Traffic Equilibrium Model Research Articles

Exploring the pre-disaster evacuation network design problem under five traffic equilibrium models

This paper explores modeling approaches for the pre-disaster Evacuation Network Design Problem (ENDP) considering different flow equilibrium conditions. We combine this problem with the modeling idea of Continuous Network Design Problem (CNDP), which we call Continuous Evacuation Network Design Problem (CENDP) in this paper. We develop five CENDP models, which are under the consideration of User Equilibrium (UE), Stochastic User Equilibrium (SUE), Boundedly Rational User Equilibrium (BRUE), and Non-equilibrium (NONE), among which we develop two types of models based on BRUE. The modeling is mainly to consider the single objective of optimizing the total evacuation time, and then to provide reasonable road expansion solutions under certain budget constraints and different equilibrium conditions. Our main motivation for developing models is to introduce various types of equilibrium conditions into models and design algorithms to solve these problems while mining for key insights. We design the corresponding five heuristic algorithms to solve models and verify the applicability of the models and algorithms by two test networks (Nguyen-Dupuis network and Sioux-Falls network). We demonstrate whether evacuation flow equilibrium need or not need to be considered in the CENDP, the applicability of different equilibrium conditions, and the correlation between the total evacuation time, the network investment cost, and the network congestion degree. Additionally, we conduct model and algorithm tests on 40 instance networks, dividing them into medium-sized networks (20 instances) and large-sized networks (20 instances). Not only do we further validate the insights obtained from the test networks, but we also expand upon them. Specifically, the main findings of this study are as follows: (1) We demonstrate that considering evacuation flow equilibrium in CENDP is essential to reduce total evacuation time, construction costs, and mitigate congestion. (2) While increased investment in road construction can meet evacuation time requirements, it is crucial to make informed decisions, as investment alone does not directly reduce total evacuation time and congestion. (3) Optimizing road evacuation time is more effective than merely increasing road capacity for reducing total evacuation time and mitigating congestion. (4) From the perspectives of total evacuation time, investment cost, and network congestion degree, the CENDP model considering user equilibrium performs better in medium-sized networks, while the CENDP model considering stochastic user equilibrium performs better in large-sized networks. Conversely, the CENDP model that does not consider flow equilibrium performs the worst across all above three metrics. Based on this, we also provide recommendations on which model to choose for different metrics. In summary, this study not only reveals the importance of different flow equilibrium conditions in evacuation network design but also provides valuable strategic recommendations for practical applications to optimize evacuation effectiveness and resource allocation.

A general coupled morning–evening traffic equilibrium model with rideshare, ride-hailing, and public transit services

We develop a general equilibrium model to capture the complex interactions between different modes, such as solo driving, public transit, as well as rideshare and ride-hailing services such as Uber and Lyft, under a joint morning and evening commute framework. Formulated as a variational inequality (VI) and equivalently as a mixed complementarity problem (MiCP), the model allows (a) travelers to switch between different transportation modes and (b) passengers from different Origin-Destination (OD) pairs to share a ride together. The computational results on the Sioux-Falls network show that our model captures the possible mode switches and the coupling effects between morning and evening commutes. Furthermore, our numerical examples demonstrate that modelling morning and evening commutes separately tends to overestimate the travelers' disutility and the average Vehicle Miles Traveled (VMT) in the network.

A Traffic Equilibrium Model for Multi-Modal Networks with Uncertain Demands

In this study, an effective travel cost (ETC) traffic equilibrium model is proposed for multi-modal networks with uncertain demands. The multi-modal networks are transformed into supernetworks and travel demands are assumed to be closed intervals. Passenger flows and travel costs are also formulated as closed intervals to capture the effects of uncertain demands. The ETC concept is introduced and regarded as a choice criterion to develop an equilibrium model which captures travellers’ travel mode and route choice behavior under interval travel costs. The model is formulated as a variational inequality problem, and the method of successive average algorithm is adapted to interval mathematics to obtain the results in the form of interval variables. Illustrative examples are also presented to demonstrate the model’s characteristics and its differences from the traditional equilibrium model, in which the expected travel cost is regarded as the choice criterion.

