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.     

Trial-and-error method for congestion pricing scheme under side-constrained probit-based stochastic user equilibrium conditions

A toll pattern that can restrict link flows on the tolled links to some predetermined thresholds is named as effective toll solution, which can be theoretically obtained by solving a side-constraint traffic assignment problem. Considering the practical implementation, this paper investigates availability of an engineering-oriented trial-and-error method for the effective toll pattern of cordon-based congestion pricing scheme, under side-constrained probit-based stochastic user equilibrium (SUE) conditions. The trial-and-error method merely requires the observed traffic counts on each entry of the cordon. A minimization model for the side-constrained probit-based SUE problem with elastic demand is first proposed and it is shown that the effective toll solution equals to the product of value of time and optimal Lagrangian multipliers with respect to the side constraints. Then, employing the Lagrangian dual formulation of the minimization method, this paper has built a convergent trial-and-error method. The trial-and-error method is finally tested by a numerical example developed from the cordon-based congestion pricing scheme in Singapore.     

Taxis in road pricing zone: should they pay the congestion charge?

This study aims at investigating the impact and feasibility of charging taxis with toll fee in the pricing zone when designing congestion pricing scheme. A bi‐level programming model is developed to compare the maximum social welfares before and after the congestion charge is imposed on taxis. The lower level is a combined network equilibrium model formulated as a variational inequality program, which considers the logit‐based mode split, route choice, elastic demand, and vacant taxi distributions. The upper level is to maximize the social welfare when toll rates vary. The bi‐level problem can be solved by the genetic algorithm, whereas the lower level is solved by the block Gauss–Seidel decomposition approach together with the method of successive averages and diagonalization algorithm. An application with numerical examples is conducted to demonstrate the effectiveness of the proposed model and algorithm and to reveal some interesting findings. Copyright © 2014 John Wiley & Sons, Ltd.     

Optimal toll locations and toll levels in congestion pricing schemes: a case study of Stockholm

As congestion pricing has moved from theoretical ideas in the literature to real-world implementation, the need for decision support when designing pricing schemes has become evident. This paper deals with the problem of finding optimal toll levels and locations in a road traffic network and presents a case study of Stockholm. The optimisation problem of finding optimal toll levels, given a predetermined cordon, and the problem of finding both optimal toll locations and levels are presented, and previously developed heuristics are used for solving these problems. For the Stockholm case study, the possible welfare gains of optimising toll levels in the current cordon and optimising both toll locations and their corresponding toll levels are evaluated. It is shown that by tuning the toll levels in the current congestion pricing cordon used in Stockholm, the welfare gain can be increased significantly, and furthermore improved by allowing a toll on a major bypass highway. It is also shown that, by optimising both toll locations and levels, a congestion pricing scheme with welfare gain close to what can be achieved by marginal social cost pricing can be designed with tolls being located on only a quarter of the tollable links.     

Solving a discrete multimodal transportation network design problem

This paper investigates the multimodal network design problem (MMNDP) that optimizes the auto network expansion scheme and bus network design scheme in an integrated manner. The problem is formulated as a single-level mathematical program with complementarity constraints (MPCC). The decision variables, including the expanded capacity of auto links, the layout of bus routes, the fare levels and the route frequencies, are transformed into multiple sets of binary variables. The layout of transit routes is explicitly modeled using an alternative approach by introducing a set of complementarity constraints. The congestion interaction among different travel modes is captured by an asymmetric multimodal user equilibrium problem (MUE). An active-set algorithm is employed to deal with the MPCC, by sequentially solving a relaxed MMNDP and a scheme updating problem. Numerical tests on nine-node and Sioux Falls networks are performed to demonstrate the proposed model and algorithm.     

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.     

