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Some Theoretical Aspects of Road Traffic Research
... A. Motivation of the study Ranging from congestion problems in traffic networks to routing problems for data packages over the internet, path optimization is ubiquitous in real-world problems. The traffic assignment problem (TAP) is one of the most prominent routing problems, with a long history dating back to the 50' [50]. In this work, we consider the integer traffic assignment problem (ITAP), a variant of TAP where the flows on edges are constrained to be integers. ...
... When the objective function is convex, ITAP is related to TAP, a problem consisting of routing paths (users) over a network in a way that minimizes the average time taken to reach the destination. The study of TAP dates back to the seminal work of Wardrop [50] who formulated a mathematical model of traffic assignment and introduced the concepts of user equilibrium and system optimum. These correspond respectively to the case where each path egoistically chooses the route that minimizes its own travel time, and the case where the route taken by each path is mandated so to minimize the total travel time of all paths. ...
... We call these the support paths. The following result by Wardrop [50] characterizes the support paths. ...
Path optimization is a fundamental concern across various real-world scenarios, ranging from traffic congestion issues to efficient data routing over the internet. The Traffic Assignment Problem (TAP) is a classic continuous optimization problem in this field. This study considers the Integer Traffic Assignment Problem (ITAP), a discrete variant of TAP. ITAP involves determining optimal routes for commuters in a city represented by a graph, aiming to minimize congestion while adhering to integer flow constraints on paths. This restriction makes ITAP an NP-hard problem. While conventional TAP prioritizes repulsive interactions to minimize congestion, this work also explores the case of attractive interactions, related to minimizing the number of occupied edges. We present and evaluate multiple algorithms to address ITAP, including a message passing algorithm, a greedy approach, simulated annealing, and relaxation of ITAP to TAP. Inspired by studies of random ensembles in the large-size limit in statistical physics, comparisons between these algorithms are conducted on large sparse random regular graphs with a random set of origin-destination pairs. Our results indicate that while the simplest greedy algorithm performs competitively in the repulsive scenario, in the attractive case the message-passing-based algorithm and simulated annealing demonstrate superiority. We then investigate the relationship between TAP and ITAP in the repulsive case. We find that, as the number of paths increases, the solution of TAP converges toward that of ITAP, and we investigate the speed of this convergence. Depending on the number of paths, our analysis leads us to identify two scaling regimes: in one the average flow per edge is of order one, and in another the number of paths scales quadratically with the size of the graph, in which case the continuous relaxation solves the integer problem closely.
... In general, these techniques basically seek to follow two principles defined by Wardrop (1952). In the first of them, the author points out that in road networks in which the user can choose between two or more different routes to make a trip of common origin and destination, the travel time for all routes converges to the same value, lower than that of any other route not commonly used. ...
... Engineering and its advancements Impact of the Charitas-Cafubá tunnel on vehicle travel time Thus, the travel time from B to A and the division of flows between the two routes after the tunnel is opened to traffic, indicate that the travel times of Sections 1 and 2 are in the process of equilibrium. Therefore, as indicated by Wardrop (1952), the flow of vehicles using the fastest route will continue to grow until the time spent by users in congestion and traffic when using "Stretch 2" will result in a travel time close to that experienced by users on "Stretch 1". ...
... It is possible to see that the travel times of the routes fluctuate according to the different divisions of the flow. According to Wardrop (1952), the variable with the greatest impact on the decision of road users in route choices is the travel time of each one, that is, the driver tends to choose the fastest one, even if it has a longer length. Therefore, users' route choices will oscillate between the two alternative paths until an equilibrium point where the travel time tends towards the same value. ...
One of the objectives of the Charitas-Cafubá Tunnel is to reduce travel time on the way to the city center of Niterói or the city of Rio de Janeiro. To measure the impact of the tunnel, a comparison of the travel time and vehicle flow of the current two existing routes after the construction of the tunnel with the single previous route was conducted. The results indicated that the travel time between the two alternatives after the construction of the tunnel approaches an equilibrium, both with a shorter travel time than the previous one. However, there was an increase in the total number of vehicles at the point of convergence between the routes. By analyzing the section after the convergence point, it is concluded that the reduction in the initial travel time provided by the Tunnel is counterbalanced by the increase in travel time in the segment after the convergence of the current routes.
... The network optimization problems have attracted much attention over the last decade for their ubiquitous appearance in real-life applications and the inherent mathematical challenges that they present, especially, in optimal transportation theory and communication networks (see [22][23][24][25][26][27][28]). Back in 1952, J.G.Wardrop (see [22]) formulated two principles of optimality of flows in networks that describe the circumstances of the user equilibrium and the system optimum. ...
... The network optimization problems have attracted much attention over the last decade for their ubiquitous appearance in real-life applications and the inherent mathematical challenges that they present, especially, in optimal transportation theory and communication networks (see [22][23][24][25][26][27][28]). Back in 1952, J.G.Wardrop (see [22]) formulated two principles of optimality of flows in networks that describe the circumstances of the user equilibrium and the system optimum. The first Wardrop principle states that the costs of all utilized links are equal and less than the costs of those unutilized links for every fixed source-destination pair. ...
