A route-based solution algorithm for dynamic user equilibrium assignments

The aim of this study is to establish a method to calculate good quality user equilibrium assignments under time varying conditions. For this purpose, it introduces a dynamic network loading method that can maintain correct flow propagation as well as flow conservation, and it shows a novel route-based solution algorithm. This novel algorithm turns out to be convenient and logically plausible compared to the conventional [Frank, M., Wolfe, P., 1956. An algorithm for quadratic programming. Naval Research Logistics Quarterly 3, 95–110] algorithm, because the former does not require evaluation of an objective function and it finds solutions maintaining correct flow propagation in the time-varying network conditions. The application of novel dynamic network loading method and solution algorithm to test networks shows that we can find high quality dynamic user equilibrium assignment. This is illustrated in an example network using the deterministic queuing model for a link performance function and associating costs and flows in a predictive way in discrete time.     

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.     

Computing dynamic user equilibria for large-scale transportation networks

In this paper, we present an approach for determining dynamic user equilibria. The method is suitable for disaggregated microscopic and mesoscopic simulation-based models. It is a modification of the convex-simplex method, which disposes with the line search step, and controls the subset of travelers to be re-routed at each step while updating the link travel times after each assignment. To guarantee finite termination, a suitable stopping criterion is adopted. The proposed method is implemented within TRANSIMS, the Transportation Analysis and Simulation System, as a two-stage process that employs a combined use of link performance functions and a microsimulator in order to design a framework suitable for application to real transportation systems. To demonstrate this capability, we apply the developed methodology to a large-scale network, Bignet, which is part of the transportation city network of Portland, Oregon; and a medium-scale network, Blacksburg, Virginia; and provide some comparative analyses. Our results exhibit that an improved distribution of travelers is obtained while consuming less than 17–33% of the effort required by the current version of TRANSIMS.     

Existence of simultaneous route and departure choice dynamic user equilibrium

This paper is concerned with the existence of the simultaneous route-and-departure choice dynamic user equilibrium (SRDC-DUE) in continuous time, first formulated as an infinite-dimensional variational inequality in Friesz et al. (1993). In deriving our existence result, we employ the generalized Vickrey model (GVM) introduced in Han et al., 2013a, Han et al., 2013b to formulate the underlying network loading problem. As we explain, the GVM corresponds to a path delay operator that is provably strongly continuous on the Hilbert space of interest. Finally, we provide the desired SRDC-DUE existence result for general constraints relating path flows to a table of fixed trip volumes without invocation of a priori bounds on the path flows.     

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.     

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.     

Modeling and solving discrete network design problem with stochastic user equilibrium

In this paper, we address the discrete network design problem, which determines the addition of new roads to existing transportation network to optimize the transportation system performance. Road users are assumed to follow the traffic assignment principle of stochastic user equilibrium. A mixed‐integer nonlinear nonconvex problem is developed to model this discrete network design problem with stochastic user equilibrium. The original problem is relaxed into a convex mixed‐integer nonlinear program, whose solution provides a lower bound of the original problem. The relaxed problem is then embedded into two proposed global optimization solution algorithms to obtain the global optimal solution of the problem. Copyright © 2016 John Wiley & Sons, Ltd.     

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

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

Global optimization method for network design problem with stochastic user equilibrium

In this paper, we consider the continuous road network design problem with stochastic user equilibrium constraint that aims to optimize the network performance via road capacity expansion. The network flow pattern is subject to stochastic user equilibrium, specifically, the logit route choice model. The resulting formulation, a nonlinear nonconvex programming problem, is firstly transformed into a nonlinear program with only logarithmic functions as nonlinear terms, for which a tight linear programming relaxation is derived by using an outer-approximation technique. The linear programming relaxation is then embedded within a global optimization solution algorithm based on range reduction technique, and the proposed approach is proved to converge to a global optimum.     

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.     

Direct formulation and algorithms for the probit-based stochastic user equilibrium traffic assignment problem

This paper proposes simple and direct formulation and algorithms for the probit-based stochastic user equilibrium traffic assignment problem. It is only necessary to account for random variables independent of link flows by performing a simple transformation of the perceived link travel time with a normal distribution. At every iteration of a Monte-Carlo simulation procedure, the values of the random variables are sampled based on their probability distributions, and then a regular deterministic user equilibrium assignment is carried out to produce link flows. The link flows produced at each iteration of the Monte-Carlo simulation are averaged to yield the final flow pattern. Two test networks demonstrate that the proposed algorithms and the traditional algorithm (the Method of Successive Averages) produce similar results and that the proposed algorithms can be extended to the computation of the case in which the random error term depends on measured travel time.     

