A link-node complementarity model and solution algorithm for dynamic user equilibria with exact flow propagations

In this paper, we propose a link-node complementarity model for the basic deterministic dynamic user equilibrium (DUE) problem with single-user-class and fixed demands. The model complements link-path formulations that have been widely studied for dynamic user equilibria. Under various dynamic network constraints, especially the exact flow propagation constraints, we show that the continuous-time dynamic user equilibrium problem can be formulated as an infinite dimensional mixed complementarity model. The continuous-time model can be further discretized as a finite dimensional non-linear complementarity problem (NCP). The proposed discrete-time model captures the exact flow propagation constraints that were usually approximated in previous studies. By associating link inflow at the beginning of a time interval to travel times at the end of the interval, the resulting discrete-time model is predictive rather than reactive. The solution existence and compactness condition for the proposed model is established under mild assumptions. The model is solved by an iterative algorithm with a relaxed NCP solved at each iteration. Numerical examples are provided to illustrate the proposed model and solution approach. We particularly show why predictive DUE is preferable to reactive DUE from an algorithmic perspective.     

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.     

Approximate network loading and dual-time-scale dynamic user equilibrium

In this paper we present a dual-time-scale formulation of dynamic user equilibrium (DUE) with demand evolution. Our formulation belongs to the problem class that Pang and Stewart (2008) refer to as differential variational inequalities. It combines the within-day time scale for which route and departure time choices fluctuate in continuous time with the day-to-day time scale for which demand evolves in discrete time steps. Our formulation is consistent with the often told story that drivers adjust their travel demands at the end of every day based on their congestion experience during one or more previous days. We show that analysis of the within-day assignment model is tremendously simplified by expressing dynamic user equilibrium as a differential variational inequality. We also show there is a class of day-to-day demand growth models that allow the dual-time-scale formulation to be decomposed by time-stepping to yield a sequence of continuous time, single-day, dynamic user equilibrium problems. To solve the single-day DUE problems arising during time-stepping, it is necessary to repeatedly solve a dynamic network loading problem. We observe that the network loading phase of DUE computation generally constitutes a differential algebraic equation (DAE) system, and we show that the DAE system for network loading based on the link delay model (LDM) of Friesz et al. (1993) may be approximated by a system of ordinary differential equations (ODEs). That system of ODEs, as we demonstrate, may be efficiently solved using traditional numerical methods for such problems. To compute an actual dynamic user equilibrium, we introduce a continuous time fixed-point algorithm and prove its convergence for effective path delay operators that allow a limited type of nonmonotone path delay. We show that our DUE algorithm is compatible with network loading based on the LDM and the cell transmission model (CTM) due to Daganzo (1995). We provide a numerical example based on the much studied Sioux Falls network.     

Dynamic user equilibrium with side constraints for a traffic network: Theoretical development and numerical solution algorithm

This paper investigates a traffic volume control scheme for a dynamic traffic network model which aims to ensure that traffic volumes on specified links do not exceed preferred levels. The problem is formulated as a dynamic user equilibrium problem with side constraints (DUE-SC) in which the side constraints represent the restrictions on the traffic volumes. Travelers choose their departure times and routes to minimize their generalized travel costs, which include early/late arrival penalties. An infinite-dimensional variational inequality (VI) is formulated to model the DUE-SC. Based on this VI formulation, we establish an existence result for the DUE-SC by showing that the VI admits at least one solution. To analyze the necessary condition for the DUE-SC, we restate the VI as an equivalent optimal control problem. The Lagrange multipliers associated with the side constraints as derived from the optimality condition of the DUE-SC provide the traffic volume control scheme. The control scheme can be interpreted as additional travel delays (either tolls or access delays) imposed upon drivers for using the controlled links. This additional delay term derived from the Lagrange multiplier is compared with its counterpart in a static user equilibrium assignment model. If the side constraint is chosen as the storage capacity of a link, the additional delay can be viewed as the effort needed to prevent the link from spillback. Under this circumstance, it is found that the flow is incompressible when the link traffic volume is equal to its storage capacity. An algorithm based on Euler’s discretization scheme and nonlinear programming is proposed to solve the DUE-SC. Numerical examples are presented to illustrate the mechanism of the proposed traffic volume control scheme.     

Multiple equilibria in a dynamic traffic network

This study provides an example in which the dynamic user equilibrium (DUE) assignment of a congested road network with bottlenecks is non-unique. In previous studies, the uniqueness of DUE assignments with the bottleneck model has been shown in limited cases such as single-origin and single-destination networks. Consequently, it is still an important issue whether or not uniqueness is a general property of DUE assignments. The present study describes a network in which multiple patterns of link travel time are found, thus providing a negative answer to this question. The network has a loopy structure with multiple bottlenecks and multiple origin-destination (OD) pairs. Given a certain demand pattern of departure times for vehicles leaving their origins, a non-convex set of equilibria with a non-unique pattern of link travel times is shown to exist.     

