Traffic Network Equilibrium Research Articles

On the Convergence of the Method of Successive Averages for Calculating Equilibrium in Traffic Networks

The traffic assignment problem aims to calculate an equilibrium route flow vector, generally by seeking a zero of an appropriate objective function. If a continuous dynamical system follows a descent direction for this objective function at each nonequilibrium route flow vector, the system converges to equilibrium. It is shown that when this dynamical system is discretized with a fixed step length, the system eventually approaches close to equilibrium provided that the objective function is continuously differentiable and that the rate of descent is bounded below. The method of successive averages is widely used in traffic assignment; it has a decreasing step size at each iteration. With the same conditions as above, it is shown that the resulting dynamical system converges to equilibrium. In the steady-state model, the necessary conditions are shown to be satisfied, provided that the route cost vector is a continuously differentiable monotone function of the route flow vector. However, continuous differentiability of the cost function is shown not to hold in the dynamic queueing model.

Optimization for Multiclass Residential Location Models with Congestible Transportation Networks

There has been extensive research on the optimal design of road congestion pricing schemes based on traffic network equilibrium models. Since travel demands are closely related to the locations of workplace and residence, it is of both theoretical and practical importance to extend the optimization approach to treat location choices as well as travel route choices. This paper addresses the problem of optimal location of housing supply and transportation network pricing in an integrated location and transportation model with heterogeneous households that have different values of travel time. In the model, housing exhibits external economies and diseconomies of scale, depending on the population density in residential zones. It is shown that in such a model the marginal cost pricing principle applied to internalize these externalities in general results in local optima of social surplus. Optimization algorithms based on sensitivity analysis are proposed to find combinations of housing supply patterns and road tolls that optimize social surplus under various conditions. A numerical example is given to illustrate the proposed method. The potential policy implications of the optimization approach are examined and compared with that of the marginal cost pricing principle.

Multiclass and multicriteria stochastic traffic network equilibrium model based on ATIS

Multiclass and multicriteria stochastic traffic network equilibrium model based on ATIS

Energy-Sustainable Traffic Signal Timings for a Congested Road Network With Heterogeneous Users

This paper proposes a novel model to address the energy-efficient traffic signal timing problem for a congested road network with heterogeneous users. In the proposed model, two types of agents, i.e., the authority and road users, are considered together with the interaction between traffic signal settings and energy policy (e.g., fuel surcharges). To model the route choice behavior of heterogeneous users, a multiclass stochastic traffic network equilibrium problem that considers vehicle delays at signalized intersections and travel demand elasticity is described and formulated as a variational inequality formulation. The authority aims to maximize social welfare of the transportation system by optimizing the traffic signal timings and fuel surcharges. A simulated-annealing-based solution algorithm is developed to solve the proposed model. The findings show that the implementation of the fuel surcharge policy can cause spatial and social inequity issues.

Dynamic Traffic Network Equilibrium System with Elastic Demands

We discuss the dynamic traffic network equilibrium system problem with elastic demands by way of a quasi-variational inequality.We introduce the equilibrium definition with elastic demands based on Wardrops principles when there are some internal relationships between different kinds of goods which transported through the same traffic network. Moreover, we also prove that the equilibrium conditions of this problem with elastic demands can be equivalently expressed as a system of evolutionary variational inequalities. By using the fixed point theory and projected dynamic system theory, we get the existence and uniqueness of the solution for this equilibrium problem with elastic demands. Finally, some numerical examples are given to illustrate our results.

A generalized $$f$$ -projection algorithm for inverse mixed variational inequalities

In this paper, a new inverse mixed variational inequality is introduced and studied in Hilbert spaces, which provides a model for the study of traffic network equilibrium control problems. An iterative algorithm involving the generalized \(f\)-projection operator for solving inverse mixed variational inequalities is constructed and the convergence of sequences generated by the algorithm is given under some suitable conditions.

Stochastic user equilibrium with combined mode in a degradable multi-modal transportation network

This paper proposes a stochastic user equilibrium assignment model for ascertaining the difference in the travel mode choice behavior in a degradable multi-modal traffic network. In the proposed model, on the basis of the actual travel behavior which often needs to transfer different traffic modals, all the travels are categorized into two classes: pure car trip travelers and combined mode trip(car-subway) travelers. The multi-modal traffic network equilibrium conditions are formulated as an equivalent fixed-point problem, and the proof of its solution existence is provided afterwards. A heuristic solution algorithm MSWA method which is the extension of the classical MSA method is introduced for solving this fixed-point problem. An example is used to illustrate the application of the proposed model and solution algorithm. Moreover, the performance comparison of the MSA and MSWA algorithm is also given to prove the precision and efficiency of MSWA method for solving this problem.

