User Equilibrium Model Research Articles

Modelling Route Choice Behaviour in a Tolled Road Network with a Time Surplus Maximisation Bi-objective User Equilibrium Model

In this paper, we propose a novel approach to model route choice behaviour in a tolled road network with a bi-objective approach, assuming that all users have two objectives: (1) minimise travel time; and (2) minimise toll cost. We assume further that users have different preferences in the sense that for any given path with a specific toll, there is a limit on the time that an individual would be willing to spend. Different users can have different preferences represented by this indifference curve between toll and time. Time surplus is defined as the maximum time minus the actual time. Given a set of paths, the one with the highest (or least negative) time surplus will be the preferred path for the individual. This will result in a bi-objective equilibrium solution satisfying the time surplus maximisation bi-objective user equilibrium (TSmaxBUE) condition. That is, for each O-D pair, all individuals are travelling on the path with the highest time surplus value among all the efficient paths between this O-D pair.We show that the TSmaxBUE condition is a proper generalisation of user equilibrium with generalised cost function, and that it is equivalent to bi-objective user equilibrium. We also present a multi-user class version of the TSmaxBUE condition and demonstrate our concepts with illustrative examples.

A Bi-level Programming Model for Multi-vehicle-type Freeway Continuous Equilibrium Network Design Problem

A bi-level programming model for optimization o( multi-vehicle-ty pe freeway continuous equilibrium network design problem is presented to set up a scientific method for road section reconstruction or extension in freeway network which integrates their positions, capacities and the corresponding network toll rates into one optimal decision-making process. The upper level program model takes the road manager (government) as the leader of the decision-making, and regards the operators' financial goals and upper limit of investment as the constraints to achieve optimum system. The low level program builds multi-vehicle­ type multi-criteria user equilibrium model considering the differences in route choice behaviors between different vehicle type us­ ers and the effects of road volume extension on the traffic distribution. Finally, a numerical example is presented to illustrate the benefit of this model. Result shows that the proposed model is efficient to make decisions on freeway reconstruction or extension project.

The two-regime transmission model for network loading in dynamic traffic assignment problems

Dynamic network loading (DNL) model is concerned with moving traffic in space and time along road network links in dynamic traffic assignment (DTA) models. DNL models strive to build in traffic realism such as modelling transient queues and spillback to upstream links, yet they need to remain computable. Most models in the literature are skewed towards either realism or computability and thus leave a wide scope for further research in arriving at a balanced model. This research proposes a new DNL model called the Two-regime transmission model (TTM) based on widely accepted first-order traffic flow theory. The TTM aims to be quick and accurate enough for planning purposes, when embedded into the framework of a DTA. The TTM considers the time-dependent density states of network links over two regimes namely, free-flowing and congested regimes, and dynamically models the time-dependent queue length, but without the need to break the link into cells. This article sets out the theoretical background necessary for developing the TTM and it also illustrates the principles with the help of a simple network serving a single OD pair. Although the numerical tests are only preliminary indicators, the TTM has been found to produce promising results, for example producing results that are apparently closer than the cell transmission model (CTM) to predicting the dissipation and formation of a queue in a homogeneous link for the same level of time discretisation. We believe that our work establishes TTM as a candidate worthy of future exploration, especially for representing plausible, first-order traffic dynamics within a dynamic user equilibrium model with a lower number of variables/side-constraints than the CTM.

Reference-Dependent Stochastic User Equilibrium with Endogenous Reference Points

We consider the application of reference-dependent consumer choice theory to traffic assignment on transportation networks. Route choice is modelled based on random utility maximisation with systematic utility embodying loss aversion for the travel time and money expenditure attributes. Stochastic user equilibrium models found in the literature have considered exogenously given reference points. The paper proposes a model where reference points are determined consistently with the equilibrium flows and travel times. The referencedependent stochastic user equilibrium (RDSUE) is defined as the condition where (i) no user can improve her utility by unilaterally changing path, (ii) each user has as reference point the current travel time and the money expenditure of one of the available paths, and (iii) if each user updates the reference point to her current path the observed path flows do not change. These conditions are formally equivalent to a multi-class stochastic equilibrium where each class is associated with a path and has as reference point the current state on the path, and the number of users in each class equals the current flow on the path. The RDSUE is formulated as a fixed point problem in the path flows. Existence of RDSUE is guaranteed under usual assumptions. A heuristic algorithm based on the method of successive averages is proposed to solve the problem. The model is illustrated by two numerical examples, one relates to a two-link network and another to the Nguyen-Dupuit network. A reference-dependent route choice model calibrated on stated preference data is used. The second example serves also to demonstrate the algorithm. The impact on the equilibrium of different assumptions on the degree of loss aversion with respect to the travel time attribute are investigated.