In this paper, we are aiming to propose a novel mathematical model that studies the dynamics of synaptic damage in terms of concentrations of toxic neuropeptides/neurotransmitters during neurotransmission processes. Our primary objective is to employ Wardrop's first and second principles within a neural network of the brain. In order to comprehensively incorporate Wardrop's first and second principles into the neural network of the brain, we introduce two novel concepts: \textit{neuropeptide's (neurotransmitter's) equilibrium} and \textit{synapses optimum}. The \textit{neuropeptide/neurotransmitter equilibrium} refers to \textit{a distribution of toxic neuropeptides/neurotransmitters that leads to uniform damage across all synaptic links}. Meanwhile, \textit{synapses optimum} is \textit{the most desirable distribution of toxic neuropeptides/neurotransmitters that minimizes the cumulative damage experienced by all synapses}. In the context of a neural network within the brain, an analogue of the price of anarchy is \textit{the price of cognition} which is \textit{the most unfavorable ratio between the overall impairment caused by toxic neuropeptide's (neurotransmitter's) equilibrium in comparison to the optimal state of synapses (synapses optimum)}. To put it differently, \textit{the price of cognition} measures \textit{the loss of cognitive ability resulting from increased concentrations of toxic neuropeptides/neurotransmitters}. Additionally, a replicator equation is proposed within this framework that leads to the establishment of the synapses optimum during the neurotransmission process.
... In this model, both choices are made simultaneously in such a way that the resulting travel costs across routes and modes lead to the stochastic user equilibrium (Daganzo and Sheffi 1977;Zhou et al. 2012), i.e., no traveler can improve her or his perceived travel costs by unilateral action. The stochastic user equilibrium relaxes the assumption of perfect knowledge of all travelers in the deterministic user equilibrium, i.e., Wardrop's first principle (Wardrop 1952). In the following, the mathematical model is explained step by step. ...
... The arbitrage condition for car drivers to use link (i, j) follows a stochastic user equilibrium (Daganzo and Sheffi 1977) that relaxes the assumption of perfect knowledge of all travelers in the deterministic user equilibrium or Wardrop's first principle (Wardrop 1952). It is formulated in this model as given in Eq. 6 based on Van Nieuwkoop et al. (2016), but modified using the perceived link travel costs C ij . ...
MobilityCoins are a tradable mobility credit (TMC) scheme variant. TMC schemes are a cap-and-trade scheme for managing mobility that are designed to limit negative externalities, e.g., congestion, of traffic. Next to having link-specific or origin–destination-specific charges for cars as in the common TMC scheme, the MobilityCoin scheme’s distinctive elements are accommodating link-specific and origin-and-destination-specific charges and incentives for all modes of transport as well as being considered a mobility currency that can be earned, saved, and spent in multiple time periods. These distinctive features of the MobilityCoin scheme does not alter the core behavioral mechanism of TMC schemes of increasing car travel costs, but these features interfere with the credit market in terms of market volume and market price that ultimately affects traffic outcomes, e.g., an uncontrolled market volume increase can lower the market price that in turns increases the attractiveness of using the car. In this paper, we develop a mathematical model of multimodal macroscopic network flows and a MobilityCoin market to investigate the impacts of charges, incentives, and multi-period budgets. The model is implemented as a single-day model with an integration of sensitivity for multi-period budgets to study how the outcomes in the transportation system change with charges, incentives, and multi-period budgets. Further, we discuss implications for the policy design of MobilityCoins schemes.
... To highlight the effects of traffic fluctuations we explicitly study a balanced transport network where the average incoming and outgoing flows at each nodes are equal. This assumption could reflect the existence of a Wardrop equilibrium [16,17] for urban traffic where the mobility paths distribute to optimize the use of the transport network. To simulate the traffic dynamics at crossing points we assume the existence of a finite transport capacity and of a maximal capacity for each node so that a displacement is possible if the number of particles in the destination nodes is smaller than the maximal capacity [18,19]. ...
... the Gibbs entropy (15) is maximal for the distribution (11) when the probabilities pi satisfy the constraints (16). The thermodynamic approach allows to characterize the statistical properties of the stationary state without considering the dynamics (7). ...
The congestion formation on a urban road network is one of the key issue for the development of a sustainable mobility in the future smart cities. In this work we propose a reductionist approach studying the stationary states of a simple transport model using of a random process on a graph, where each node represents a location and the weight links give the transition rates to move from one node to another that represent the mobility demand. Each node has a finite transport capacity and a maximum load capacity and we assume that the average. In the approximation of the single step process we are able to analytically characterize the traffic load distribution on the single nodes, using a local Maximum Entropy Principle. Our results explain how the congested nodes emerge when the total traffic load increases in analogous way to a percolation transition where the appearance of a congested node is a independent random event, However, using numerical simulations, we show that in the more realistic case of the synchronous dynamics for the nodes, there are entropic forces that introduce correlation among the node state and favor the clustering of the empty and congested nodes. Our aim is to highlight universal properties of the congestion formation and, in particular, to understand the role traffic load fluctuations as a possible precursor of congestion in a transport network.
... Then, let denote the truck traffic flow on arc ∈ . Under Wardrop's first principle [16] (i.e., user equilibrium), our objective is to find a flow distribution over the network such that the total travel cost (or time) is minimized, and no user can decrease their travel cost by unilaterally changing paths. All the notations used in the model are listed in Table 1. ...