A Reliability-Based Traffic Equilibrium Model with Boundedly Rational Travelers Considering Acceptable Arrival Thresholds

This paper examines the effects of boundedly rational decision characteristics on travelers’ route choice behavior. The concept of boundedly rational confidence level (BRCL) is redefined, which is the probability that a trip arrives between the acceptable earliest arrival time and the acceptable latest arrival time on the shortest travel time budget (TTB). Mathematically, the acceptable boundedly rational arrival thresholds are proposed. Then, a reliability-based boundedly rational traffic equilibrium model (R-BRTE) considering both travel time reliability and acceptable arrival thresholds is developed. Moreover, the equivalent variational inequality problem and uniqueness of solution on the proposed model are proved. A route-based solution algorithm is used to solve the proposed R-BRTE model. Numerical results present the important decision ideas of the proposed model. The results demonstrate that travelers’ bounded rationality has a great impact on their route choice behavior and network performance.

System-level impacts of en-route information sharing considering adaptive routing

Connected and automated vehicles (CAVs) and infrastructure-to-vehicle (I2V) communication have great potential to improve traffic safety and mobility at intersection and road segment levels. However, since transportation is an interconnected network, local transportation state change could lead to broader impacts by rerouting. In this study, we focus on investigating the impacts of en-route locational information updates on the system-level safety and mobility considering adaptive routing of CAVs. We propose a novel two-stage stochastic traffic equilibrium model to characterize the equilibrium traffic patterns considering adaptive routing behavior when en-route traffic information is provided. The proposed methodology is tested using the Orlando transportation network with real traffic and accident data. Different I2V information sharing strategies are analyzed to better understand the impacts of information sharing locations of I2V on transportation network safety and mobility, through which we generate insights on the optimal design of traffic information sharing systems.

On Solving a Class of Continuous Traffic Equilibrium Problems and Planning Facility Location Under Congestion

This paper presents methods to obtain analytical solutions to a class of continuous traffic equilibrium problems, where continuously distributed customers from a bounded two-dimensional service region seek service from one of several discretely located facilities via the least congested travel path. We show that under certain conditions, the traffic flux at equilibrium, which is governed by a set of partial differential equations, can be decomposed with respect to each facility and solved analytically. This finding paves the foundation for an efficient solution scheme. Closed-form solution to the equilibrium problem can be obtained readily when the service region has a certain regular shape, or through an additional conformal mapping if the service region has an arbitrary simply connected shape. These results shed light on some interesting properties of traffic equilibrium in a continuous space. This paper also discusses how service facility locations can be easily optimized by incorporating analytical formulas for the total generalized cost of spatially distributed customers under congestion. Examples of application contexts include gates or booths for pedestrian traffic, as well as launching sites for air vehicles. Numerical examples are used to show the superiority of the proposed optimization framework, in terms of both solution quality and computation time, as compared with traditional approaches based on discrete mathematical programming and partial differential equation solution methods. An example with the metro station entrances at the Beijing Railway Station is also presented to illustrate the usefulness of the proposed traffic equilibrium and location design models.

Accelerating the gradient projection algorithm for solving the non-additive traffic equilibrium problem with the Barzilai-Borwein step size

The non-additive traffic equilibrium problem (NaTEP) overcomes the inadequacies of the additivity assumption in traditional traffic equilibrium models by relaxing the cost incurred on each path that is not a simple sum of the link costs on that path. The computation of the NaTEP heavily depends on the efficiency of the step size determination. This paper aims to accelerate the path-based gradient projection (GP) algorithm for solving the NaTEP using the Barzilai-Borwein (BB) step size scheme. The GP algorithm with the BB step size scheme uses the solution information of the last two iterations to determine a suitable step size and avoids extra evaluations of the mapping value. The proposed algorithm only needs to perform one-time projection onto the nonnegative orthant at each iteration. Two approaches with and without column generation are considered in the GP algorithm implementation. A non-additive shortest path algorithm is adopted for the column generation approach. Numerical results on four transportation networks demonstrate the superior efficiency and robustness of the GP algorithm with the BB step size scheme over the self-adaptive GP method.