An innovative freight traffic assignment model for multimodal networks

Although there is a considerable number of freight traffic assignment models available, they still tend to lag behind their passenger counterparts in terms of maturity of their scientific knowledge. In order to contribute to the improvement of this state of affairs, the authors developed an innovative traffic assignment model, intended to model large networks with a strategic level of planning, which considers both road and rail transport modes and is not very demanding in terms of data. The model applies two different assignment techniques for the two types of cargo considered, with its main innovative feature being the fact that it takes into account both capacity constraints and a variable perception of costs by users, while being much simpler and lighter than a stochastic user equilibrium model. The simultaneous consideration of both of those factors distinguishes it from the traditionally used traffic assignment techniques, namely all or nothing, equilibrium or stochastic (multi-flow) techniques, none of which considers those factors simultaneously.

Dynamic user equilibrium based on a hydrodynamic model

In this paper we present a continuous-time network loading procedure based on the Lighthill–Whitham–Richards model proposed by Lighthill and Whitham (1955) and Richards (1956). A system of differential algebraic equations (DAEs) is proposed for describing traffic flow propagation, travel delay and route choices. We employ a novel numerical apparatus to reformulate the scalar conservation law as a flow-based partial differential equation (PDE), which is then solved semi-analytically with the Lax–Hopf formula. This approach allows for an efficient computational scheme for large-scale networks. We embed this network loading procedure into the dynamic user equilibrium (DUE) model proposed by Friesz et al. (1993). The DUE model is solved as a differential variational inequality (DVI) using a fixed-point algorithm. Several numerical examples of DUE on networks of varying sizes are presented, including the Sioux Falls network with a significant number of paths and origin–destination pairs (OD).The DUE model presented in this article can be formulated as a variational inequality (VI) as reported in Friesz et al. (1993). We will present the Kuhn–Tucker (KT) conditions for that VI, which is a linear system for any given feasible solution, and use them to check whether a DUE solution has been attained. In order to solve for the KT multiplier we present a decomposition of the linear system that allows efficient computation of the dual variables. The numerical solutions of DUE obtained from fixed-point iterations will be tested against the KT conditions and validated as legitimate solutions.

Optimal Tolling Strategies and Concession Period Required for Build-Operate-Transfer Road Projects in Network with Demand Uncertainty by Vehicle Types

In view of uncertainty of traffic demand by vehicle types during the concession period, the toll charges of these BOT roads should be time-dependent and varied for different vehicle types by time of the concession period so as to maximize the system performance.It can be formulated as a bi-level programming problem. At the upper level, the objective is to maximize the social surplus combining of consumer surplus and the investor’s net profit. With taking account the demand elasticity in respect to the toll variability and incorporating the demand uncertainty into the revenue-cost constraint, the lower level is a multi-class reliability-based stochastic user equilibrium model. A genetic solution algorithm is adopted for solving the bi-level programming problem. A numerical example is presented to illustrate the applications of the proposed model and solution algorithm and some conclusions are drawn.

Stochastic user Equilibrium with Reference-dependent Route Choice and Endogenous Reference Points

We consider the application of reference-dependent consumer choice theory to traffic assignment on transportation networks. Route choice is modeled based on random utility maximisation with systematic utility embodying loss aversion for the travel time and money expenditure attributes. Stochastic user equilibrium models found in the literature have considered exogenously given reference points. The paper proposes a model where reference points are determined consistently with the equilibrium flows and travel times. The reference-dependent stochastic user equilibrium (RDSUE) is defined as the condition where (i) no user can improve her utility by unilaterally changing path, (ii) the reference points are the current states, and (iii) if each user updates the reference point to her current path the observed path flows do not change. These conditions are formally equivalent to a multi-class stochastic equilibrium where each class is associated with a path and has as reference point the current state on the path, and the number of users in each class equals the current flow on the path. The RDSUE is formulated as a fixed point problem in the path flows. Existence of RDSUE is guaranteed under usual assumptions. The model is illustrated by an application to a two link network which uses a reference-dependent route choice model calibrated on stated preference data. The impact on the equilibrium of different assumptions on the degree of loss aversion with respect to the travel time attribute are investigated.

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.