... Equation (15) follows the same logic as Equation (14). Equations (16)(17) define the feasibility and non-negativity of variables. ...
This study presents a path-based network equilibrium model for long-haul electric trucks (e-trucks), navigating the U.S. highway network. Emphasizing e-trucks' unique operational characteristics, the model, built on a background passenger car traffic, integrates essential travel costs, including travel and charging times, as well as mandated rest periods. The Method of Successive Averages (MSA) is utilized in our algorithm. A comparative analysis with conventional diesel trucks, using a streamlined version of the U.S. national highway network, reveals extended travel times for electric trucking, primarily due to charging requirements. Empirical findings show that e-trucks experience a significant increase in total travel time, about 55.9% more than their diesel counterparts. This delay, largely due to prolonged charging (including waiting) times, could potentially be offset through strategic infrastructure development and policy interventions.
... The purpose of this paper is to demonstrate that backpressure, using either sQ-based policies or sb-based policies, may cause additional delays to traffic at equilibrium; where all flow is on quickest routes. (See Wardrop (1952). ) We do this first by comparing the two control policies above (P 0 and P 0 + BP) in the simple example network shown in Fig. 1. ...
... The simplest models for estimating this are route choice models where demand is fixed. Wardrop (1952) is credited with the first clear statement that the total OD flow is likely to distribute itself over the links of a network so that: ...
... In 1952, Wardrop developed two traffic assignment principles [102]. The first principle states that the travel times between each Origin-Destination (OD) pair on all routes that are used are equal and less than the travel times that a single vehicle would encounter on any route that is not used. ...
Frequent and escalating natural disasters pose an increasing threat to society and the environment. Effective disaster management strategies are crucial to mitigate their impact. This paper reviews recent methodologies for large-scale evacuation planning, a key element in risk reduction. A systematic analysis of 100 articles and conference proceedings in evacuation planning, focusing on human factors/behavior modeling and evacuation routing optimization, reveals that Agent-Based Simulation (ABS) is commonly used to predict human factors/behaviors. Heuristics/meta-heuristics and traffic assignment techniques dominate evacuation routing planning, often aiming to identify the shortest evacuation path. While evacuation decisions and route choice are extensively studied, optimization approaches frequently lack integration with human factors/behavior modeling. This review underscores the need for further research to enhance evacuation planning by integrating human factors/behavior and optimization methodologies for increased effectiveness and efficiency.
... From the problem description, a link-based network design model (M1) is proposed based on the classical traffic assignment problem (Beckmann et al., 1956;Gao et al., 2005;Peeta & Ziliaskopoulos, 2001;Wardrop, 1952). ...
Global pandemics restrict long‐haul mobility and international trade. To restore air traffic, a policy named “travel bubble” was implemented during the recent COVID‐19 pandemic, which seeks to re‐establish air connections among specific countries by permitting unrestricted passenger travel without mandatory quarantine upon arrival. However, travel bubbles are prone to bursting for safety reasons, and how to develop an effective restoration plan through travel bubbles is under‐explored. Thus, it is vital to learn from COVID‐19 and develop a formal framework for implementing travel bubble therapy for future public health emergencies. This article conducts an analytical investigation of the air travel bubble problem from a network design standpoint. First, a link‐based network design problem is established with the goal of minimizing the total infection risk during air travel. Then, based on the relationship between origin‐destination pairs and international candidate links, the model is reformulated into a path‐based one. A Lagrangian relaxation‐based solution framework is proposed to determine the optimal restored international air routes and assign the traffic flow. Finally, computational experiments on both hypothetical data and real‐world cases are conducted to examine the algorithm's performance. The results demonstrate the effectiveness and efficiency of the proposed model and algorithm. In addition, compared to a benchmark strategy, it is found that the infection risk under the proposed travel bubble strategy can be reduced by up to 45.2%. More importantly, this work provides practical insights into developing pandemic‐induced air transport recovery schemes for both policymakers and aviation operations regulators.
... This paper considers the traffic assignment, the last step of the four-step urban planning process, which can predict the traffic volume on the links. User equilibrium and system optimum are two widely used principles for the traffic assignment, and this paper studies the user equilibrium proposed in [1]. Nowadays, this principle is still valid, and some extensions have been made (e.g., [2][3][4]). ...
This study proposes a target-oriented method to study travelers’ route choice behavior under travel time variability, and discusses the resulted equilibrium flow patterns. Both travel time reliability and travel time unreliability are considered in this new method, and accordingly, there are two targets. The first one is target for travel time to ensure travel time reliability, and based on this target, another one is target for excess delay to mitigate travel time unreliability. In this model, travel time and excess delay (i.e., the random vector) are stochastically correlated with each other, which is modeled with the copula function based on Sklar’s Theorem, and the exact form of the copula is obtained by the proved comonotonicity relationship of this random vector. The target interaction, i.e., the complementarity relationship, is also modeled based on the utility functions, the meaning of which is that travelers have the will to make more targets achieved so as to obtain more utility. Furthermore, with this model, this paper formulates the user equilibrium as a variational inequality problem to study the long-term effect of the route choice behavior, and solves it with the method of successive average. Finally, numerical testings on the traffic network are conducted to show the convergence of the solution algorithm, and to illustrate the impact of targets on the equilibrium results. Results show that the flow change can be five times more than that with less risk-averse travelers.
