Traffic Network Equilibrium Research Articles

The projection algorithm for inverse quasi-variational inequalities with applications to traffic assignment and network equilibrium control problems

We consider a first-order dynamical system for solving inverse quasi-variational inequalities in finite dimensional spaces. Through an explicit time discrete version of the proposed dynamical system, we investigate the linear convergence of a projection algorithm under suitable conditions of parameters. Moreover, we investigate the application of inverse quasi-variational inequalities in traffic assignment problem and network equilibrium control problem. The numerical experiments for these practical problems confirm the linear convergence of the theoretical part. In particular, the obtained results provide a positive answer to an open question posted by S. Dey and S. Reich in Optimization, DOI: 10.1080/02331934.2023.2173525 (2023).

Optimization of regulation and fiscal policies for urban residential land use and traffic network management

Residential land use density regulation is an important policy instrument for reducing the local external diseconomies caused by concentration of population and buildings in urban areas. But density regulation also affects the distribution of population across the whole urban space, which determines the travel demands hence traffic flows over the transportation network in the city. In this paper, we formulate a model integrating the residential location choice equilibrium, traffic network equilibrium and the housing market equilibrium, subject to floor area ratio (FAR) regulation, a representative practical strategy for density regulation. The restriction of FAR regulation on housing supply is formulated as a development coordination cost function dependent on the level of FAR that is actually realized under the designated upper FAR limit. Complementary location-specific tax/subsidy policies, and road pricing schemes, are also addressed in a unified manner.The mathematical structure of the model is explored, and efficient algorithms are developed for solving the equilibrium equations, and for the optimization of FAR regulations and the pecuniary strategies.

Stability analysis for set-valued inverse mixed variational inequalities in reflexive Banach spaces

This work is devoted to the analysis for a new class of set-valued inverse mixed variational inequalities (SIMVIs) in reflexive Banach spaces, when both the mapping and the constraint set are perturbed simultaneously by two parameters. Several equivalence characterizations are given for SIMVIs to have nonempty and bounded solution sets. Based on the equivalence conditions, under the premise of monotone mappings, the stability result for the SIMVIs is obtained in the reflexive Banach space. Furthermore, to illustrate the results, an example of the traffic network equilibrium control problem is provided at the end of this paper. The results presented in this paper generalize and extend some known results in this area.

Robust Multi-Criteria Traffic Network Equilibrium Problems with Path Capacity Constraints

With the progress of society and the diversification of transportation modes, people are faced with more and more complicated travel choices, and thus, multi-criteria route choosing optimization problems have drawn increased attention in recent years. A number of multi-criteria traffic network equilibrium problems have been proposed, but most of them do not involve data uncertainty nor computational methods. This paper focuses on the methods for solving robust multi-criteria traffic network equilibrium problems with path capacity constraints. The concepts of the robust vector equilibrium and the robust vector equilibrium with respect to the worst case are introduced, respectively. For the robust vector equilibrium, an equivalent min–max optimization problem is constructed. A direct search algorithm, in which the step size without derivatives and redundant parameters, is proposed for solving this min–max problem. In addition, we construct a smoothing optimization problem based on a variant version of ReLU activation function to compute the robust weak vector equilibrium flows with respect to the worst case and then find robust vector equilibrium flows with respect to the worst case by using the heaviside step function. Finally, extensive numerical examples are given to illustrate the excellence of our algorithms compared with existing algorithms. It is shown that the proposed min–max algorithm may take less time to find the robust vector equilibrium flows and the smoothing method can more effectively generate a subset of the robust vector equilibrium with respect to the worst case.

Cascading failure in coupled networks of transportation and power grid

The large-scale application of electric vehicles (EVs) makes the power grid and transportation network deeply coupled, resulting in the propagation of random disturbances and failures between the two systems. This paper proposes an analysis method of cascading failure in the coupled networks of transportation networks and power grids. The cascading failure model of a power grid with large-scale EV charging loads is established by using a direct current power flow based on the frequency response (DFR). The behavior characteristics of EVs and transportation states under failure conditions are analyzed by combining the travel decisions of users with the traffic network equilibrium. Considering the coupling relationship between the power grid and transportation networks at charging station nodes, the load–capacity model is adopted to describe the propagation of failures between the two networks. Finally, the propagation characteristics of failures considering the primary frequency modulation and rationality degree of EVs are simulated and analyzed. The simulation results show that the application of EVs makes the power grid and transportation mutually dependent, and the primary frequency modulation characteristic and bounded rationality of EVs can weaken the effect of cascading failure in the coupled networks.

