On the flexibility of using marginal distribution choice models in traffic equilibrium

Traffic equilibrium models are fundamental to the analysis of transportation systems. The stochastic user equilibrium (SUE) model which relaxes the perfect information assumption of the deterministic user equilibrium is one such model. The aim of this paper is to develop a new user equilibrium model, namely the MDM-SUE model, that uses the marginal distribution model (MDM) as the underlying route choice model. In this choice model, the marginal distributions of the path utilities are specified but the joint distribution is not. By focusing on the joint distribution that maximizes expected utility, we show that MDM-SUE exists and is unique under mild assumptions on the marginal distributions. We develop a convex optimization formulation for the MDM-SUE. For specific choices of marginal distributions, the MDM-SUE model recreates the optimization formulation of logit SUE and weibit SUE. Moreover, the model is flexible since it can capture perception variance scaling at the route level and allows for modeling different user preferences by allowing for skewed distributions and heavy tailed distributions. The model can also be generalized to incorporate bounded support distributions and discrete distributions which allows to distinguish between used and unused routes within the SUE framework. We adapt the method of successive averages to develop an efficient approach to compute MDM-SUE traffic flows. In our numerical experiments, we test the ability of MDM-SUE to relax the assumption that the error terms are independently and identically distributed random variables as in the logit models and study the additional modeling flexibility that MDM-SUE provides on small-sized networks as well as on the large network of the city of Winnipeg. The results indicate that the model provides both modeling flexibility and computational tractability in traffic equilibrium.     

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.     

Comparative analysis of transit assignment: evidence from urban railway system in the Tokyo Metropolitan Area

This paper empirically compares the performance of six traffic assignment methods using the same empirical dataset of route choice. Multinomial logit (MNL), structured multinomial probit (SMNP), user equilibrium (UE), logit-based stochastic user equilibrium (SUE), probit-based SUE, and all-or-nothing (AON) assignment methods are applied to the comparative analysis. The investigated methods include those with three types of error components in their cost functions and two types of flow dependencies. Four methods of generating the route choice set are also compared for use as stochastic traffic assignment methods. The revealed preference data of urban rail route choice in the Tokyo Metropolitan Area are used for the case analysis. The empirical case analysis shows that probit-based SUE provides the best accuracy but requires the longest computation time. It also shows that the heuristics used to generate the choice set influence the method’s accuracy, while the incorporation of route commonality and in-vehicle congestion significantly improves its accuracy. Finally, the implications for practical rail planning are discussed on the basis of the analysis results.     

Optimal joint distance and time toll for cordon-based congestion pricing

This paper addresses the optimal toll design problem for the cordon-based congestion pricing scheme, where both a time-toll and a nonlinear distance-toll (i.e., joint distance and time toll) are levied for each network user’s trip in a pricing cordon. The users’ route choice behaviour is assumed to follow the Logit-based stochastic user equilibrium (SUE). We first propose a link-based convex programming model for the Logit-based SUE problem with a joint distance and time toll pattern. A mathematical program with equilibrium constraints (MPEC) is developed to formulate the optimal joint distance and time toll design problem. The developed MPEC model is equivalently transformed into a semi-infinite programming (SIP) model. A global optimization method named Incremental Constraint Method (ICM) is designed for solving the SIP model. Finally, two numerical examples are used to assess the proposed methodology.     

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.     

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.     

Impact analysis of cordon-based congestion pricing on mode-split for a bimodal transportation network

This paper investigates the impact of cordon-based congestion pricing scheme on the mode-split of a bimodal transportation network with auto and rail travel modes. For any given toll-charge pattern, its impact on the mode-split can be estimated by solving a combined mode-split and traffic-assignment problem. Using a binary logit model for the mode-split, the combined problem is converted into a traffic-assignment problem with elastic demand. Probit-based stochastic user equilibrium (SUE) principle is adopted for this traffic-assignment problem, and a continuously distributed value of time (VOT) is assumed to convert the toll charges and transit fares into time-units. This combined mode-split and traffic-assignment problem is then formulated as a fixed-point model, which can be solved by a convergent Cost Averaging method. The combined mode-split and traffic-assignment problem is then used to analyze a multimodal toll design problem for cordon-based congestion pricing scheme, with the aim of increasing the mode-share of public transport system to a targeted level. Taking the fixed-point model as a constraint, the multimodal toll design problem is thus formulated as a mathematical programming with equilibrium constraints (MPEC) model. A genetic algorithm (GA) is employed to solve this MPEC model, which is then numerical validated by a network example.     

On the degradation of performance for traffic networks with oblivious users

We consider the problem of characterizing user equilibria and optimal solutions for routing in a given network. We extend the known models by considering users oblivious to congestion in the following sense: While in the typical user equilibrium setting the users follow a strategy that minimizes their individual cost by taking into account the (dynamic) congestion due to the current routing pattern, an oblivious user ignores congestion altogether; instead, he or she decides his routing on the basis of cheapest routes on a network without any flow whatsoever. These cheapest routes can be, for example, the shortest paths in the network without any flow. This model tries to capture the fact that a certain percentage of travelers base their route simply on the distances they observe on a map, without thinking (or knowing, or caring) about the delays experienced on this route due to their fellow travelers. In this work we study the effect of such users using as the measure of network performance its price of anarchy, i.e., the ratio of the total latency experienced by the users (oblivious or not) at equilibrium over the social optimum.     