... To express the user equilibrium flow pattern [36], the current study applies the complementarity conditions as follows: ...
The dedicated lanes management policy is a possible solution to the issues arising from the coexistence of human-driving vehicles (HDVs) and connected and autonomous vehicles (CAVs) in traffic. Although numerous studies have been conducted on the network deployment problem of CAV-dedicated lanes, the safety implications of CAV and its dedicated lanes are ignored. This study proposes a mathematical approach to optimize the deployment of CAV-dedicated lanes incorporating efficiency and safety concerns. An integrated framework is developed based on headway distributions to systematically evaluate the efficiency and safety performance of the road network. The platoon intensity index is utilized to model the platooning effect of CAVs on traffic safety and efficiency. A safety performance estimation method is proposed to account for the potential collision risk of mixed traffic flow and heterogeneity in car-following behavior. A bi-level programming model is adopted to solve the optimal deployment problem. The upper-level model is formulated as a bi-objective model to minimize the total travel time and the safety risk. The lower-level model describes the multi-class user equilibrium state of the CAV-HDV mixed traffic flow on the network. A genetic algorithm is utilized to solve the bi-level programming model and obtain the Pareto-optimal solution set. Two numerical studies are conducted to validate the proposed model and algorithm. The results revealed that the optimal deployment plan can significantly improve the road network’s safety and efficiency performance, whereas higher platoon intensities have a negative impact on traffic safety and efficiency for certain headway settings. Moreover, the results highlighted a trade-off between efficiency and safety in the optimal deployment problem, which may help decision-makers choose the optimal deployment plan based on the road network design needs.
... Selfish routing, first erected as a principle by Wardrop (1952), is a canonical description of how a crowd of selfminded drivers minimizing their own travel time decide on their paths across a road network prone to congestion. The resulting equilibria, termed User Equilibria (UE), stem from independent non-cooperative actions, yet, as Beckmann et al. (1956) revealed, can be captured as minima of a convex function: the potential of the game (Monderer and Shapley, 1996;Sandholm, 2001). ...
Boundedly Rational User Equilibria (BRUE) capture situations where all agents on a transportation network are electing the fastest option up to some time indifference, and serve as a relaxation of User Equilibria (UE), where each agent exactly minimizes their travel time. We study how the social cost under BRUE departs from that of UE in the context of static demand and stochastic costs, along with the implications of BRUE on the optimal signaling scheme of a benevolent central planner. We show that the average excess time is sublinear in the maximum time indifference of the agents, though such aggregate may hide disparity between populations and the sublinearity constant depends on the topology of the network. Regarding the design of public signals, even though in the limit where agents are totally indifferent, it is optimal to not reveal any information, there is in general no trend in how much information is optimally disclosed to agents. What is more, an increase in information disclosed may either harm or benefit agents as a whole.
... One of the natural ways to deal with problems with a large number of agents that have been developped in different fields of research is to replace such complex models with relatively simpler ones with a continuum of infinitesimal players. This kind of approximations have appeared in one-step games at least since two seminal papers by Wardrop [56] and Schmeidler [53], but for a long time have not been introduced to dynamic game models. The situation has changed since a series of papers by Lasry and Lions [41,42] and by Huang, Caines and Malhamé [36][37][38] where models of non-cooperative differential games with a continuum of identical players have been introduced. ...
In the paper we present a model of discrete-time mean-field game with several populations of players. Mean-field games with multiple populations of the players have only been studied in the literature in the continuous-time setting. The main results of this article are the first stationary and Markov mean-field equilibrium existence theorems for discrete-time mean-field games of this type. We consider two payoff criteria: β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-discounted payoff and total payoff. The results are provided under some rather general assumptions on one-step reward functions and individual transition kernels of the players. In addition, the results for total payoff case, when applied to a single population, extend the theory of mean-field games also by relaxing some strong assumptions used in the existing literature.
... In most transportation planning research, the basic premise is to choose a route according to Wardrop's frst principle (i.e., user equilibrium; UE), which states that a user does not use a route with a longer travel time than other users with the same origin and destination [2]. Te trafc pattern in which all users follow Wardrop's frst principle has one unique solution (i.e., UE solution) under several conditions [3]. ...
Connected and automated vehicles can reduce the traffic congestion level of the entire network through platoon-driving technologies compared to human-driven vehicles. One promising approach to enhancing platoon-driving technology’s efficiency is deploying dedicated lanes or roads for connected and automated vehicles. Since asymmetric interactions between different vehicle types increase road congestion, it is necessary to distinguish routes for efficient traffic management. However, the traditional traffic assignment problem, which uses only user equilibrium as a constraint with no difference in travel time between users, could not be proposed as a globally optimal solution because it generates an infinite number of locally optimal solutions. Recent studies have attempted to overcome the limitations by considering the sum of system-wide travel times as an additional constraint. Their research sought to help propose optimal deployment strategies through the lowest total travel time solution (best-case) or design robust transport planning strategies through the highest total travel time solution (worst-case). However, past studies have not focused on the possibility of the best/worst case appearing in reality. This study focused on the relationship between the two solutions pointed out in past studies and traffic patterns likely to appear in reality. This study interprets the Karush–Kun–Tucker condition of the static traffic assignment problem, considering the asymmetric interaction, and proposes a solution algorithm using discrete dynamics. The proposed algorithm extends the most widely used method in transportation planning research, which can overcome the limitations of asymmetric interaction problems through simple variations. The proposed algorithm can reliably derive two solutions, and entropy theory shows that both solutions are unlikely to appear in reality without additional policies such as dedicated lanes or roads.