Traffic Network Equilibrium Problems with Capacity Constraints of Arcs and Linear Scalarization Methods

Traffic network equilibrium problems with capacity constraints of arcs are studied. A (weak) vector equilibrium principle with vector‐valued cost functions, which are different from the ones in the work of Lin (2010), and three kinds of parametric equilibrium flows are introduced. Some necessary and sufficient conditions for a (weak) vector equilibrium flow to be a parametric equilibrium flow are derived. Relationships between a parametric equilibrium flow and a solution of a scalar variational inequality problem are also discussed. Some examples are given to illustrate our results.

Network impacts of a road capacity reduction: Empirical analysis and model predictions

In spite of their widespread use in policy design and evaluation, relatively little evidence has been reported on how well traffic equilibrium models predict real network impacts. Here we present what we believe to be the first paper that together analyses the explicit impacts on observed route choice of an actual network intervention and compares this with the before-and-after predictions of a network equilibrium model. The analysis is based on the findings of an empirical study of the travel time and route choice impacts of a road capacity reduction. Time-stamped, partial licence plates were recorded across a series of locations, over a period of days both with and without the capacity reduction, and the data were ‘matched’ between locations using special-purpose statistical methods. Hypothesis tests were used to identify statistically significant changes in travel times and route choice, between the periods of days with and without the capacity reduction. A traffic network equilibrium model was then independently applied to the same scenarios, and its predictions compared with the empirical findings. From a comparison of route choice patterns, a particularly influential spatial effect was revealed of the parameter specifying the relative values of distance and travel time assumed in the generalised cost equations. When this parameter was ‘fitted’ to the data without the capacity reduction, the network model broadly predicted the route choice impacts of the capacity reduction, but with other values it was seen to perform poorly. The paper concludes by discussing the wider practical and research implications of the study’s findings.

An efficient algorithm for solving supply chain network equilibria and equivalent supernetwork based traffic network equilibria

This paper is concerned with the algorithm of the supply chain network equilibrium model and its equivalent supernetwork based traffic network equilibrium model with elastic demands. Both models are further written as nonlinear complementarity problems. Semismooth least squares reformulations of the complementarity problems are presented and their convergence properties are investigated. Considering the drawbacks of Quasi-Newton method (using the Fischer-Burmeister function), a semi-smooth Levenberg-Marquardt-type method is proposed to solve the problems. Numerical examples show that the Levenberg-Marquardt-type method can solve the supply chain network equilibrium model and its equivalent supernetwork based traffic network equilibrium model significantly, and is more efficient than the Quasi Newton method and the modified projection method. Furthermore, the Levenberg-Marquardt-type method with the equivalent supernetwork based complementarity formulation can be implemented faster than with the supply chain network equilibrium complementarity formulation.

Traffic network equilibrium with capacity constraints and generalized Wardrop equilibrium

The traffic network equilibrium problem with capacity constraints is investigated with the help of generalized Lagrangian theory of variational inequality in our paper. It is proved to be equivalent to the generalized Wardrop equilibrium solution. This result is very useful in transportation analysis since the study in Wardrop equilibrium problem is relatively mature. Furthermore, some new concepts are offered.

Dynamic Traffic Network Equilibrium System

AbstractWe discuss the dynamic traffic network equilibrium system problem. We introduce the equilibrium definition based on Wardrop's principles when there are some internal relationships between different kinds of goods which transported through the same traffic network. Moreover, we also prove that the equilibrium conditions of this problem can be equivalently expressed as a system of evolutionary variational inequalities. By using the fixed point theory and projected dynamic system theory, we get the existence and uniqueness of the solution for this equilibrium problem. Finally, a numerical example is given to illustrate our results.

On the regularity of retarded equilibria in time-dependent traffic equilibrium problems

The paper is divided into two parts. In the first part, the retarded link formulation traffic equilibrium problem is presented and a variational inequality formulation is provided. In the second part, we consider a class of strictly monotone evolutionary variational inequalities and present a regularity result for this class. Then, we apply the result to equilibrium problems with data depending explicitly on time. The regularity results allow us to provide a discretization procedure for the calculation of the approximated solution to the retarded time-dependent traffic network equilibrium problem.

