System optimal dynamic traffic assignment: Properties and solution procedures in the case of a many-to-one network

Thanks to its high dimensionality and a usually non-convex constraint set, system optimal dynamic traffic assignment remains one of the most challenging problems in transportation research. This paper identifies two fundamental properties of the problem and uses them to design an efficient solution procedure. We first show that the non-convexity of the problem can be circumvented by first solving a relaxed problem and then applying a traffic holding elimination procedure to obtain the solution(s) of the original problem. To efficiently solve the relaxed problem, we explore the relationship between the relaxed problems based on different traffic flow models (PQ, SQ, CTM) and a minimal cost flow (MCF) problem for a special space-expansion network. It is shown that all the four problem formulations produce the same minimal system cost and share one common solution which does not involve inside queues in the network. Efficient solution algorithms such as the network simplex method can be applied to solve the MCF problem and identify such an optimal traffic pattern. Numerical examples are also presented to demonstrate the efficiency of the proposed solution procedure.     

A bi-level model for GHG emission charge based on a continuous distribution of travelers’ value of time (VOT)

Transportation is an important source of greenhouse gas (GHG) emissions. In this paper, we develop a bi-level model for GHG emission charge based on continuous distribution of the value of time (VOT) for travelers. In the bi-level model framework, a policy maker (as the leader) seeks an optimal emission charge scheme, with tolls differentiated across travel modes (e.g., bus, motorcycles, and cars), to achieve a given GHG reduction target by shifting the proportions of travelers taking different modes. In response, travelers (as followers) will adjust their travel modes to minimize their total travel cost. The resulting mode shift, hence the outcome of the emission charge policy, depends on travelers’ VOT distribution. For the solution of the bi-level model, we integrate a differential evolution algorithm for the upper level and the “all or nothing” traffic assignment for the lower level. Numerical results from our analysis suggest important policy implications: (1) in setting the optimal GHG emission charge scheme for the design of transportation GHG emission reduction targets, policy makers need to be equipped with rigorous understanding of travelers’ VOT distribution and the tradeoffs between emission reduction and system efficiency; and (2) the optimal emission charge scheme in a city depends significantly on the average value of travelers’ VOT distribution—the optimal emission charge can be designed and implemented in consistency with rational travel flows. Further sensitivity analysis considering various GHG reduction targets and different VOT distributions indicate that plausible emission toll schemes that encourage travelers to choose greener transportation modes can be explored as an efficient policy instrument for both transportation network performance improvement and GHG reduction.     

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.     

Network traffic flow optimization under performance constraints

In this paper, a model-based perimeter control policy for large-scale urban vehicular networks is proposed. Assuming a homogeneously loaded vehicle network and the existence of a well-posed Network Fundamental Diagram (NFD), we describe a protected network throughout its aggregated dynamics including nonlinear exit flow characteristics. Within this framework of constrained optimal boundary flow gating, two main performance metrics are considered: (a) first, connected to the NFD, the concept of average network travel time and delay as a performance metric is defined; (b) second, at boundaries, we take into account additional external network queue dynamics governed by uncontrolled inflow demands. External queue capacities in terms of finite-link lengths are used as the second performance metric. Hence, the corresponding performance requirement is an upper bound of external queues. While external queues represent vehicles waiting to enter the protected network, internal queue describes the protected network’s aggregated behavior.By controlling the number of vehicles joining the internal queue from the external ones, herewith a network traffic flow maximization solution subject to the internal and external dynamics and their performance constraints is developed. The originally non-convex optimization problem is transformed to a numerically efficiently convex one by relaxing the performance constraints into time-dependent state boundaries. The control solution can be interpreted as a mechanism which transforms the unknown arrival process governing the number of vehicles entering the network to a regulated process, such that prescribed performance requirements on travel time in the network and upper bound on the external queue are satisfied. Comparative numerical simulation studies on a microscopic traffic simulator are carried out to show the benefits of the proposed method.     

Fuzzy modeling approach to the rail freight car inventory problem

Techniques to improve freight car fleet use are of considerable interest to the railroad industry. In this paper, we present a fuzzy inventory control approach applied to the sizing of empty cars on a rail network. We address the problem of deciding the optimal inventory level and the optimal ordering quantity for a rail freight car fleet system in which demand and travel time are uncertain variables represented as triangular fuzzy numbers. Based on the fuzzy economic order quantity (EOQ) formula, a modified fuzzy EOQ model is set up and the optimal policy is developed using the signed distance method to defuzzify the fuzzy total cost. Computational results made for the Serbian rail network case verify the proposed model as well as the efficiency of the approach.     