... Assuming that K i is the set of paths between OD pair i , f k is the flow of path k , t a and T k are the travel time functions of link a and path k , respectively, u i is the equilibrium travel time between OD pair i, and ak is a binary variable that equals 1 if link a lies on path k and 0 otherwise. The so-called Wardropian equilibrium conditions (Wardrop 1952) can be formulated in terms of the variables f k , u i , and v a as the following sub-problem (SP): ...
The high cost of conventional surveys has motivated researchers to develop methods for adjusting a prior Origin–destination (OD) matrix from easily available traffic counts. The gradient method is a mathematical programming approach widely used for the OD matrix adjustment problem (ODMAP). However, this method easily gets trapped in local optima due to the non-convexity of the problem. Moreover, validation of the gradient solutions against predefined target matrices shows the method has considerable difficulty with estimating the sum of the OD matrix elements. Particle swarm optimization (PSO) is a metaheuristic which is getting lots of attention for its global search ability, but is less accurate in local search. The proposed algorithm hybridizes PSO with the gradient method, considering that the combination of good local convergence properties and effective global search makes an excellent algorithm for the ODMAP. Comparison of the results for a small and a real-life network demonstrates that the hybrid algorithm provides higher convergence properties and achieves more accurate solutions than its constituent parts working alone.
... A fundamental characteristic of collective travel behavior within Agent-Based Models (ABMs) is the propensity for the system to self-organize into an equilibrium state. The concept of traffic reaching a state of equilibrium was first introduced by Knight (1924) and subsequently formalized by Wardrop (1952), who developed the mathematical principles governing the first and second principles of traffic equilibrium-the User Equilibrium (UE) and System Optimum (SO) (Patriksson, 2015). ...
Automated Vehicles (AVs) are poised to disrupt travel patterns and the sustainability of transportation networks. Conventional methods for studying these changes, such as stated preference surveys and agent-based simulations, have limitations. Serious games offer a promising alternative , providing a controlled and engaging environment for investigating travel behavior. In our study, 200 participants, grouped into sessions of 10, engaged in a competitive serious game simulating 50 daily choices of travel mode and departure time across three automated options. Two scenarios were examined: one with recurring congestion and another with nonrecurring congestion. Automated transit had fixed schedules, while private and shared rides could adapt to a congested bottleneck. Results revealed that ridesharing dominated, reaching 60% mode share under recurring congestion, displacing transit, and a comparative equilibrium emerged between shared and private rides. In the nonrecurring congestion scenario, ridesharing dropped to 37%, and a comparable multimodal equilibrium developed. Participants rarely achieved the optimal score, attaining a maximum of 88% of its potential. This study highlights a policy paradox: un-regulated AV traffic can reduce transit use, exacerbate recurring congestion, yet necessitate increased transit investment to address nonrecurring congestion, confirming the Downs-Thomson paradox. Creating appealing mass transit alternatives is imperative to ensure efficiency and sustainability in the era of automated mobility.
... The notion of a user equilibrium (UE) is a commonly used equilibrium construct in strategic TA applications. According to Wardrop's first principle, in UE travellers selfishly keep updating their route until they can no longer reduce their travel time, which results in a situation where all used routes have the same minimum travel time (Wardrop, 1952). ...
In a macroscopic assignment model, traffic flows are distributed onto the network by means of a network loading model. The network loading propagates flows along links via a link model and through junctions or intersections via a node model. Most of the travel time delays are caused by queues forming at junctions or intersections, especially in urban networks. Therefore, the efficiency and accuracy of the underlying node model is paramount in capturing these delays (and flows). Existing link-based macroscopic node models make the simplifying assumption that first-in-first-out (FIFO) holds at the link level, which is often unrealistic when a link has multiple approach lanes near an intersection or junction. In this work we propose to relax this assumption such that FIFO holds at the movement level. We do so by developing several model extensions. First, a novel lane-based formulation of the node model is proposed. Secondly, we formulate an equilibrium problem and a general solution algorithm to allocate sending flows to lanes. This allows us to explicitly consider approach lane configurations that contain important information about the layout of an intersection or junction. We show that the conventional link-based node model is a special case of our newly proposed model in case each approach lane on an incoming link allows all possible movements. Various numerical examples are provided, demonstrating the capabilities of the proposed extensions to the node model.
... where is the space-mean speed, flow ( ) is the temporal representation of several vehicles passing a stationary object, and density ( ) is the spatial representation of a number of vehicles in unit length. Various measurement methods were used to define traffic flow variables in spatial-temporal measurements [1,2]. Edie's generalized definition of traffic stream characteristics addressed all the challenges and controversies of fundamental parameter estimation in a given region for homogeneous traffic conditions [2,3]. ...
