road traffic assignment

The Geography of Transport Systems

The spatial organization of transportation and mobility

A.8 – Route Selection and Traffic Assignment

Author: dr. jean-paul rodrigue.

Transportation seeks to minimize the effort of moving passengers and freight between locations. A component of this effort involves route selection.

1. Route Selection

Human beings are natural effort minimizers , notably when it involves moving around. When given the opportunity, they will always try to choose the shortest path to go from one place to another. This behavior commonly characterizes pedestrians. When possible, a pedestrian will walk over a lawn, zigzag by cars in a parking lot, or cross a street sideways between intersections if the route selected enables them to reach a destination faster.

Transportation, as an economic activity, replicates this process of minimization, notably by trying to minimize the friction of distance between locations. Shorter times and lower costs are looked upon by all transport users, from individuals managing their own mobility to multinational corporations managing complex supply chains. For an individual, it is often only a matter of convenience, but for a corporation, it is strategically important as a direct monetary cost is involved. Under such circumstances, numerous methods have been developed to deal with the complex issue of route selection. One such classic application is the “ traveling salesperson ” problem, where the shortest route has to be selected from a set of possible paths.

Route selection has two major dimensions:

• Construction . Involves activities related to the setting of transport networks, such as road and rail construction where a physical path has to be traced. Among the primary considerations are factors such as distance and topography.
• Operation . Concerns the management of flows in a network. This is the most common route selection activity since it considers routes as fixed entities and seeks an optimal path considering existing constraints.

2. Evaluating the Route Selection Process

The choice of linking a location to another, and more importantly, the path selected, is part of a route selection process that respects a set of constraints. Although route selection varies by mode, the underlying principles remain similar; in its most simple form, a route selection process (R) tries to respect these general constraints:

R = f(min C : max E)

Route selection tries to find or use a path minimizing costs (C) and maximizing efficiency (E). There are two major dimensions of this function:

• Cost minimization . A good route selection should minimize the overall costs of the transport system. This implies construction as well as operating costs. The most direct route is not necessarily the least expensive, notably if rugged terrain is concerned, but a direct route is usually selected. It also implies that route selection must be the least damageable to the environment if environmental consequences are considered.
• Efficiency maximization . A route must support economic activities by providing a level of accessibility, thus fulfilling the needs of regional development. Even if a route is longer and thus more expensive to build and operate, it might provide better services for an area. Its efficiency is thus increased at the expense of higher costs. In numerous instances, roads were constructed more for political reasons than for meeting economic considerations.

Route selection is consequently a compromise between the cost of a transport service and its efficiency. Sometimes, there are no compromises, as the most direct route is the most efficient. At other times, a compromise is very difficult to establish as cost and efficiency are inversely proportional.

3. Traffic Assignment

Contemporary transportation networks are intensively used and congested to various degrees, notably road transportation systems in urban areas. Less known is the spatial logic behind the generation, attraction, and distribution of traffic on a network. There are two important concepts related to understanding traffic in transport systems:

• The transport demand between places must either be known or estimated. For instance, the gravity model offers a methodology to assess potential flows between locations if a set of attributes are known, such as respective distances and emission and attraction variables.
• The transport supply between places must also either be known or estimated. This involves establishing a set of paths between places that are generating and attracting movements. This includes the geometric definition of transport networks with the graph theory.

However, a fundamental concept is absent: how traffic is distributed in a transport network when its structure, capacity, and spatial demand are known.

A traffic assignment problem is traffic distribution in a network considering a demand between a set of locations and the transport supply of the network. Assignment methods are looking to model the distribution of traffic in a network according to a set of constraints, notably related to transport capacity, time, and cost.

Purchasing an airplane ticket is a classic traffic assignment example. For instance, a potential traveler wishes to go from city A to city B at a specific date and time. A query to a reservation system will offer a set of choices (paths) along with a price quote for each path. The traveler will likely choose the least expensive path, which may not necessarily be a direct path and may involve a transfer at an intermediate airport C. When tens of thousands of travelers make these daily decisions, assigning passengers to paths (air service) becomes a very complex task for airlines and their reservation (traffic assignment) system. On the other hand, airline companies use these decisions to adjust their transport supply (mainly flights) to match the demand as closely as possible. This type of problem can be solved using optimization methods.

4. Traffic and its Properties

Traffic is the number of units passing on a link in a given period of time, and it is commonly represented by Q(a,b), that is, the amount of traffic passing on the a,b link (between a and b). Units can be vehicles, passengers, tons of freight, etc. Because of the characteristics of transportation networks, there are two major types of traffic flows:

• Uninterrupted traffic . Traffic regulated by vehicle-vehicle interactions and interactions between vehicles and the transport infrastructure. The most common example of uninterrupted traffic is a highway.
• Interrupted traffic . Traffic regulated by an external means, such as a traffic signal, often creates queuing. Under interrupted flow conditions, vehicle-vehicle interactions and vehicle-infrastructure interactions play a less important part. The most common example of interrupted traffic in urban circulation is regulated by traffic signals such as lights and stop signs.

Traffic is not a spatial interaction as an interaction represents movements between locations (origins and destinations), while traffic represents movements on network links. Traffic could be similar to an interaction when the transport network is equal to the set of Origin / Destination (O/D) pairs, but this is very unlikely.

• Traffic is represented in a graph (network) by its value ; the number of any units flowing (cars, people, tons, etc.). The intensity of the traffic is proportional to the load of the network.
• Traffic is also represented in a graph by its assignment ; how the traffic is distributed on a graph according to supply and demand.

Traffic is assigned on a network according to a sequence of links where every link has its value and direction where several conditions must be satisfied:

• The graph must have nodes where traffic can be generated and attracted. These nodes are generally associated with centroids in an O-D matrix.
• The minimal (l(a,b)) and maximal (k(a,b)) capacities of every link must be respected. k(a,b) is the transport supply on the link (a,b).
• Transport demand must be respected. The O/D matrix has equal inputs and outputs (closed system).
• There is a conservation of the traffic at every node that is not an origin or a destination.

There are also two general network traffic measures: maximum load and load.

Maximum Load (ML): Number of traffic units a network can support at any time. The maximal load is the summation of the capacity of all links.

Load (L): Number of traffic units that a network supports while fulfilling a transport demand. Load is the summation of the traffic of all links.

When the load of a network reaches the maximum load, congestion is reached.

5. Traffic Maximization and Costs Minimization

Traffic in a transportation network can be represented from two perspectives, traffic maximization and costs minimization . Traffic maximization involves the determination of the maximal transport demand that a network or a section of a network can support between its nodes.

It involves maximizing traffic for all links, where the traffic on links must be equal to or lower than the link’s capacity. For simple networks, this procedure can be solved heuristically .

Cost minimization involves determining the minimal transport costs considering a known demand. Transport costs on a link are expressed by g(Q(a,b)) and the minimization function by:

This equation aims to minimize the summation of transport costs (global cost) of each link subject to capacity constraints. Again, for simple networks, the procedure can be solved heuristically . Several types of costs are involved in the minimization procedure:

• The global cost is the sum of transport costs for every link of a network, considering the demand.
• The average cost expresses the transport cost per unit in a network considering the demand (global cost/load). It often varies with the demand.
• The marginal cost expresses the costs incurred to transport a supplementary unit in a network considering an existing demand. The more a network is congested, the higher the marginal cost.

Bibliography

• Cambridge Systematics (2019) Quick Response Freight Methods, USDOT, Federal Highway Administration, Office of Planning and Environment Technical Support Services for Planning Research.

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• Data Descriptor
• Open access
• Published: 29 March 2024

A unified dataset for the city-scale traffic assignment model in 20 U.S. cities

• Xiaotong Xu   ORCID: orcid.org/0000-0001-7577-6194 1 ,
• Zhenjie Zheng 1 ,
• Zijian Hu 1 ,
• Kairui Feng   ORCID: orcid.org/0000-0001-8978-2480 2 &
• Wei Ma 1 , 3  

Scientific Data volume  11 , Article number:  325 ( 2024 ) Cite this article

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Metrics details

• Engineering

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.

