Dynamic Stochastic User Equilibrium Research Articles

A nonlinear equation system approach to the dynamic stochastic user equilibrium simultaneous route and departure time choice problem

In dynamic stochastic user equilibrium simultaneous route and departure time choice (DSUE-SRDTC) problems, route travel costs can be non-monotone even if route travel times are monotone with respect to route flows. As a result, the mapping function of the variational inequality (VI) problems for the DSUE-SRDTC problems can be non-monotone, and many existing solution algorithms developed for the DSUE-SRDTC problems do not guarantee convergence under this non-monotone condition. This paper formulates the DSUE-SRDTC problem with fixed demand as a system of nonlinear equations. The mapping function of the proposed system of nonlinear equations is defined by a dynamic route choice problem, which can also be formulated as a VI problem with a strictly monotone mapping function under some assumptions. This property enables that the solution algorithm for the DSUE-SRDTC problem can avoid the requirement of the monotonicity of the route travel cost functions for the convergence of the solution procedure. A backtracking inexact Broyden–Fletcher–Goldfarb–Shanno (BFGS) method is adopted to solve the system of nonlinear equations, and iterative methods are developed to generate an initial solution for the BFGS method and solve the dynamic route choice problem. Finally, numerical examples are set up to show that the proposed method outperforms many existing algorithms for solving the DSUE-SRDTC problem in terms of guaranteeing solution convergence.

Better Convergence for Dynamic Traffic Assignment Methods

An essential feature in many of the dynamic traffic assignment methods that are used to study planning and traffic management problems is to compute the dynamic user equilibrium (DUE). In this paper the focus is on efficiently computing the deterministic and stochastic DUE. Both these types of equilibria are found using an iterative solution procedure. An important step in the solution algorithm for deterministic assignments is the choice for the contraction factor which determines how fast the related quadratic programming problem is solved. For stochastic assignments the step size to determine the starting point for the next iteration is important for convergence speed. As we show, the contraction factor and the step size heavily affect both the convergence efficiency and stability. We discuss various fixed contraction factors, a number of step size adjustments proposed by others, and a newly proposed dynamically adjusted step size. The solution method is evaluated on two transport networks of different scale. The comparative analysis suggests that, particularly for larger networks where equilibrium is harder to obtain, a dynamic adjusted contraction factor for deterministic assignments or a dynamic step size for stochastic assignments is preferable as it consistently converges considerably faster, because it does not suffer from a decreasing convergence rate.

Approximating dynamic equilibrium conditions with macroscopic fundamental diagrams

Real-time coordinated traffic management strategies that benefit from parsimonious models with aggregated network dynamics, provide a new generation of smart hierarchical strategies to improve network capacity and performance. However, this raises the question of route choice behavior in case of heterogeneous urban networks, where different parts of the city are subject to different types of control. Traffic equilibrium phenomena have not been thoroughly investigated in these models. Approximate traffic equilibrium conditions can be integrated within the parsimonious traffic models to develop regional routing strategies, while detailed route choice strategies can be incorporated at a later stage in a hierarchical framework. In this study, we develop an aggregated and approximate dynamic traffic assignment (DTA) procedure to be incorporated in the macroscopic fundamental diagram (MFD) dynamics, and establish dynamic stochastic user equilibrium (DSUE) conditions. The methodology consists of two main components; stochastic network loading and a fixed-point solution method. Loading procedure is designed to handle stochastic components in the model such as trip length uncertainty, variation of speeds across the links, perception error of travelers. The results taken from this procedure are averaged through the well-known method of successive averages (MSA) to reach fixed-point solution for the system. Real-time route guidance strategies can be revisited towards a “system of systems” approach.

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.

Dynamic pricing, heterogeneous users and perception error: Probit-based bi-criterion dynamic stochastic user equilibrium assignment

A probit-based bi-criterion dynamic stochastic user equilibrium (BDSUE) model is presented to capture path choice behavior of heterogeneous users with distinct values of time (VOT) and different perception of travel costs in response to pricing and congestion in a transportation network. Across the population of travelers, the VOT is represented by a continuously distributed random variable, and path travel cost perception errors are multivariate normally distributed. The BDSUE problem is formulated as a fixed point problem in the infinite dimensional space, and solved by a column generation framework which embeds (i) a parametric analysis method (PAM) to transform the continuous problem to the finite dimensional space by finding breakpoints that partition the entire range of VOT into subintervals and define a multi-class dynamic stochastic user equilibrium (MDSUE) problem; (ii) a column generation algorithm to augment a feasible path set for each user class; (iii) a probit-based stochastic path flow updating scheme to solve a restricted MDSUE problem defined by the set of feasible paths using an averaging method; and (iv) dynamic network loading using a particle-based traffic simulator to capture traffic dynamics and determine experienced travel times for a given path flow pattern. Numerical experiments on a medium size network are conducted to explore convergence of the solution algorithm and to illustrate heterogeneous user responses to dynamic tolls.

