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3.6: 3-6 Route Choice

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Page ID 48085

David Levinson et al.
Associate Professor (Engineering) via Wikipedia

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Route assignment , route choice , or traffic assignment concerns the selection of routes (alternative called paths) between origins and destinations in transportation networks. It is the fourth step in the conventional transportation forecasting model, following Trip Generation, Destination Choice, and Mode Choice. The zonal interchange analysis of trip distribution provides origin-destination trip tables. Mode choice analysis tells which travelers will use which mode. To determine facility needs and costs and benefits, we need to know the number of travelers on each route and link of the network (a route is simply a chain of links between an origin and destination). We need to undertake traffic (or trip) assignment. Suppose there is a network of highways and transit systems and a proposed addition. We first want to know the present pattern of travel times and flows and then what would happen if the addition were made.

Link Performance Function

The cost that a driver imposes on others is called the marginal cost. However, when making decisions, a driver only faces his own cost (the average cost) and ignores any costs imposed on others (the marginal cost).

\[AverageCost=\dfrac{S_T}{Q}\]
\[MarginalCost=\dfrac{\delta S_T}{\delta Q}\]

where $S_T$ is the total cost, and $Q$ is the flow.

BPR Link Performance Function

Suppose we are considering a highway network. For each link there is a function stating the relationship between resistance and volume of traffic. The Bureau of Public Roads (BPR) developed a link (arc) congestion (or volume-delay, or link performance) function, which we will term S a (Q a )

\[S_a(Q_a)=t_a(1+0.15\dfrac ({Q_a}{c_a})^4)\]

t a = free-flow travel time on link a per unit of time

Q a = flow (or volume) of traffic on link a per unit of time (somewhat more accurately: flow attempting to use link a )

c a = capacity of link a per unit of time

S a (Q a ) is the average travel time for a vehicle on link a

There are other congestion functions. The CATS has long used a function different from that used by the BPR, but there seems to be little difference between results when the CATS and BPR functions are compared.

Can Flow Exceed Capacity?

On a link, the capacity is thought of as “outflow.” Demand is inflow.

If inflow > outflow for a period of time, there is queueing (and delay).

For Example, for a 1 hour period, if 2100 cars arrive and 2000 depart, 100 are still there. The link performance function tries to represent that phenomenon in a simple way.

Wardrop's Principles of Equilibrium

User Equilibrium

Each user acts to minimize his/her own cost, subject to every other user doing the same. Travel times are equal on all used routes and lower than on any unused route.

System optimal

Each user acts to minimize the total travel time on the system.

Price of Anarchy

The reason we have congestion is that people are selfish. The cost of that selfishness (when people behave according to their own interest rather than society's) is the price of anarchy .

The ratio of system-wide travel time under User Equilibrium and System Optimal conditions.

For a two-link network with linear link performance functions (latency functions), Price of Anarchy is < 4/3.

Is this too much? Should something be done, or is 33% waste acceptable? [The loss may be larger/smaller in other cases, under different assumptions, etc.]

Conservation of Flow

An important factor in road assignment is the conservation of flow. This means that the number of vehicles entering the intersection (link segment) equals the number of vehicles exiting the intersection for a given period of time (except for sources and sinks).

Similarly, the number of vehicles entering the back of the link equals the number exiting the front (over a long period of time).

Auto assignment

Long-standing techniques.

The above examples are adequate for a problem of two links, however real networks are much more complicated. The problem of estimating how many users are on each route is long standing. Planners started looking hard at it as freeways and expressways (motorways) began to be developed. The freeway offered a superior level of service over the local street system and diverted traffic from the local system. At first, diversion was the technique. Ratios of travel time were used, tempered by considerations of costs, comfort, and level of service.

The Chicago Area Transportation Study (CATS) researchers developed diversion curves for freeways versus local streets. There was much work in California also, for California had early experiences with freeway planning. In addition to work of a diversion sort, the CATS attacked some technical problems that arise when one works with complex networks. One result was the Moore algorithm for finding shortest paths on networks.

The issue the diversion approach didn’t handle was the feedback from the quantity of traffic on links and routes. If a lot of vehicles try to use a facility, the facility becomes congested and travel time increases. Absent some way to consider feedback, early planning studies (actually, most in the period 1960-1975) ignored feedback. They used the Moore algorithm to determine shortest paths and assigned all traffic to shortest paths. That’s called all or nothing assignment because either all of the traffic from i to j moves along a route or it does not.

The all-or-nothing or shortest path assignment is not trivial from a technical-computational view. Each traffic zone is connected to n - 1 zones, so there are numerous paths to be considered. In addition, we are ultimately interested in traffic on links. A link may be a part of several paths, and traffic along paths has to be summed link by link.

An argument can be made favoring the all-or-nothing approach. It goes this way: The planning study is to support investments so that a good level of service is available on all links. Using the travel times associated with the planned level of service, calculations indicate how traffic will flow once improvements are in place. Knowing the quantities of traffic on links, the capacity to be supplied to meet the desired level of service can be calculated.

Heuristic procedures

To take account of the affect of traffic loading on travel times and traffic equilibria, several heuristic calculation procedures were developed. One heuristic proceeds incrementally. The traffic to be assigned is divided into parts (usually 4). Assign the first part of the traffic. Compute new travel times and assign the next part of the traffic. The last step is repeated until all the traffic is assigned. The CATS used a variation on this; it assigned row by row in the O-D table.

The heuristic included in the FHWA collection of computer programs proceeds another way.

Step 0: Start by loading all traffic using an all or nothing procedure.
Step 1: Compute the resulting travel times and reassign traffic.
Step 2: Now, begin to reassign using weights. Compute the weighted travel times in the previous two loadings and use those for the next assignment. The latest iteration gets a weight of 0.25 and the previous gets a weight of 0.75.
Step 3. Continue.

These procedures seem to work “pretty well,” but they are not exact.

Frank-Wolfe algorithm

Dafermos (1968) applied the Frank-Wolfe algorithm (1956, Florian 1976), which can be used to deal with the traffic equilibrium problem.

Equilibrium Assignment

To assign traffic to paths and links we have to have rules, and there are the well-known Wardrop equilibrium (1952) conditions. The essence of these is that travelers will strive to find the shortest (least resistance) path from origin to destination, and network equilibrium occurs when no traveler can decrease travel effort by shifting to a new path. These are termed user optimal conditions, for no user will gain from changing travel paths once the system is in equilibrium.

The user optimum equilibrium can be found by solving the following nonlinear programming problem

\[min \displaystyle \sum_{a} \displaystyle\int\limits_{0}^{v_a}S_a(Q_a)\, dx\]

subject to:

\[Q_a=\displaystyle\sum_{i}\displaystyle\sum_{j}\displaystyle\sum_{r}\alpha_{ij}^{ar}Q_{ij}^r\]

\[sum_{r}Q_{ij}^r=Q_{ij}\]

\[Q_a\ge 0, Q_{ij}^r\ge 0\]

where $Q_{ij}^r$ is the number of vehicles on path r from origin i to destination j . So constraint (2) says that all travel must take place: i = 1 ... n; j = 1 ... n

$\alpha_{ij}^{ar}$= 1 if link a is on path r from i to j ; zero otherwise.

So constraint (1) sums traffic on each link. There is a constraint for each link on the network. Constraint (3) assures no negative traffic.

Transit assignment

There are also methods that have been developed to assign passengers to transit vehicles. In an effort to increase the accuracy of transit assignment estimates, a number of assumptions are generally made. Examples of these include the following:

All transit trips are run on a set and predefined schedule that is known or readily available to the users.
There is a fixed capacity associated with the transit service (car/trolley/bus capacity).

Solve for the flows on Links a and b in the Simple Network of two parallel links just shown if the link performance function on link a :

$S_a=5+2*Q_a$

and the function on link b :

$S_b=10+Q_b$

where total flow between the origin and destination is 1000 trips.