Analysis of Traffic Flow Equilibrium in Urban Road Network Based on Floating Vehicle Data

Based on the floating vehicle data, this paper analyzes the equilibrium of urban road network traffic flow. This can guide traffic flow distribution and provide reliable basis for traffic control. This paper starts with the quantitative analysis of the traffic network equilibrium, on the basis of verifying the validity of the floating car data, divides the urban road network into regions, and constructs an analysis model of the traffic flow equilibrium of the urban road network. The urban road traffic distribution model is constructed in accordance with the number of road traffic segments. On this basis, gini coefficient index is introduced to judge the road network flow balance, which is used to analyze the balance of each sub-region. By means of traffic guidance, signal control and other traffic control means, the traffic flow in each sub-area is balanced, and the traffic flow in the whole road network becomes balanced.

Evaluation and Application of Port Industrial Upgrading Based on Global Value Chain of Complex Network

Wang, Z.P., 2020. Evaluation and application of port industrial upgrading based on global value chain of complex network. In: Al-Tarawneh, O. and Megahed, A. (eds.), Recent Developments of Port, Marine, and Ocean Engineering. Journal of Coastal Research, Special Issue No. 110, pp. 231–234. Coconut Creek (Florida), ISSN 0749-0208.This paper transforms cluster escalation into complex networks where social networks, explicit and implicit knowledge networks interact with each other based on the level of knowledge and social relationship, then the relationship between variational inequality theory and optimum works out an equilibrium status of network and complex network, so as to explore how the network traffic and complex network equilibrium are subjected to change with the cluster scale. In the end, the cluster has realized its scalability.

Refugee migration networks and regulations: a multiclass, multipath variational inequality framework.

In this paper, we take up the timely topic of the modeling, analysis, and solution of refugee migration networks. We construct a general, multiclass, multipath model, determine the governing equilibrium conditions, and provide alternative variational inequality formulations in path flows and in link flows. We also demonstrate how governmental imposed regulations associated with refugees can be captured via constraints. We provide qualitative properties and then establish, via a supernetwork transformation, that the model(s) are isomorphic to traffic network equilibrium models with fixed demands. Illustrative examples are given, along with numerical examples, inspired by a refugee crisis from Mexico to the United States, which are solved using the Euler method embedded with exact equilibration. The work sets the foundation for the development of additional models and algorithms and also provides insights as to who wins and who loses under certain refugee regulations.

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.

Convergence study on the logarithmic-quadratic proximal regularization of strictly contractive Peaceman–Rachford splitting method with larger step-size

Recently, a strictly contractive Peaceman–Rachford splitting method with logarithmic-quadratic proximal regularization (SPRSM-LQP) was proposed for solving two-block separable convex minimization model. In practical applications, however, the smaller step-size should be strongly avoided. So we actually have the desire of seeking larger step-size whenever possible in order to accelerate the numerical performance. In this paper, we combine Fortin and Glowinski's accelerating techniques with the SPRSM-LQP. Thus a new algorithm with larger step-size is proposed. Under the same assumptions as the SPRSM-LQP, we establish the global convergence of its larger step-size counterpart. Moreover, preliminary numerical results show that the proposed method on a traffic network equilibrium problem is reliable and more efficient with larger step-size.

On the stability of approximate solutions to set-valued equilibrium problems

In this paper, we consider set-valued equilibrium problems in Hausdorff locally convex topological vector spaces. Based on linear scalarization techniques for sets, we study sufficient conditions for the stability of approximate solutions to such problems. Variational inequalities with equilibrium constraints and weak traffic network equilibrium problems are also discussed as applications of the main results.

Traffic network equilibrium problems with demands uncertainty and capacity constraints of arcs by scalarization approaches

This paper focuses on the vector traffic network equilibrium problem with demands uncertainty and capacityconstraints of arcs, in which, the demands are not exactly known and assumed as a discrete set that contains finite scenarios.For different scenario, the demand may be changed, which seems much more reasonable in practical programming.By using the linear scalarization method, we introduce several definitions of parametric equilibriumflows and reveal their mutual relations. Meanwhile, the relationships between the scalar variational inequality as well as the (weak) vector equilibrium flows are explored, meanwhile, some necessary and sufficient conditions that ensure the (weak) vector equilibrium flows are also considered.Additionally, by means of nonlinear scalarization functionals, twokinds of equilibrium principles are derived. All of the derived conclusions contain the demands uncertainty and capacity constraints of arcs, thus the resultsproposed in this paper improved some existing works. Finally, some numericalexamples are employed to show the merits of the improved conclusions.