Modeling and investigating the properties of fundamental diagrams (FDs) in mixed traffic, which encompasses heterogeneous with non-lane-based flow, has been one of the emerging research areas in the past few years. The main challenges in modeling are: estimating accurate steady-state (ss) points based on empirical observations and properly representing FDs in mixed traffic conditions. The first part of this work uses the traditional discretization approach and the optimal time-space window to apply Edie’s generalized definitions to estimate the traffic flow variables and the steady states. The second part of the work involves a trajectory shear mapping method to estimate less-scattered FDs. Finally, the shape of the FDs are determined, and their properties are studied by developing area occupancy-based (ao) normalized flow and speed models. Empirical observations from multiple locations show that the power-law relationships seem to be the best fit based on the ao-based representation of the fundamental parameter that indicates the possibility of universality in the FDs from the mixed traffic conditions.
... Since vehicles on the traffic network have distinct starting and ending points, each vehicle completes an origin-destination (o-d) process during its journey. This is represented in the traffic network as the corresponding traffic flow, denoted as Additionally, the user route selection scheme must adhere to the Wardrop equilibrium principle, which involves achieving both user equilibrium (UE) and system optimization (SO) for the entire transportation system [12]. ...
To address energy and nature-related issues, the development of new energy vehicles has been rapid in recent years, gradually replacing traditional vehicles. This paper specifically focuses on the uncertainty of traffic flow for new energy vehicles within the transport system. We analyze the consequence of output from various equipment units in the energy system and their optimized scheduling. We construct an optimization scheduling model for integrating the electric-gas network and the transportation network, known as the energy-transport coupling system (ETCS). By incorporating the uncertainty of traffic flow and utilizing a data-driven approach to generate a set of uncertain traffic flow, we conduct a case study using an improved ETCS consisting of an IEEE-33 node distribution network, a 7-node natural gas network, and a 12-node transportation network to validate the scheduling method. The results demonstrate that the uncertainty of traffic flow does indeed affect the output of equipment units in the ETCS. Furthermore, through optimized scheduling, we can achieve positive interactions between different systems.
... The first pioneering work on the development of delay models (9,20) was done by Wardrop and Clayton. 10 Shortly after, it was noticed that the delay estimates are not realistic or accurate due to the underlying model 11 ...
Accurate estimation of delay is crucial for efficient traffic signal operations. Estimation of delay in a real-time manner using tradition loop detectors requires advanced detectors (in addition to stop-bar detection). In cases when this detection layout is not in place, delay estimates are approximated with a lower accuracy. Video detection is one of the most frequently deployed detection systems at signalized intersections in recent years. In most cases video detection operates in the same way as traditional inductive loops. However, when coupled with computer vision algorithms, these detection systems could be used to retrieve additional information (e.g., vehicular arrivals and departures) that cannot be taken out from the conventional systems. Although present for several decades, video detection data were not frequently examined for delay estimation purposes. In this study, we proposed a novel delay estimation model which can be developed with only data from stop-bar video detectors. Relevant data were collected from 11 signalized intersections at downtown Chattanooga, TN and processed to create needed inputs for model development. With a use of multigene genetic programming the authors developed a delay model that outperforms the accuracy of the multi-regression model. Furthermore, authors evaluated the developed model by comparison with the other benchmark delay models, such as HCM and approach delay model. It was found that the developed MGGP delay model outperforms benchmark models for a wide range of traffic and signal operation conditions.
... In CTEAP, each passenger attempts to select a hyperpath with the minimum generalised cost (Wardrop 1952). According to the above definitions, the solution of CTEAP can be characterised by the following user equilibrium (UE) conditions: ...
This paper studied the capacitated transit equilibrium assignment problem (CTEAP), which particularly accounts for the capacity effect that impacts passengers' route choices in a transit network. Specifically, we impose strict capacity constraints on line segments to ensure that the passenger flow does not exceed the line capacity. By introducing associated Lagrange multipliers, we derived the generalised hyperpath cost function and established an equivalent variational inequality formulation. A solution framework is developed to solve CTEAP based on the Method of Multipliers. To handle destroyed Cartesian product structures, we transformed the master problem into a sequence of uncapacitated subproblems, which can be tackled by two modified Newton-type hyperpath-based algorithms. Numerical analyses were performed for a small Gentile network and a large Shenzhen transit network to evaluate the impacts of the capacity constraint on passenger flows and computation costs. Our results demonstrate the superiority of the proposed CTEAP model to prevent over-loaded flows, and show the promise of applying the hyperpath-based algorithm in the implementation of real-world networks.
... The delay costs for the travellers opting for an alternative route is assumed to be the same as if they would have travelled through the station. This assumption is based on the concept of user equilibrium, in which the travel costs for each individual for both routes (travelling through the station or using another route) should be equal (Wardrop, 1952). The equilibrium is possible if the alternate routes are known by the users and if the access restriction to the platforms is implemented on a long period, so that the users are used to it, which is the case here. ...