Time (Cost) is equal on all used routes so $S_a=S_b$

And we have Conservation of flow so, $Q_a+Q_b=Q_o=Q_d=1000$

$5+2*(1000-Q_b)=10+Q_b$

$1995=3Q_b$

$Q_b=665;Q_a=335$

An example from Eash, Janson, and Boyce (1979) will illustrate the solution to the nonlinear program problem. There are two links from node 1 to node 2, and there is a resistance function for each link (see Figure 1). Areas under the curves in Figure 2 correspond to the integration from 0 to a in equation 1, they sum to 220,674. Note that the function for link b is plotted in the reverse direction.

$S_a=15(1+0.15(\dfrac{Q_a}{1000})^4)$

$S_b=20(1+0.15(\dfrac{Q_a}{3000})^4)$

$Q_a+Q_b=8000$

Show graphically the equilibrium result.

At equilibrium there are 2,152 vehicles on link a and 5,847 on link b . Travel time is the same on each route: about 63.

Figure 3 illustrates an allocation of vehicles that is not consistent with the equilibrium solution. The curves are unchanged, but with the new allocation of vehicles to routes the shaded area has to be included in the solution, so the Figure 3 solution is larger than the solution in Figure 2 by the area of the shaded area.

Assume the traffic flow from Milwaukee to Chicago, is 15000 vehicles per hour. The flow is divided between two parallel facilities, a freeway and an arterial. Flow on the freeway is denoted $Q_f$, and flow on the two-lane arterial is denoted $Q_a$.

The travel time (in minutes) on the freeway ($C_f$) is given by:

$C_f=10+Q_f/1500$

$C_a=15+Q_a/1000$

Apply Wardrop's User Equilibrium Principle, and determine the flow and travel time on both routes.

The travel times are set equal to one another

$C_f=C_a$

$10+Q_f/1500=15+Q_a/1000$

The total traffic flow is equal to 15000

$Q_f+Q_a=15000$

$Q_a=15000-Q_f$

$10+Q_f/1500=15+(15000-Q_f)/1000$

Solve for $Q_f$

$Q_f=60000/5=12000$

$Q_a=15000-Q_f=3000$

Thought Questions

How can we get drivers to consider their marginal cost?
Alternatively: How can we get drivers to behave in a “System Optimal” way?

Sample Problems

Given a flow of six (6) units from origin “o” to destination “r”. Flow on each route ab is designated with Qab in the Time Function. Apply Wardrop's Network Equilibrium Principle (Users Equalize Travel Times on all used routes)

A. What is the flow and travel time on each link? (complete the table below) for Network A

Link Attributes

B. What is the system optimal assignment?

C. What is the Price of Anarchy?

What is the flow and travel time on each link? Complete the table below for Network A:

These four links are really 2 links O-P-R and O-Q-R, because by conservation of flow Qop = Qpr and Qoq = Qqr.

By Wardrop's Equilibrium Principle, the travel time (cost) on each used route must be equal. Therefore $C_{opr}=C_{oqr}$

OR $25+6*Q_{opr}=20+7*Q_{oqr}$

$5+6*Q_{opr}=7*Q_{oqr}$

$Q_{oqr}=5/7+6*Q_{opr}/7$

By the conservation of flow principle

$Q_{oqr}+Q_{opr}=6$

$Q_{opr}=6-Q_{oqr}$

By substitution

\Q_{oqr}=5/7+6/7(6-Q_{oqr})=41/7-6*Q_{oqr}/7\)

$13*Q_{oqr}=41$

$Q_{oqr}=41/13=3.15$

$Q_{opr}=2.84$

$42.01=25+6(2.84)$

$42.05=20+7(3.15)$

Check (within rounding error)

or expanding back to the original table:

User Equilibrium: Total Delay = 42.01 * 6 = 252.06

What is the system optimal assignment?

Conservation of Flow:

$Q_{opr}+Q_{oqr}=6$

$TotalDelay=Q_{opr}(25+6*Q_{oqr})+Q_{oqr}(20+7*Q_{oqr})$

$25Q_{opr}+6Q_{opr}^2+(6_Q_{opr})(20+7(6-Q_{opr}))$

$25Q_{opr}+6Q_{opr}^2+(6_Q_{opr})(62-7Q_{opr}))$

$25Q_{opr}+6Q_{opr}^2+372-62Q_{opr}-42Q_{opr}+7Q_{opr}^2$

$13Q_{opr}^2-79Q_{opr}+372$

Analytic Solution requires minimizing total delay

$\deltaC/\deltaQ=26Q_{opr}-79=0$

$Q_{opr}=79/26-3.04$

$Q_{oqr}=6-Q_{opr}=2.96$

And we can compute the SO travel times on each path

$C_{opr,SO}=25+6*3.04=43.24$

$C_{opr,SO}=20+7*2.96=40.72$

Note that unlike the UE solution, $C_{opr,SO}\g C_{oqr,SO}$

Total Delay = 3.04(25+ 6*3.04) + 2.96(20+7*2.96) = 131.45+120.53= 251.98

Note: one could also use software such as a "Solver" algorithm to find this solution.

What is the Price of Anarchy?

User Equilibrium: Total Delay =252.06 System Optimal: Total Delay = 251.98

Price of Anarchy = 252.06/251.98 = 1.0003 < 4/3

The Marcytown - Rivertown corridor was served by 3 bridges, according to the attached map. The bridge over the River on the route directly connecting Marcytown and Citytown collapsed, leaving two alternatives, via Donkeytown and a direct. Assume the travel time functions Cij in minutes, Qij in vehicles/hour, on the five links routes are as given.

Marcytown - Rivertown Cmr = 5 + Qmr/1000

Marcytown - Citytown (prior to collapse) Cmc = 5 + Qmc/1000

Marcytown - Citytown (after collapse) Cmr = ∞

Citytown - Rivertown Ccr = 1 + Qcr/500

Marcytown - Donkeytown Cmd = 7 + Qmd/500

Donkeytown - Rivertown Cdr = 9 + Qdr/1000

Also assume there are 10000 vehicles per hour that want to make the trip. If travelers behave according to Wardrops user equilibrium principle.

A) Prior to the collapse, how many vehicles used each route?

Route A (Marcytown-Rivertown) = Ca = 5 + Qa/1000

Route B (Marcytown-Citytown-Rivertown) = Cb = 5 + Qb/1000 + 1 + Qb/500 = 6 + 3Qb/1000

Route C (Marcytown-Donkeytown-Rivertown)= Cc = 7 + Qc/500 + 9 + Qc/1000 = 16 + 3Qc/1000

At equilibrium the travel time on all three used routes will be the same: Ca = Cb = Cc

We also know that Qa + Qb + Qc = 10000

Solving the above set of equations will provide the following results:

Qa = 8467;Qb = 2267;Qc = −867

We know that flow cannot be negative. By looking at the travel time equations we can see a pattern.

Even with a flow of 0 vehicles the travel time on route C(16 minutes) is higher than A or B. This indicates that vehicles will choose route A or B and we can ignore Route C.

Solving the following equations:

Route A (Marcytown-Rivertown) = Ca = 5 + Qa /1000

Route B (Marcytown-Citytown-Rivertown) = Cb = 6 + 3Qb /1000

Qa + Qb = 10000

We can the following values:

Qa = 7750; Qb = 2250; Qc = 0

B) After the collapse, how many vehicles used each route?

We now have only two routes, route A and C since Route B is no longer possible. We could solve the following equations:

Route C (Marcytown- Donkeytown-Rivertown) = Cc = 16 + 3Qc /1000

Qa+ Qc= 10000

But we know from above table that Route C is going to be more expensive in terms of travel time even with zero vehicles using that route. We can therefore assume that Route A is the only option and allocate all the 10,000 vehicles to Route A.