Multi-class percentile user equilibrium with flow-dependent stochasticity

Travelers often reserve a buffer time for trips sensitive to arrival time in order to hedge against the uncertainties in a transportation system. To model the effects of such behavior, travelers are assumed to choose routes to minimize the percentile travel time, i.e. the travel time budget that ensures their preferred probability of on-time arrival; in doing so, they drive the system to a percentile user equilibrium (UE), which can be viewed as an extension of the classic Wardrop equilibrium. The stochasticity in the supply of transportation are incorporated by modeling the service flow rate of each road segment as a random variable. Such stochasticity is flow-dependent in the sense that the probability density functions of these random variables, from which the distribution of link travel time are constructed, are specified endogenously with flow-dependent parameters. The percentile route travel time, obtained by directly convolving the link travel time distributions in this paper, is not available in closed form in general and has to be numerically evaluated. To reveal their structural properties, percentile UE solutions are examined in special cases and verified with numerical results. For the general multi-class percentile UE traffic assignment problem, a variational inequality formulation is given and solved using a route-based algorithm. The algorithm makes use of the diagonal elements in the Jacobian of percentile route travel time, which is approximated through recursive convolution. Preliminary numerical experiments indicate that the algorithm is able to achieve highly precise equilibrium solutions.     

General stochastic user equilibrium traffic assignment problem with link capacity constraints

This paper addresses a general stochastic user equilibrium (SUE) traffic assignment problem with link capacity constraints. It first proposes a novel linearly constrained minimization model in terms of path flows and then shows that any of its local minimums satisfies the generalized SUE conditions. As the objective function of the proposed model involves path‐specific delay functions without explicit mathematical expressions, its Lagrangian dual formulation is analyzed. On the basis of the Lagrangian dual model, a convergent Lagrangian dual method with a predetermined step size sequence is developed. This solution method merely invokes a subroutine at each iteration to perform a conventional SUE traffic assignment excluding link capacity constraints. Finally, two numerical examples are used to illustrate the proposed model and solution method.     

Algorithms for logit-based stochastic user equilibrium assignment

The paper proposes an efficient algorithm for determining the stochastic user equilibrium solution for logit-based loading. The commonly used Method of Successive Averages typically has a very slow convergence rate. The new algorithm described here uses Williams’ result [ Williams, (1977) On the formation of travel demand models and economic evaluation measures of user benefit. Environment and Planning 9A(3), 285–344] which enables the expected value of the perceived travel costs S_rs to be readily calculated for any flow vector x. This enables the value of the Sheffi and Powell, 1982 objective function [Sheffi, Y. and Powell, W. B. (1982) An algorithm for the equilibrium assignment problem with random link times. Networks 12(2), 191–207], and its gradient in any specified search direction, to be calculated. It is then shown how, at each iteration, an optimal step length along the search direction can be easily estimated, rather than using the pre-set step lengths, thus giving much faster convergence. The basic algorithm uses the standard search direction (towards the auxiliary solution). In addition the performance of two further versions of the algorithm are investigated, both of which use an optimal step length but alternative search directions, based on the Davidon–Fletcher–Powell function minimisation method. The first is an unconstrained and the second a constrained version. Comparisons are made of all three versions of the algorithm, using a number of test networks ranging from a simple three-link network to one with almost 3000 links. It is found that for all but the smallest network the version using the standard search direction gives the fastest rate of convergence. Extensions to allow for multiple user classes and elastic demand are also possible.     

Complementarity formulations for the cell transmission model based dynamic user equilibrium with departure time choice, elastic demand and user heterogeneity

In this paper we formulate the dynamic user equilibrium problem with an embedded cell transmission model on a network with a single OD pair, multiple parallel paths, multiple user classes with elastic demand. The formulation is based on ideas from complementarity theory. The travel time is estimated based on two methods which have different transportation applications: (1) maximum travel time and (2) average travel time. These travel time functions result in linear and non-linear complementarity formulations respectively. Solution existence and the properties of the formulations are rigorously analyzed. Extensive computational experiments are conducted to demonstrate the benefits of the proposed formulations on various test networks.     

