Stochastic User Equilibrium Model Research Articles

Multiclass Stochastic User Equilibrium Model with Elastic Demand

This paper proposes a multiclass, multidelay stochastic user equilibrium model with elastic demand (MC-MDSUE-ED) that explicitly considers both systematic delay and accidental delay in the route choice decision process. This new model hypothesizes that for each user class and each origin–destination (O-D) pair, no traveler can reduce his or her “multi-delay travel time,” defined as the systematic delay plus accidental delay, by unilaterally changing paths. The actual travel demand for each user class between each O-D pair satisfies its elastic demand function. Travel time budget and robust optimization theory are used to compute the systematic delay and accidental delay, respectively. The MC-MDSUE-ED conditions compose a cobweb model. A self-adaptive bisection algorithm that converts the problem with elastic demand to a series of problems with fixed demand was developed for solving the cobweb model. A path-based quasi method of successive averages was used to solve the MC-MDSUE model with fixed demand. Numerical examples illustrate the essential ideas of the proposed model and the applicability of the proposed solution algorithm.

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.

Continuous Network Design Based on the Paired Combinatorial Logit Stochastic User Equilibrium Model

In order to design the traffic network more accurately, the bi-level programming model for the continuous network design problem based on the paired combinatorial Logit stochastic user equilibrium model is proposed in this study. In the model, the paired combinatorial Logit stochastic user equilibrium model which is used to characterize the route choice behaviors of the users is adopted in the lower level model, and the minimum summation of the system total costs and investment amounts is used in the upper objective function. The route-based self-regulated averaging (SRA) algorithm is designed to solve the stochastic user equilibrium model and the genetic algorithm (GA) is designed to get the optimal solution of the upper objective function. The effectiveness of the proposed combining algorithm which contains GA and SRA is verified by using a simple numerical example. The solutions of the bi-level models which use the paired combinatorial Logit stochastic user equilibrium model in the lower level model with different demand levels are compared. Finally, the impact of the dispersion coefficient parameter which influences the decision results of the network design problem is analyzed.

Upper Bounds on Inefficiency of Taxed Stackelberg Network with Cross- Nested Logit Assignment

Upper bound of efficiency loss is a valuable issue for transport network design and planning. This paper ini- tially explores it in a taxed stochastic traffic network whose equilibrium flow pattern is deduced by a cross-nested Logit (CNL) flow assignment model, and a centrally controlling Stackelberg strategy. With the assumptions of separability, nondecreasingness, and convexity of the link time function and the fixed origin-destination (OD) demand of network, the equivalent variational inequality for a CNL-based stochastic user equilibrium (CNL-SUE) model is established and first used to obtain upper bounds on Stackelberg network inefficiency. Further, for low-degree link time function such as Bu- reau of Public Roads and the affine forms, their inefficiency upper bounds are analyzed with some meaningful results.

Is equilibrium in transport pure Nash, mixed or Stochastic?

The classical theory of transport equilibrium is based on the Wardrop’s first principle that describes a Nash User Equilibrium (UE), where in no driver can unilaterally change routes to improve his/her travel times. A growing number of economic laboratory experiments aiming at testing Nash-Wardrop equilibrium have shown that the Pure Strategy Nash Equilibrium (PSNE) is not able to explain the observed strategic choices well. In addition even though Mixed Strategy Nash Equilibrium (MSNE) has been found to fit better the observed aggregate choices, it does not explain the variance in choices well. This study analyses choices made by users in three different experiments involving strategic interactions in endogenous congestion to evaluate equilibrium prediction. We compare the predictions of the PSNE, MSNE and Stochastic User Equilibrium (SUE). In SUE, the observed variations in choices are assumed to be due to perception errors. The study proposes a method to iteratively estimate SUE models on choice data with strategic interactions. Among the three sets of experimental data the SUE approach was found to accurately predict the average choices, as well as the variances in choices. The fact that the SUE model was found to accurately predict variances in choices, suggests its applicability for transport equilibrium models that attempt to evaluate reliability in transportation systems. This finding is fundamental in the effort to determining a behaviourally consistent paradigm to model equilibrium in transport networks. The study also finds that Fechner error which is the inverse of the scale parameter in the SUE model is affected by the group sizes and the complexity of the cost function. In fact, the larger group sizes and complexity of cost functions increased the variability in choices. Finally, from an experimental design standpoint we show that it is not possible to estimate a noise parameter associate to Fechner error in the case when the choices are equally probable.