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Background & summary.

City-scale traffic data, including traffic flow, speed, and density on every road segment of the entire road network, are foundational inputs and building blocks for modern urban research. These traffic datasets offer an overview of urban mobility, facilitating a better understanding of traffic conditions and travelers’ behaviors in a city. Utilizing the city-scale traffic data, policymakers could develop appropriate transport policies and strategies to mitigate traffic congestion 1 , 2 . Additionally, the traffic data can also be used to evaluate the noise and air pollution caused by vehicles in urban areas 3 , 4 , 5 , which are important in enhancing public health and environmental conditions 6 , 7 , 8 . Furthermore, it assists in formulating energy-efficient traffic management and control strategies that can substantially reduce energy consumption 9 , 10 , 11 . In view of this, it is of great importance to produce and publish open-access traffic datasets on a city scale to support related studies in interdisciplinary research.

However, it is challenging to directly collect the traffic data on every road segment on the entire road network. This is because the traffic data are typically collected from various traffic sensors (e.g., loop detectors, CCTV cameras), which are usually insufficient to cover the entire network due to the associated high installation and maintenance costs. For instance, there are over 30,000 links on the road network of Hong Kong, but less than 10% of the links (i.e., 2,800) are equipped with volume detectors 12 . Moreover, data missing or data measurement errors are inevitable problems due to various factors such as sensor failures, software malfunctions, and weak communication signal transmission 13 , 14 . For example, existing studies indicate that approximately 30% of the freeway sensors in California Performance Measurement System (PeMS: https://pems.dot.ca.gov/ ) are not working properly, resulting in data missing 15 , 16 . More importantly, directly observing the traffic conditions may not be sufficient since the underlying mechanism of the traffic dynamics is not reflected. For example, a reduction in traffic speed indicates congestion, while it is still not clear how the congestion is formed 17 .

To address the above challenges, many urban planning or transport departments utilize traffic modeling techniques to estimate the city-scale traffic data in a generative manner. Specifically, the traffic assignment model 18 , which is a mature model that has been studied extensively in the transportation field, is adopted to estimate the city-scale traffic states. The input of the traffic assignment model only includes the Origin-Destination (OD) demand information and network structure, both of which are public and openly available. Then, the model outputs the city-scale traffic dataset. Traffic assignment models utilize OD data to predict traffic flow and route choices for individual travelers, relying on either predefined or data-driven behavioral models. By modeling the interactions between travelers’ behaviors and traffic congestion, the traffic assignment model searches for the equilibrium condition that mimics real-world traffic conditions. Traffic assignment models can often serve as the primary tool for local governments to assess the potential impact of changes in land use or road network expansions on both local and global traffic conditions. These models are indispensable because they inherently focus on optimizing travel decisions for local residents, aligning with their individual preferences. This capability enables the model to predict changes in agent-level behavior in situations that may not be fully reflected in the available data. Moreover, traffic assignment models demonstrate robust predictive capabilities for estimating future traffic conditions. For example, Metropolitan Planning Organizations (MPOs) in urban areas of the United States would utilize travel survey data, such as the National Household Travel Survey (NHTS: https://nhts.ornl.gov/ ), to produce traffic data for each local urban area that represent residents’ travel patterns 19 . However, these traffic assignment models and data are usually maintained by public agencies and generally not available to most researchers or the public due to difficulties in information sharing or privacy concerns 20 , 21 . Furthermore, the data used in traffic assignment models are under the ownership of various institutions and lack standardization in terms of their structures, granularity, and output formats. As a result, the data are restricted to a few researchers and it is challenging to access the necessary data for traffic assignment models across cities from official sources. Given the above, there is still a notable absence of city-scale traffic datasets that include multiple major cities within one geographic and cultural region, adhere to consistent standards, collect and validate information on a uniform scale, provide comprehensive data types, and meet high-quality standards for public availability.

Although there are a few publicly available datasets 22 , 23 concerning urban areas (see Table  1 ), the reliability and completeness of these datasets limit their applications across broader urban studies, especially in fields like energy, environment, and public health 24 , 25 . The limitations come from the following aspects: First, the existing traffic datasets typically cover some important traffic segments for a single city rather than a city-scale traffic dataset for multiple cities. Second, these current datasets often lack the necessary input, including road network data and corresponding OD data, directly usable for traffic assignment models. Third, these datasets often suffer from incomplete data types and lack of timely updating, resulting in limited convenience when utilizing them. In other words, these datasets are often collected by different researchers or volunteers several years ago, leading to a lack of uniformity in the data types and formats, as well as infrequent updates and maintenance. Fourth, these datasets frequently lack comprehensive validation across multiple variables or fail to offer adequate tools for predicting traffic features from behavioral data. For example, a dataset that includes OD numbers may result in unrealistic traffic flow predictions when attempting to utilize a traffic assignment model. In light of these mentioned facts, currently, there is no unified and well-validated traffic dataset available for multiple cities that covers the entire urban road network at a citywide scale, which hinders the feasibility of conducting comprehensive urban studies across cities to unearth novel discoveries.

To facilitate convenient access to citywide traffic assignment models and data for researchers from different domains besides transportation fields, this study provides a unified traffic dataset for traffic assignment models in 20 representative U.S. cities, with populations ranging from 0.3 million to over 8.8 million. Specifically, we first obtain the input of the model by fusing multiple open public data sources, including OpenStreetMap, The Longitudinal Employer-Household Dynamics Origin-Destination Employment Statistics (LODES), Waze, and TomTom. Then, we employ a grid-search method to fine-tune the parameters and generate the final traffic dataset for each city. The real world’s average travel time and traffic speed serve as validation criteria to ensure a reliable and effective traffic dataset for multiple cities. The validation results demonstrate that our approach can successfully produce the dataset with an average correlation coefficient of 0.79 for average travel time and an average error of 5.16% and 10.47% for average travel speed between real-world data and our data. Finally, we upload the validated traffic dataset and the code used in this study to a public repository.

To sum up, we utilize the static traffic assignment model, leveraging annually aggregated statistical data and open public data sources, to offer a city-scale traffic dataset for macroscopic urban research. It is worth noting that the approach provided in this study can also be applied to other cities. A comprehensive workflow of processing multi-source open public datasets to acquire this dataset is provided in Fig.  1 .

The workflow of obtaining unified and validated traffic datasets from multi-source open public datasets.

Creating a unified traffic dataset in multiple cities involves four main procedures: (1) the identification of representative cities; (2) the acquisition of corresponding input data from multi-source open public datasets; (3) the fusion of the obtained data; and (4) the implementation of traffic assignment, along with parameters calibration. The main procedures are illustrated accordingly below.

Identification of representative cities

In this study, we select a total of 20 representative cities in the United States and generate corresponding traffic datasets using the proposed approach. To ensure diversity and exemplarity among the selected cities, we primarily consider factors such as geographic location, urban scale, topography, and traffic conditions during the commute. Our selection includes a range of cities, including megacities like New York City, as well as several large cities such as Chicago and Philadelphia. We also included smaller but equally representative cities such as Honolulu. The topography of these cities also varies widely. For example, New York and San Francisco are separated by several rivers and rely on critical bridges and tunnels for commuting, while Las Vegas and Phoenix have relatively flat and continuous terrain, with surface transportation playing a predominant role.

Basic information of the 20 representative cities in the United States is given in Table  2 . The population and land area data in the year 2020 are sourced from the U.S. Census Bureau ( https://www.census.gov/ ) while the congestion ranking information in the year 2022 is from TomTom ( https://www.tomtom.com/traffic-index/ranking/ ). Their geospatial distribution is shown in Fig.  2 .

The geospatial distribution of 20 representative U.S. cities.

Data acquisition

The road network structure and travel demand are two crucial inputs for traffic assignment. In this study, we derive these data from public open-source datasets. This section provides a brief overview of the data acquisition procedures.