Robust optimization of distance-based tolls in a network considering stochastic day to day dynamics

This paper investigates the nonlinear distance-based congestion pricing in a network considering stochastic day-to-day dynamics. After an implementation/adjustment of a congestion pricing scheme, the network flows in a certain period of days are not on an equilibrium state, thus it is problematic to take the equilibrium-based indexes as the pricing objective. Therefore, the concept of robust optimization is taken for the congestion toll determination problem, which takes into account the network performance of each day. First, a minimax model which minimizes the maximum regret on each day is proposed. Taking as a constraint of the minimax model, a path-based day to day dynamics model under stochastic user equilibrium (SUE) constraints is discussed in this paper. It is difficult to solve this minimax model by exact algorithms because of the implicity of the flow map function. Hence, a two-phase artificial bee colony algorithm is developed to solve the proposed minimax regret model, of which the first phase solves the minimal expected total travel cost for each day and the second phase handles the minimax robust optimization problem. Finally, a numerical example is conducted to validate the proposed models and methods.     

Some developments in equilibrium traffic assignment

A network optimization problem is formulated which yields a probabilistic equilibrated traffic assignment incorporating congestion effects and which as a special case, reduces to a user optimized equilibrium solution. In the resulting model, path choice is determined by a logit formula in which path costs are functions of the assigned flows. The article also demonstrates the similarity between some fixed demand incremental methods of traffic assignment and the minimization problem associated with computing the user equilibrium assignment.     

Bounded rationality and irreversible network change

A network change is said to be irreversible if the initial network equilibrium cannot be restored by revoking the change. The phenomenon of irreversible network change has been observed in reality. To model this phenomenon, we develop a day-to-day dynamic model whose fixed point is a boundedly rational user equilibrium (BRUE) flow. Our BRUE based approach to modeling irreversible network change has two advantages over other methods based on Wardrop user equilibrium (UE) or stochastic user equilibrium (SUE). First, the existence of multiple network equilibria is necessary for modeling irreversible network change. Unlike UE or SUE, the BRUE multiple equilibria do not rely on non-separable link cost functions, which makes our model applicable to real-world large-scale networks, where well-calibrated non-separable link cost functions are generally not available. Second, travelers’ boundedly rational behavior in route choice is explicitly considered in our model. The proposed model is applied to the Twin Cities network to model the flow evolution during the collapse and reopening of the I-35 W Bridge. The results show that our model can to a reasonable level reproduce the observed phenomenon of irreversible network change.     

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.     

Route Choice Models Used in the Stochastic User Equilibrium Problem: A Review

Several route choice models are reviewed in the context of the stochastic user equilibrium problem. The traffic assignment problem has been extensively studied in the literature. Several models were developed focusing mainly on the solution of the link flow pattern for congested urban areas. The behavioural assumption governing route choice, which is the essential part of any traffic assignment model, received relatively much less attention. The core of any traffic assignment method is the route choice model. In the wellknown deterministic case, a simple choice model is assumed in which drivers choose their best route. The assumption of perfect knowledge of travel costs has been long considered inadequate to explain travel behaviour. Consequently, probabilistic route choice models were developed in which drivers were assumed to minimize their perceived costs given a set of routes. The objective of the paper is to review the different route choice models used to solve the traffic assignment problem. Focus is on the different model structures. The paper connects some of the route choice models proposed long ago, such as the logit and probit models, with recently developed models. It discusses several extensions to the simple logit model, as well as the choice set generation problem and the incorporation of the models in the assignment problem.     

Trial-and-error method for congestion pricing scheme under side-constrained probit-based stochastic user equilibrium conditions

A toll pattern that can restrict link flows on the tolled links to some predetermined thresholds is named as effective toll solution, which can be theoretically obtained by solving a side-constraint traffic assignment problem. Considering the practical implementation, this paper investigates availability of an engineering-oriented trial-and-error method for the effective toll pattern of cordon-based congestion pricing scheme, under side-constrained probit-based stochastic user equilibrium (SUE) conditions. The trial-and-error method merely requires the observed traffic counts on each entry of the cordon. A minimization model for the side-constrained probit-based SUE problem with elastic demand is first proposed and it is shown that the effective toll solution equals to the product of value of time and optimal Lagrangian multipliers with respect to the side constraints. Then, employing the Lagrangian dual formulation of the minimization method, this paper has built a convergent trial-and-error method. The trial-and-error method is finally tested by a numerical example developed from the cordon-based congestion pricing scheme in Singapore.     