Parking permits management and optimal parking supply considering traffic emission cost

Given a many-to-one bi-modal transportation network where each origin is connected to the destination by a bottleneck-constrained highway and a parallel transit line, we investigate the parking permit management methods to minimize traffic time cost and traffic emission cost simultaneously. More importantly, the optimal supply of parking spots is also discussed in the policies of parking permit. First, we derive the total travel costs and emission costs for the two cases of sufficient and insufficient parking spot provisions at the destination. Second, we propose a bi-objective model and solve the Pareto optimal parking permit distribution, given a certain level of parking supply. Third, we investigate the optimal parking supply in the policy of parking permit distribution, with the objectives of minimizing both total travel cost and traffic emission. Fourth, we provide a model of optimizing parking supply, in the policy of free trading of parking permits. Finally, the numerical examples are presented to illustrate the effectiveness of these schemes, and the numerical results show that restricting parking supply at the city center could be efficient to reduce traffic emission.     

A dynamic route choice model for public transport networks with boarding queues

Abstract

The concepts of optimal strategy and hyperpath were born within the framework of static frequency-based public transport assignment, where it is assumed that travel times and frequencies do not change over time and no overcrowding occurs. However, the formation of queues at public transport stops can prevent passengers from boarding the first vehicle approaching and can thus lead to additional delays in their trip. Assuming that passengers know from previous experience that for certain stops/lines they will have to wait for the arrival of the 2nd, 3rd, …, k-th vehicle, they may alter their route choices, thus resulting in a different assignment of flows across the network. The aim of this paper is to investigate route choice behaviour changes as a result of the formation and dispersion of queues at stops within the framework of optimal travel strategies. A new model is developed, based on modifications of existing algorithms.     

Reducing average comprehensive travel cost by rationally allocating trips to different travel modes

To study the effect of different transport policies on reducing the average comprehensive travel cost (CTC) of all travel modes, by increasing public transport modal share and decreasing car trips, an optimization model is developed based on travel cost utility. A nested logit model is applied to analyze trip modal split. A Genetic Algorithm is then used to determine the implementation of optimal solutions in which various transport policies are applied in order to reduce average CTC. The central urban region of Beijing is selected as the study area in this research. Different policies are analyzed for comparison, focusing on their optimal impacts on minimizing the average CTC utility of all travel modes by rationally allocating trips to different travel modes in the study area. It is found that the proposed optimization model provides a reasonable indication of the effect of policies applied.     

Determining structural route components from GPS traces

Analysis of GPS traces shows that people often do not use the least cost path through the transportation network while making trips. This leads to the question which structural path characteristics can be used to construct realistic route choice sets for use in traffic simulation models. In this paper, we investigate the hypothesis that, for utilitarian trips, the route between origin and destination consists of a small number of concatenated least cost paths. The hypothesis is verified by analyzing routes extracted from large sets of recorded GPS traces which constitute revealed preference information. Trips have been extracted from the traces and for each trip the path in the transportation network is determined by map matching. This is followed by a path decomposition phase for which the algorithm constitutes the first contribution of this paper. There are multiple ways to split a given path in a directed graph into a minimal number of subpaths of minimal cost. By calculating two specific path splittings, it is possible to identify subsets of the vertices (splitVertexSuites) that can be used to generate every possible minimum path splitting by taking one vertex from each such subset. As a second contribution, we show how the extracted information is used in microscopic travel simulation. The distribution for the size of the minimum decomposition, extracted from the GPS traces, can be used in constrained enumeration methods for route choice set generation. The sets of vertices that can act as boundary vertices separating consecutive route parts contain way points (landmarks) having a particular meaning to their user. The paper explains the theoretical aspects of route splitting as well as the process to extract splitVertexSuites from big data. It reports statistical distributions extracted from sets of GPS traces for both multimodal person movements and unimodal car trips.     

A planning model for multiple‐mode transportation system operations

The problem of generating a set of “good” transportation alternatives during the early and intermediate stages of transportation planning is addressed in this paper. A linear programming model of a multi‐modal transportation system is developed. The model is run interactively to determine optimal operating levels for all modes for various transport policy decisions. The model described is a component of a composite network generation model incorporating dynamic changes. The linear programming component determines optimal operating policies for given points in time. The composite model incorporates these in a dynamic programming framework to determine optimal staged investment policies over several time periods.     

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.     