Road networks

First, the road network structures of the 20 cities are generated from the OpenStreetMap (OSM: https://www.openstreetmap.org/ ) database, which is an open-source mapping platform that provides crowd-sourced road network geographic information, including network topology, road attributes, and connectivity information. By leveraging OSM data, researchers gain convenient access to a comprehensive and up-to-date depiction of the network structure, which facilitates the research in urban studies 26 , 27 , 28 , 29 . The road attributes are also sourced from OSM. After the implementation of cleaning and integration procedures, these processed data can serve as the input for the traffic assignment. A summary of the road network data is given in Table  3 .

Specifically, we employ a Python package named osmnx 30 ( https://github.com/gboeing/osmnx ) to download the OSM data. We then use another Python package called osm2gmns 31 ( https://github.com/jiawlu/OSM2GMNS ) to extract the nodes and links on the road network from the OSM data and save them into separate CSV files in GMNS format 32 , 33 . We use five main link types including ‘motorway’, ‘trunk’, ‘primary’, ‘secondary’, and ‘tertiary’ to implement the traffic assignment. For each link type, we initiate the corresponding road attributes, including parameters such as road capacity, speed limits, the number of lanes, and so on. For the nodes, each node represents the intersection between two links and contains a unique identifier along with latitude and longitude information. By establishing the connectivity between nodes and links through their corresponding relationships, the network topology and road attributes can be constructed. We use the graphing functions of osmnx to visualize the constructed road networks of 20 representative U.S. cities in Fig.  3 .

Road networks of 20 representative U.S. cities extracted from OpenStreetMap.

Travel demand

We then estimate the travel demand, another essential input data for traffic assignment, using the data from the LODES dataset ( https://lehd.ces.census.gov/data/lodes/ ) provided by the U.S. Census Bureau. The LODES dataset includes commuting data for the workforce in all states across the United States over multiple years, which have been widely used in existing studies 34 . LODES data collection involves employers reporting employee details to state workforce agencies, including work and home locations. The U.S. Census Bureau collaborates with state agencies to process and anonymize this data. It’s then used to create Origin-Destination (OD) pairs. This dataset, at the finest granularity of block level, documents the block code for both workplace census and residence census, along with the corresponding total number of jobs. Essentially, the LODES dataset provides an excellent representation of the trip distributions of the U.S. working population that can be used to construct the OD matrix. In this study, we mainly focus on producing the traffic dataset for the year 2019 and the commuting OD data in that year are collected. Moreover, the data collection process is performed at the block level, resulting in the OD data between blocks.

Travel time and speed

We collect data from two open-source dataset platforms, namely TomTom ( https://www.tomtom.com/traffic-index/ranking/ ) and Waze ( https://www.waze.com/live-map/ ), as two indicators of travel time and average speed respectively for our dataset validation. The detailed procedures of data collection can be found in the subsequent sec:Technical ValidationTechnical Validation section.

Data fusion

In this section, we integrate the road network data and OD data to unify the data format. Since the origins and destinations in the OD matrix are not associated with network nodes, it is infeasible to directly take these data as input for the traffic assignment. Therefore, we need to establish a connection between network nodes and blocks. After establishing the connection, we can employ the traffic assignment model to identify appropriate travel paths and allocate traffic flow to the respective links.

To be specific, we begin by aggregating the OD data from its minimum granularity at the block level to a higher level, namely, the tract level. According to the United State Bureau 35 , 36 , 37 , blocks are statistical units with small areas, generally defined to contain between 600 and 3,000 people, whereas tracts composed of multiple blocks are relatively larger and typically have a population size ranging from 1,200 to 8,000 people. In order to achieve a balance between computational complexity and accuracy, we consider tracts as an ideal basic unit for the traffic assignment, which is similar to the existing studies 38 , 39 . This implies that we use the tract as a Traffic Analysis Zone (TAZ) in the traffic assignment model.

Then, the geographical location of each TAZ is determined as the average coordinates of all the blocks within a tract. These TAZs (also called centroids) are generated and stored in the existing node file labeled with a unique identifier. Finally, we generate connectors to bridge the TAZs and network nodes. These connectors can be regarded as a special type of links that are generated from each TAZ center to their neighbor links. Moreover, these connectors are incorporated into the existing links labeled with a unique identifier. As a result, the commuting trips could start from the origin TAZ, traverse a connector to access the nearby road network, choose a suitable path, and then use another connector to reach the destination TAZ.

Traffic assignment

In this section, we use the traffic assignment model to produce the dataset based on the User Equilibrium (UE) 40 . To be specific, we formulate the UE using an optimization model and calibrate four categories of parameters used in the model. Using the network structure and OD demand as input, the model would output the traffic flow, speed, and density on each link. Moreover, we mainly focus on the static traffic assignment and do not consider the influence of temporal variations on traffic conditions.

User equilibrium

All travelers naturally make decisions to minimize their own travel costs (either travel time or equivalent monetary value). Wardrop’s First Principle 41 posits that when every traveler seeks to minimize their individual travel costs, traffic flow eventually stabilizes. In this equilibrium state, the travel costs on all utilized paths become equal and minimized. Meanwhile, the travel costs on unused paths for any given OD pair are greater than or equal to those on the used paths. In other words, a steady-state traffic condition is reached only when no traveler can improve his or her travel time by unilaterally changing routes. The satisfaction of Wardrop’s first principle is commonly referred to as User Equilibrium (UE).

The physical transport network including road segments and intersections in an urban area can be represented as a graph structure G ( N , A ) containing a link set A and a node set N . For each link α ∈ A , it has the link flow x a and the link travel cost t a respectively. For each node r , s ∈ N , it is defined as the TAZ that generates or attracts traffic demand. Therefore, the mathematical formulation of the traffic assignment model under the UE condition 42 can be expressed as follows:

where t a ( x a ) denotes the link performance function that indicates the travel cost on link a when the traffic flow is x a . \({f}_{k}^{rs}\) represents the traffic flow on path k connecting origin r and destination s . q rs indicates the number of trips from origin r to destination s . \({\delta }_{ka}^{rs}\) is a binary variable indicates whether link a is part of path k between origin r and destination s . Equation ( 2 ) imposes the flow conservation constraints. Equation ( 3 ) expresses the relationship between link flow and path flow. Please refer to the book Urban Transportation Networks 40 for details.

Once the traffic flow on each link is determined, the total travel time, denoted as \({c}_{k}^{rs}\) , for a specific path k can be calculated by summing the travel time of each link along this path, which can be formulated as follows:

Although the above optimization model has been proven to be a strict convex problem with a unique solution for traffic flow on links 40 , the computational cost of finding the optimal solution would significantly increase when dealing with large-scale city road networks. To alleviate the computational burden, a bi-conjugate Frank-Wolfe algorithm 43 , 44 is employed to find the optimal solution. In order to enable convenient usage of the provided dataset by users from various disciplines and allow them to easily modify the core parameter settings of the traffic assignment process according to their research needs, we employ two traffic modeling platforms to generate the final dataset. Subsequent users can either directly view the dataset in a no-code format or quickly adjust parameters through a low-code approach to conduct scenario testing under different scenarios. Specifically, a commercial software (named TransCAD ) and an open-source Python package for transportation modeling (named AequilibraE ) are utilized simultaneously in this study. For both platforms, the maximum assignment iteration time and the convergence criteria are set to 500 and 0.001, respectively. The results of the traffic assignment model in 20 U.S. cities are shown in Fig.  4 .

Results of the traffic assignment model in 20 representative U.S. cities.

Parameters calibration

The traffic conditions on the network are influenced by many factors related to traffic supply and demand. Consequently, the traffic assignment model would be impacted and output different results. Since the disturbances in the transport system are nonlinear and challenging to quantify, it is difficult to establish a deterministic mapping relationship between various influencing factors and the results of the traffic assignment model. Therefore, we adopt a grid-search approach to calibrate four common categories of factors that are closely related to the traffic assignment model. We determine the final model by continuously fine-tuning various parameters associated with the traffic assignment model until the transport system reaches the UE condition. In this study, we introduce four categories of factors including road attributes, travel demand, impedance function, and turn penalty, as outlined below.

