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.     

Impact analysis of cordon-based congestion pricing on mode-split for a bimodal transportation network

This paper investigates the impact of cordon-based congestion pricing scheme on the mode-split of a bimodal transportation network with auto and rail travel modes. For any given toll-charge pattern, its impact on the mode-split can be estimated by solving a combined mode-split and traffic-assignment problem. Using a binary logit model for the mode-split, the combined problem is converted into a traffic-assignment problem with elastic demand. Probit-based stochastic user equilibrium (SUE) principle is adopted for this traffic-assignment problem, and a continuously distributed value of time (VOT) is assumed to convert the toll charges and transit fares into time-units. This combined mode-split and traffic-assignment problem is then formulated as a fixed-point model, which can be solved by a convergent Cost Averaging method. The combined mode-split and traffic-assignment problem is then used to analyze a multimodal toll design problem for cordon-based congestion pricing scheme, with the aim of increasing the mode-share of public transport system to a targeted level. Taking the fixed-point model as a constraint, the multimodal toll design problem is thus formulated as a mathematical programming with equilibrium constraints (MPEC) model. A genetic algorithm (GA) is employed to solve this MPEC model, which is then numerical validated by a network example.     

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.     

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.     

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.     

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.     

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.     

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.     

Optimization of transport investment and pricing policies: the role of transport pricing in network design

This paper investigates the role of transport pricing in network design and describes two facts about flow pattern in a transportation system. The first, illustrated by an example of Braess paradox, is that adding a new link to the network does not necessarily minimize the total travel time. The second is that introducing of appropriate toll pricing may reduce not only the total network time but also the travel time for each individual traveller. It follows with the investigations of different system objectives and different pricing policies (only toll pricing and distance‐based pricing are considered), and shows how they affect the system performance and flow pattern. Lastly, a systematic optimization process is proposed for integrated planning of transport network and pricing policies.     

Second best toll pricing within the framework of bounded rationality

The network design problem is usually formulated as a bi-level program, assuming the user equilibrium is attained in the lower level program. Given boundedly rational route choice behavior, the lower-level program is replaced with the boundedly rational user equilibria (BRUE). The network design problem with boundedly rational route choice behavior is understudied due to non-uniqueness of the BRUE. In this study, thus, we mainly focus on boundedly rational toll pricing (BR-TP) with affine link cost functions. The topological properties of the lower level BRUE set are first explored. As the BRUE solution is generally non-unique, urban planners cannot predict exactly which equilibrium flow pattern the transportation network will operate after a planning strategy is implemented. Due to the risk caused by uncertainty of people’s reaction, two extreme scenarios are considered: the traffic flow patterns with either the minimum system travel cost or the maximum, which is the “risk-prone” (BR-TP-RP) or the “risk-averse” (BR-TP-RA) scenario respectively. The upper level BR-TP is to find an optimal toll minimizing the total system travel cost, while the lower level is to find the best or the worst scenario. Accordingly BR-TP can be formulated as either a min –min or a min –max program. Solution existence is discussed based on the topological properties of the BRUE and algorithms are proposed. Two examples are accompanied to illustrate the proposed methodology.     

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.     

Examining the scaling effect and overlapping problem in logit-based stochastic user equilibrium models

The purpose of this paper is to examine the scaling effect and overlapping problem in a route choice context using the logit-based stochastic user equilibrium (SUE) principle to explicitly account for the congestion effect. Numerical experiments are performed on nine models: the deterministic user equilibrium model, the multinomial logit SUE model with and without scaling, the C-logit SUE model with and without scaling, the path-size logit SUE model with and without scaling, and the paired combinatorial logit SUE model with and without scaling. Sensitivity analysis is conducted to examine the effects of route sets, congestion levels, dispersion intensities, and network asymmetries. A real transportation network in the City of Winnipeg, Canada is also used to compare the network equilibrium flow allocations of different SUE models. The results of the sensitivity analysis and the Winnipeg network reveal that both scaling effect and overlapping problem can have a significant impact on the network equilibrium flow allocations.     

Traffic rationing and pricing in a linear monocentric city

This paper presents a simple spatial equilibrium model for a linear monocentric city to investigate the effects of rationing and pricing on morning commuters' travel cost and modal choice behavior in each location. Under rationing and pricing, every day in the morning peak hour, each commuter is classified as either “free” or “rationed”. “Free” commuters are allowed to use the highway without paying the toll, whereas “rationed” commuters can avoid the toll only if they travel by transit. Each day, a fraction of commuters are rationed in their free use of the highway, and the rationing fractions are determined systematically so that everyone is equally rationed in a given period. It is found that Pareto‐improving rationing and pricing scheme might be obtained as a combination of the rationing degree and the toll associated with rationing. Extension to the rationing and pricing scheme with cordon and park‐and‐ride service has been made. Cordon and park‐and‐ride might help in improving the efficiency of rationing and pricing strategy although remains its Pareto‐improving property. Copyright © 2012 John Wiley & Sons, Ltd.     