If we actually solve the problem using the above set of equations, you will get the following results:

Qa = 10250; Qc = -250

which again indicates that route C is not an option since flow cannot be negative.

C) After the collapse, public officials want to reduce inefficiencies in the system, how many vehicles would have to be shifted between routes? What is the “price of anarchy” in this case?

TotalDelayUE =(15)(10,000)=150,000

System Optimal

TotalDelaySO =(Qa)(5+Qa/1000)+(Qc)(16+3Qc/1000)

Using Qa + Qc = 10,000

TotalDelaySO =(Qa2)/250−71Qa+460000

Minimize total delay ∂((Qa2)/250 − 71Qa + 460000)/∂Qa = 0

Qa/125−7 → Qa = 8875 Qc = 1125 Ca = 13,875 Cc = 19,375

TotalDelaySO =144938

Price of Anarchy = 150,000/144,938 = 1.035

$C_T$ - total cost
$C_k$ - travel cost on link $k$
$Q_k$ - flow (volume) on link $k$

Abbreviations

VDF - Volume Delay Function
LPF - Link Performance Function
BPR - Bureau of Public Roads
UE - User Equilbrium
SO - System Optimal
DTA - Dynamic Traffic Assignment
DUE - Deterministic User Equilibrium
SUE - Stochastic User Equilibrium
AC - Average Cost
MC - Marginal Cost
Route assignment, route choice, auto assignment
Volume-delay function, link performance function
User equilibrium
Conservation of flow
Average cost
Marginal cost

External Exercises

Use the ADAM software at the STREET website and try Assignment #3 to learn how changes in network characteristics impact route choice.

Additional Questions

1. If trip distribution depends on travel times, and travel times depend on the trip table (resulting from trip distribution) that is assigned to the road network, how do we solve this problem (conceptually)?

2. Do drivers behave in a system optimal or a user optimal way? How can you get them to move from one to the other.

3. Identify a mechanism that can ensure the system optimal outcome is achieved in route assignment, rather than the user equilibrium. Why would we want such an outcome? What are the drawbacks to the mechanism you identified?

4. Assume the flow from Dakotopolis to New Fargo, is 5300 vehicles per hour. The flow is divided between two parallel facilities, a freeway and an arterial. Flow on the freeway is denoted $Q_f$, and flow on the two-lane arterial is denoted $Q_r$. The travel time on the freeway $C_f$ is given by:

$C_f=5+Q_f/1000$

The travel time on the arterial (Cr) is given by

$C_r=7+Q_r/500$

(a) Apply Wardrop's User Equilibrium Principle, and determine the flow and travel time on both routes from Dakotopolis to New Fargo.

(b) Solve for the System Optimal Solution and determine the flow and travel time on both routes.

5. Given a flow of 10,000 vehicles from origin to destination traveling on three parallel routes. Flow on each route A, B, or C is designated with $Q_a$, $Q_b$, $Q_c$ in the Time Function Respectively. Apply Wardrop's Network Equilibrium Principle (Users Equalize Travel Times on all used routes), and determine the flow on each route.

$T_A=500+20Q_A$

$T_B=1000+10Q_B$

$T_C=2000+30Q_C$

How does average cost differ from marginal cost?
How do System Optimal and User Equilibrium travel time differ?
Why do we want people to behave in an SO way?
How can you get people to behave in an SO way?
Who was John Glen Wardrop?
What are Wardrop’s Two Principles?
What does conservation of flow require in route assignment?
Can Variable Message Signs be used to encourage System Optimal behavior?
What is freeflow travel time?
If a problem has more than two routes, where does the extra equation come from?
How can you determine if a route is unused?
What is the difference between capacity and flow
Draw a typical volume-delay function for a deterministic, static user equilibrium assignment.
Can Q be negative?
What is route assignment?
Is it important that the output travel times from route choice be consistent with the input travel times for destination choice and mode choice? Why?

The Geography of Transport Systems

The spatial organization of transportation and mobility

Traffic Assignment Problem

Traffic assignment problems usually consider two dimensions.

Generation and attraction . A place of origin generates movements that are bound (attracted) to a place of destination. The relationship between traffic generation and attraction is commonly labeled as spatial interaction. The above example considers one origin/generation and destination/attraction, but the majority of traffic assignment problems consider several origins and destinations.
Path selection . Traffic assignment considers which paths are to be selected and the amount of traffic using these paths (if more than one unit). For simple problems, a single path will be selected, while for complex problems, several paths could be used. Factors behind the choice of traffic assignment may include cost, time, or the number of connections.

13 Last Step of Four Step Modeling (Trip Assignment Models)

Chapter overview.

Chapter 13 presents trip assignment, the last step of the Four-Step travel demand Model (FSM). This step determines which paths travelers choose for moving between each pair of zones. Additionally, this step can yield numerous results, such as traffic volumes in different transportation corridors, the patterns of vehicular movements, total vehicle miles traveled (VMT) and vehicle travel time (VTT) in the network, and zone-to-zone travel costs. Identification of the heavily congested links is crucial for transportation planning and engineering practitioners. This chapter begins with some fundamental concepts, such as the link cost functions. Next, it presents some common and useful trip assignment methods with relevant examples. The methods covered in this chapter include all-or-nothing (AON), user equilibrium (UE), system optimum (SO), feedback loop between distribution and assignment (LDA), incremental increase assignment, capacity restrained assignment, and stochastic user equilibrium assignment.

Learning Objectives

Describe the reasons for performing trip assignment models in FSM and relate these models’ foundation through the cost-function concept.
Compare static and dynamic trip assignment models and infer the appropriateness of each model for different situations.
Explain Wardrop principles and relate them to traffic assignment algorithms.
Complete simple network traffic assignment models using static models such as the all-or-nothing and user equilibrium models.
Solve modal split analyses manually for small samples using the discrete choice modeling framework and multinominal logit models.

Introduction

In this chapter, we continue the discussion about FSM and elaborate on different methods of traffic assignment, the last step in the FSM model after trip generation, trip distribution, and modal split. The traffic assignment step, which is also called route assignment or route choice , simulates the choice of route selection from a set of alternatives between the origin and the destination zones (Levinson et al., 2014). The first three FSM steps determine the number of trips produced between each zone and the proportion completed by different transportation modes. The purpose of the final step is to determine the routes or links in the study area that are likely to be used. For example, when updating a Regional Transportation Plan (RTP), traffic assignment is helpful in determining how much shift or diversion in daily traffic happens with the introduction an additional transit line or extension a highway corridor (Levinson et al., 2014). The output from the last step can provide modelers with numerous valuable results. By analyzing the results, the planner can gain insight into the strengths and weaknesses of different transportation plans. The results of trip assignment analysis can be:

The traffic flows in the transportation system and the pattern of vehicular movements.
The volume of traffic on network links.
Travel costs between trip origins and destinations (O-D).
Aggregated network metrics such as total vehicle flow, vehicle miles traveled (VMT) , and vehicle travel time (VTT).
Zone-to-zone travel costs (travel time) for a given level of demand.
Modeled link flows highlighting congested corridors.
Analysis of turning movements for future intersection design.
Determining the Origin-Destination (O-D) pairs using a specific link or path.
Simulation of the individual choice for each pair of origins and destinations (Mathew & Rao, 2006).

Link Performance Function

Building a link performance function is one of the most important and fundamental concepts of the traffic assignment process. This function is usually used for estimating travel time, travel cost, and speed on the network based on the relationship between speed and travel flow. While this function can take different forms, such as linear, polynomial , exponential , and hyperbolic , one of the most common functions is the link performance function which represents generalized travel costs (United States Bureau of Public Roads, 1964). This equation estimates travel time on a free-flow road (travel with speed limit) adding a function that exponentially increases travel time as the road gets more congested. The road volume-to-capacity ratio can represent congestion (Meyer, 2016).