Modeling user adoption of advanced traveler information systems: dynamic evolution and stationary equilibrium

This paper models the growth rate and the saturation market penetration level for advanced traveler information system (ATIS) products/services with heterogeneous drivers. The price of using and the benefit gained from ATIS services are considered two key factors in explaining the growth of adoption of ATIS products. The information benefit is measured as the travel time saving between equipped and unequipped drivers and evaluated by a mixed stochastic and deterministic network equilibrium model. A modified logistic type growth model is adopted to describe the cumulative adoption of ATIS products over years. The final stationary equilibrium level of ATIS market penetration is so determined that the value of the information provided will decline to the point at which no new users will find it advantageous to purchase that service. The endogenous growth and stationary equilibrium model of market penetration of ATIS services is useful for forecasting the growth process and the impacts on the system performance of ATIS.     

Robust Wardrop’s user equilibrium assignment under stochastic demand and supply: Expected residual minimization approach

Various models of traffic assignment under stochastic environment have been proposed recently, mainly by assuming different travelers’ behavior against uncertainties. This paper focuses on the expected residual minimization (ERM) model to provide a robust traffic assignment with an emphasis on the planner’s perspective. The model is further extended to obtain a stochastic prediction of the traffic volumes by the technique of path choice approach. We show theoretically the existence and the robustness of the ERM solution. In addition, we employ an improved solution algorithm for solving the ERM model. Numerical experiments are carried out to illustrate the characteristics of the proposed model, by comparing with other existing models.     

A dynamic traffic assignment model for a continuum transportation system

A predictive continuum dynamic user-optimal (PDUO-C) model is formulated in this study to investigate the dynamic characteristics of traffic flow and the corresponding route-choice behavior of travelers within a region with a dense urban road network. The modeled region is arbitrary in shape with a single central business district (CBD) and travelers continuously distributed over the region. Within this region, the road network is represented as a continuum and travelers patronize a two-dimensional continuum transportation system to travel to the CBD. The PDUO-C model is solved by a promising solution algorithm that includes elements of the finite volume method (FVM), the finite element method (FEM), and the explicit total variation diminishing Runge-Kutta (TVD-RK) time-stepping method. A numerical example is given to demonstrate the utility of the proposed model and the effectiveness of the solution algorithm in solving this PDUO-C problem.     

A cell-based Merchant-Nemhauser model for the system optimum dynamic traffic assignment problem

A cell-based variant of the Merchant-Nemhauser (M-N) model is proposed for the system optimum (SO) dynamic traffic assignment (DTA) problem. Once linearized and augmented with additional constraints to capture cross-cell interactions, the model becomes a linear program that embeds a relaxed cell transmission model (CTM) to propagate traffic. As a result, we show that CTM-type traffic dynamics can be derived from the original M-N model, when the exit-flow function is properly selected and discretized. The proposed cell-based M-N model has a simple constraint structure and cell network representation because all intersections and cells are treated uniformly. Path marginal costs are defined using a recursive formula that involves a subset of multipliers from the linear program. This definition is then employed to interpret the necessary condition, which is a dynamic extension of the Wardrop’s second principle. An algorithm is presented to solve the flow holding back problem that is known to exist in many discrete SO-DTA models. A numerical experiment is conducted to verify the proposed model and algorithm.     

A methodology to estimate travel time using dynamic traffic assignment (DTA) under incident conditions

This paper presents results from a research case study that examined the distribution of travel time of origin–destination (OD) pairs on a transportation network under incident conditions. Using a transportation simulation dynamic traffic assignment (DTA) model, incident on a transportation network is executed under normal conditions, incident conditions without traveler information availability, and incident conditions assuming that users had perfect knowledge of the incident conditions and could select paths to avoid the incident location. The results suggest that incidents have a different impact on different OD pairs. The results confirm that an effective traveler information system has the potential to ease the impacts of incident conditions network wide. Yet it is also important to note that the use of information may detriment some OD pairs while benefiting other OD pairs. The methodology demonstrated in this paper provides insights into the usefulness of embedding a fully calibrated DTA model into the analysis tools of a traffic management and information center.     

An improved Dial's algorithm for logit-based traffic assignment within a directed acyclic network

Abstract

Dial's algorithm is one of the most effective and popular procedures for a logit-type stochastic traffic assignment, as it does not require path enumeration over a network. However, a fundamental problem associated with the algorithm is its simple definition of ‘efficient paths’, which sometimes produces unrealistic flow patterns. In this paper, an improved algorithm based on the route extension coefficient is proposed in order to circumvent this problem, in which ‘efficient paths’ simultaneously consider link travel cost and minimum travel cost. Path enumeration is still not required and a similar computing efficiency with the original algorithm is guaranteed. A limitation of the algorithm is that it can only be applied to a directed acyclic network because a topological sorting algorithm is used to decide the order of the sequential calculation. A numerical example based on the Beijing subway network illustrates the effectiveness of the proposed algorithm. It is found that it is able to exclude most unrealistic paths, but include all reasonable paths when compared with path enumeration and the original Dial's algorithm.     