The high concentration of people in train stations during rush hour poses a significant risk to passenger comfort and safety. Recent technological advancements in real-time crowd counting and pedestrian microsimulations have made it feasible to create responsive gateways for managing passenger flow to prevent overcrowding situations. This paper proposes a simulation-based cost-benefit analysis, utilizing the Real Options methodology commonly employed to assess flexible infrastructure, to estimate the costs and benefits associated with implementing responsive gateways at the London Bridge Station, while considering uncertainties in passenger flow and the interests of various stakeholders. Results from the demonstrative example indicate that the responsive gateways offer a better service than the static ones mainly due to the expected reduction in the risks related to safety, travel time and comfort that compensate for a higher cost of installation and maintenance. Further research is envisioned to enhance the modeling of passenger flow uncertainties and expand the objective function to encompass additional externalities. The simulation-based cost-benefit analysis proposed in this paper can be extended to a broader range of intelligent transportation systems, considering uncertainties and their diverse impacts on society. Thus, this paper presents a tool that contributes to a more transparent and objective investment decision-making process, particularly concerning the development of smart cities.
... Traffic assignment problems (TAPs) play a vital role in urban transportation planning, and it has been widely used as a tool to predict the path choice of commuters (Jiang and Nielsen 2022;Akuh et al. 2023). Therein, the user equilibrium (UE) condition has been widely investigated (Wardrop 1952;Sheffi and Powell 1982;Bar-Gera 2002;Gentile 2014;Wang 2022;Zhang et al. 2023a), while it follows the assumption that each traveler has accurate perceptions of the transportation network condition; namely, each traveler seeks to choose the shortest path. With practical consideration, it is recognized as an unrealistic situation. ...
This paper presents two novel algorithmic frameworks to address the logit-based stochastic user equilibrium traffic assignment problem (SUE-TAP). Following the different variant of the gradient projection (termed as GP2) algorithm, we propose an improved GP2 algorithm (IGP) for the SUE-TAP. This study initially presents a smart approach for determining the allocation of more or less effort to specific origin–destination (OD) pairs. Subsequently, the TAP can be decomposed by different OD pairs, whereas the proposed IGP algorithm is designed based on the serial scheme (i.e., the Gauss–Seidel method). Therefore, a new parallel algorithm P-IGP is proposed, which integrates the block coordinate descent (BCD) method and the IGP algorithm. In specific, the independent OD pairs can be separated into several blocks, and the OD-based restricted subproblems within each block can be solved in parallel. Then, we outline the entire process of implementing the P-IGP algorithm to address the SUE-TAP. Several numerical experiments are conducted to verify the proposed algorithms. The results reveal that the proposed IGP algorithm demonstrates significantly speeder convergence in comparison to the traditional GP2 algorithm, achieving a remarkable acceleration of approximately 12%. Furthermore, the performance of the P-IGP algorithm surpasses that of the proposed IGP algorithm, and it can further achieve a notable 4–5-fold enhancement in convergence efficiency.
We propose a novel mathematical model that studies the dynamics of synaptic damage in terms of concentrations of toxic neuropeptides (neurotransmitters) during neurotransmission processes. Our objective is to employ “Wardrop’s first and second principles” within a neural network of the brain. Complete manifestations of Wardrop’s first and second principles within a neural network of the brain are presented through the introduction of two novel concepts: neuropeptide’s (neurotransmitter’s) equilibrium and synapses optimum. In the context of a neural network within the brain, an analogue of the price of anarchy is the price of cognition which is the most unfavorable ratio between the overall impairment caused by toxic neuropeptide’s (neurotransmitter’s) equilibrium in comparison to the optimal state of synapses (synapses optimum). Finally, we also propose an iterative algorithm (neurodynamics) in which the synapses optimum is eventually established during the neurotransmission process. We envision that this mathematical model can serve as a source of motivation to instigate novel experimental and computational research avenues in the fields of artificial neural networks and contemporary neuroscience.
Analysing bus travel time is essential for providing valuable information to users for effective journey planning. This study aimed to analyse bus travel time in Klang Valley, Malaysia, which includes the urban centre of Kuala Lumpur as well as the surrounding suburban areas. A two-part study is presented here. The findings on superiority of proposed mixture model, namely the Burr XII mixture model as a new statistical distribution to fit travel time distribution based on day-to-day variability under spatial (number of signalized intersections, link length, route type) and temporal aggregation (15 minutes aggregated, peak/non-peak period) are validated in the previous publication. Based on the results, the study was deepened to analyse the effects of some factors on travel time variability. As an extension, factors and indicators influencing multimodality travel time patterns on all the bus sections are examined in this paper. This research explores the impact of several factors outlining the usage of Burr XII mixture regression model, particularly the quantile regression analysis on three distinct routes, each exhibiting unique characteristics, originating from both suburban and urban areas. The results have demonstrated that the factors vary depending on the time of day and coverage area. By gaining an understanding of and effectively addressing the crucial elements that influence travel time, the overall effectiveness of bus systems can be enhanced.