While transportation planners now recognize that intersection delays contribute to link delays, the following sections will focus on the traditional function. Equation (1) is the most common and general formula for the link performance function.

$t=t_o[1+\alpha\left(\frac{x}{k}\right)\beta]$

t and x are the travel time and vehicle flow;
t 0 is the link free flow travel time;
k is the link capacity;
α and β are parameters for specific type of links and calibrated using the field data. In the absence of any field data, it is usually assumed = 0.15, and β= 4.0.

α and β are the coefficients for this formula and can take different values (model parameters). However, most studies and planning practices use the same value for them. These values can be locally calibrated for the most efficient results.

Figure 13.1 demonstrates capacity as the relationship between flow and travel time. In this plot, the travel time remains constant as vehicle volumes increase until the turning point , which indicates that the link’s volume is approaching its capacity.

This figure shows the exponential relationship between travel time and flow of traffic,

The following example shows how the link performance function helps us to determine the travel time according to flow and capacity.

Performance Function Example

Assume the traffic volume on a path between zone i and j was 525. The travel time recorded on this path is 15 minutes. If the capacity of this path would be 550, then calculate the new travel time for future iteration of the model.

Based on the link performance function, we have:

Now we have to plug in the numbers into the formula to determine the new travel time:

$t=15[1+\0.15\left(\frac{525}{550}\right)\4]=16.86$

Traffic Assignment Models

Typically, traffic assignment is calculated for private cars and transit systems independently. Recall that the impedance function differs for drivers and riders, and thus simulating utility maximization behavior should be approached differently. For public transit assignment, variables such as fare, stop or transfer, waiting time, and trip times define the utility (equilibrium) (Sheffi, 1985). For private car assignment, however, in some cases, the two networks are related when public buses share highways with cars, and congestion can also affect the performance.

Typically, private car traffic assignment models the path choice of trip makers using:

algorithms like all-or-nothing
user equilibrium
system optimum assignment

Of the assignment models listed above, user equilibrium is widely adopted in the U.S. (Meyer, 2016). User equilibrium relies on the premise that travelers aim to minimize their travel costs. This algorithm achieves equilibrium when no user can decrease their travel time or cost by altering their travel path.

incremental
capacity-restrained
iterative feedback loop
Stochastic user equilibrium assignment
Dynamic traffic assignment

All-or-nothing Model

Through the all-or-nothing (AON) assignment, it is assumed that the impedance of a road or path between each origin and destination is constant and equal to the free-flow level of service. This means that the traffic time is not affected by the traffic flow on the path. The only logic behind this model is that each traveler uses the shortest path from his or her origin to the destination, and no vehicle is assigned to other paths (Hui, 2014). This method is called the all-or-nothing assignment model and is the simplest one among all assignment models. This method is also called the 0-1 assignment model, and its advantage is its simple procedure and calculation. The assumptions of this method are:

Congestion does not affect travel time or cost, meaning that no matter how much traffic is loaded on the route, congestion does not take place.
Since the method assigns one route to any travel between each pair of OD, all travelers traveling from a particular zone to another particular zone choose the same route (Hui, 2014).

To run the AON model, the following process can be followed:

Step 0: Initialization. Use free flow travel costs Ca=Ca(0) , for each link a on the empty network. Ɐ
Step 1: Path finding. Find the shortest path P for each zonal pair.
Step 2: Path flows assigning. Assign both passenger trips (hppod) and freight trips (hfpod) in PCEs from zonal o to d to path P.
Step 3: Link flows computing. Sum the flows on all paths going through a link as total flows of this link.

Example 2 illustrates the above-mentioned process for the AON model

All-or-nothing Example

Table 13.1 shows a trip distribution matrix with 4 zones. Using the travel costs between each pair of them shown in Figure 13.2, assign the traffic to the network. Load the vehicle trips from the trip distribution table shown below using the AON technique. After assigning the traffic, illustrate the links and the traffic volume on each on them.

Table 13.1 Trip Distribution Results.

This photo shows the hypothetical network and travel time between zones: 1-2: 5 mins 1-4: 10 min 4-2: 4 mins 3-2: 4 mins 3-4: 9 mins

To solve this problem, we need to find the shortest path among all alternatives for each pair of zones. The result of this procedure would be 10 routes in total, each of which bears a specific amount of travels. For instance, the shortest path between zone 1 and 2 is the straight line with 5 min travel time. All other routes like 1 to 4 to 2 or 1 to 4 to 3 to 2 would be empty from travelers going from zone 1 to zone 2. The results are shown in Table 13.2.

As you can see, some of the routes remained unused. This is because in all-or-nothing if a route has longer travel time or higher costs, then it is assumed it would not be used at all.

User Equilibrium

The next method for traffic assignment is called User Equilibrium (UE). The rule or algorithm is adapted from the well-known Wardrop equilibrium (1952) conditions (Correa & Stier-Moses, 2011). In this algorithm, it is assumed that travelers will always choose the shortest path, and equilibrium conditions are realized when no traveler is able to decrease their travel impedance by changing paths (Levinson et al., 2014).

As we discussed, the UE method is based on the first principle of Wardrop : “for each origin- destination (OD) pair, with UE, the travel time on all used paths is equal and less than or equally to the travel time that would be experienced by a single vehicle on any unused path”( Jeihani Koohbanani, 2004, p. 10). The mathematical format of this principle is shown in equation (3):

For a given OD pair, the UE condition can be expressed in equation (3):

$fk\left(ck-u\right)=0:\forall k$

This model assumes that all paths have equal travel time. Additionally, the model includes the following general assumptions:

The users possess all the knowledge needed about different paths.
The users have perfect knowledge of the path cost.
Travel time in a route is subject to change only by the cost flow function of that route.
Travel times increases as we load travel into the network (Mathew & Rao, 2006).

Hence, the UE assignment comes to an optimization problem that can be formulated using equation (4):

$Minimize\ Z=\sum_{a}\int_{0}^{Xa}ta\left(xa\right)dx$

k is the path x a equilibrium flow in link a t a travel time on link a f k rs flow on path connecting OD pairs q rs trip rate between and δ a, k rs is constraint function defined as 1 if link a belongs to path k and 0 otherwise

Example 3 shows how the UE method can be applied for the traffic assignment step. This example is a very simple network consisting of two zones with two possible paths between them.

UE Example

This photo shows the hypothetical network with two possible paths between two zones 1: 5=4x_1 2: 3+2x_2 (to power of two)

In this example, t 1 and t 2 are travel times measured by min on each route, and x 1 and x 2 are traffic flows on each route measured by (Veh/Hour).

Using the UE method, assign 4,500 Veh/Hour to the network and calculate travel time on each route after assignment, traffic volume, and system total travel time.

According to the information provided, total flow (X 1 +X 2 ) is equal to 4,500 (4.5).

First, we need to check, with all traffic assigned to one route, whether that route is still the shortest path. Thus we have:

T 1 (4.5)=23min

T 2 (0)=3min

if all traffic is assigned to route 2:

T 1 (0)=3min

T 2 (4.5)=43.5 min

Step 2: Wardrope equilibrium rule: t 1 =t 2 5+4x 1 =3+ 2x 2 2 and we have x 1 =4.5-x 2

Now the equilibrium equation can be written as: 6 + 4(4.5 − x2)=4+ x222

x 1 = 4.5 − x 2 = 1.58

Now the updated average travel times are: t 1 =5+4(1.58)=11.3min and T 2 =3+2(2.92)2=20.05min

Now the total system travel time is:

Z(x)=X 1 T 1 (X 1 )+X 2 T 2 (X 2 )=2920 veh/hr(11.32)+1585 veh/hr(20.05)=33054+31779=64833 min

System Optimum Assignment

One traffic assignment model is similar to the previous one and is called system optimum (SO). The second principle of the Wardrop defines the model’s logic. Based on this principle, drivers’ rationale for choosing a path is to minimize total system costs with one another to minimize total system travel time (Mathew & Rao, 2006). Using the SO traffic assignment, one can solve various problems, such as optimizing the departure time for a single commuting route, minimizing the total travel time from multiple origins to a single destination, or minimizing travel time in stochastic time-dependent O-D flows from several origins to a single destination ( Jeihani & Koohbanani, 2004).