Road attributes

We categorize the entire road network into three major types, namely expressways, arterial highways, and local roads. Capacity and free flow speed of each road type are two parameters identified to be calibrated. Based on the experimental results, the appropriate range of road capacity for expressways is between 1800 veh/h/lane and 2200 veh/h/lane, while the range for free flow speed is from 65 km/h to 90 km/h. In the case of highways, the corresponding capacity value falls within the range of 1500 veh/h/lane to 2000 veh/h/lane, and the free flow speed value ranges from 40 km/h to 65 km/h. As for local roads, their capacity varies from 600 veh/h/lane to 1500 veh/h/lane, while the suitable speed ranges between 25 km/h and 45 km/h. The detailed information for each type of road can be found in Table  4 .

The OD travel demand is another significant factor influencing the outcome of the traffic assignment. In this study, we aim to simulate the traffic conditions during the peak hours. As mentioned above, the OD demand matrix is derived from the total number of jobs in the United States in 2019, generated from LODES datasets. Although it is reasonable to assume that commuting travel accounts for the majority during peak hours, such demand cannot reflect the actual traffic conditions. Therefore, it is necessary to adjust the initial OD demand, considering variations in transport modes, travel departure time, and carpooling availability during commuting to work. To address this issue, we introduce an OD multiplier to estimate the actual traffic demand during the commuting time. We find that stable results can be obtained when the parameter ranges from 0.55 to 0.65. We show the travel demand and the percentage of internal travel within each TAZ in Fig.  5 .

Total travel demand and the percentage of internal travel demand for 20 U.S. cities.

Link performance function

The link performance function, also known as the impedance function or volume delay function, refers to the relationship between travel time and traffic flow on a road. Typically, travel time increases non-linearly with the increase in traffic flow, which also significantly affects the traffic assignment. One of the most commonly adopted functions in the literature is called the Bureau of Public Roads (BPR) function 45 , which is expressed as follows:

In the function above, t indicates the actual travel time on the road while t 0 represents the free flow travel time on the corresponding road. v and c are the traffic flow and capacity of the road, respectively. α and β are parameters needed to be fine-tuned. We find that the results are satisfactory when parameter α ranges from 0.15 to 0.6 while parameter β changes from 1.2 to 3. The specific values of parameters for each city are provided in Table  5 .

Turn penalty

The turning delay at intersections is also a significant factor that should not be dismissed. When vehicles pass through road intersections, their speed typically decreases, either due to signal control or the necessity to make turns. However, this behaviour cannot be adequately represented in solving traffic assignment problems. To ensure that the results of the traffic assignment model are in accordance with real-world scenarios, we uniformly set corresponding parameters for all junctions to simulate the turning delay effects. In other words, the turn penalty parameters are an average value for the turning delay at all intersections in the road network and these intersection types include signal-controlled intersections, roundabouts, yield or stop intersections, and others. Specifically, the time delay for right turns varies between 0 and 0.25 minutes, while the penalty for making a left turn ranges from 0 to 0.35 minutes. The delay for through traffic is between 0 and 0.15 minutes. U-turn is prohibited in the traffic assignment simulation. The specific parameter setting is demonstrated in Table  5 .

Data Records

We share the traffic dataset on a public repository (Figshare 46 ). In this dataset, each folder, named after the city, contains the input and output of the traffic assignment model specific to that city. We elaborate on the details as follows:

This folder contains all the input data required for the traffic assignment model, namely the OD demand data and network data. The network data contains both node and link files in a CSV format. The data in this file folder specifically includes the following contents:

the initial network data obtained from OSM

the visualization of the OSM data

processed node/link/od data

The detailed meanings of the fields contained in different input data are given in Table  6 .

TransCAD results

This folder contains all the input data required for the traffic assignment model in TransCAD, as well as the corresponding output data. The data in this file folder specifically includes the following contents:

cityname.dbd: geographical network database of the city supported by TransCAD

cityname_link.shp/cityname_node.shp: network data supported by the GIS software, which can be imported into TransCAD manually

od.mtx: OD matrix supported by TransCAD

LinkFlows.bin/LinkFlows.csv: results of the traffic assignment model by TransCAD

ShortestPath.mtx/ue_travel_time.csv: the travel time (in minutes) between OD pairs by TransCAD

The detailed meanings of the fields contained in output data generated from TransCAD are given in Table  7 .

AequilibraE results

This folder contains all the input data required for the traffic assignment model in AequilibraE, as well as the corresponding output data. The data in this file folder specifically includes the following contents:

cityname.shp: shapefile network data of the city support by QGIS or other GIS software

od_demand.aem: OD matrix supported by AequilibraE

network.csv: the network file used for traffic assignment in AequilibraE

assignment_result.csv: results of the traffic assignment model by AequilibraE

The detailed meanings of the fields contained in output data generated from AequilibraE are given in Table  8 .

Technical Validation

To ensure the consistency between the traffic assignment model’s output and real-world traffic conditions, we conduct validation using two different public open sources of traffic data. Specifically, the travel time between different OD pairs and the overall average travel speed are employed as two validation indicators to ensure the reliability and accuracy of the provided dataset. The validation results are shown in Tables  9 , 10 and we can see that the provided dataset for each city is accurate and valid.

Travel time

In examining the travel time metric, we obtain the travel time between different OD pairs both from traffic assignment models and map service providers. As for the model side, the travel time under both UE and free flow conditions are calculated respectively using traffic assignment models. First, under UE conditions, the travel time between different OD pairs could be generated by summing the link travel time determined by the corresponding assigned traffic flow along the shortest path as shown in Eq. ( 5 ). Then, under free flow conditions, the travel time between OD pairs is the travel time associated with the shortest path, disregarding congestion on road segments. Furthermore, the average value of Travel Time (in minutes) under UE conditions (UETT) as well as free flow conditions (FFTT) for all OD pairs can be expressed as follows:

where \({c}_{ue}^{rs}\) and \({c}_{ff}^{rs}\) denote the travel time between origin r and destination s under the UE and free flow conditions respectively. Additionally, the difference as well as the ratio between these two types of travel time give the average travel delay (in minutes) and delay factor for each city.

In terms of the real-world data for validation, since nowadays many map service providers have the capability to offer travel time estimates between two location points at different departure times based on users’ historical navigation records, in this study, we choose Waze as the data source to obtain the actual travel time between each OD pair by using its WazeRouteCalculator API ( https://github.com/kovacsbalu/WazeRouteCalculator ) with Python code.

The results of travel time are shown in Table  9 . It can be seen that Honolulu experiences the least travel time under free flow conditions, at about 8.70 minutes, while Minneapolis has the shortest average travel time during commuting hours, at about 10.25 minutes. Minneapolis also has the lowest delay travel time among all cities, merely 0.47 minutes, indicating that the commuting travel time in this city is almost the same as the travel time under free flow conditions. In contrast, New York City experiences significant delays, with a delay time of 24.47 minutes, revealing that the travel time during peak periods in New York is more than double that of the free flow condition. In terms of the delay factor, New York City has the highest value, reaching 2.24, followed by Chicago with a value of 1.65. Minneapolis and Pittsburgh have the lowest delay factor values, both at 1.05.

To evaluate the results, we use the Pearson Correlation Coefficient (PCC) 47 to measure the correlation between the actual travel time and the travel time produced by our model. The PCC r xy is defined as follows:

where r xy denotes the Pearson’s Correlation Coefficient. x i and y i are the individual sample points indexed with i . n represents the sample size.