Analyzing the preference of route choice not only facilitates the understanding of individuals' decision‐making behavior, but also provides valuable information for improving traffic management strategies. As the layout of the road network, the variability of individual preferences and the spatial distribution of origins and destinations all play a role in route choice, it is a great challenge to reveal the interplay of such numerous complex factors. In this paper, we propose RouteVis, an interactive visual analytics system that enables traffic analysts to gain insight into what factors drive individuals to choose a specific route. To uncover the relationship between route choice and influencing factors, we design a quantitative analytical framework that supports analysts in conducting closed‐loop analysis of various factors, i.e., data preprocessing, route identification, and the quantification of influence and contribution. Furthermore, given the multidimensional and spatio‐temporal characteristics of the analysis results, we customize a set of coordinated views and visual designs to provide an intuitive presentation of the factors affecting people's travels, thus freeing analysts from tedious repetitive tasks and significantly enhancing work efficiency. Two typical usage scenarios and expert feedback on the system's functionality demonstrate that RouteVis can greatly enhance the capabilities of understanding the travel status.
In the area of Solid Waste Management, transportation of the collected waste is a critical aspect considering the substantial time spent by garbage trucks on public roads. Studies have reported that transporting garbage has challenges related to public exposure and aesthetics. This study presents a generalised bi-objective formulation for the optimal routing of garbage trucks from transfer stations to recycling sites/landfills considering the trade-off between public exposure and aesthetic loss and constraining the operating cost. The formulation uses the novel link capacity function to account for the delay at traffic signals and the mix of trucks and cars on link performance. The proposed formulation is solved using the weighted sum and ε-constraint methods and applied to a realistic case study of the City of Chicago, USA. The Pareto Front obtained for the bi-objective formulation offers diverse trade-off solutions catering to distinct performance metrics. The results highlight the disparity across the solutions; the solution (P0.95 on Pareto Front) for minimum operating cost (or travel time or distance travelled) is very different from the solution (P0.4 on Pareto Front) for aesthetic cost and public exposure. The parametric study indicated that a modest operating budget may suffice for achieving aesthetic benefits, but minimising public exposure requires a higher operating budget. Finally, the proposed framework is adaptable to address various challenges pertaining to waste transportation, thereby serving as a valuable tool for evaluating policies and practices geared towards sustainability objectives.
Pavement segments are functionally interdependent under traffic equilibrium, leading to interdependent maintenance and rehabilitation (M&R) decisions for different segments, but it has not received significant attention in the pavement management community yet. This study developed a maintenance optimization model for interdependent pavement networks based on the simultaneous network optimization (SNO) framework and a multi‐agent reinforcement learning algorithm. The established model was demonstrated on a highway pavement network in the real‐world, compared to a previously built two‐stage bottom‐up (TSBU) model. The results showed that, compared to TSBU, SNO produced a 3.0% reduction in total costs and an average pavement performance improvement of up to 17.5%. It prefers concentrated M&R schedules and tends to take more frequent preventive maintenance to reduce costly rehabilitation. The results of this research are anticipated to provide practitioners with quantitative estimates of the possible impact of ignoring segment interdependencies in M&R planning.
City-scale traffic data, such as traffic flow, speed, and density on every road segment, are the foundation of modern urban research. However, accessing such data on a city scale is challenging due to the limited number of sensors and privacy concerns. Consequently, most of the existing traffic datasets are typically limited to small, specific urban areas with incomplete data types, hindering the research in urban studies, such as transportation, environment, and energy fields. It still lacks a city-scale traffic dataset with comprehensive data types and satisfactory quality that can be publicly available across cities. To address this issue, we propose a unified approach for producing city-scale traffic data using the classic traffic assignment model in transportation studies. Specifically, the inputs of our approach are sourced from open public databases, including road networks, traffic demand, and travel time. Then the approach outputs comprehensive and validated citywide traffic data on the entire road network. In this study, we apply the proposed approach to 20 cities in the United States, achieving an average correlation coefficient of 0.79 in average travel time and an average relative error of 5.16% and 10.47% in average travel speed when compared with the real-world data.
The increasing scale of electric vehicles (EVs) and their stochastic charging behavior have resulted in a growing coupling between the transportation network and the grid. Consequently, effective smart management in the EV-transportation network-grid integration system has become paramount. This paper presents a comprehensive review of the current state of the art in system modeling and optimization approaches for the smart management of this coupled system. We begin by introducing the types of EVs that impact the transportation and grid systems through their charging behavior, along with an exploration of charging levels. Subsequently, we delve into a detailed discussion of the system model, encompassing EV charging load forecasting models and transportation-grid coupling models. Furthermore, optimization technologies are analyzed from the perspectives of system planning and EV charging scheduling. By thoroughly reviewing these key scientific issues, the latest theoretical techniques and application results are presented. Additionally, we address the challenges and provide future outlooks for research in modeling and optimization, aiming to offer insights and inspiration for the development and design of the EV-transportation network-grid integration system.
The growing importance of strategic supply chain management in light of intermodal transportation networks has attracted increased attention from researchers in recent decades. The study of intermodal logistic services has become urgent for multiple branches of science. This work is focused on hub operation pricing under an equilibrium freight flow assignment model. We formulate this model as a non-linear optimization problem and show that its solution corresponds to the equilibrium assignment pattern in an intermodal transportation network. We analyze the sensitivity of the average purchase cost and hub load to different strategies of hub operation pricing. To this end, we obtain the equilibrium freight flow assignment pattern in an explicit form for the network with one consumer-supplier pair, a single layer of hubs, and affine performance functions. The findings of the paper can give fresh managerial insights; in particular, we show risks arising within available pricing strategies.
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