A Multi-mode, Multi-class Dynamic Network Model With Queues For Advanced Transportation Information Systems

In this paper we propose a composite Variational Inequality formulation for modeling multimode, multi-class stochastic dynamic user equilibrium problem in recurrent congestion networks with queues. The modes typically refer to different vehicle types such as passenger cars, trucks, and buses sharing the same road space. Each vehicle type has its own characteristics, such as free flow speed, vehicle size. We extend single mode deterministic point model to multimode deterministic point model for modeling the asymmetric interactions among various modes. Meanwhile, each mode of travelers is classified into two classes. Class 1 is equipped travelers following stochastic dynamic user-equilibrium with less uncertainty of travel cost, class 2 is unequipped travelers following stochastic dynamic user-equilibrium with more uncertainty of travel cost. A solution algorithm based on stochastic dynamic network loading for logit-based simultaneous route and departure time choices is adopted. Finally a numerical example is presented in a simple network.

Dynamic departure time and stochastic user equilibrium assignment

This paper investigates a logit-based combined departure time and dynamic stochastic user equilibrium assignment (DDSUE) problem and shows the existence of DDSUE condition as well as exploring its properties: a novel invariant quantity of DDSUE assignments is established. The relations between daily volume of traffic and cost incurred from travelling are also investigated. In order to calculate the equilibrium, we propose a solution algorithm that can solve the problem directly, and without appealing to an equivalent mathematical programme. Since the model is based on paths where all feasible paths are enumerated in advance, we propose an efficient method that works within a reasonable path set. Through two numerical examples, we show the DDSUE and some valuable results from the model such as relation between departure flow (or path inflow) and departure cost (or path cost). Sensitivity analysis of travel volumes with respect to travel costs and model parameters is also undertaken.

Temporal-Spatial Microassignment and Sequencing of Travel Demand with Activity-Trip Chains

A stochastic temporal–spatial microassignment and activity sequencing model for activity–trip chains is presented. In this model, trip-chain patterns are defined by the respective locations of destinations in the chain, preferred arrival times at these destinations, and the activity durations at the intermediate destinations; they are given as input to the model. A stochastic dynamic user equilibrium problem is formulated and solved for this purpose. In this problem, drivers simultaneously seek to determine their departure time, route choice, and sequence of their intermediate activities at the origin to minimize their perceived travel cost. This perceived cost is typically a function of the travel time and the schedule delay at the intermediate and final destinations. The model is presented through a study of the relative efficiency of carpooling and trip-chaining travel behavior in a network context. In that example, the performance of travelers who have the option to carpool and chain trips is compared with that of households with single-occupant and individual trip-based travel. Several measures of travel performance, including travel distance, travel time, and schedule delay, are considered for that comparison.

Dynamic traffic modelling and dynamic stochastic user equilibrium assignment for general road networks

This paper investigates the requirements of dynamic traffic modelling, and proposes the deterministic queuing model as a plausible link performance function to describe the relationship between inflows, outflows, and link travel costs in time-varying condition. Then, it explains how we can perform logit-based stochastic network loading for general road networks in the dynamic case. In particular, this paper shows how to perform dynamic stochastic network loadings for many-to-many origin–destination pairs, and what should be considered to maintain correct flow propagation in the network loading process. Next, this paper shows how the stochastic dynamic user equilibrium (SDUE) assignment problem can be solved without direct evaluation of the objective function. For this purpose, a quadratic interpolation, the method of successive average, and the pure network loading method are adopted at the line-search step in the solution algorithm. Numerical examples show that the present SDUE assignment model with a quadratic interpolation gives rise to a convergent solution with good quality whilst needing less computation time. Furthermore, it is found that the predictive cost-flow association (Proceedings of the European Transport Conferences, Seminar F, P434, 1999, p. 79) is preferable to the reactive one because the former can produce consistent assignment patterns regardless of the size of dispersion parameter θ in the logit model for route choice.