Modeling absolute and relative cost differences in stochastic user equilibrium problem

This paper aims to develop a hybrid closed-form route choice model and the corresponding stochastic user equilibrium (SUE) to alleviate the drawbacks of both Logit and Weibit models by simultaneously considering absolute cost difference and relative cost difference in travelers’ route choice decisions. The model development is based on an observation that the issues of absolute and relative cost differences are analogous to the negative exponential and power impedance functions of the trip distribution gravity model. Some theoretical properties of the hybrid model are also examined, such as the probability relationship among the three models, independence from irrelevant alternatives, and direct and indirect elasticities. To consider the congestion effect, we provide a unified modeling framework to formulate the Logit, Weibit and hybrid SUE models with the same entropy maximization objective but with different total cost constraint specifications representing the modelers’ knowledge of the system. With this, there are two ways to interpret the dual variable associated with the cost constraint: shadow price representing the marginal change in the entropy level to a marginal change in the total cost, and dispersion/shape parameter representing the travelers’ perceptions of travel costs. To further consider the route overlapping effect, a path-size factor is incorporated into the hybrid SUE model. Numerical examples are also provided to illustrate the capability of the hybrid model in handling both absolute and relative cost differences as well as the route overlapping problem in travelers’ route choice decisions.     

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.     

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.     

Road Pricing models with maintenance cost

According to the Federal Highway Administration of the United States, maintenance expenditure takes up more than 25% of road revenue disbursement and this percentage has been increasing gradually. The reason for the increment in maintenance cost is that there lacks incentives for road users to take this cost component into their driving behavior. That is, different classes of vehicles should be levied different levels of congestion tax due to the different degrees of damage on the highway if a road pricing policy is implemented. This paper intends to incorporate this concept into road pricing literature by introducing two types of vehicles. After the analysis of the problem, we find that different types of vehicles should be charged different tolls. The toll includes not only the travel delay cost of one's own vehicle and the other types of vehicles, but also the marginal maintenance cost that is dependent on the traffic flow. A set of numerical examples is provided to demonstrate the theoretical analyses. The result shows that both the welfare and cost coverage rate will increase when the road pricing mechanism takes the maintenance cost factor into account.     

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.     

Traffic congestion pricing methodologies and technologies

This paper reviews the methods and technologies for congestion pricing of roads. Congestion tolls can be implemented at scales ranging from individual lanes on single links to national road networks. Tolls can be differentiated by time of day, road type and vehicle characteristics, and even set in real time according to current traffic conditions. Conventional toll booths have largely given way to electronic toll collection technologies. The main technology categories are roadside-only systems employing digital photography, tag & beacon systems that use short-range microwave technology, and in-vehicle-only systems based on either satellite or cellular network communications. The best technology choice depends on the application. The rate at which congestion pricing is implemented, and its ultimate scope, will depend on what technology is used and on what other functions and services it can perform.     

State-dependent congestion pricing with reference-dependent preferences

Demand and capacity fluctuations are common for roads and other congestible facilities. With ongoing advances in pricing technology and ways of communicating information to prospective users, state-dependent congestion pricing is becoming practical. But it is still rare or nonexistent in many potential applications. One explanation is that people dislike uncertainty about how much they will pay. To explore this idea, a model of reference-dependent preferences is developed based on Köszegi and Rabin (2006). Using a facility yields an “intrinsic” utility and a “gain-loss” utility measured relative to the probability distribution over states of utility outcomes. Two types of preferences are analyzed: bundled preferences in which gains and losses are perceived for overall utility, and unbundled preferences in which gains and losses are perceived separately for the toll and other determinants of utility.Tolls are chosen to maximize total expected utility plus revenues. With bundled preferences the toll is set above the Pigouvian level when usage conditions are good, and below it when conditions are bad, in order to reduce fluctuations in utility. With unbundled preferences the direction of toll adjustment is less clear and depends on whether supply or demand is variable. For both types of preferences tolls are sensitive to the strength of gain-loss utility. If gain-loss utility is moderately strong, a state-independent toll can be optimal.