A Strategic Markovian Traffic Equilibrium Model for Capacitated Networks

In the realm of traffic assignment over a network involving rigid arc capacities, the aim of the present work is to generalize the model of Marcotte, Nguyen, and Schoeb [Marcotte P, Nguyen S, Schoeb A (2004) A strategic flow model of traffic assignment in static capacitated networks. Oper. Res. 52(2):191–212.] by casting it within a stochastic user equilibrium framework. The strength of the proposed model is to incorporate two sources of stochasticity stemming, respectively, from the users’ imperfect knowledge regarding arc costs (represented by a discrete choice model) and the probability of not accessing saturated arcs. Moreover, the arc-based formulation extends the Markovian traffic equilibrium model of Baillon and Cominetti [Baillon JB, Cominetti R ( 2008 ) Markovian traffic equilibrium. Math. Programming 111(1-2):33–56.] through the explicit consideration of capacities. This paper is restricted to the case of acyclic networks, for which we present solution algorithms and numerical experiments.

The Effect of Regret-Based Risky Route Choice on the Traffic Equilibrium for Emergency Evacuation

Following the research on human decision-making under risk and uncertainty, the purpose of this paper is to analyze evacuees’ risky route decision behavior and its effect on traffic equilibrium. It examines the possibility of applying regret theory to model travellers’ regret-taking behavior and network equilibrium in emergency context. By means of modifying the utility function in expected utility theory, a regret-based evacuation traffic equilibrium model is established, accounting for the evacuee’s psychological behavior of regret aversion and risk aversion. Facing two parallel evacuation routes choice situation, the effect of evacuees’ risk aversion and regret aversion on traffic equilibrium is numerically investigated as well as the road capacity reduction from natural disaster. The findings reveal that evacuees prefer the riskless route with the lower travel time as the increase of the regret aversion degree. The equilibrium tends to be achieved when more evacuees choose the safer route jointly affected by risk aversion and regret aversion. Moreover, an optimization model for disaster occurring possibility is formulated to assess the traffic system performance for evacuation management. These findings are helpful for understanding how the regret aversion and risk aversion influence traffic equilibrium.

The Online Charging and Discharging Scheduling Potential of Electric Vehicles Considering the Uncertain Responses of Users

A method for evaluating the charging and discharging scheduling potential of electric vehicles in case of the occasional operation of a power grid is proposed, which describes the variations in the users’ responses. Considering the social relationships and mutual learning among users, an evolutionary game model for the optimal response in a complex network is established to study the response rule of the users for the scheduling measures. Wardrop's first traffic equilibrium model is applied to calculate the shortest time for the vehicles to arrive at the charging station from their response sites. Finally, the scheduling potential at a charging station is effectively evaluated considering the influence of the type of activities that the users participate in and their credits. Taking a test traffic network and the traffic network of Beijing third ring road as examples, the responses of the electric vehicle users are studied, and the scheduling potential of electric vehicles in different scenarios is verified. Although an increased incentive can improve the response level of the electric vehicles, it may lead to a reduction in the scheduling capacity, delays in the charging and discharging starting time, and a shortened scheduling duration.

A faster path-based algorithm with Barzilai-Borwein step size for solving stochastic traffic equilibrium models

Step size determination (also known as line search) is an important component in effective algorithmic development for solving the traffic assignment problem. In this paper, we explore a novel step size determination scheme, the Barzilai-Borwein (BB) step size, and adapt it for solving the stochastic user equilibrium (SUE) problem. The BB step size is a special step size determination scheme incorporated into the gradient method to enhance its computational efficiency. It is motivated by the Newton-type methods, but it does not need to explicitly compute the second-order derivative. We apply the BB step size in a path-based traffic assignment algorithm to solve two well-known SUE models: the multinomial logit (MNL) and cross-nested logit (CNL) SUE models. Numerical experiments are conducted on two real transportation networks to demonstrate the computational efficiency and robustness of the BB step size. The results show that the BB step size outperforms the current step size strategies, i.e., the Armijo rule and the self-regulated averaging scheme.