Robust optimization of distance-based tolls in a network considering stochastic day to day dynamics

This paper investigates the nonlinear distance-based congestion pricing in a network considering stochastic day-to-day dynamics. After an implementation/adjustment of a congestion pricing scheme, the network flows in a certain period of days are not on an equilibrium state, thus it is problematic to take the equilibrium-based indexes as the pricing objective. Therefore, the concept of robust optimization is taken for the congestion toll determination problem, which takes into account the network performance of each day. First, a minimax model which minimizes the maximum regret on each day is proposed. Taking as a constraint of the minimax model, a path-based day to day dynamics model under stochastic user equilibrium (SUE) constraints is discussed in this paper. It is difficult to solve this minimax model by exact algorithms because of the implicity of the flow map function. Hence, a two-phase artificial bee colony algorithm is developed to solve the proposed minimax regret model, of which the first phase solves the minimal expected total travel cost for each day and the second phase handles the minimax robust optimization problem. Finally, a numerical example is conducted to validate the proposed models and methods.     

Network user equilibrium of battery electric vehicles considering flow-dependent electricity consumption

This paper addresses the equilibrium traffic assignment problem involving battery electric vehicles (BEVs) with flow-dependent electricity consumption. Due to the limited driving range and the costly/time-consuming recharging process required by current BEVs, as well as the scarce availability of battery charging/swapping stations, BEV drivers usually experience fear that their batteries may run out of power en route. Therefore, when choosing routes, BEV drivers not only try to minimize their travel costs, but also have to consider the feasibility of their routes. Moreover, considering the potential impact of traffic congestion on the electricity consumption of BEVs, the feasibility of routes may be determined endogenously rather than exogenously. A set of user equilibrium (UE) conditions from the literature is first presented to describe the route choice behaviors of BEV drivers considering flow-dependent electricity consumption. The UE conditions are then formulated as a nonlinear complementarity model. The model is further formulated as a variational inequality (VI) model and is solved using an iterative solution procedure. Numerical examples are provided to demonstrate the proposed models and solution algorithms. Discussions of how to evaluate and improve the system performance with non-unique link flow distribution are offered. A robust congestion pricing model is formulated to obtain a pricing scheme that minimizes the system travel cost under the worst-case tolled flow distribution. Finally, a further extension of the mathematical formulation for the UE conditions is provided.     

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.     

Private road competition and equilibrium with traffic equilibrium constraints

Toll road competition is one of the important issues under a build-operate-transfer (BOT) scheme, which is being encountered nowadays in many cities. When there are two or more competing firms and each firm operates a competitive toll road, their profits are interrelated due to the competitors' choices and demand inter-dependence in the network. In this paper we develop game-theoretic approaches to the study of the road network, on which multiple toll roads are operated by competitive private firms. The strategic interactions and market equilibria among the private firms are analyzed both in determining their supply (road capacity) and price (toll level) over the network. The toll road competition problems in general traffic equilibrium networks are formulated as an equilibrium program with equilibrium constraints or bi-level variational inequalities. Heuristic solution methods are proposed and their convergences are demonstrated with simple network examples. It is shown that private pricing and competition can be both profitable and welfare-improving.     

A dynamic cordon pricing scheme combining the Macroscopic Fundamental Diagram and an agent-based traffic model

Pricing is considered an effective management policy to reduce traffic congestion in transportation networks. In this paper we combine a macroscopic model of traffic congestion in urban networks with an agent-based simulator to study congestion pricing schemes. The macroscopic model, which has been tested with real data in previous studies, represents an accurate and robust approach to model the dynamics of congestion. The agent-based simulator can reproduce the complexity of travel behavior in terms of travelers’ choices and heterogeneity. This integrated approach is superior to traditional pricing schemes. On one hand, traffic simulators (including car-following, lane-changing and route choice models) consider travel behavior, i.e. departure time choice, inelastic to the level of congestion. On the other hand, most congestion pricing models utilize supply models insensitive to demand fluctuations and non-stationary conditions. This is not consistent with the physics of traffic and the dynamics of congestion. Furthermore, works that integrate the above features in pricing models are assuming deterministic and homogeneous population characteristics. In this paper, we first demonstrate by case studies in Zurich urban road network, that the output of a agent-based simulator is consistent with the physics of traffic flow dynamics, as defined by a Macroscopic Fundamental Diagram (MFD). We then develop and apply a dynamic cordon-based congestion pricing scheme, in which tolls are controlled by an MFD. And we investigate the effectiveness of the proposed pricing scheme. Results show that by applying such a congestion pricing, (i) the savings of travel time at both aggregated and disaggregated level outweigh the costs of tolling, (ii) the congestion inside the cordon area is eased while no extra congestion is generated in the neighbor area outside the cordon, (iii) tolling has stronger impact on leisure-related activities than on work-related activities, as fewer agents who perform work-related activities changed their time plans. Future work can apply the same methodology to other network-based pricing schemes, such as area-based or distance-traveled-based pricing. Equity issues can be investigated more carefully, if provided with data such as income of agents. Value-of-time-dependent pricing schemes then can also be determined.     