Since the turning penalties are not incorporated in the traffic assignment algorithm of AequilibraE, the parameter settings in TransCAD and AequilibraE are not identical. Consequently, results of the two platforms are not entirely consistent. Considering the more comprehensive parameter settings in TransCAD, we adopt the results of TransCAD as the primary benchmark. We perform PCC analysis between Waze and TransCAD, as well as between TransCAD and AequilibraE, with the evaluation results presented in Table  9 .

From the correlation analysis, we can find that all correlation coefficients R 2 are greater than 0.7, which confirms the accuracy and reliability of the results to some extent. We also visualize the correlation coefficient for each city in Fig.  6 . It can be seen that the simulated travel time is consistent with the travel time in the real world.

Correlation analysis results between Waze and TransCAD.

Average speed

The overall average speed of the entire road network is another important indicator for validation. In this study, we use the speed data collected from TomTom Traffic Index as the actual speed to validate our model. We first calculate the average link-based speed of our model through dividing Vehicle Hours Travelled (VHT) by Vehicle Kilometers Travelled (VKMT). Then, the average OD-based speed values are derived from the ratio of distance to travel time between each OD pair. The Mean Absolute Percent Errors (MAPE) and Mean Absolute Errors (MAE) for both the link-based speed and the OD-based speed are used to measure the reliability of our model:

where y i is the actual observed value, \({\widehat{y}}_{i}\) is the predicted value, and n is the number of samples.

The results are summarized in Table  10 . We find that the average MAPE and MAE values for the link-based speed metric are 5.16% and 1.77 km/h, respectively. Moreover, the average MAPE and MAE values for the OD-based speed indicator are 10.47% and 3.82 km/h, respectively. This implies that our approach can produce satisfactory and reliable results.

Network traffic impact on model performance

To validate the effectiveness and robustness of our model across cities, we further investigate how traffic conditions of a city affect the model performance. The MAE and MAPE values for link-based average speed metrics obtained in Table  10 are used to evaluate the model performance. The traffic conditions are characterized by two different indicators. One is the ratio of the total OD travel demand to the number of links for the entire road network, which can characterize the average OD demand and represent the traffic conditions of a city. The other is the average speed (km/h) in rush hour obtained from TomTom (refer to Table  10 ). If the values of average traffic demand are large, it reveals a congested city network experiencing substantial traffic demand, exemplified by cities like New York and San Francisco. Conversely, a small value suggests a city road network with low traffic demand, as observed in cities like Atlanta and Dallas. We can draw similar conclusions with respect to the average traffic speed.

The results are shown in Fig.  7 . The red dashed line represents the linear regression trendline that has been fitted to the data points. The R 2 values of Fig.  7a and Fig.  7b are 0.0049 and 0.0218, respectively. This implies that there is no evident relationship between the model performance and the varying traffic demand of the network. Similarly, the R 2 values of Fig.  7c and Fig.  7d are 0.0212 and 0.0177, respectively. This suggests that the model performance is not affected by the varying traffic speeds in different cities. In summary, the proposed model exhibits low sensitivity to variations in city traffic conditions and achieves satisfactory performance across cities.

The model performance in relation to different traffic conditions for 20 U.S. cities. ( a ) The MAPE values (%) regarding the average OD demand for different cities. ( b ) The MAE values (km/h) regarding to the average OD demand for different cities. ( c ) The MAPE values (%) regarding the average speed for different cities. ( d ) The MAE values (km/h) regarding the average speed for different cities.

Usage Notes

The acquisition of OD data is crucial in performing the traffic assignment and producing the citywide traffic dataset. In this study, we utilize the commuting OD data (LODES) provided by the U.S. Census Bureau to generate the OD matrix. For cities in other countries, OD data can be substituted with alternative open data sources, such as OD data provided by TomTom ( https://developer.tomtom.com/od-analysis/documentation/product-information/introduction ).

Moreover, we use the average traffic time and average travel speed between different OD pairs in the real world to validate the results of our approach, ensuring its reliability and accuracy. If additional data sources are available, such as traffic flow data obtained from traffic detectors, we can also use the corresponding data to further evaluate the effectiveness of the provided dataset.

It is worth noting that the provided dataset is mainly used for macroscopic urban research and policy development across interdisciplinary studies. In view of this, the given dataset provides full spatial coverage of the entire road network, unlike existing traffic datasets that focus on specific areas. Hence, the provided traffic dataset and existing traffic datasets complement each other, which can better facilitate research in urban studies. Specifically, the full spatial coverage of the provided dataset makes it valuable for comprehensive macroscopic urban research and policy development, making a notable contribution to the literature, such as public transport planning, road expansions, the determination of bus routes, the estimation of the transport-related environmental impact and so on. In contrast, existing traffic datasets (e.g., PeMS) may exhibit incomplete spatial coverage, making them less suitable for the aforementioned macroscopic urban studies. Actually, the datasets containing fine-grained temporal information are more suitable for investigating regional traffic dynamics by leveraging the spatiotemporal relationship between the traffic data, such as traffic prediction, spatiotemporal propagation of shockwaves, calibration of fundamental diagrams, traffic data imputation, and so on.

In this study, the provided dataset lacks fine-grained temporal information due to the limited availability of input data. To fully understand dynamic traffic patterns, it is essential to consider both spatial and temporal dimensions within the traffic data. Consequently, developing a dynamic traffic assignment model that effectively captures the spatiotemporal interdependencies of traffic data is important. Moreover, employing daily traffic data for more fine-grained validation would enhance further urban research.

Code availability

The guidelines for data retrieval and utilization have been uploaded to GitHub 48 . The specific contents comprise:

1. Input data introduction.ipynb : A brief introduction and data demonstration about the input data for the traffic assignment process in the dataset.

2. A guide for TransCAD users.md : It is a guide for users who want to view and modify the dataset in the Graphical User Interface (GUI) of TransCAD.

3. AequilibraE_assignmnet.py : A Python code file for users who want to get access to the traffic assignment results by using the AqeuilibraE.

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Acknowledgements

The work described in this paper was supported by the National Natural Science Foundation of China (No. 52102385), grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. PolyU/25209221 & PolyU/15206322), and a grant from Dean’s Reserve at the Hong Kong Polytechnic University (Project No. P0034271). The authors would like to thank Prof. Xuesong Zhou for providing constructive suggestions and active discussions regarding the data.

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Xiaotong Xu, Zhenjie Zheng, Zijian Hu & Wei Ma

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Kairui Feng

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X.X. conceived the study, curated data, developed methodology, conducted experiment and wrote the original draft. Z.Z. conceived the study, developed methodology, coded for the data acquisition, reviewed and edited writing. Z.H. coded for the data acquisition. K.F. conceived the study, contributed to the original data, reviewed and edited writing. W.M. conceived the study, acquired funding, developed methodology and supervised the study. All authors reviewed and agreed on the final manuscript.

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Xu, X., Zheng, Z., Hu, Z. et al. A unified dataset for the city-scale traffic assignment model in 20 U.S. cities. Sci Data 11 , 325 (2024). https://doi.org/10.1038/s41597-024-03149-8

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Network assignment

What is Network Assignment?

Role of Network Assignment in Travel Forecasting

Overview of Methods for Traffic Assignment for Highways

All-or-nothing Assignments

Incremental assignment

Brief History of Traffic Equilibrium Concepts

Calculating Generalized Costs from Delays

Challenges for Highway Traffic Assignment

Transit Assignment

Latest Developments

Page categories

Topic Circles

Trip Based Models

More pages in this category:

# what is network assignment.

In the metropolitan transportation planning and analysis, the network assignment specifically involves estimating travelers’ route choice behavior when travel destinations and mode of travel are known. Origin-destination travel demand are assigned to a transportation network in order to estimate traffic flows and network travel conditions such as travel time. These estimated outputs from network assignment are compared against observed data such as traffic counts for model validation .