Unified closed-form expression of logit and weibit and its extension to a transportation network equilibrium assignment

This study proposes a generalized multinomial logit model that allows heteroscedastic variance and flexible utility function shape. The novelty of our approach is that the model is theoretically derived by applying a generalized extreme-value distribution to the random component of utility, while retaining its closed-form expression. In addition, the weibit model, in which the random utility is assumed to follow the Weibull distribution, is a special case of the proposed model. This is achieved by utilizing the q-generalization method developed in Tsallis statistics. Then, our generalized logit model is incorporated into a transportation network equilibrium model. The network equilibrium model with a generalized logit route choice is formulated as an optimization problem for uncongested networks. The objective function includes Tsallis entropy, a type of generalized entropy. The generalization of the Gumbel and Weibull distributions, logit and weibit models, and network equilibrium model are formulated within a unified framework with q-generalization or Tsallis statistics.     

A review of models for the temporal distribution of peak traffic demand

The existing models for predicting the temporal distribution of peak traffic demand are reviewed in this paper. Based on their assumptions and concepts, the models are classified into three groups as follows: deterministic user equilibrium (DUE), stochastic user equilibrium (SUE) and system optimum (SO). The merits and weaknesses of each group are discussed with regards to the validity of their assumptions and computational problems associated with them. Most of the models were developed for the single origin-destination pair with simple network system which is an oversimplification of a practical problem involving multiple origin-destination pairs with complex network systems. This is a major limitation when considering the application of the existing models to real life problems. Directions for future research are then proposed.     

General stochastic user equilibrium traffic assignment problem with link capacity constraints

This paper addresses a general stochastic user equilibrium (SUE) traffic assignment problem with link capacity constraints. It first proposes a novel linearly constrained minimization model in terms of path flows and then shows that any of its local minimums satisfies the generalized SUE conditions. As the objective function of the proposed model involves path‐specific delay functions without explicit mathematical expressions, its Lagrangian dual formulation is analyzed. On the basis of the Lagrangian dual model, a convergent Lagrangian dual method with a predetermined step size sequence is developed. This solution method merely invokes a subroutine at each iteration to perform a conventional SUE traffic assignment excluding link capacity constraints. Finally, two numerical examples are used to illustrate the proposed model and solution method.     

Network equilibrium for congested multi‐mode networks with elastic demand

This paper proposes an elastic demand network equilibrium model for networks with transit and walking modes. In Hong Kong, the multi‐mode transit system services over 90% of the total journeys and the demand on it is continuously increasing. Transit and walking modes are related to each other as transit passengers have to walk to and from transit stops. In this paper, the multi‐mode elastic‐demand network equilibrium problem is formulated as a variational inequality problem where the combined mode and route choices are modeled in a hierarchical logit structures and the total travel demand for each origin‐destination pair is explicitly given by an elastic demand function. In addition, the capacity constraint for transit vehicles and the effects of bi‐directional flows on walkways are considered in the proposed model. All these congestion effects are taken into account for modeling the travel choices. A solution algorithm is developed to solve the multi‐mode elastic‐demand network equilibrium model. It is based on a Block Gauss‐Seidel decomposition approach coupled with the method of successive averages. A numerical example is used to illustrate the application of the proposed model and solution algorithm.     

Modeling the evolutions of day‐to‐day route choice and year‐to‐year ATIS adoption with stochastic user equilibrium

Suppose that in an urban transportation network there is a specific advanced traveler information system (ATIS) which acts for reducing the drivers' travel time uncertainty through provision of pre‐trip route information. Because of the imperfect information provided, some travelers are not in compliance with the ATIS advice although equipped with the device. We thus divide all travelers into three groups, one group unequipped with ATIS, another group equipped and in compliance with ATIS advice and the third group equipped but without compliance with the advice. Each traveler makes route choice in a logit‐based manner and a stochastic user equilibrium with multiple user classes is reached for every day. In this paper, we propose a model to investigate the evolutions of daily path travel time, daily ATIS compliance rate and yearly ATIS adoption, in which the equilibrium for every day's route choice is kept. The stability of the evolution model is initially analyzed. Numerical results obtained from a test network are presented for demonstrating the model's ability in depicting the day‐to‐day and year‐to‐year evolutions.     

Intermodal hub-and-spoke network design: Incorporating multiple stakeholders and multi-type containers

This paper develops a mathematical program with equilibrium constraints (MPEC) model for the intermodal hub-and-spoke network design (IHSND) problem with multiple stakeholders and multi-type containers. The model incorporates a parametric variational inequality (VI) that formulates the user equilibrium (UE) behavior of intermodal operators in route choice for any given network design decision of the network planner. The model also uses a cost function that is capable of reflecting the transition from scale economies to scale diseconomies in distinct flow regimes for carriers or hub operators, and a disutility function integrating actual transportation charges and congestion impacts for intermodal operators. To solve the MPEC model, a hybrid genetic algorithm (HGA) embedded with a diagonalization method for solving the parametric VI is proposed. Finally, the comparative analysis of the HGA and an exhaustive enumeration algorithm indicates a good performance of the HGA in terms of computational time and solution quality. The HGA is also applied to solve a large-scale problem to show the applicability of the proposed model and algorithm.