Rationing and pricing strategies for congestion mitigation: Behavioral theory,econometric model,and application in Beijing

Some travel demand management policies such as road pricing have been widely studied in literature. Rationing policies, including vehicle ownership quota and vehicle usage restrictions, have been implemented in several megaregions to address congestion and other negative transportation externalities, but not well explored in literature. Other strategies such as Vehicle Mileage Fee have not been well accepted by policy makers, but attract growing research interest. As policy makers face an increasing number of policy tools, a theoretical framework is needed to analyze these policies and provide a direct comparison of their welfare implications such as efficiency and equity. However, such a comprehensive framework does not exist in literature. To bridge this gap, this study develops an analytical framework for analyzing and comparing travel demand management policies, which consists of a mathematical model of joint household vehicle ownership and usage decisions and welfare analysis methods based on compensating variation and consumer surplus. Under the assumptions of homogenous users and single time period, this study finds that vehicle usage rationing performs better when relatively small percentages of users (i.e. low rationing ratio) are rationed off the roads and when induced demand elasticity resulting from congestion mitigation is low. When the amount of induced demand exceeds a certain level, it is shown analytically that vehicle usage restrictions will always cause welfare losses. When the policy goal is to reduce vehicle travel by a fixed portion, road pricing provides a larger welfare gain. The performance of different policies is influenced by network congestion and congestibility. This paper further generalizes the model to consider heterogenous users and demonstrates how it can be applied for policy analysis on a real network after careful calibration.     

A multi-commodity flow network design problem

There exist systems which can be usefully described by a network containingarcs through which a commodity of one type flows. This paper is concerned with finding a solution procedure for a particular multi-commodity flow network design problem. The problem is to identify a set of arcs in the network such that if travel is prohibited in them all flow travels by feasible paths and its total cost is minimal. The total flow in each arc may not exced its capacity, which is a known constant. Each arc and each node of the network has a non-negative constant unit traversal cost. Between each pair of distinct nodes there is a given non-negative rate of flow from the first vertex to the second which may be split up among a number of paths according to some constant traversal cost flow assignment process. The optimality criterion is the total traversal cost of all flow, which is to be minimized. Previous work on network design problems of this type is surveyed. The principal contribution of this paper is the presentation of a solution procedure for the above problem based on branch and bound enumeration. An illustrative numerical example is included. Computational experience gained in using the procedure with a FORTRAN IV program on an IBM 370 is favourable.     

Network-wide adaptive tolling for connected and automated vehicles

This article proposes Δ-tolling, a simple adaptive pricing scheme which only requires travel time observations and two tuning parameters. These tolls are applied throughout a road network, and can be updated as frequently as travel time observations are made. Notably, Δ-tolling does not require any details of the traffic flow or travel demand models other than travel time observations, rendering it easy to apply in real-time. The flexibility of this tolling scheme is demonstrated in three specific traffic modeling contexts with varying traffic flow and user behavior assumptions: a day-to-day pricing model using static network equilibrium with link delay functions; a within-day adaptive pricing model using the cell transmission model and dynamic routing of vehicles; and a microsimulation of reservation-based intersection control for connected and autonomous vehicles with myopic routing. In all cases, Δ-tolling produces significant benefits over the no-toll case, measured in terms of average travel time and social welfare, while only requiring two parameters to be tuned. Some optimality results are also given for the special case of the static network equilibrium model with BPR-style delay functions.     

Travel mental budgeting under road toll: An investigation based on user equilibrium

With the approach of introducing the conceptions of mental account and mental budgeting into the process of travelers’ route choice, we try to identify why the usages of tolled roads are often overestimated. Assuming that every traveler sets a mental account for his/her travel to keep track of their expense and keep out-of-pocket spending under control, it addresses these questions such that “How much money can I spend on the travel?” and “What if I spend too much?”. Route tolls that exceed the budget are much more unacceptable compared to those within budget due to the non-fungibility of money between different accounts. A simple network with two nodes and two routes is analyzed firstly, the analytical solutions are obtained and the optimal road tolls supporting the user equilibrium as a system optimum are also derived. The proposed model is then extended to a generalized network. The multiclass user equilibrium conditions with travel mental budgeting are formulated into an equivalent variational inequality (VI) problem and an equivalent minimization problem. Through analyses with numerical examples, it is found that the main reason that the usages of high tolled roads are often overestimated is due to the fact that travelers with low and moderate out-of-pocket travel budget perceive a much higher travel cost than their actual cost on the high tolled roads.     