Coordinated Perimeter Control for Multiregion Heterogeneous Networks Based on Optimized Transfer Flows

Exploring efficient control strategies for heterogeneously congested urban networks remains a big research challenge. The theory of macroscopic fundamental diagram (MFD) provides a new perspective for network-wide congestion control decisions. This paper proposes a coordinated perimeter control strategy for multiregion heterogeneous networks based on optimized transfer flows. First, a two-layer network partitioning method is presented to capture spatial heterogeneity dynamics of urban networks. For this partition, traffic flow equilibrium model based on MFD and multiagent based hierarchical traffic management scheme are built. Then, an improved multinomial logit model is developed for deriving optimized transfer flows among multiple congested regions. A coordinated perimeter control strategy using model predictive control is further proposed, which is aimed at tracking desired accumulations of each congested region. As a case study, the proposed control strategy is applied to the downtown network of Jinan, China, using simulation analysis. The results demonstrate that it can achieve balanced network flow distribution and increased mobility.

A Two-Class Stochastic Network Equilibrium Model under Adverse Weather Conditions

Adverse weather condition is one of the inducements that lead to supply uncertainty of an urban transportation system, while travelers’ multiple route choice criteria are the nonignorable reason resulting in demand uncertainty. This paper proposes a novel stochastic traffic network equilibrium model considering impacts of adverse weather conditions on roadway capacity and route choice criteria of two-class mixed roadway travellers on demand modes, in which the two-class route choice criteria root in travelers’ different network information levels (NILs). The actual route travel time (ARTT) and perceived route travel time (PRTT) are considered as the route choice criteria of travelers with perfect information (TPI) and travelers with bounded information (TBI) under adverse weather conditions, respectively. We then formulate the user equilibrium (UE) traffic assignment model in a variational inequality problem and propose a solution algorithm. Numerical examples including a small triangle network and the Sioux Falls network are presented to testify the validity of the model and to clarify the inner mechanism of the two-class UE model under adverse weather conditions. Managerial implications and applications are also proposed based on our findings to improve the operation efficiency of urban roadway network under adverse weather conditions.

Distributionally Robust Markovian Traffic Equilibrium

In a Markovian traffic equilibrium model, users move toward their destinations by a sequence of successive link choices using a discrete choice model at each node, taking congestion into account. Although a convex optimization formulation is available to compute the equilibrium flows for a continuous distribution of link utilities, practical applications have thus far been mainly restricted to the multinomial logit model and its variants. In this paper, we relax the assumption of a complete joint distribution of link utilities to only knowledge on the marginal distributions and propose a new convex optimization formulation for a distributionally robust Markovian traffic equilibrium. The formulation is provably efficiently solvable and has the flexibility of allowing for general marginal distributions, thus capturing different types of nonidentical, skewed, and heavy-tailed distributions at the link level.

Bilevel dynamic continuum model for housing allocation and transportation emission problems in an urban city

In this study, we examine housing allocation and transportation emission problems in an urban city. Housing development patterns have a significant influence on travel behavior and related transportation emissions. We consider a hypothetical city with one central business district in which the road network is assumed to be sufficiently dense to be viewed as a continuum. We establish a bilevel dynamic continuum model to describe the relationships among housing provision, the transportation system, and traffic-related emissions. In the lower-level subprogram, housing allocation is considered in a simultaneous departure time and route-choice dynamic traffic user equilibrium model, in which travelers choose routes and departure times that satisfy dynamic user-optimal principles. In the upper-level subprogram, the total emissions are minimized by optimizing the housing allocation. The finite-volume method, the projection method, the method of successive averages, and the Frank–Wolfe method are applied to solve the presented model based on unstructured meshes. A numerical experiment for an urban area with one central business district is given to demonstrate the effectiveness of the model and the numerical algorithm.

Discrete-time dynamic road congestion pricing under stochastic user optimal principle

Road pricing is believed to be an effective instrument for alleviating roadway congestion. Most existing road pricing schemes are developed based on traditional static traffic equilibrium models, in which a fixed toll can be obtained to support the corresponding traffic flow pattern as an equilibrium. However, static models cannot consider the evolution process of traffic flow caused by the day-to-day fluctuations of road users’ route choices. Under practical traffic conditions, the multiple traffic equilibria may exist (e.g., due to the asymmetric travel cost function). Indeed, the fixed road pricing scheme derived from the equilibrium model cannot guarantee that the dynamic traffic system can converge to the desired equilibrium state from any initial traffic state. This study, assuming that travelers follow the stochastic user optimal principle, develops a day-to-day dynamic road pricing scheme that can drive the traffic dynamic system to converge to a given stochastic user equilibrium (SUE) even when the traffic system has multiple SUE states. The characteristic of this dynamic road pricing scheme is verified by rigorous proof and numerical tests in this study.