Examining the scaling effect and overlapping problem in logit-based stochastic user equilibrium models

The purpose of this paper is to examine the scaling effect and overlapping problem in a route choice context using the logit-based stochastic user equilibrium (SUE) principle to explicitly account for the congestion effect. Numerical experiments are performed on nine models: the deterministic user equilibrium model, the multinomial logit SUE model with and without scaling, the C-logit SUE model with and without scaling, the path-size logit SUE model with and without scaling, and the paired combinatorial logit SUE model with and without scaling. Sensitivity analysis is conducted to examine the effects of route sets, congestion levels, dispersion intensities, and network asymmetries. A real transportation network in the City of Winnipeg, Canada is also used to compare the network equilibrium flow allocations of different SUE models. The results of the sensitivity analysis and the Winnipeg network reveal that both scaling effect and overlapping problem can have a significant impact on the network equilibrium flow allocations.

Stochastic Traffic Equilibrium Based on Travel Time Robust Reliability

An assumption that pervades the current reliable traffic equilibrium problem is that probability distributions of the origin-destination (O-D) demand or/and link capacities are known explicitly. However, these distributions are difficult to be accurately obtained. This paper relaxes this assumption. It only needs to know the first m moments of travel demand (where m is a positive integer associated with the formulation of link cost function), and then applies two worse-case Value-at-Risk (WVaR) and Conditional value-at-risk (CVaR) risk measures to define robust percentile travel time (RPTT) and robust mean-excess travel time (RMETT) and prove that this two kinds travel time is equal under general distribution. By incorporating the defined travel time and travelers' perception error, the robust percentile stochastic user equilibrium (RPSUE) or robust mean-excess stochastic traffic equilibrium model (RMESTE) is proposed, which is formulated as an equivalent route-based variational inequality. Conditions are established guaranteeing existence of this equilibrium. A heuristic solution problem is introduced to solve the variational inequalities problem. A numerical example is used to illustrate the applications of the proposed model and the solution algorithm.

Incorporating transfer reliability into equilibrium analysis of railway passenger flow

Passenger’s transfer route choice behavior is one of the prominent research topics in the field of railway transportation. Existing traffic assignment approaches do not properly account for passenger’s expectation for transfer reliability. In this study, the transfer reliability is explicitly defined and a multi-class user equilibrium model is established, given which passengers choose the minimal-cost path based on their expected reliability thresholds. In particular, a path-based traffic assignment algorithm which combines a k-shortest path algorithm and the method of successive averages is proposed. The validity of the proposed approach is verified by an illustrative example. Using the proposed modeling approach, it is possible to determine the passenger’s collective route choice behavior based on the user equilibrium pattern. Moreover, the railway timetables can be evaluated and optimized based on the cost-based level of service estimation.

C-logit stochastic user equilibrium model: formulations and solution algorithm

This article considers the stochastic user equilibrium (SUE) problem with the route choice model based on the C-logit function. The C-logit model has a simple closed-form analytical probability expression and requires relatively lower calibration efforts and represents a more realistic route choice behaviour compared with the multinomial logit model. This article proposes two versions of the C-logit SUE model that captures the route similarity using different attributes in the commonality factors. The two versions differ with respect to the independence assumption between cost and flow. The corresponding stochastic traffic equilibrium models are called the length-based and congestion-based C-logit SUE models, respectively. To formulate the length-based C-logit SUE model, an equivalent mathematical programming formulation is proposed. For the congestion-based C-logit SUE model, we provide two equivalent variational inequality formulations. To solve the proposed formulations, a new self-adaptive gradient projection algorithm is developed. The proposed formulations and new solution algorithm are tested in two well-known networks. Numerical results demonstrate the validity of the formulations and solution algorithm.

Numerical simulation of a continuum model for bi-directional pedestrian flow

An algorithm for an extended reactive dynamic user equilibrium model of pedestrian counterflow as a continuum is developed. It is based on a cell-centered high-resolution finite volume scheme with a fast sweeping method for an Eikonal-type equation on an orthogonal grid. A high-order total variation diminishing Runge–Kutta method is adopted for the time integration of semi-discrete equations. The numerical results demonstrate the rationality of the model and efficiency of the algorithm. Some crowd pedestrian flow phenomena, such as dynamic lane formation in bi-directional flow, are observed which are helpful for a global comprehension of pedestrian dynamics. Also, the model can be utilized with different potential applications.