Network assignment is a mathematical problem which is solved by a solution algorithm through the use of computer. It is usually resolved as a travel cost optimization problem for each origin-destination pair on a model network. For every origin-destination pair, a path is selected that typically minimizes travel costs. The simplest kind of travel cost is travel time from beginning to end of the trip. A more complex form of travel cost, called generalized cost, may include combinations of other costs of travel such as toll cost and auto operating cost on highway networks. Transit networks may include within generalized cost weights to emphasize out-of-vehicle time and penalties to represent onerous tasks. Usually, monetary costs of travel, such as tolls and fares, are converted to time equivalent based on an estimated value of time. The shortest path is found using a path finding algorithm .

The surface transportation network can include the auto network, bus network, passenger rail network, bicycle network, pedestrian network, freight rail network, and truck network. Traditionally, passenger modes are handled separately from vehicular modes. For example, trucks and passenger cars may be assigned to the same network, but bus riders often are assigned to a separate transit network, even though buses travel over roads. Computing traffic volume on any of these networks first requires estimating network specific origin-destination demand. In metropolitan transportation planning practice in the United States, the most common network assignments employed are automobile, truck, bus, and passenger rail. Bicycle, pedestrian, and freight rail network assignments are not as frequently practiced.

# Role of Network Assignment in Travel Forecasting

The urban travel forecasting process is analyzed within the context of four decision choices:

• Personal Daily Activity
• Locations to Perform those Activities
• Mode of Travel to Activity Locations, and
• Travel Route to the Activity Locations.

Usually, these four decision choices are named as Trip Generation , Trip Distribution , Mode Choice , and Traffic Assignment. There are variations in techniques on how these travel decision choices are modeled both in practice and in research. Generalized cost, which is typically in units of time and is an output of the path-choice step of the network assignment process, is the single most important travel input to other travel decision choices, such as where to travel and by which mode. Thus, the whole urban travel forecasting process relies heavily on network assignment. Generalized cost is also a major factor in predicting socio-demographic and spatial changes. To ensure consistency in generalized cost between all travel model components in a congested network, travel cost may be fed back to the earlier steps in the model chain. Such feedback is considered “best practice” for urban regional models. Outputs from network assignment are also inputs for estimating mobile source emissions as part of a review of metropolitan area transportation plans, a requirement under the Clean Air Act Amendments of 1990 for areas not in attainment of the National Ambient Air Quality Standard.

# Overview of Methods for Traffic Assignment for Highways

This topic deals principally with an overview of static traffic assignment. The dynamic traffic assignment is discussed elsewhere.

There are a large number of traffic assignment methods, but they all have at their core a procedure called “all-or-nothing” (AON) traffic assignment. All-or-nothing traffic assignment places all trips between an origin and destination on the shortest path between that origin and destination and no trips on any other possible path (compare path finding algorithm for a step-by-step introduction). Shortest paths may be determined by a well-known algorithm by Dijkstra; however, when there are turn penalties in the network a different algorithm, called Vine building , must be used instead.

# All-or-nothing Assignments

The simplest assignment algorithm is the all-or-nothing traffic assignment. In this algorithm, flows from every origin to every destination are assigned using the path finding algorithm , and travel time remains unchanged regardless of travel volumes.

All-or-nothing traffic assignment may be used when delays are unimportant for a network. Another alternative to the user-equilibrium technique is the stochastic traffic assignment technique, which assumes variation in link level travel time.

One of the earliest, computationally efficient stochastic traffic assignment algorithms was developed by Robert Dial. [1] More recently the k-shortest paths algorithm has gained popularity.

The biggest disadvantage of the all-or-nothing assignment and the stochastic assignment is that congestion cannot be considered. In uncongested networks, these algorithms are very useful. In congested conditions, however, these algorithm miss that some travelers would change routes to avoid congestion.

# Incremental assignment

The incremental assignment method is the simplest way to (somewhat rudimentary) consider congestion. In this method, a certain share of all trips (such as half of all trips) is assigned to the network. Then, travel times are recalculated using a volume-delay function , or VDF. Next, a smaller share (such as 25% of all trips) is assigned based using the revised travel times. Using the demand of 50% + 25%, travel times are recalculated again. Next, another smaller share of trips (such as 10% of all trips) is assigned using the latest travel times.

A large benefit of the incremental assignment is model runtime. Usually, flows are assigned within 5 to 10 iterations. Most user-equilibrium assignment methods (see below) require dozens of iterations, which increases the runtime proportionally.

In the incremental assignment, the first share of trips is assigned based on free-flow conditions. Following iterations see some congestion, on only the very last trip to be assigned will consider true congestion levels. This is reasonable for lightly congested networks, as a large number of travelers could travel at free-flow speed.

The incremental assignment works unsatisfactorily in heavily congested networks, as even 50% of the travel demand may lead to congestion on selected roads. The incremental assignment will miss the fact that a portion of the 50% is likely to select different routes.

# Brief History of Traffic Equilibrium Concepts

Traffic assignment theory today largely traces its origins to a single principle of “user equilibrium” by Wardrop [2] in 1952. Wardrop’s “first” principle simply states (slightly paraphrased) that at equilibrium not a single driver may change paths without incurring a greater travel impedance . That is, any used path between an origin and destination must have a shortest travel time between the origin and destination, and all other paths must have a greater travel impedance. There may be multiple paths between an origin and destination with the same shortest travel impedance, and all of these paths may be used.

Prior to the early 1970’s there were many algorithms that attempted to solve for Wardrop’s user equilibrium on large networks. All of these algorithms failed because they either did not converge properly or they were too slow computationally. The first algorithm to be able to consistently find a correct user equilibrium on a large traffic network was conceived by a research group at Northwestern University (LeBlanc, Morlok and Pierskalla) in 1973. [3] This algorithm was called “Frank-Wolfe decomposition” after the name of a more general optimization technique that was adapted, and it found the minimum of an “objective function” that came directly from theory attributed to Beckmann from 1956. [4] The Frank-Wolfe decomposition formulation was extended to the combined distribution/assignment problem by Evans in 1974. [5]

A lack of extensibility of these algorithms to more realistic traffic assignments prompted model developers to seek more general methods of traffic assignment. A major development of the 1980s was a realization that user equilibrium traffic assignment is a “variational inequality” and not a minimization problem. [6] An algorithm called the method of successive averages (MSA) has become a popular replacement for Frank-Wolfe decomposition because of MSA’s ability to handle very complicated relations between speed and volume and to handle the combined distribution/mode-split/assignment problem. The convergence properties of MSA were proven for elementary traffic assignments by Powell and Sheffi and in 1982. [7] MSA is known to be slower on elementary traffic assignment problems than Frank-Wolfe decomposition, although MSA can solve a wider range of traffic assignment formulations allowing for greater realism.

A number of enhancements to the overall theme of Wardop’s first principle have been implemented in various software packages. These enhancements include: faster algorithms for elementary traffic assignments, stochastic multiple paths, OD table spatial disaggregation and multiple vehicle classes.

# Calculating Generalized Costs from Delays

Equilibrium traffic assignment needs a method (or series of methods) for calculating impedances (which is another term for generalized costs) on all links (and nodes) of the network, considering how those links (and nodes) were loaded with traffic. Elementary traffic assignments rely on volume-delay functions (VDFs), such as the well-known “BPR curve” (see NCHRP Report 365), [8] that expressed travel time as a function of link volume and link capacity. The 1985 US Highway Capacity Manual (and later editions through 2010) made it clear to transportation planners that delays on large portions of urban networks occur mainly at intersections, which are nodes on a network, and that the delay on any given intersection approach relates to what is happening on all other approaches. VDFs are not suitable for situations where there is conflicting and opposing traffic that affects delays. Software for implementing trip-based models are now incorporating more sophisticated delay relationships from the Highway Capacity Manual and other sources, although many MPO forecasting models still use VDFs, exclusively.