Intermodal hub-and-spoke network design: Incorporating multiple stakeholders and multi-type containers

This paper develops a mathematical program with equilibrium constraints (MPEC) model for the intermodal hub-and-spoke network design (IHSND) problem with multiple stakeholders and multi-type containers. The model incorporates a parametric variational inequality (VI) that formulates the user equilibrium (UE) behavior of intermodal operators in route choice for any given network design decision of the network planner. The model also uses a cost function that is capable of reflecting the transition from scale economies to scale diseconomies in distinct flow regimes for carriers or hub operators, and a disutility function integrating actual transportation charges and congestion impacts for intermodal operators. To solve the MPEC model, a hybrid genetic algorithm (HGA) embedded with a diagonalization method for solving the parametric VI is proposed. Finally, the comparative analysis of the HGA and an exhaustive enumeration algorithm indicates a good performance of the HGA in terms of computational time and solution quality. The HGA is also applied to solve a large-scale problem to show the applicability of the proposed model and algorithm.     

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.     

Modeling intermodal equilibrium for bimodal transportation system design problems in a linear monocentric city

This paper investigates the intermodal equilibrium, road toll pricing, and bus system design issues in a congested highway corridor with two alternative modes - auto and bus - which share the same roadway along this corridor. On the basis of an in-depth analysis of the demand and supply sides of the bimodal transportation system, the mode choice equilibrium of travelers along the continuum corridor is first presented and formulated as an equivalent variational inequality problem. The solution properties of the bimodal continuum equilibrium formulation are analytically explored. Two models, which account for different infrastructure/system regulatory regimes (public and private), are then proposed. In the public regulatory model, the road toll location and charge level are simultaneously optimized together with the bus service fare and frequency. In the private regulatory model, the fare and frequency of bus services, which are operated by a profit-driven private operator, are optimized for exogenously given toll pricing schemes. Finally, an illustrative example is given to demonstrate the application of the proposed models. Sensitivity analysis of residential/household distribution along the corridor is carried out together with a comparison of four different toll pricing schemes (no toll, first best, distance based, and location based). Insightful findings are reported on the interrelationships among modal competition, market regulatory regimes, toll pricing schemes, and urban configurations as well as their implications in practice.     

Congestion pricing and roadspace rationing: an application to the San Francisco Bay Bridge corridor

This paper presents an empirical application of a congestion–alleviation strategy that C. Daganzo [Transportation Research B 29 (1995) 139–154] proposed as a “hybrid between rationing and pricing”. This strategy is applied to the San Francisco Bay Bridge corridor, in search of a practical and Pareto-improving solution to the Bridge's congestion. The work relies on a mode-split model for work trips across four different income groups residing in 459 origin zones, and it applies an equilibrium analysis based on Bridge performance. Results indicate that modal utilities (and thus choices) are sensitive to the specific combination of toll and rationing rate, as well as to the Bridge's travel-time (or performance) function, and the length of the congested section. Though no combination of tolls plus rationing rates was found to benefit all groups of travelers studied, further investigations may improve upon these results by refining some of the assumptions made here.     