A Cumulative Perceived Value-Based Dynamic User Equilibrium Model Considering the Travelers’ Risk Evaluation on Arrival Time

This paper presents a cumulative perceived value-based dynamic user equilibrium model by applying the prospect theory to formulate the travelers’ risk evaluation on arrival time. The network uncertainty caused by link exit capacity degradation is incorporated into the analysis. The model which considers departure time and route choices simultaneously is expressed by a variational inequality in a discrete time space. Numerical results show that the travelers’ risk preference indeed has big influence on flow distribution. Our study constitutes a deepening of cognition in developing more realistic dynamic traffic assignment technologies.

Highway Alignment Optimization Using Cost-Benefit Analysis Under User Equilibrium

Usually, selection of a highway alignment depends on an economical route that minimizes alignment sensitive costs, such as construction cost, user cost, right-of-way cost, and earthwork cost. Most of the available highway alignment optimization algorithms do not consider traffic assignment and distribution of traffic as a result of the new road network consisting of the new alignment as well as other pre-existing alignments. Constructing a new highway will ease the traffic in the existing road network. Based on Wardrop’s principle, the users will choose a route that will minimize their travel-time. Users will unilaterally shift to the available routes for their benefit and thus, traffic flow will attain equilibrium. Theoretically, the equilibrium of traffic flow between the existing highway and the newly designed highway alternative can be achieved by a user equilibrium model. A new methodology is developed in this paper to optimize a new three-dimensional highway alignment based on the existing highway alignment system information using a cost-benefit analysis approach. The results are quite promising for new road design and bypass construction since benefit maximization and cost minimization is performed simultaneously while attaining user equilibrium.

Stochastic Traffic Assignment Model Based on Travel Time Reliability

Considering the un-deterministic transportation networks, the paper proposes the change of the route choice decisions under the stochastic transportation networks. The route choice behavior is described as a choice for a time shortest route which is subject to a time-reliability level. The paper also considered this new route choice behavior in the stochastic user equilibrium model, and proposed stochastic user equilibrium model based on the optimized reliability travel time route choice behavior in the stochastic networks. The equivalence and uniqueness of the solution of the model are demonstrated. Numerical results of a small network show that the proposed model can reflect the real traveler’s route choice behavior in stochastic transportation networks.

The Application of Stochastic User Equilibrium Model to Traffic Diverging during Highway Expanding Construction

Stochastic User Equilibrium (SUE) model is equivalent to the problem of Stochastic User Equilibrium, and the solution is unique, so SUE model could be applied to traffic diverging during highway expanding construction. In this paper, SUE model is used with the expanding construction of Tongji highway(G010). Firstly transverse interference and parameters of travel time are adjusted according to the actual situation. And then the linear growth rule is used to predict the current theory traffic, based on which we use SUE model to divert the traffic in Tongji highway(G010) during expanding construction. Through the comparison between the calculation results and the actual traffic, we have validated the reliability of the calculation results and the rationality of the model.

Comparison of Methods for Path Flow Reassignment for Dynamic User Equilibrium

Models to describe or predict of time-varying traffic flows and travel times on road networks are usually referred to as dynamic traffic assignment (DTA) models or dynamic user equilibrium (DUE) models. The most common form of algorithms for DUE consists of iterating between two components namely dynamic network loading (DNL) and path inflow reassignment or route choice. The DNL components in these algorithms have been investigated in many papers but in comparison the path inflow reassignment component has been relatively neglected. In view of that, we investigate various methods for path inflow reassignment that have been used in the literature. We compare them numerically by embedding them in a DUE algorithm and applying the algorithm to solve DUE problems for various simple network scenarios. We find that the choice of inflow reassignment method makes a huge difference to the speed of convergence of the algorithms and, in particular, find that ‘travel time responsive’ reassignment methods converge much faster than the other methods. We also investigate how speed of convergence is affected by the extent of congestion on the network, by higher demand or lower capacity. There appears to be much scope for further improving path inflow reassignment methods.

A continuous-time linear complementarity system for dynamic user equilibria in single bottleneck traffic flows

This paper formally introduces a linear complementarity system (LCS) formulation for a continuous-time, multi-user class, dynamic user equilibrium (DUE) model for the determination of trip timing decisions in a simplified single bottleneck model. Existence of a Lipschitz solution trajectory to the model is established by a constructive time-stepping method whose convergence is rigorously analyzed. The solvability of the time-discretized subproblems by Lemke’s algorithm is also proved. Combining linear complementarity with ordinary differential equations and being a new entry to the mathematical programming field, the LCS provides a computational tractable framework for the rigorous treatment of the DUE problem in continuous time; this paper makes a positive contribution in this promising research venue pertaining to the application of differential variational theory to dynamic traffic problems.