A practical method to estimate the benefits of improved road network reliability: an application to departing air passengers

This paper develops and applies a practical method to estimate the benefits of improved reliability of road networks. We present a general methodology to estimate the scheduling costs due to travel time variability for car travel. In contrast to existing practical methods, we explicitly consider the effect of travel time variability on departure time choices. We focus on situations when only mean delays are known, which is typically the case when standard transport models are used. We first show how travel time variability can be predicted from mean delays. We then estimate the scheduling costs of travellers, taking into account their optimal departure time choice given the estimated travel time variability. We illustrate the methodology for air passengers traveling by car to Amsterdam Schiphol Airport. We find that on average planned improvements in network reliability only lead to a small reduction in access costs per trip in absolute terms, mainly because most air passengers drive to the airport outside peak hours, when travel time variability tends to be low. However, in relative terms the reduction in access costs due to the improvements in network reliability is substantial. In our case we find that for every 1 Euro reduction in travel time costs, there is an additional cost reduction of 0.7 Euro due to lower travel time variability, and hence lower scheduling costs. Ignoring the benefits from improved reliability may therefore lead to a severe underestimation of the total benefits of infrastructure improvements.     

Agent-based network transmission model using the properties of macroscopic fundamental diagram

The field of research that has recently come to the fore is the perimeter control, which aims to control traffic demand for a large urban area prior to controlling internal flow inside the area. Such control concept needs to be tested by simulations, hence, it is necessary to develop a model that can appropriately estimate the network-wide flow dynamics. In this paper, agent-based network transmission model (ANTM) is proposed for describing the aggregated flow dynamics over an urban area of multiple large-scale networks. The proposed model is the combination of the cell transmission model (CTM), macroscopic fundamental diagram (MFD), and agent concept. The CTM-based simulation is adopted for the simplicity considering the computation requirements for real-time feasibility. The MFD concept is applied for representing the network properties, and a new approach is taken particularly for estimating network outflow affected by both demand patterns and boundary capacity. The agent concept is applied for representing drivers’ travel behaviors. The model is compared with microscopic simulations and shows reasonable accuracy for large areas. In addition, various travel direction choice behaviors are applicable to this model. Various perimeter control policies are applicable as well, thus, the proposed model can be a useful tool for testing various control methods, in terms of reducing the congestion in urban areas.     

Dynamic pricing for managed lanes with multiple entrances and exits

Priced managed lanes are increasingly being used to better utilize the existing capacity of the roadway to relieve congestion and offer reliable travel time to road users. In this paper, we investigate the optimization problem for pricing managed lanes with multiple entrances and exits which seeks to maximize the revenue and minimize the total system travel time (TSTT) over a finite horizon. We propose a lane choice model where travelers make online decisions at each diverge point considering all routes on a managed lane network. We formulate the problem as a deterministic Markov decision process and solve it using the value function approximation (VFA) method for different initializations. We compare the performance of the toll policies predicted by the VFA method against the myopic revenue policy which maximizes the revenue only at the current timestep and two heuristic policies based on the measured densities on the managed and general purpose lanes (GPLs). We test the results on four different test networks. The primary findings from our research suggest the usefulness of the VFA method for determining dynamic tolls. The best-found objective value from the method at its termination is better than other heuristics for all test networks with average improvements in the objective ranging between 10% and 90% for revenue maximization and 0–27% for TSTT minimization. Certain VFA initializations obtain best-found toll profiles within first 5–50 iterations which warrants computational time savings. Our findings also indicate that the revenue-maximizing optimal policies follow the “jam-and-harvest” behavior where the GPLs are pushed towards congestion in the earlier time steps to generate higher revenue in the later time steps, a characteristic not observed for the policies minimizing TSTT.     

Combining driver response with cyclical queue optimisation over selected corridors

A case study located in Auckland, New Zealand, was used to quantify the magnitude of savings that may result if the SCATS adaptive traffic control system contains an explicitly combined queue estimation and offset adjustment on a cycle‐by‐cycle basis. A validated SATURN traffic model was used to evaluate five scenarios that represent the short‐run and long‐run efficiency gains resulting from progressive signal adaption with an objective of queue minimisation on the main corridors. Optimisation was applied both area‐wide, and on selected arterial corridors, using a combined split/offset optimisation routine with responsive driver behaviour to achieve a network‐wide and corridor‐specific efficiency gain. The modelling heuristic evaluates the efficiency of both the Equisat and P₀ optimisation policies that would mimic a more progressive adaption of signals under SCATS. Results for the long‐run area‐wide optimisation can produce network‐wide travel‐time savings in the order of 20% and a reduction in transient queues of 28% if only selected corridors are optimised, with a 5% reduction in journey time over an average 8‐min journey. Copyright © 2016 John Wiley & Sons, Ltd.