Advanced pricing and rationing policies for large scale multimodal networks

The applying of simplified schemes, such as cordon pricing, as second-best solution to the toll network design problem is investigated here in the context of multiclass traffic assignment on multimodal networks. To this end a suitable equilibrium model has been developed, together with an efficient algorithm capable of simulating large scale networks in quite reasonable computer time. This model implements the theoretical framework proposed in a previous work on the toll optimization problem, where the validity of marginal cost pricing for the context at hand is stated. Application of the model to the real case of Rome shows us, not only that on multimodal networks a relevant share (up to 20%) of the maximum improvements in terms of social welfare achievable with marginal cost pricing can in fact be obtained through cordon pricing, but also that in practical terms rationing is a valid alternative to pricing, thus getting around some of the relevant questions (theoretical, technical, social) the latter raises. As a result we propose a practical method to analyze advanced pricing and rationing policies differentiated for user categories, which enables us to compare alternative operative solutions with an upper bound on social welfare based on a solid theoretical background.     

Joint road toll pricing and capacity development in discrete transport network design problem

The paper demonstrates a method to determine road network improvements that also involve the use of a road toll charge, taking the perspective of the government or authority. A general discrete network design problem with a road toll pricing scheme, to minimize the total travel time under a budget constraint, is proposed. This approach is taken in order to determine the appropriate level of road toll pricing whilst simultaneously addressing the need for capacity. The proposed approach is formulated as a bi-level programming problem. The optimal road capacity improvement and toll level scheme is investigated with respect to the available budget levels and toll revenues.     

A quantum evolutionary algorithm for the second-best congestion pricing problem in urban traffic networks

This paper investigates the congestion pricing problem in urban traffic networks. A first-best strategy, a second-best strategy for toll leveling in closed cordons and a second-best strategy for determining both toll levels and toll points are considered. The problem is known to be a mixed integer programming model and formulated as a bi-level optimization problem, with an objective of maximizing the social welfare. A method is presented to solve the problem, based on a novel metaheuristic algorithm, namely quantum evolutionary algorithm (QEA). To verify the proposed method, the widely used genetic algorithm (GA) is also applied to solve the problem. The problem is solved for a medium-size urban traffic network and the results of the QEA are compared against the conventional GA. Computational results show that the QEA outperforms the GA in solution quality.     

Efficiency and equity comparison of cordon- and area-based road pricing schemes using a trip-chain equilibrium model

This paper compares performances of cordon- and area-road pricing regimes on their social welfare benefit and equity impact. The key difference between the two systems is that the cordon charges travellers per crossing whereas the area scheme charges the travellers for an entry permit (e.g. per day). For the area licensing scheme, travellers may decide to pay or not to pay the toll depending on the proportion between their travel costs for the whole trip-chains during a valid period of the area license and the toll level. A static trip-chain equilibrium based model is adopted in the paper to provide a better evaluation of the area-based tolls on trip-chain demands. The paper proposes a modified Gini coefficient taking in account assumptions of revenue re-distribution to measure the spatial equity impact. The model is tested with the case study of the Utsunomiya city in Japan. The results demonstrate a higher level of optimal tolls and social welfare benefits of the area-based schemes compared to those of the cordon-based schemes. Different sizes of the charging boundary have significant influences on the scheme benefits. The tests also show an interesting result on the non-effect of the boundary design (for both charging types) on their equity impacts. However, when comparing between charging regimes it is clear that the area schemes generate more inequitable results.     

Design of income-equitable toll prices for high occupancy toll lanes in a single toll facility

Income inequity potentially exists under high occupancy toll (HOT) lanes whereby higher-income travelers may reap the benefits of the facility. An income-based multi-toll pricing approach is proposed for a single HOT lane facility in a network to maximize simultaneously the toll revenue and address the income equity concern, while ensuring a minimum level-of-service on the HOT lanes and that the toll prices do not exceed pre-specified thresholds. The problem is modeled as a bi-level optimization formulation. The upper level model maximizes revenue for the tolling authority subject to pre-specified upper bounds on tolls. The lower level model solves the stochastic user equilibrium problem. An agent-based solution approach is used to determine the toll prices by considering the tolling authority and commuters as agents. Results from numerical experiments indicate that a multi-toll pricing scheme is more equitable and can yield higher revenues compared to a single toll price scheme across travelers.