Traffic equilibrium with a continuously distributed bound on travel weights: the rise of range anxiety and mental account

A new traffic network equilibrium problem with continuously distributed bounds on path weights is introduced in this paper, as an emerging modeling tool for evaluating traffic networks in which the route choice behavior of individual motorists is subject to some physical or psychological upper limit of a travel weight. Such a problem may arise from at least two traffic network instances. First, in a traffic network serving electric vehicles, the driving range of these vehicles is subject to a distance constraint formed by onboard battery capacities and electricity consumption rates as well as network-wide battery-recharging opportunities, which cause the range anxiety issue in the driving population. Second, in a tolled traffic network, while drivers take into account both travel time and road toll in their route choice decisions, many of them implicitly or explicitly set a budget constraint in their mental account for toll expense, subject to their own income levels and other personal and household socio-economic factors. In both cases, we model the upper limit of the path travel weight (i.e., distance or toll) as a continuously distributed stochastic parameter across the driving population, to reflect the diverse heterogeneity of vehicle- and/or motorist-related travel characteristics. For characterizing this weight-constrained network equilibrium problem, we proposed a convex programming model with a finite number of constraints, on the basis of a newly introduced path flow variable named interval path flow rate. We also analyzed the problem’s optimality conditions for the case of path distance limits, and studied the existence of optimal tolls for the case of path toll limits. A linear approximation algorithm was further developed for this complex network equilibrium problem, which encapsulates an efficient weight-constrained k-minimum time path search procedure to perform the network loading. Numerical results obtained from conducting quantitative analyses on example networks clearly illustrate the applicability of the modeling and solution methods for the proposed problem and reveal the mechanism of stochastic weight limits reshaping the network equilibrium.

Robust traffic assignment model: Formulation, solution algorithms and empirical application

ABSTRACTThe deterministic traffic assignment problem based on Wardrop's first criterion of traffic network utilization has been widely studied in the literature. However, the assumption of deterministic travel times in these models is restrictive, given the large degree of uncertainty prevalent in urban transportation networks. In this context, this paper proposes a robust traffic assignment model that generalizes Wardrop's principle of traffic network equilibrium to networks with stochastic and correlated link travel times and incorporates the aversion of commuters to unreliable routes.The user response to travel time uncertainty is modeled using the robust cost (RC) measure (defined as a weighted combination of the mean and standard deviation of path travel time) and the corresponding robust user equilibrium (UE) conditions are defined. The robust traffic assignment problem (RTAP) is subsequently formulated as a Variational Inequality problem. To solve the RTAP, a Gradient Projection algorithm is proposed, which involves solving a series of minimum RC path sub-problems that are theoretically and practically harder than deterministic shortest path problems. In addition, an origin-based heuristic is proposed to enhance computational performance on large networks. Numerical experiments examine the computational performance and convergence characteristics of the exact algorithm and establish the accuracy and efficiency of the origin-based heuristic on various real-world networks. Finally, the proposed RTA model is applied to the Chennai road network using empirical data, and its benefits as a normative benchmark are quantified through comparisons against the standard UE and System Optimum (SO) models.

Painlevé–Kuratowski stability of approximate efficient solutions for perturbed semi-infinite vector optimization problems

This paper is concerned with the stability of semi-infinite vector optimization problems (SIVOP) under functional perturbations of both objective functions and constraint sets. First, we establish the Berge-lower semicontinuity and Painleve–Kuratowski convergence of the constraint set mapping. Then, using the obtained results, we obtain sufficient conditions of Painleve–Kuratowski stability for approximate efficient solution mapping and approximate weakly efficient solution mapping to the (SIVOP). Furthermore, an application to the traffic network equilibrium problems is also given.

Developing architecture of a traveler information system for dynamic equilibrium in traffic networks

Having criticized the current architecture of Advanced Traveler Information Systems (ATISs), this work discusses a new base of requirements to develop a new paradigm for traffic information systems. It mainly integrates three dimensions within a traffic system; drivers’ pattern of behavior and preferences, urban traffic desires, and capabilities of traffic information service providers. Based on the above, functional segments from several related backgrounds are brought together to structure a new architecture, called Interactive Traveler Information System (ITIS). The main interactive feature of this new architecture is a two-way communication track between drivers and the traffic information system provider—in fact, a decision on choosing a road at a particular time for an individual will be made based on the utility of both sides. This new configuration consists of driver-side smartphone application, centric traffic prediction, and decision-maker units, which will shape a new approach of decision-making processes. These all work together to satisfy the designated goal of ITIS that it is preserving the Wardrop equilibrium condition in the traffic network level. Finally, we concentrate on a comparison study, which shows a differentiation between performance of the proposed ITIS and the current ATIS model in a real situation. This has been done with simulations of analogical scenarios. The most noticeable advantage of the proposed architecture is not being limited to a saturation limit, and the positive effect of increasing system penetration in the performance of the newly introduced information system. In conclusion, new research subjects are suggested to be carried out.

A Bio-Inspired Approach to Traffic Network Equilibrium Assignment Problem.