Elastic demand with weibit stochastic user equilibrium flows and application in a motorised and non-motorised network

In this paper, we propose a new elastic demand (ED) stochastic user equilibrium (SUE) model with an application to the combined modal split and traffic assignment (CMSTA) problem. This new model, called the path-size weibit (PSW) SUE model with ED, is derived based on the Weibull distribution, which does not require the identically distributed assumption typically imposed in the multinomial logit (MNL) model with the Gumbel distribution. In addition, a path-size factor is included to correct the choice probabilities of routes that are not truly independent (i.e. another assumption typically required in the MNL model). Equivalent mathematical programming (MP) formulation of the PSW-SUE-ED model is developed to simultaneously consider both travel choice and route choice. The travel choice is determined based on the ED function that explicitly considers the network level of service based on the logarithmic expected perceived cost of the Weibull distribution to determine the travel demand, while the route choice accounts for both route overlapping and non-identical perception variance with respect to different trip lengths. Qualitative properties of the proposed MP formulation are rigorously proved. A path-based partial linearisation algorithm combined with a self-regulated averaging line search strategy is developed for solving the PSW-SUE-ED model and its application to the CMSTA problem. Numerical examples are also provided to demonstrate the features of the proposed PSW-SUE-ED model as well as a real-case study in a bi-modal network with motorised and non-motorised mode choices.

Development of an enhanced route choice model based on cumulative prospect theory

Route choice models have important theoretical value and practical significance for the systematic analysis of urban transportation. The method utilized to determine the reference point of travel time by considering the restrictions on reserved travel time of travelers is improved in this study. On this basis, a route choice model based on cumulative prospect theory (CPT) is proposed. A stochastic user equilibrium model based on CPT that integrates the Wardrop equilibrium principle is also developed. The data analysis results indicate that CPT provides a better description of risk attitudes than expected utility theory (EUT). The improved reference point of travel time can capture the demand of reliable travel time from different travelers. Furthermore, the numerical results of the stochastic user equilibrium model based on CPT are in agreement with the route choice behavior observed in an actual road network.

A Stochastic User Equilibrium Model and Optimal Congestion Pricing with Prospect Theory

In spite of the fact that expected utility theory is a main method to model people's travel choice behavior under uncertainty, the assumption that each decision maker (DM) is completely rational limits its wide application. This paper applies prospect theory, which can appropriately describe DMs’ bounded rational behavior, to route choice behavior analysis and develops an improved stochastic user equilibrium model and finally analyzes optimal congestion pricing. We find that the prospect in prospect theory could be regarded as a special utility in some degree. Based on this result we assume that the prospect of each route is constituted by a fixed term and a random one, and then proposed the concept of perceived prospect. Further assumption is that the random variable follows some certain normal distribution and then a probit model is put forward. Considering the complexity of probit model, Monte Carlo simulation is brought into the solution algorithm of MSA. A numerical example is given to show the impact of uncertainty, including subjective and objective uncertainty that travelers have to deal with. Congestion pricing is also taken into account to express the influence of traffic management policy on traffic flow on a network. At last, an approximately optimal charging standard is proposed through comparing one-link charging and two-links charging. The results prove the effectiveness of congestion pricing. Further research shows that there are 1.91% and 2.71% drops of total travel time and total travel cost respectively when achieving the optimal charging standard.

Modeling Demand Elasticity and Route Overlapping in Stochastic User Equilibrium through Paired Combinatorial Logit Model

In this paper an equivalent mathematical programming formulation is provided for modeling demand elasticity and route overlapping in the stochastic user equilibrium (SUE) problem. The elastic demand establishes the equilibrium between supply function and demand function on the basis of microeconomics. Because the elasticity of demand is an important factor in predicting the future demand pattern and avoiding the potential biased assessment in transportation planning, the elasticity of travel demand must be modeled endogenously. The route overlapping problem is handled by the paired combinatorial logit (PCL) model while retaining the analytical tractability of the logit choice probability function. The PCL SUE model with elastic demand (PCL-SUE-ED) explicitly models the elasticity of travel demand and the effect of route overlapping on travel choice and route choice simultaneously. A path-based partial linearization algorithm is also developed for solving the PCL-SUE-ED model. In addition, a self-regulated averaging line search strategy is incorporated into the algorithm to minimize the computational efforts required to determine a suitable step size that guarantees convergence. Numerical results are provided to examine the features of the PCL-SUE-ED model as well as the efficiency of the path-based partial linearization algorithm.

Unconstrained weibit stochastic user equilibrium model with extensions

This study provides an unconstrained minimization program as an alternative formulation for the multinomial weibit (MNW) stochastic user equilibrium (SUE) model that explicitly considers the heterogeneous perception variances with respect to different trip lengths under congested conditions. Qualitative properties of the unconstrained minimization program are given to establish the equivalency and uniqueness of the MNW-SUE solution. The advantage of the unconstrained minimization programming formulation is that it allows the development of a link-based algorithm, which obviates path storage and enumeration. The methodological contributions lie in the derivation of the expected perceived travel cost (or the satisfaction function) that enables the development of an unconstrained MNW-SUE minimization program and a link-based stochastic loading mechanism combined with recent advances in line search strategies in the link-based algorithm. Numerical examples are also provided to illustrate the features of the MNW-SUE model and the link-based algorithm along with several extensions for future research.