One other traffic assignment model similar to the previous one is called system optimum (SO) in which the second principle of the Wardrop defines the logic of the model. Based on this principle, drivers’ rationale for choosing a path is to minimize total system costs with one another in order to minimize total system travel time (Mathew & Rao, 2006). Using the SO traffic assignment, problems like optimizing departure time for a single commuting route, minimizing total travels from multiple origins to one destination, or minimizing travel time in stochastic time-dependent OD flows from several origins to a single destination can be solved (Jeihani Koohbanani, 2004).

The basic mathematical formula for this model that satisfies the principle of the model is shown in equation (5):

$minimize\ Z=\sum_{a}{xata\left(xa\right)}$

In example 4, we will use the same network we described in the UE example in order to compare the results for the two models.

In that simple two-zone network, we had:

T 1 =5+4X 1 T2=3+2X 2 2

Now, based on the principle of the model we have:

Z(x)=x 1 t 1 (x 1 )+x 2 t 2 (x 2 )

Z(x)=x 1 (5+4x 1 )+x 2 (3+2x 2 2 )

Z(x)=5x 1 +4x 1 2 +3x 2 +2x 2 3

From the flow conservation. we have: x 1 +x 2 =4.5 x 1 =4.5-x 2

Z(x)=5(4.5-x 2 )+4(4.5-x 2 )2+4x 2 +x 2 3

Z(x)=x 3 2 +4x 2 2 -27x 2 +103.5

In order to minimize the above equation, we have to take derivatives and equate it to zero. After doing the calculations, we have:

Based on our finding, the system travel time would be:

T 1 =5+4*1.94=12.76min T 2 =3+ 2(2.56)2=10.52 min

And the total travel time of the system would be:

Z(x)=X 1 T 1 (X 1 )+X 2 T 2 (X 2 )=1940 veh/hr(12.76)+2560 veh/hr(10.52)=24754+26931=51685 min

Incremental Increase model

Incremental increase is based on the logic of the AON model and models a process designed with multiple steps. In each step or level, a fraction of the total traffic volume is assigned, and travel time is calculated based on the allocated traffic volume. Through this incremental addition of traffic, the travel time of each route in step (n) is the updated travel time from the previous step (n-1) (Rojo, 2020).

The steps for the incremental increase traffic assignment model are:

Finding the shortest path between each pair of O-Ds (Origin Destination).
Assigning a portion of the trips according to the matrix (usually 40, 30, 20 and 10 percent to the shortest path).
Updating the travel time after each iteration (each incremental increase).
Continuing until all trips are assigned.
Summing the results.

The example below illustrates the implementation process of this method.

A hypothetical network accommodates two zones with three possible links between them. Perform an incremental increase traffic assignment model for assigning 200 trips between the two zones with increments of: 30%, 30%, 20%, 20%. (The capacity is 50 trips.)

Incremental Increase Example

This photo shows the hypothetical network with two possible paths between two zones 1: 6 mins 2: 7 mins 3: 12 mins

Step 1 (first iteration): Using the method of AON, we now assign the flow to the network using the function below:

$t=to[1+\alpha\left(\frac{x}{k}\right)\beta]$

Since the first route has the shortest travel time, the first 30% of the trips will be assigned to route 1. The updated travel time for this path would be:

$t=6\left[1+0.15\left(\frac{60}{50}\right)4\right]=7.86$

And the remaining route will be empty, and thus their travel times are unchanged.

Step 2 (second iteration): Now, we can see that the second route has the shortest travel time, with 30% of the trips being assigned to this route, and the new travel time would be:

$t=7\left[1+0.15\left(\frac{60}{50}\right)4\right]=9.17$

Step 3 (third iteration): In the third step, the 20% of the remaining trips will be assigned to the shortest path, which in this case is the first route again. The updated travel time for this route is:

$t=7.86\left[1+0.15\left(\frac{40}{50}\right)4\right]=8.34$

Step 4 (fourth iteration): In the last iteration, the remaining 10% would be assigned to first route, and the time is:

$t=8.34\left[1+0.15\left(\frac{40}{50}\right)4\right]=8.85$

Finally, we can see that route 1 has a total of 140 trips with a 8.85 travel time, the second route has a total of 60 trips with a 9.17 travel time, and the third route was never used.

Capacity Restraint Assignment

So far, all the presented algorithms or rules have considered the model’s link capacity. The flow is assigned to a link based on travel time as the only factor. In this model, after each iteration, the total number of trips is compared with the capacity to observe how much increase in travel time was realized by the added volume. In this model, the iteration stops if the added volume in step (n) does not change the travel time updated in step (n-1). With the incorporation of such a constraint, the cost or performance function would be different from the cost functions discussed in previous algorithms (Mathew & Rao, 2006). Figure 13.6 visualizes the relationship between flow and travel time with a capacity constraint.

This figure shows the exponential relationship between travel time and flow of traffic with capacity line.

Based on this capacity constraint specific to each link, the α, β can be readjusted for different links such as highways, freeways, and other roads.

Feedback Loop Model (Combined Traffic Assignment and Trip Distribution)

The feedback loop model defines an interaction between the trip distribution route choice step with several iterations. The model allows travelers to change their destination if a route is congested. For example, the feedback loop models that the traveler has a choice of similar destinations, such as shopping malls, in the area. In other words, in a real-world situation, travelers usually simultaneously decide about their travel characteristics (Qasim, 2012).

The chart below shows how the combination of these two modes can take place:

This photo shows the feedback loop in FSM.

Equation (6), shown below for this model, ensures convergence at the end of the model is:

$Min\funcapply\sum_a\hairsp\int_0^{p_a+f_a}\hairsp C_a(x)dx+\frac{1}{\zeta}\sum_o\hairsp\sum_d\hairsp T^{od}\left(\ln\funcapply T^{od}-K\right)$

where C a (t) is the same as previous

P a , is total personal trip flows on link a,

f a ; is total freight trip flows on link a,

T od is the total flow from node o to node d,

p od is personal trip from node o to node d,

F od is freight trip from node o to node d,

ζ is a parameter estimated from empirical data,

K is a parameter depending on the type of gravity model used to calculate T od , Evans (1976) proved that K’ equals to 1 for distribution using doubly constrained gravity model and it equals to 1 plus attractiveness for distribution using singly constrained model. Florian et al. (1975) ignored K for distribution using a doubly constrained gravity model because it is a constant.

Stochastic User Equilibrium Traffic Assignment

Stochastic user equilibrium traffic assignment is a sophisticated and more realistic model in which the level of uncertainty regarding which link should be used based on a measurement of utility function is introduced. This model performs a discrete choice analysis through a logistic model. Based on the first Wardrop principle, this model assumes that all drivers perceive the costs of traveling in each link identically and choose the route with minimum cost. In stochastic UE, however, the model allows different individuals to have different perceptions about the costs, and thus, they may choose non-minimum cost routes (Mathew & Rao, 2006). In this model, flow is assigned to all links from the beginning, unlike previous models, which is closer to reality. The probability of using each path is calculated with the following logit formula shown in equation (7):

$Pi=\frac{e^{ui}}{\sum_{i=1}^{k}e^{ui}}$

P i is the probability of using path i

U i is the utility function for path i

In the following, an example of a simple network is presented.