Day-to-day stationary link flow pattern

Day-to-day traffic dynamics aim at modeling driver’s day-to-day learning and behavior adjustment process and providing insight on how the traffic flow pattern evolves over time. A number of continuous-time path flow dynamics have been proposed and studied in the literature whose stationary path flow pattern coincides with user equilibrium. However, the application prospects of these findings are limited by the fact that it might be difficult for transportation management centers to measure and monitor day-to-day dynamic path flow pattern. On the other hand, thanks to the fast development of Intelligent Transportation Systems (ITS), the day-to-day link flow pattern can be more easily monitored and analyzed. Thus, it is not only of theoretical interest but also with implement viability to explore equilibrium in the perspective of link flow dynamics. This paper carries on the study originated by [Zhang, D., Nagurney A., Wu, J., 2001. On the equivalence between stationary link flow patterns and traffic network equilibria. Transportation Research, 35B, 731–748] in investigating the relationship between a stationary link flow (SLF) pattern and traffic equilibrium, and makes significant developments on the subject in the sense that we prove that five existing route choice adjustment processes in the literature are all special cases of a so-called rational behavior adjustment process (RBAP) and thus this RBAP is mild and reasonable and can serve as a general framework of modeling driver’s rational behavior.

Multiclass, Two-criteria Traffic Network Equilibrium Models and Vector Variational Inequalities

In this article, multiclass, the two-criteria traffic network equilibrium model is dissussed. The relations between the vector and the weak vector variational inequalities and an equilibrium principle of the model are established, respectively.

A Multiclass and Multicriteria Stochastic Traffic Network Equilibrium Variational Inequality Model with Fixed Demand

AbstractBased on variational inequality theory, a multiclass and multicriteria stochastic traffic network equilibrium model with fixed demand was obtained under the assumption that the weights of criteria are class-dependent and the cost functions are unsymmetrical. The qualitative analysis, such as existence and uniqueness, was studied. Since generalized road-transit-cost is affected each other by road flow, MSA (Method of Successive Averages) algorithm was extended to solve the variational inequality problem of this traffic network equilibrium. The process of this algorithm and an example were put forward.

Hölder continuity of solutions to elastic traffic network models

This paper aims to study stability and sensitivity analysis for quasi-variational inequalities which model traffic network equilibrium problems with elastic travel demand. In particular, we provide a Holder stability result under parametric perturbations.

Convergence to Equilibrium in Dynamic Traffic Networks when Route Cost Is Decay Monotone

This paper addresses the issue of convergence to equilibrium in dynamic traffic assignment. Within-day time is considered to be a continuous variable, so that traffic flows and costs are functions of within-day time. Flow propagates along routes connecting origin-destination (OD) pairs with the demand for travel between each OD pair considered to be rigid (fixed from day to day although it can vary within day). Day-to-day time is also modelled as continuous with the day-to-day dynamical system derived naturally from the usual dynamical user equilibrium (DUE) condition. This paper focuses on the bottleneck model, which has deterministic vertical queueing at bottleneck link exits when flow exceeds capacity. A new property called decay monotonicity is introduced. The link delay (and hence link cost) function is shown to be a decay monotone function of link flow provided that the link capacity is continuously differentiable and positive. In a restricted version of the single bottleneck per route case, it is shown that link cost decay monotonicity implies route cost decay monotonicity. Decay monotonicity of the route cost function is shown to be sufficient for convergence to equilibrium of the dynamical system.

REGULARITY RESULTS FOR TIME-DEPENDENT VARIATIONAL AND QUASI-VARIATIONAL INEQUALITIES AND APPLICATION TO THE CALCULATION OF DYNAMIC TRAFFIC NETWORK

The aim of this paper is to consider time-dependent variational and quasi-variational inequalities and to study under which assumptions the continuity of solutions with respect to time can be ensured. Making an appropriate use of the set convergence in Mosco's sense, we are able to prove continuity results for strongly monotone variational and quasi-variational inequalities. The continuity results allow us to provide a discretization procedure for the calculation of solutions to the variational inequalities and, as a consequence, we can solve the time-dependent traffic network equilibrium problem.

Forecasting Travel on Congested Urban Transportation Networks: Review and Prospects for Network Equilibrium Models

The historical origins, development and application of traffic network equilibrium models with variable origin–destination flows are examined in this review. The original formulation and analysis of this model during 1952–1955 by Beckmann, McGuire and Winsten at the Cowles Commission for Research in Economics is the starting point for the review. Subsequent research building upon the original formulation is described, and contrasted with urban travel forecasting practice stemming from the same period. Additional model developments through the end of the twentieth century are then summarized, followed by a brief discussion of travel forecasting practice and software development stimulated by the original formulation.