Finding an equilibrium state of the traffic assignment plays a significant role in the design of transportation networks. We adapt the path finding mathematical model of slime mold Physarum polycephalum to solve the traffic equilibrium assignment problem. We make three contributions in this paper. First, we propose a generalized Physarum model to solve the shortest path problem in directed and asymmetric graphs. Second, we extend it further to resolve the network design problem with multiple source nodes and sink nodes. At last, we demonstrate that the Physarum solver converges to the user-optimized (Wardrop) equilibrium by dynamically updating the costs of links in the network. In addition, convergence of the developed algorithm is proved. Numerical examples are used to demonstrate the efficiency of the proposed algorithm. The superiority of the proposed algorithm is demonstrated in comparison with several other algorithms, including the Frank-Wolfe algorithm, conjugate Frank-Wolfe algorithm, biconjugate Frank-Wolfe algorithm, and gradient projection algorithm.

Day-to-day dynamical evolution of network traffic flow with elastic demand

Network traffic flow is an aggregated result of a huge number of travelers' route choices, which is influenced by the travelers' choice behaviors. So day-to-day traffic flow is not static, but presents a complex and tortuous day-to-day dynamic evolution process. Studying day-to-day dynamic evolution of network traffic flow, we can not only know whether the traffic network equilibrium can be reached and how the process is achieved, but also can know what phenomenon will occur in the evolution of network traffic flow if the equilibrium is not reached. In a real traffic system, taking day as scale unit, the day-to-day network traffic demand is variable and changes with everyday's traffic network state. The travelers' route choices are also influenced by the previous day's behaviors and network state. Then, will the day-to-day network traffic flow evolution be stable? If it is unstable, when will bifurcation and chaos occur? In this paper we discuss the day-to-day dynamic evolution of network traffic flow with elastic demand in a simple two-route network. The dynamic evolution model of network traffic flow with elastic demand is formulated. Based on a nonlinear dynamic theory, the existence and uniqueness of the fixed point of dynamic evolution model are proved, and an equilibrium stability condition for the dynamic evolution of network traffic flow with elastic demand is derived. Then, the evolution of network traffic flow is investigated through numerical experiments by changing the three parameters associated with travelers, which are the sensitivity of travelers' travel demand to travel cost, the randomness of travelers' route choices, and travelers' reliance on the previous day's actual cost. Our findings are as follows. Firstly, there are three kinds of final states in the evolution of network traffic flow: stability and convergence to equilibrium, periodic motion and chaos. The final state of the network traffic flow evolution is related to the above three parameters. It is found that under certain conditions the bifurcation diagram of the network traffic flow evolution reveals a complicated phenomenon of period doubling bifurcation to chaos, and then period-halving bifurcation. Meanwhile, the chaotic region is interspersed with odd periodic windows. Moreover, the more sensitive to cost the travelers' travel demand the more likely the system evolution is to be stable. The smaller the randomness of travelers' route choices, the less likely the system evolution is to be stable. The lower the degree of travelers' reliance on the previous day's actual cost, the more likely the system evolution is to be stable.

An empirical interpolation approach to reduced basis approximations for variational inequalities

ABSTRACTVariational inequalities (VIs) are pervasive in mathematical modelling of equilibrium and optimization problems in engineering and science. Examples of applications include traffic network equilibrium problems, financial equilibrium, obstacle problems, lubrication phenomena and many others. Since these problems are computationally expensive to solve, we focus here on the development of model order reduction techniques, in particular the reduced basis technique. Reduced basis techniques for the approximation of solutions to elliptic VIs have been developed in the last few years. These methods apply to VIs of the so-called first kind, i.e. problems that can be equivalently described by a minimization of a functional over a convex set. However, these recent approaches are inapplicable to VIs of the so-called second kind, i.e. problems that involve minimization of a functional containing non-differentiable terms. In this article, we evaluate the feasibility of using the reduced basis method (RBM) combined with the empirical interpolation method (EIM) to treat VIs. In the proposed approach, the problem is approximated using a penalty or barrier method, and EIM is then applied to the penalty or barrier term. Numerical examples are presented to assess the performance of the proposed method, in particular the accuracy and computational efficiency of the approximation. Although the numerical examples involve only VIs of the first kind, we also evaluate the feasibility of using the RBM combined with the EIM to treat VIs of the second kind.

Semicontinuity and convergence for vector optimization problems with approximate equilibrium constraints

This paper mainly intends to present some semicontinuity and convergence results for perturbed vector optimization problems with approximate equilibrium constraints. We establish the lower semicontinuity of the efficient solution mapping for the vector optimization problem with perturbations of both the objective function and the constraint set. The constraint set is the set of approximate weak efficient solutions of the vector equilibrium problem. Moreover, upper Painlevé–Kuratowski convergence results of the weak efficient solution mapping are showed. Finally, some applications to the optimization problems with approximate vector variational inequality constraints and the traffic network equilibrium problems are also given. Our main results are different from the ones in the literature.