Stochastic User Equilibrium Example

There is a flow of 200 trips between two points and their possible path, each of which has a travel time specified in Figure 13.7.

This photo shows the hypothetical network with two possible paths between two zones 1: 21 mins 2: 23 mins 3: 26 mins

Using the mentioned logit formula for these paths, we have:

$P1=\frac{e^{-21i}}{e^{-21i}+e^{-23}+e^{-26i}}=0.875$

Based on the calculated probabilities, the distribution of the traffic flow would be:

Q 1 =175 trips

Q 2 =24 trips

Q 3 =1 trips

Dynamic Traffic Assignment

Recall the first Wardrop principle, in which travelers are believed to choose their routes with the minimum cost. Dynamic traffic assignment is based on the same rule, but the difference is that delays result from congestion. In this way, not only travelers’ route choice affects the network’s level of service, but also the network’s level of service affects travelers’ choice. However, it is not theoretically proven that an equilibrium would result under such conditions (Mathew & Rao, 2006).

Today, various algorithms are developed to solve traffic assignment problems. In any urban transportation system, travelers’ route choice and different links’ level of service have a dynamic feedback loop and affect each other simultaneously. However, a lot of these rules are not present in the models presented here. In real world cases, there can be more than thousands of nodes and links in the network, and therefore more sensitivity to dynamic changes is required for a realistic traffic assignment (Meyer, 2016). Also, the travel demand model applies a linear sequence of the four steps, which is unlike reality. Additionally, travelers may have only a limited knowledge of all possible paths, modes, and opportunities and may not make rational decisions.

In this last chapter of landuse/transportation modeling book, we reviewed the basic concepts and principles of traffic assignment models as the last step in travel demand modeling. Modeling the route choice and other components of travel behavior and demand for transportation proven to be very challenging and can incorporate multiple factors. For instance, going from AON to incremental increase assignment, we factor in the capacity and volume (and resulting delays) relationship in the assignment to make more realistic models. Multiple-time-period assignments for multiple classes, separate specification of facilities like high-occupancy vehicle (HOV) and high-occupancy toll (HOT) lanes; and, independent transit assignment using congested highway travel times to estimate a bus ridership assignment, are some of the new extensions and variation of algorithms that take into account more realities within transportation network. A new prospect in traffic assignment models that adds several capabilities for such efforts is emergence of ITS such as data that can be collected from connected vehicles or autonomous vehicles. Using these data, perceived utility or impedances of different modes or infrastructure from individuals perspective can be modeled accurately, leading to more accurate assignment models, which are crucial planning studies such as growth and land use control efforts, environmental studies, transportation economies, etc.

Route choice is the process of choosing a certain path for a trip from a very large choice sets.

Regional Transportation Plan is long term planning document for a region’s transportation usually updated every five years.

Vehicles (VMT) is the aggregate number of miles deriven from in an area in particular time of day.

Total vehicle travel time is the aggregate amount of time spent in transportation usually in minutes.

Link performance function is function used for estimating travel time, travel cost, and speed on the network based on the relationship between speed and travel flow.

Hyperbolic function is a function used for linear differential equations like calculating distances and angels in hyperbolic geometry.

Free-flow road is situation where vehicles can travel with the maximum allowed travel speed.

Algorithms like all-or-nothing an assignment model where we assume that the impedance of a road or path between each origin and destination is constant and is equal to free-flow level of service, meaning that the traffic time is not affected by the traffic flow on the path.

Capacity-restrained is a model which takes into account the capacity of a road compared to volume and updates travel times.

User equilibrium is a traffic assignment model where we assume that travelers will always choose the shortest path and equilibrium condition would be realized when no traveler is able to decrease their travel impedance by changing paths.

System optimum assignment is an assignment model based on the principle that drivers’ rationale for choosing a path is to minimize total system costs with one another in order to minimize total system travel time.

Static user-equilibrium assignment algorithm is an iterative traffic assignment process which assumes that travelers chooses the travel path with minimum travel time subject to constraints.
Iterative feedback loop is a model that iterates between trip distribution and route choice step based on the rational that if a path gets too congested, the travel may alter travel destination.

First principle of Wardrop is the assumption that for each origin-destination (OD) pair, with UE, the travel time on all used paths is equal and less than or equally to the travel time that would be experienced by a single vehicle on any unused path.

System optimum (SO) is a condition in trip assignment model where total travel time for the whole area is at a minimum.

Stochastic time-dependent OD is a modeling framework where generation and distribution of trips are randomly assigned to the area.

Incremental increase is AON-based model with multiple steps in each of which, a fraction of the total traffic volume is assigned, and travel time is calculated based on the allocated traffic volume.

Stochastic user equilibrium traffic assignment employs a probability distribution function that controls for uncertainties when drivers compare alternative routes and make decisions.

Dynamic traffic assignment is a model based on Wardrop first principle in which delays resulted from congestion is incorporated in the algorithm.

Key Takeaways

In this chapter, we covered:

Traffic assignment is the last step of FSM, and the link cost function is a fundamental concept for traffic assignment.
Different static and dynamic assignments and how to perform them using a simplistic transportation network.
Incorporating stochastic decision-making about route choice and how to solve assignment problems with regard to this feature.

Prep/quiz/assessments

Explain what the link performance function is in trip assignment models and how it is related to link capacity.
Name a few static and dynamic traffic assignment models and discuss how different their rules or algorithms are.
How does stochastic decision-making on route choice affect the transportation level of service, and how it is incorporated into traffic assignment problems?
Name one extension of the all-or-nothing assignment model and explain how this extension improves the model results.

Correa, J.R., & Stier-Moses, N.E.(2010).Wardrope equilibria. In J.J. Cochran( Ed.), Wiley encyclopedia of operations research and management science (pp.1–12). Hoboken, NJ: John Wiley & Sons. http://dii.uchile.cl/~jcorrea/papers/Chapters/CS2010.pdf

Hui, C. (2014). Application study of all-or-nothing assignment method for determination of logistic transport route in urban planning. Computer Modelling & New Technologies , 18 , 932–937. http://www.cmnt.lv/upload-files/ns_25crt_170vr.pdf

Jeihani Koohbanani, M. (2004). Enhancements to transportation analysis and simulation systems (Unpublished Doctoral dissertation, Virginia Tech). https://vtechworks.lib.vt.edu/bitstream/handle/10919/30092/dissertation-final.pdf?sequence=1&isAllowed=y

Levinson, D., Liu, H., Garrison, W., Hickman, M., Danczyk, A., Corbett, M., & Dixon, K. (2014). Fundamentals of transportation . Wikimedia. https://upload.wikimedia.org/wikipedia/commons/7/79/Fundamentals_of_Transportation.pdf

Mathew, T. V., & Rao, K. K. (2006). Introduction to transportation engineering. Civil engineering–Transportation engineering. IIT Bombay, NPTEL ONLINE, Http://Www. Cdeep. Iitb. Ac. in/Nptel/Civil% 20Engineering .

Meyer, M. D. (2016). Transportation planning handbook . John Wiley & Sons.

Qasim, G. (2015). Travel demand modeling: AL-Amarah city as a case study . [Unpublished Doctoral dissertation , the Engineering College University of Baghdad]

Rojo, M. (2020). Evaluation of traffic assignment models through simulation. Sustainability , 12 (14), 5536. https://doi.org/10.3390/su12145536

Sheffi, Y. (1985). Urban transportation networks: Equilibrium analysis with mathematical programming method . Prentice-Hall. http://web.mit.edu/sheffi/www/selectedMedia/sheffi_urban_trans_networks.pdf

US Bureau of Public Roads. (1964). Traffic assignment manual for application with a large, high speed computer . U.S. Department of Commerce, Bureau of Public Roads, Office of Planning, Urban Planning Division.

https://books.google.com/books/about/Traffic_Assignment_Manual_for_Applicatio.html?id=gkNZAAAAMAAJ

Wang, X., & Hofe, R. (2008). Research methods in urban and regional planning . Springer Science & Business Media.