Inefficiency of Stackelberg Stochastic Traffic Network with Tax Scheme

Inefficiency upper bounds are explored in stochastic traffic network. Equilibrium flow pattern therein is deduced by a central Stackelberg strategy and tax schemes imposed on each link.. The equivalent variational inequality (VI) for Logit-based stochastic user equilibrium (SUE) model is established and first used to obtain upper bounds on Stackelberg network inefficiency under the assumption of separable, nondecreasing, and convex link time function and of fixed network origin-destination (OD) demand. For typical Bureau of Public Roads (BPR) functions and its affine forms, the upper bounds of their inefficiency are investigated with some meaningful results.

A Comparative Study of Two Alternative Methods for the Path-Based Logit Stochastic User Equilibrium Problem

in this paper, a computational study of two alternative methods for the path-based logit stochastic user equilibrium model is conducted. The two methods under investigation are the Jacobi gradient projection method and the Gauss-Seidel gradient projection method. We compare the two methods on the Sioux Falls network. Numerical results indicate that for the path-based logit SUE problem, Jacobi gradient projection method is more efficient than Gauss-Seidel gradient projection method.

Computation and application of the paired combinatorial logit stochastic user equilibrium problem

The paired combinatorial logit (PCL) model is one of the recent extended logit models adapted to resolve the overlapping problem in the route choice problem, while keeping the analytical tractability of the logit choice probability function. However, the development of efficient algorithms for solving the PCL model under congested and realistic networks is quite challenging, since it has large-dimensional solution variables as well as a complex objective function. In this paper, we examine the computation and application of the PCL stochastic user equilibrium (SUE) problem under congested and realistic networks. Specifically, we develop an improved path-based partial linearization algorithm for solving the PCL SUE problem by incorporating recent advances in line search strategies to enhance the computational efficiency required to determine a suitable stepsize that guarantees convergence. A real network in the city of Winnipeg is applied to examine the computational efficiency of the proposed algorithm and the robustness of various line search strategies. In addition, in order to acquire the practical implications of the PCL SUE model, we investigate the effectiveness of how the PCL model handles the effects of congestion, stochasticity, and similarity in comparison with the multinomial logit stochastic traffic equilibrium problem and the deterministic traffic equilibrium problem.

A path-size weibit stochastic user equilibrium model

The aim of this paper is to develop a path-size weibit (PSW) route choice model with an equivalent mathematical programming (MP) formulation under the stochastic user equilibrium (SUE) principle that can account for both route overlapping and route-specific perception variance problems. Specifically, the Weibull distributed random error term handles the identically distributed assumption such that the perception variance with respect to different trip lengths can be distinguished, and a path-size factor term is introduced to resolve the route overlapping issue by adjusting the choice probabilities for routes with strong couplings with other routes. A multiplicative Beckmann’s transformation (MBec) combined with an entropy term are used to develop the MP formulation for the PSW-SUE model. A path-based algorithm based on the partial linearization method is adopted for solving the PSW-SUE model. Numerical examples are also provided to illustrate features of the PSW-SUE model and its differences compared to some existing SUE models as well as its applicability on a real-size network.

C-logit stochastic user equilibrium model with elastic demand

Modeling the elasticity of travel demand in network equilibrium analysis has several important transportation applications. In this paper, we provide a mathematical programming formulation for the C-logit stochastic user equilibrium problem with elastic demand (CL-SUE-ED) in the route domain. The proposed model is capable of explicitly modeling the elasticity of travel demand and the effect of route overlapping on travel choice and route choice simultaneously. Some qualitative properties of the model, including the equivalency and uniqueness of the solution, are also rigorously proved. To solve the CL-SUE-ED model, a partial linearization method is developed to handle the elastic demand and route overlapping considerations. In addition, a self-regulated averaging stepsize scheme is adopted to smartly determine the stepsize while avoiding evaluating the complex objective function. Numerical examples are also provided to demonstrate the features of the proposed model and solution algorithm.

A Path-size Weibit Stochastic User Equilibrium Model

The aim of this paper is to develop a path-size weibit (PSW) route choice model with an equivalent mathematical programming (MP) formulation under the stochastic user equilibrium (SUE) principle that can account for both route overlapping and route-specific perception variance problems. Specifically, the Weibull distributed random error term handles the identically distributed assumption such that the perception variance with respect to different trip lengths can be distinguished, and a path-size factor term is introduced to resolve the route overlapping issue by adjusting the choice probabilities for routes with strong couplings with other routes. A multiplicative Beckmann's transformation (MBec) combined with an entropy term are used to develop the MP formulation for the PSW-SUE model. A path-based algorithm based on the partial linearization method is adopted for solving the PSW-SUE model. Numerical examples are also provided to illustrate features of the PSW-SUE model and its differences compared to some existing SUE model as well as its applicability on a real-size network.

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.

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.