# Challenges for Highway Traffic Assignment

Numerous practical and theoretical inadequacies pertaining to Static User Equilibrium network assignment technique are reported in the literature. Among them, most widely noted concerns and challenges are:

• Inadequate network convergence;
• Continued use of legacy slow convergent network algorithm, despite availability of faster solution methods and computers;
• Non-unique route flows and link flows for multi-class assignments and for assignment on networks that include delays from opposing and conflicting traffic;
• Continued use of VDFs , when superior delay estimation techniques are available;
• Unlikeness of a steady-state network condition;
• Impractical assumption that all drivers have flawless route information and are acting without bias;
• Every driver travels at the same congested speed, no vehicle traveling on the same link overtakes another vehicle;
• Oncoming traffic does not affect traffic flows;
• Interruptions, such as accidents or inclement weather, are not represented;
• Traffic does not form queues;
• Continued use of multi-hour time periods, when finer temporal detail gives better estimates of delay and path choice.

# Transit Assignment

Most transit network assignment in implementation is allocation of known transit network specific demand based on routes, vehicle frequency, stop location, transfer point location and running times. Transit assignments are not equilibrium, but can be either all-or-nothing or stochastic. Algorithms often use complicated expressions of generalized cost which include the different effects of waiting time, transfer time, walking time (for both access and egress), riding time and fare structures. Estimated transit travel time is not directly dependent on transit passenger volume on routes and at stations (unlike estimated highway travel times, which are dependent on vehicular volumes on roads and at intersection). The possibility of many choices available to riders, such as modes of access to transit and overlaps in services between transit lines for a portion of trip segments, add further complexity to these problems.

# Latest Developments

With the increased emphasis on assessment of travel demand management strategies in the US, there have been some notable increases in the implementation of disaggregated modeling of individual travel demand behavior. Similar efforts to simulate travel route choice on dynamic transportation network have been proposed, primarily to support the much needed realistic representation of time and duration of roadway congestion. Successful examples of a shift in the network assignment paradigm to include dynamic traffic assignment on a larger network have emerged in practice. Dynamic traffic assignments are able to follow UE principles. An even newer topic is the incorporation of travel time reliability into path building.

# References

Dial , Robert Barkley, Probabilistic Assignment; a Multipath Traffic Assignment Model Which Obviates Path Enumeration, Thesis (Ph.D.), University of Washington, 1971. ↩︎

Wardrop, J. C., Some Theoretical Aspects of Road Traffic Research, Proceedings, Institution of Civil Engineers Part 2, 9, pp. 325–378. 1952. ↩︎

LeBlanc, Larry J., Morlok, Edward K., Pierskalla, William P., An Efficient Approach to Solving the Road Network Equilibrium Traffic Assignment Problem, Transportation Research 9, 1975, 9, 309–318. ↩︎

(opens new window) ) ↩︎

Evans, Suzanne P., Derivation and Analysis of Some Models for Combining Trip Distribution and Assignment, Transportation Research, Vol 10, pp 37–57 1976. ↩︎

Dafermos, S.C., Traffic Equilibrium and Variational Inequalities, Transportation Science 14, 1980, pp. 42-54. ↩︎

Powell, Warren B. and Sheffi, Yosef, The Convergence of Equilibrium Algorithms with Predetermined Step Sizes, Transportation Science, February 1, 1982, pp. 45-55. ↩︎

(opens new window) ). ↩︎

← Mode choice Dynamic Traffic Assignment →

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Review of traffic assignment and future challenges.

1. Introduction

2.1. problem statement.

• Traffic supply: Road network and/or public transportation services as well as their corresponding behaviors.
• Traffic demand: A flow matrix indicating the demand volume between each origin–destination ( o – d ) pair.

2.2. User Criteria

• The monetary costs, such as the costs of fuel consumption, vehicle’s depreciation and maintenance, toll, and transport fare.
• The non-monetary costs, such as the travel time and user’s perception of comfort and convenience.

2.3. Mode of Transport

2.4. illustrative example.

• In road segment a 1 , the capacity, Q m a x 1 = 1800 vph, and the travel time behave as follows: c 1 ( r 1 ) = 10 1 + 2 r 1 1800 2 . (1)
• In road segment a 2 , the capacity, Q m a x 2 = 3600 vph, and the travel time behave as follows: c 2 ( r 2 ) = 20 1 + 2 r 2 3600 2 . (2)
• Shortest path (All-or-nothing): Without considering Equations ( 1 ) and ( 2 ) as though both roads are devoid of traffic, all vehicles follow road a 1 because it is the shortest path. However, since travel time increases with traffic flow, an all-or-nothing assignment cannot be adopted to predict road use. In this figure, the solution of the all-or-nothing assignment is presented by a green circle. According to Equation ( 1 ), the users spend more than an hour to go from o to d .
• Optimal assignment (Centralized): If we consider that energy consumption is proportional to travel time, the manager of a fleet of the 3000 vehicles has a strong incentive to minimize the average travel time. The red curve in Figure 2 presents the average travel time. The optimal solution is represented by the red circles. One can note that, with this solution, the vehicles taking road a 1 spend less travel time than the ones using road a 2 .
• Fair assignment (Decentralized): In this configuration, each vehicle optimizes its itinerary, so that there is no better solution. The fair assignment is represented by the yellow circle.

3. Traffic Assignment: Theoretical Background

3.1. static traffic assignment ( s t a ), 3.1.1. user equilibrium ( u e ): definition and formulation of the problem.

• For each origin–destination pair, o–d, the generalized costs of each utilized route are less than or equal to those of alternative (unused) routes.
• If multiple routes are used for an o–d pair, their generalized costs are equal.

3.1.2. Stochastic User Equilibrium ( S U E )

• All reasonable options can be chosen, even if their probability of selection is very low (In [ 20 ], the author describes the concept)
• If two options have the same cost, the probability of selection is the same.
• The probability of choosing options depends on their costs: a route with a higher cost has a lower probability of being chosen.
• The user of the S U E model must have some control over the probability of diverting routes.

3.1.3. Approaches to Solving Static Assignment

3.2. Dynamic Traffic Assignment ( D T A )

• A traffic model in which congestion (travel time) varies over time.
• A time-varying demand.
• An equilibrium that is based on experienced travel cost, not instantaneous travel cost.

3.2.1. Time-Dependent User Equilibrium

• For each o–d pair and for each departure time interval , the routes taken by users exhibit a generalized cost (experienced travel time) that is both equal and minimized to the extent possible .
• No user can unilaterally reduce their experienced generalized cost (travel time).

3.2.2. Resolving Approaches

• Initialization: In the majority of studies, the network is initialized using the all-or-nothing assignment based on the computation of the shortest path. In the context of S T A , the shortest path is computed with an empty network. Consequently, the all-or-nothing assignment associates a route with each origin–destination ( o – d ) pair throughout the entire temporal horizon of the study. In the case of dynamic assignment, the travel time (generalized cost) for a given route, k o d , varies based on the departure time intervals of vehicles. Indeed, with each new time interval, it is necessary to update the generalized costs induced by vehicles already assigned in previous time intervals. Thus, the search for the shortest path occurs progressively as the network fills up.
• Iteration: Recall that, at each iteration, a flow of vehicles is shifted from a costly route to a less costly one until a convergence criterion is met. In the context of D T A , this flow shift occurs for all time intervals, T , within the study interval at each iteration. Several approaches exist in the literature for determining the direction and quantity of the traffic shift. The most classical approach is based on the Frank–Wolfe algorithm [ 30 , 83 ]. Other more efficient algorithms have been proposed since, including the gradient projection algorithm [ 41 ] and the method of successive averages ( M S A ) [ 79 , 84 , 85 ] to compute the generalized cost. Approaches based on meta-heuristics have also been proposed.
• Evaluation of Travel Times: The time it takes to travel a road segment can vary from when a vehicle starts its journey to when it is moving on that segment. Drivers base their route decisions on the time spent actively driving on the segment. Thus, it is crucial to predict this travel time accurately. In advanced methods, travel times are assessed by adding up the times for each segment, forming a travel time chain. The estimated travel time for each segment is based on when the vehicle is expected to reach that specific segment [ 78 ].
• Stopping Criterion: Recall that, in S T A , there is a unique equilibrium solution in terms of traffic flow for each segment. This solution is reached when Algorithm 1 converges. Unlike S T A , the convergence of vehicle routes in D T A does not imply that the network has reached a dynamic equilibrium. The convergence problem becomes more complex when the granularity of the traffic model is high. It is widely reported that microscopic models are intractable.