Polynomial is distribution that involves the non-negative integer powers of a variable.

Hyperbolic function is a function that the uses the variable values as the power to the constant of e.

A point on the curve where the derivation of the function becomes either maximum or minimum.

all-or-nothing is an assignment model where we assume that the impedance of a road or path between each origin and destination is constant and is equal to free-flow level

Incremental model is a model that the predictions or estimates or fed into the model for forecasting incrementally to account for changes that may occur during each increment.

Iterative feedback loop is a model that iterates between trip distribution and route choice step based on the rational that if a path gets too congested, the travel may alter travel destination

Wardrop equilibrium is a state in traffic assignment model where are drivers are reluctant to change their path because the average travel time is at a minimum.

second principle of the Wardrop is a principle that assumes drivers’ rationale for choosing a path is to minimize total system costs with one another in order to minimize total system travel time

Stochastic time-dependent OD is a modeling framework where generation and distribution of trips are randomly assigned to the area

feedback loop model is type of dynamic traffic assignment model where an iteration between route choice and traffic assignment step is peformed, based on the assumption that if a particular route gets heavily congested, the travel may change the destination (like another shopping center).

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Dynamic traffic assignment: model classifications and recent advances in travel choice principles

Dynamic Traffic Assignment (DTA) has been studied for more than four decades and numerous reviews of this research area have been conducted. This review focuses on the travel choice principle and the classification of DTA models, and is supplementary to the existing reviews. The implications of the travel choice principle for the existence and uniqueness of DTA solutions are discussed, and the interrelation between the travel choice principle and the traffic flow component is explained using the nonlinear complementarity problem, the variational inequality problem, the mathematical programming problem, and the fixed point problem formulations. This paper also points out that all of the reviewed travel choice principles are extended from those used in static traffic assignment. There are also many classifications of DTA models, in which each classification addresses one aspect of DTA modeling. Finally, some future research directions are identified.

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Dynamic Traffic Assignment

Early Experiences

Current Practices

Research Needs

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Network Assignment

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(opens new window) is a hot topic in travel forecasting.

# Background

Traditional user equilibrium highway assignment models predict the effects of congestion and the routing changes of traffic as a result of that congestion. They neglect, however, many of the details of real-world traffic operations, such as queuing, shock waves, and signalization. Currently, it is common practice to feed the results of user equilibrium traffic assignments into dynamic network models as a mechanism for evaluating these policies. The simulation models themselves, however, do not predict the routing of traffic, and therefore are unable to account for re-routing owing to changes in congestion levels or policy, and can be inconsistent with the routes determined by the assignment. Dynamic network models overcome this dichotomy by combining a time-dependent shortest path algorithm with some type of simulation (often meso or macroscopic) of link travel times and delay. In doing so it allows added reality and consistency in the assignment step, as well as the ability to evaluate policies designed to improve traffic operations. These are some of the main benefits of dynamic network models .

DTA models can generally be classified by how they model link or intersection delay. Analytical DTA models treat it in the same manner as static equilibrium assignment models, with no explicit representation of signals. Link capacity functions, often similar or identical to those used in static assignment, are used to calculate link travel times. Analytical models have been widely used in research and for real-time control system applications. Simulation-based DTA models include explicit representation of traffic control devices. Such models require detailed signal parameters to include phasing, cycle length, and offsets for each signal in the network. Delay is calculated for each approach, with vehicles moving from one link to the next only if available downstream capacity is available. The underlying traffic model is often different, but at the network level such models behave in a similar fashion.

Demand is specified in the form of origin–destination matrices for short time intervals, typically 15 minutes each. Trips are typically randomly loaded onto the network during each time interval. As with traffic microsimulation models, adequate downstream capacity must be present to load the trips onto the network. The shortest paths through time and space are found for each origin–destination pair, and flows loaded to these paths. A generalized flowchart of the process is shown below.

As with static assignment models, the process shown above is iteratively solved until a stable solution is reached. The memory and computing requirements of DTA, however, are orders of magnitude larger than for static assignment, reducing the number of iterations and paths that can be kept in memory. Instead of a single time period, as with static assignment, DTA models must store data for each time interval as well. A three-hour static assignment would involve only one time interval. A DTA model of the same period, however, might require 12 intervals, each 15 minutes in duration. These are all in addition to the memory requirements imposed by the number of user classes and zones.

# Early Experiences

Research into DTA dates back several decades, but was largely limited to academics working on its formulation and theoretical aspects. DTA overcomes the limitations of static assignment models, although at the cost of increased data requirements and computational burden. Moreover, software platforms capable of solving the DTA problem for large urban systems and experience in their use are recent developments.

(opens new window) has been successfully applied to a large subarea of Calgary and to analyses of the Rue Notre-Dame in Montreal. Although user group presentations of both applications have been made, and reported very encouraging results, the work is currently unpublished and inaccessible except through contact with the developers.

(opens new window) . The network from the Atlanta Regional Commission (ARC) regional travel model formed the starting point for the DTA network. Intersections were coded, centroid connectors were re-defined, and network coding errors were corrected. A signal synthesizer derived locally optimal timing parameters for more than 2,200 signalized intersections in the network. Trip matrices from the ARC model were divided into 15-minute intervals for the specification of demand. Approximately 40 runs of the model were required to diagnose coding and software errors. Unfortunately, the execution time for the model was approximately one week per run. The resulting model eventually validated well to observed conditions; however, the length of time required to render it operational and the run time required prevented it from being used in studies as originally intended. Subsequent work by the developer has resulted in substantial reductions in run time, but this remains a significant issue that must be overcome before such models can be more widely used.

# Current Practices

# research needs.

A number of cities are currently testing DTA models, but are not far enough along in their work to share even preliminary results. At least a dozen such cases are known to be in varying stages of planning or execution, suggesting that the use of DTA models in planning applications is about to expand dramatically. However, in addition to the issue of long run times, a number of other issues must be addressed before such models are likely to be widely adopted:

Criteria for the validation of such models have not been widely accepted. The paucity of traffic counts in most urban areas, and especially at 15, 30, or 60 minute intervals, is a significant barrier to definitive assessment of these models.

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Highway Network:

[Solution Shown Below]

The all-or-nothing technique simply assumes that all of the traffic between a particular origin and destination will take the shortest path (with respect to time). For example, all of the 200 vehicles that travel between nodes 1 and 4 will travel via nodes 1-5-4. The tables shown below indicate the routes that were selected for loading as well as the total traffic volume for each link in the system after all of the links were loaded.

Traffic assignment

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R. J. Salter 2

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Previously the estimation of generated trip ends has been discussed together with the distribution of trips between the traffic zones. Modal split methods also have been reviewed in which the proportion of trips by the varying travel modes are determined. At this stage the number of trips and their origins and destinations are known but the actual route through the transportation system is unknown. This process of determining the links of the transportation system on which trips will be loaded is known as traffic assignment.

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Salter, R.J. (1996). Traffic assignment. In: Highway Traffic Analysis and Design. Palgrave, London. https://doi.org/10.1007/978-1-349-13423-6_9

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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

2935 Accesses

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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.

A fine resolution dataset of accessibility under different traffic conditions in European cities

Defining a city — delineating urban areas using cell-phone data

City-scale holographic traffic flow data based on vehicular trajectory resampling

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

The Department of Civil and Environmental Engineering, Princeton University, Princeton, 08544, USA

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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IMAGES

Mod 6, Part 1: Traffic Assignment (Introduction)
PPT
Traffic Assignment Process.
PPT
Chapter 9: Traffic Assignment Model: Methods with examples
PPT

VIDEO

👮🏼‍♂️ A typical day for a traffic police inspector🚓 👮🏼‍♂️Обычный день дорожного инспектора 🚓
Assignment problem
english assignment ( change in meaning)
English assignment, speech about obeying traffic rules (HASAN JEFFRI)
What’s the assignment?
What will traffic look like during the ACM Awards Thursday night?