The discrete-time second-best dynamic road pricing scheme

Congestion pricing is widely recognized as an efficient instrument for alleviating road congestion. Most of existing road pricing scheme are developed based on traffic equilibrium model. The equilibrium can be looked as a traffic state desired by traffic management authorities. However, when the traffic system has multiple equilibria if the initial traffic state falls beyond the attraction domain, it may not converge to the desired equilibrium under the pricing based on the traffic equilibrium through a day-to-day adjustment process (Bie and Lo, 2010). In this study, we aim to develop a more practical second-best dynamic road pricing scheme implemented on a subset of links, which can drive the traffic evolution towards the desired second-best traffic user equilibrium state from any initial traffic state. This second-best dynamic pricing scheme has the following characteristics: (i) the dynamic pricing is discrete-time scheme, as opposed to most of existing continuous dynamic pricing scheme; (ii) the derivation of the dynamic pricing is applicable to very general day-to-day traffic dynamic model; (iii) the dynamic pricing can direct the traffic system to converge to the desired equilibrium from any initial traffic state even multiple equilibria exist. This study also presents rigorous proofs and numerical tests to verify above these characteristics of our dynamic pricing scheme.

A link-based mean-excess traffic equilibrium model under uncertainty

Traffic equilibrium models under uncertainty characterize travelers’ route choice behaviors under travel time variability. In this paper, we develop a link-based mean-excess traffic equilibrium (L-METE) model by integrating the sub-additivity property and complete travel time variability characterization of mean-excess travel time (METT), and the computationally tractable additive route cost structure of the conventional user equilibrium (UE) problem. Compared to the majority of relevant models formulated in the route domain, the link-based modeling has two desirable features on modeling flexibility and algorithmic development. First, it avoids the normal route travel time distribution assumption (uniformly imposed for all routes) that inherits from the Central Limit Theorem in most route-based models, permitting the use of any suitable link travel time distributions from empirical studies. Second, the additive route cost structure makes the L-METE model solvable by readily adapting existing UE algorithms without the need of storing/enumerating routes while avoiding the computationally demanding nonadditive shortest path problem and route flow allocations in route-based models, which is a significant benefit for large-scale network applications under uncertainty.

Integrated traffic-transit stochastic equilibrium model with park-and-ride facilities

We propose an Integrated Stochastic Equilibrium model that considers both private automobile traffic and transit networks to incorporate the interactions between these two modes in terms of travel time and generalized costs. In addition, in the general version of the model, travelers are allowed to switch from personal vehicles to mass transit at specific locations in a park-and-ride scheme. The assignment for traffic equilibrium is based on the Markovian Traffic Equilibrium model of Baillon and Cominetti (2008), whereas the equilibrium of the transit network is represented by the Stochastic Transit Equilibrium model of Cortés et al. (2013). Stochastic travel decisions are made at the node level, thereby avoiding the enumeration of routes or strategies and incorporating various perception and uncertainty issues. We propose a Method-of-Successive-Averages algorithm to calculate an Integrated Stochastic Equilibrium and conduct numerical experiments to highlight the effect of stochasticity on equilibrium flows and travel times. Our experiments show that higher stochasticity implies greater dispersion of equilibrium flows and longer expected travel times. Results on a real network with mode combination and park and ride facilities provide insights regarding the use of park and ride in terms of number and location, potential modal share of the combined mode option under different circumstances, and travel time impact due to the implementation of such park and ride facilities in a real setting.

A traffic equilibrium model with paid-parking search

This paper describes a Wardropian traffic flow model, integrated with a search model for paid parking. The occupancy rate influences the probability of finding on-street (kerbside) or off-street (garage) parking spaces. We formulate the model as a mixed complementarity problem, which can be solved without a complete enumeration of all possible paths, and the use of readily available software. We analyse different parking policies in Zurich and find that changing the parking fee structure will lead to high efficiency gains. Incorporating household heterogeneity is not critical for the overall efficiency effects, but shows important regressive, distributional effects. .