Revisiting Jiang’s dynamic continuum model for urban cities

Jiang et al. (Jiang, Y.Q., Wong, S.C., Ho, H.W., Zhang, P., Liu, R.X., Sumalee, A., 2011. A dynamic traffic assignment model for a continuum transportation system. Transportation Research Part B 45 (2), 343–363) proposed a predictive continuum dynamic user-optimaDUO-l to investigate the dynamic characteristics of traffic flow and the corresponding route-choice behavior of travelers. Their modeled region is a dense urban city that is arbitrary in shape and has a single central business district (CBD). However, we argue that the model is not well posed due to an inconsistency in the route-choice strategy under certain conditions. To overcome this inconsistency, we revisit the PDUO-C problem, and construct an improved path-choice strategy. The improved model consists of a conservation law to govern the density, in which the flow direction is determined by the improved path-choice strategy, and a Hamilton–Jacobi equation to compute the total travel cost. The simultaneous satisfaction of both equations can be treated as a fixed-point problem. A self-adaptive method of successive averages (MSA) is proposed to solve this fixed-point problem. This method can automatically determine the optimal MSA step size using the least squares approach. Numerical examples are used to demonstrate the effectiveness of the model and the solution algorithm.     

A combined activity/travel choice model for congested road networks with queues

This paper presents a combined activity/travel choice model and proposes a flow-swapping method for obtaining the model's dynamic user equilibrium solution on congested road network with queues. The activities of individuals are characterized by given temporal utility profiles. Three typical activities, which can be observed in morning peak period, namely at-home activity, non-work activity on the way from home to workplace and work-purpose activity, will be considered in the model. The former two activities always occur together with the third obligatory activity. These three activities constitute typical activity/travel patterns in time-space dimension. At the equilibrium, each combined activity/travel pattern, in terms of chosen location/route/departure time, should have identical generalized disutility (or utility) experienced actually. This equilibrium can be expressed as a discrete-time, finite-dimensional variational inequality formulation and then converted to an equivalent "zero-extreme value" minimization problem. An algorithm, which iteratively adjusts the non-work activity location, corresponding route and departure time choices to reach an extreme point of the minimization problem, is proposed. A numerical example with a capacity constrained network is used to illustrate the performance of the proposed model and solution algorithm.     

Stochastic cell transmission model (SCTM): A stochastic dynamic traffic model for traffic state surveillance and assignment

The paper proposes a first-order macroscopic stochastic dynamic traffic model, namely the stochastic cell transmission model (SCTM), to model traffic flow density on freeway segments with stochastic demand and supply. The SCTM consists of five operational modes corresponding to different congestion levels of the freeway segment. Each mode is formulated as a discrete time bilinear stochastic system. A set of probabilistic conditions is proposed to characterize the probability of occurrence of each mode. The overall effect of the five modes is estimated by the joint traffic density which is derived from the theory of finite mixture distribution. The SCTM captures not only the mean and standard deviation (SD) of density of the traffic flow, but also the propagation of SD over time and space. The SCTM is tested with a hypothetical freeway corridor simulation and an empirical study. The simulation results are compared against the means and SDs of traffic densities obtained from the Monte Carlo Simulation (MCS) of the modified cell transmission model (MCTM). An approximately two-miles freeway segment of Interstate 210 West (I-210W) in Los Ageles, Southern California, is chosen for the empirical study. Traffic data is obtained from the Performance Measurement System (PeMS). The stochastic parameters of the SCTM are calibrated against the flow-density empirical data of I-210W. Both the SCTM and the MCS of the MCTM are tested. A discussion of the computational efficiency and the accuracy issues of the two methods is provided based on the empirical results. Both the numerical simulation results and the empirical results confirm that the SCTM is capable of accurately estimating the means and SDs of the freeway densities as compared to the MCS.     

A multiclass cell transmission model for shared human and autonomous vehicle roads

Autonomous vehicles have the potential to improve link and intersection traffic behavior. Computer reaction times may admit reduced following headways and increase capacity and backwards wave speed. The degree of these improvements will depend on the proportion of autonomous vehicles in the network. To model arbitrary shared road scenarios, we develop a multiclass cell transmission model that admits variations in capacity and backwards wave speed in response to class proportions within each cell. The multiclass cell transmission model is shown to be consistent with the hydrodynamic theory. This paper then develops a car following model incorporating driver reaction time to predict capacity and backwards wave speed for multiclass scenarios. For intersection modeling, we adapt the legacy early method for intelligent traffic management (Bento et al., 2013) to general simulation-based dynamic traffic assignment models. Empirical results on a city network show that intersection controls are a major bottleneck in the model, and that the legacy early method improves over traffic signals when the autonomous vehicle proportion is sufficiently high.