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.     

Competition and disruption in a dynamic urban supply chain

Rapid changes and complexities in business environments have stressed the importance of interactions between partners and competitors, leading supply chains to become the most important element of contemporary business environments. There is a concomitant need for foresight in describing supply chain performance in all operating environments, including those involving punctuated disruptions. Furthermore, the urban metropolis is now widely recognized to be an environment which is especially vulnerable to supply chain disruptions and for which integrated supply chain decisions can produce very substantial net benefits. Accordingly, this paper presents a dynamic supply chain network model formulated as a differential variational inequality; the model is fashioned to allow consideration of supply chain disruption threats to producers, freight carriers, and retail enterprises. The DVI is solved using a fixed-point algorithm, and a simple numerical example, introduced to illustrate how the impacts of supply chain disruptions may be quantified, is presented.     

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.     

Improving the efficiency of repeated dynamic network loading through marginal simulation

Currently, the applicability of macroscopic Dynamic Network Loading (DNL) models for large-scale problems such as network-wide traffic management, reliability and vulnerability studies, network design, traffic flow optimization and dynamic origin–destination (OD) estimation is computationally problematic. The main reason is that these applications require a large number of DNL runs to be performed. Marginal DNL simulation, introduced in this paper, exploits the fact that the successive simulations often exhibit a large overlap. Through marginal simulation, repeated DNL simulations can be performed much faster by approximating each simulation as a variation to a base scenario. Thus, repetition of identical calculations is largely avoided. The marginal DNL algorithm that is presented, the Marginal Computation (MaC) algorithm, is based on first order kinematic wave theory. Hence, it realistically captures congestion dynamics. MaC can simulate both demand and supply variations, making it useful for a wide range of DNL applications. Case studies on different types of networks are presented to illustrate its performance.     

Multiclass fuzzy user equilibrium with endogenous membership functions and risk‐taking behaviors

Over the last decades, several approaches have been proposed in the literature to incorporate users' perceptions of travel costs, their bounded rationality, and risk‐taking behaviors into network equilibrium modeling for traffic assignment problem. While theoretically advanced, these models often suffer from high complexity and computational cost and often involve parameters that are difficult to estimate. This study proposes an alternative approach where users' imprecise perceptions of travel times are endogenously constructed as fuzzy sets based on the probability distributions of random link travel times. Two decision rules are proposed accordingly to account for users' heterogeneous risk‐taking behaviors, that is, optimistic and pessimistic rules. The proposed approach, namely, the multiclass fuzzy user equilibrium, can be formulated as a link‐based variational inequality model. The model can be solved efficiently, and parameters involved can be either easily estimated or treated as factors for calibration against observed traffic flow data. Numerical examples show that the proposed model can be solved efficiently even for a large‐scale network of Mashhad, Iran, with 2538 links and 7157 origin–destination pairs. The example also illustrates the calibration capability of the proposed model, highlighting that the model is able to produce much more accurate flow estimates compared with the Wardropian user equilibrium model. Copyright © 2016 John Wiley & Sons, Ltd.     

Formulating the within-day dynamic stochastic traffic assignment problem from a Bayesian perspective

This study proposes a formulation of the within-day dynamic stochastic traffic assignment problem. Considering the stochastic nature of route choice behavior, we treat the solution to the assignment problem as the conditional joint distribution of route traffic, given that the network is in dynamic stochastic user equilibrium. We acquire the conditional joint probability distribution using Bayes’ theorem. A Metropolis–Hastings sampling scheme is developed to estimate the characteristics (e.g., mean and variance) of the route traffic. The proposed formulation has no special requirements for the traffic flow models and user behavior models, and so is easily implemented.     

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.     

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.     

From user equilibrium to system optimum: a literature review on the role of travel information,bounded rationality and non-selfish behaviour at the network and individual levels

Travel information continues to receive significant attention in the field of travel behaviour research, as it is expected to help reduce congestion by directing the network state from a user equilibrium towards a more efficient system optimum. This literature review contributes to the existing literature in at least two ways. First, it considers both the individual perspective and the network perspective when assessing the potential effects of travel information, in contrast to earlier studies. Secondly, it highlights the role of bounded rationality as well as that of non-selfish behaviour in route choice and in response to information, complementing earlier reviews that mostly focused on bounded rationality only. It is concluded that information strategies should be tailor-made to an individual's level of rationality as well as level of selfishness in order to approach system-optimal conditions on the network level. Moreover, initial ideas and future research directions are provided for assessing the potential of travel information in order to improve network efficiency of existing road networks.