition on the equilibria, and (4) to develop a heuristic algorithm that requires neither a path enumeration nor a storage of path-specific flow and cost information. Computational results are presented for a simple test network with 4 arcs, 3 nodes, and 2 origin–destination pairs over the time interval of 120 periods.     

Boundedly rational user equilibria (BRUE): Mathematical formulation and solution sets

Boundedly rational user equilibria (BRUE) represent traffic flow distribution patterns where travellers can take any route whose travel cost is within an ‘indifference band’ of the shortest path cost. Those traffic flow patterns satisfying the above condition constitute a set, named the BRUE solution set. It is important to obtain all the BRUE flow patterns, because it can help predict the variation of the link flow pattern in a traffic network under the boundedly rational behavior assumption. However, the methodology of constructing the BRUE set has been lacking in the established literature. This paper fills the gap by constructing the BRUE solution set on traffic networks with fixed demands. After defining ε-BRUE, where ε is the indifference band for the perceived travel cost, we formulate the ε-BRUE problem as a nonlinear complementarity problem (NCP), so that a BRUE solution can be obtained by solving a BRUE–NCP formulation. To obtain the BRUE solution set encompassing all BRUE flow patterns, we propose a methodology of generating acceptable path set which may be utilized under the boundedly rational behavior assumption. We show that with the increase of the indifference band, the acceptable path set that contains boundedly rational equilibrium flows will be augmented, and the critical values of indifference band to augment these path sets can be identified by solving a family of mathematical programs with equilibrium constraints (MPEC) sequentially. The BRUE solution set can then be obtained by assigning all traffic demands to the acceptable path set. Various numerical examples are given to illustrate our findings.     

Walking infrastructure design assessment by continuous space dynamic assignment modeling

For the planning and design of walking infrastructure, characterized by the fact that the pedestrians can choose their paths freely in two‐dimensional space, applicability of traditional discrete network models is limited. This contribution puts forward an approach for user‐optimal dynamic assignment in continuous time and space for analyzing for instance walking infrastructure in a two‐dimensional space. Contrary to network‐based approaches, the theory allows the traffic units to choose from an infinite non‐countable set of paths through the considered space. The approach first determines the continuous paths using a path choice model. Then, origin‐destination flows are assigned and traffic conditions are calculated. The approach to determine a user‐optimal assignment is heuristic and consists of a sequence of all‐or‐nothing assignments. An application example is presented, showing dynamic user equilibrium traffic flows through a realistic transfer station. The example is aimed at illustrating the dynamic aspects of the modeling approach, such as anticipation on expected flow conditions, and predicted behavior upon catching or missing a connection.     

Exploration of day-to-day route choice models by a virtual experiment

This paper examines existing day-to-day models based on a virtual day-to-day route choice experiment using the latest mobile Internet technologies. With the realized day-to-day path flows and path travel times in the experiment, we calibrate several well-designed path-based day-to-day models that take the Wardrop’s user equilibrium as (part of) their stationary states. The nonlinear effects of path flows and path time differences on path switching are then investigated. Participants’ path preferences, time-varying sensitivity, and learning behavior in the day-to-day process are also examined. The prediction power of various models with various settings (nonlinear effects, time-varying sensitivity, and learning) is compared. The assumption of “rational behavior adjustment process” in Yang and Zhang (2009) is further verified. Finally, evolutions of different Lyapunov functions used in the literature are plotted, and no obvious diversity is observed.