COMMENTS

What is Traffic Assignment
Traffic assignment is a key element in the urban travel demand forecasting process. The traffic assignment model predicts the network flows that are associated with future planning scenarios, and generates estimates of the link travel times and related attributes that are the basis for benefits estimation and air quality impacts. The traffic ...
Route assignment
Route assignment, route choice, or traffic assignment concerns the selection of routes (alternatively called paths) between origins and destinations in transportation networks.It is the fourth step in the conventional transportation forecasting model, following trip generation, trip distribution, and mode choice.The zonal interchange analysis of trip distribution provides origin-destination ...
Traffic Networks: Dynamic Traffic Routing, Assignment, and ...
Traffic assignment is defined as the basic problem of finding the link flows given anorigin‐destination trip matrix and a set of link or marginal link travel times, as illustrated inFig. ... Given that trafficstream space‐mean speed can be related to traffic streamflow and density, a unique speed‐flow‐densityrelationship (in the ...
A.8
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. ... This includes the geometric definition of transport networks with the ...
3.6: 3-6 Route Choice
Step 1: Compute the resulting travel times and reassign traffic. Step 2: Now, begin to reassign using weights. Compute the weighted travel times in the previous two loadings and use those for the next assignment. The latest iteration gets a weight of 0.25 and the previous gets a weight of 0.75. Step 3.
PDF Tra c Assignment
CHAPTER 10. TRAFFIC ASSIGNMENT NPTEL May 7, 2007 Di erentiate the above equation to zero, and solving for x1 and then x2 leads to the solution x1 = 5.3,x2= 6.7 which gives Z(x) = 327.55 10.6 Other assignment methods Let us discuss brie y some other assignments like incremental assignment, capacity restraint assignment,
Traffic Assignments to Transportation Networks
Traffic assignment to uncongested networks is based on the assumption that cost does not depend on traffic flow. Therefore, traffic path flows and link flows are obtained from path choice probabilities that are themselves computed from flow-independent link performance attributes and costs (Cascetta 2009).The all-or-nothing assignment for uncongested networks is based on the following assumptions:
Traffic Assignment Problem
Traffic assignment problems usually consider two dimensions. Generation and attraction. A place of origin generates movements that are bound (attracted) to a place of destination. The relationship between traffic generation and attraction is commonly labeled as spatial interaction. The above example considers one origin/generation and ...
Traffic Assignment: A Survey of Mathematical Models and Techniques
The traditional transportation planning process [1, 2] has the following four stages, having traffic assignment as one of the four stages:1. Trip Generation: Trip generation involves estimating the number of trips generated at each origin node and/or the number of trips attracted to each destination node.This estimation is performed based on surveys conducted and generally uses a model that ...
Traffic Assignment in Practice: Overview and Guidelines for Users
This paper presents an overview of the elements of traffic assignment and a synthesis of the problems that may be encountered in applying traffic‐assignment models in practice. The elements include preparing the transportation network, establishing the origin‐destination (OD) demands, identifying a traffic‐assignment technique ...
Review of Traffic Assignment and Future Challenges
The problem of traffic assignment consists of determining the routes taken by the users of transportation infrastructure. This problem has been the subject of numerous studies, particularly in analyzing scenarios for developing road infrastructure and pricing strategies. This paper reviews the major progress in the field. Accordingly, it shows that the evolution of intelligent transportation ...
Dynamic Traffic Assignment: A Primer
This work presents an overview of developing and deploying a metropolitan area dynamic assignment model (MADAM) for the Sydney region, and focuses on the role of dynamic traffic assignment models, lessons learned during the current deployment, and a overview of the calibration process and the model outputs. Expand. 2.
Last Step of Four Step Modeling (Trip Assignment Models
The traffic assignment step, which is also called route assignment or route choice, simulates the choice of route selection from a set of alternatives between the origin and the destination zones (Levinson et al., 2014). The first three FSM steps determine the number of trips produced between each zone and the proportion completed by different ...
Traffic Assignment
1 Introduction. The process of allocating given set of trip interchanges to the specified transportation system is usually refered to as traffic assignment. The fundamental aim of the traffic assignment process is to reproduce on the transportation system, the pattern of vehicular movements which would be observed when the travel demand ...
PDF Introduction to Dynamic Traffic Assignment
CE 392 D Dynamic Traffic Assignment. Three reasons: (1) there are some important parallels; but (2) where STA and DTA di er, they do so very intentionally. STA and DTA make the same behavioral assumption: drivers want to reach their destination in the shortest time possible. Route Travel Times. Route Choices.
PDF TRAFFIC ASSIGNMENT
Significance of traffic assignment. Represents the "basic" level of what we mean by "traffic conditions". Essential to make planning, operational, renewal, and policy decisions. Provides "feedback" to trip distribution and mode split steps of the 4-step model. Provides input to assess and influence energy and environmental impacts.
Dynamic traffic assignment: model classifications and recent advances
Dynamic Traffic Assignment (DTA) has been studied for more than four decades and numerous reviews of this research area have been conducted. This review focuses on the travel choice principle and the classification of DTA models, and is supplementary to the existing reviews. The implications of the travel choice principle for the existence and uniqueness of DTA solutions are discussed, and the ...
Dynamic Traffic Assignment
Dynamic Traffic Assignment. Dynamic network assignment models (also referred to as dynamic traffic assignment models or DTA) capture the changes in network performance by detailed time-of-day, and can be used to generate time varying measures of this performance. They occupy the middle ground between static macroscopic traffic assignment and ...
Data-Driven Traffic Assignment: A Novel Approach for Learning Traffic
The traffic assignment problem (TAP) is one of the key components of transportation planning and operations. It is used to determine the traffic flow of each link of a transportation network for a given travel demand based on modeling the interactions among traveler route choices and the congestion that results from their travel over the network (Sheffi 1985).
Dynamic traffic assignment: A review of the ...
Traffic assignment models are crucial for traffic flow and travel time forecasting in long-term transportation planning and project appraisal, as well as in short-term traffic operation management and control. ... Total travel delay and mean excess emission (CO) exposure: Simulation-based single-objective GA: Common cycle length, offsets, green ...
Traffic Assignment
Traffic Assignment. Assign the vehicle trips shown in the following O-D trip table to the network, using the all-or-nothing assignment technique. To summarize your results, list all of the links in the network and their corresponding traffic volume after loading. The all-or-nothing technique simply assumes that all of the traffic between a ...
PDF Traffic assignment
Traffic assignment Previously the estimation of generated trip ends has been discussed together with the distribution of trips between the traffic zones. Modal split methods also have ... Cl is the mean disutility on route 1 (usually travel time or generalised travel cost), C2 is the mean disutility on route 2,
A unified dataset for the city-scale traffic assignment model ...
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 ...

Margin Size

3.6: 3-6 Route Choice

Link Performance Function

BPR Link Performance Function

Can Flow Exceed Capacity?

Wardrop's Principles of Equilibrium

Price of Anarchy

Conservation of Flow

Auto assignment

Heuristic procedures

Frank-Wolfe algorithm

Equilibrium Assignment

Transit assignment

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Traffic Assignment Problem

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13 Last Step of Four Step Modeling (Trip Assignment Models)

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Traffic Assignment Models

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All-or-nothing Example

User Equilibrium

UE Example

System Optimum Assignment

Incremental Increase Example

Capacity Restraint Assignment

Feedback Loop Model (Combined Traffic Assignment and Trip Distribution)

Stochastic User Equilibrium Traffic Assignment

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Dynamic traffic assignment: model classifications and recent advances in travel choice principles

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