4. Extended Traffic Assignment

4.1. traffic assignment for alternative modes, 4.2. environmental concerns and traffic control, 4.3. paradigm shift through multiagent-based approaches, 5. discussion, 5.1. intelligent transportation systems, 5.2. promising directions.

• Mixed traffic digital twin : The cohabitation of a multitude of systems would require either new macroscopic and mesoscopic traffic models or the use of microscopic models coupled with an agent-based approach [ 76 , 169 ]. The latter option appears better suited to accommodate the diverse objectives of road users, especially in the context of mixed traffic: human drivers with navigation systems, human drivers familiar with the road, autonomous shuttles, optimized freight transport, etc. Furthermore, employing microscopic models could enable us to address the challenge of finding an equilibrium between diverse equitable routes and the costs induced by conflicts. However, it is worth noting that the use of microscopic traffic models for computing the travel times is resource-intensive in terms of computation resources and time. Another option is to investigate the implementation of digital twins [ 170 , 171 ] with different levels of granularity of the traffic entities. This digital twin should be capable of assimilating data from the cloud, ranging from macroscopic quantities such as road flow and occupancy rates to more detailed user profiles at the vehicle level, including departure times and itineraries. Harnessing A I techniques is crucial [ 172 ] for generating a comprehensible map of road usage for decision-makers (e.g., identifying saturated intersections, assessing environmental impact, improving safety, and offering the forecasts needed by cities). More importantly, A I can also be used to propose solutions (e.g., the necessity for new transit services and a new toll policy). As emphasized throughout this article, the problem is undeniably intricate. Nevertheless, the substantial advancements in machine learning technologies, data analysis techniques, and cloud computing hold the promise of highly innovative approaches. The data has previously been employed for a range of tasks related to traffic assignment, including estimating the origin–destination ( o – d ) matrix [ 173 , 174 , 175 ], evaluating travel times [ 176 ], and calibrating models and traffic assignment results [ 174 , 177 , 178 , 179 , 180 ]. This utilization encompasses the data of cell phones, information systems, and magnetic loops. The development of the mixed traffic digital twin aims to transcend these contributions, representing a significant stride towards exploring advanced artificial-intelligence ( A I ) techniques for traffic management. An illustrative foundation for data-driven traffic assignment is established in the study presented in [ 181 ]. Instead of assuming user behavior, the authors showcased the feasibility of precisely estimating road demand by directly learning flow patterns from the available data. This innovative approach lays the groundwork for incorporating A I into traffic forecasting. The extension introduces additional parameters, such as traffic control strategies and new transportation scenarios. Moreover, the mixed traffic digital twin can be designed to be proactive, not only highlighting problems, such as those proposed in ref. [ 182 ], but also proposing solutions to the decision-makers. To this end, the mixed traffic digital twin may be trained by using “classical traffic” assignment approaches.
• New generation of navigation systems : The evolving uses of C A V and associated regulatory systems will likely challenge current traffic assignment approaches. These vehicles will need to plan their routes in real time based on received demands and communicate their estimated arrival times as accurately as possible. Although Wardrop’s equilibrium assumption does not allow for the perfect rationalization of vehicle usage, it remains desirable for fairness reasons. Indeed, it seems evident that no user would want to take a shuttle service that significantly takes longer than other shuttles without financial compensation. However, current traffic assignment techniques do not allow for real-time route planning. Computation times increase when considering vehicle behavior and traffic regulation details with high resolution. Currently, traditional traffic assignment approaches fall short in delivering real-time itineraries. Simultaneously, there exists a lack of consensus in the literature regarding the impact of contemporary navigation systems. Several authors have underscored adverse effects, as indicated in refs. [ 183 , 184 , 185 , 186 ]. This issue pertains to the challenge of efficiently assigning C A V on a constantly evolving road network, taking into consideration real-time traffic conditions. One crucial aspect of the problem is determining optimal routes for vehicles based on real-time traffic information. The commonly used shortest path search method, which calculates the fastest routes between two points on a road network, does not effectively solve the real-time traffic assignment problem. The main drawback of this method is its lack of responsiveness to constantly changing traffic conditions. Figure 4 and Figure 5 depict the adverse effects of such an approach on route selection. The first figure illustrates how vehicles can become trapped when new vehicles are rerouted to an alternative path with smoother traffic, highlighting the necessity for a thorough evaluation of intersection times. Meanwhile, the second figure demonstrates how a vehicle may be misdirected due to the absence of traffic that is not yet present, emphasizing the critical importance of accurately estimating upcoming traffic conditions. New strategies with simple rules must be defined to guide vehicles in real time. These rules should be capable of providing both efficient and fair routes while fostering smooth traffic flow. Some studies already address these issues by proposing itinerary reservations [ 187 , 188 ]. Among these studies, some focus on road booking in order to not exceed their capacity [ 189 , 190 , 191 , 192 ]. Others are inclined towards intersection reservations [ 149 , 193 , 194 , 195 ] to alleviate costs associated with conflicts arising from diversified routes. However, these approaches are relatively recent and deserve to garner broader attention within the community to receive more feedback on microscopic models of large cities incorporating innovative strategies for sharing road infrastructures with more transparency.

6. Conclusions

Author contributions, institutional review board statement, informed consent statement, data availability statement, conflicts of interest, abbreviations.

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Elimadi, M.; Abbas-Turki, A.; Koukam, A.; Dridi, M.; Mualla, Y. Review of Traffic Assignment and Future Challenges. Appl. Sci. 2024 , 14 , 683. https://doi.org/10.3390/app14020683

Elimadi M, Abbas-Turki A, Koukam A, Dridi M, Mualla Y. Review of Traffic Assignment and Future Challenges. Applied Sciences . 2024; 14(2):683. https://doi.org/10.3390/app14020683

Elimadi, Manal, Abdeljalil Abbas-Turki, Abder Koukam, Mahjoub Dridi, and Yazan Mualla. 2024. "Review of Traffic Assignment and Future Challenges" Applied Sciences 14, no. 2: 683. https://doi.org/10.3390/app14020683

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Principles of Wardrop for Traffic Assignment in a Road Network

• First Online: 27 November 2019

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• Alexander Krylatov   ORCID: orcid.org/0000-0002-6634-1313 5 , 6 ,
• Victor Zakharov   ORCID: orcid.org/0000-0002-2743-3880 6 &
• Tero Tuovinen   ORCID: orcid.org/0000-0001-7787-3836 7  

Part of the book series: Springer Tracts on Transportation and Traffic ((STTT,volume 15))

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In this chapter is devoted to user equilibrium and system optimum of Wardrop. Discussion on the mathematical formulation of traffic assignment problems with regard to their meaning is available in the Sect.  2.1 . The specification of necessary basic statements completes this discussion further. The dual traffic assignment problem with travel times between all origins and destinations as dual variables is considered in the Sect.  2.2 . The practical significance of such dual formulation is shown to become evident due to the wide spread of online traffic services. The route-flow assignment problem and link-flow assignment problem are reduced to fixed-point problems with explicit operators in the Sect.  2.3 and Sect.  2.4 respectively. Proofs of corresponding theorems are fully given.

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Alexander Krylatov

Faculty of Applied Mathematics and Control Processes, Saint Petersburg State University, Saint Petersburg, Russia

Alexander Krylatov & Victor Zakharov

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Krylatov, A., Zakharov, V., Tuovinen, T. (2020). Principles of Wardrop for Traffic Assignment in a Road Network. In: Optimization Models and Methods for Equilibrium Traffic Assignment. Springer Tracts on Transportation and Traffic, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-030-34102-2_2

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