traffic assignment routing

school Campus Bookshelves
menu_book Bookshelves
perm_media Learning Objects
login Login
how_to_reg Request Instructor Account
hub Instructor Commons
Download Page (PDF)
Download Full Book (PDF)
Periodic Table
Physics Constants
Scientific Calculator
Reference & Cite
Tools expand_more
Readability

selected template will load here

This action is not available.

3.6: 3-6 Route Choice

Last updated
Save as PDF
Page ID 48085

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

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.

Traffic assignment: A survey of mathematical models and techniques

Civil and Urban Engineering
Center for Interacting Urban Networks

Research output : Chapter in Book/Report/Conference proceeding › Chapter

This chapter presents the fundamentals of the theory and techniques of traffic assignment problem. It first presents the steady-state traffic assignment problem formulation which is also called static assignment, followed by Dynamic Traffic Assignment (DTA), where the traffic demand on the network is time varying. The static assignment problem is shown in a mathematical programming setting for two different objectives to be satisfied. The first one where all users experience same travel times in alternate used routes is called user-equilibrium and another setting called system optimum in which the assignment attempts to minimize the total travel time. The alternate formulation uses variational inequality method which is also presented. Dynamic travel routing problem is also reviewed in the variational inequality setting. DTA problem is shown in discrete and continuous time in terms of lumped parameters as well as in a macroscopic setting, where partial differential equations are used for the link traffic dynamics. A Hamilton–Jacobi- based travel time dynamics model is also presented for the links and routes, which is integrated with the macroscopic traffic dynamics. Simulation-based DTA method is also very briefly reviewed. This chapter is taken from the following Springer publication and is reproduced here, with permission and with minor changes: Pushkin Kachroo, and Neveen Shlayan, “Dynamic traffic assignment: A survey of mathematical models and technique,” Advances in Dynamic Network Modeling in Complex Transportation Systems (Editor: Satish V. Ukkusuri and Kaan Özbay) Springer New York, 2013. 1-25.

Publication series

Asjc scopus subject areas.

Control and Systems Engineering
Automotive Engineering
Aerospace Engineering
Industrial and Manufacturing Engineering

Access to Document

10.1007/978-3-319-69231-9_2

Fingerprint

Mathematical models Engineering & Materials Science 100%
Travel time Engineering & Materials Science 56%
Time varying networks Engineering & Materials Science 25%
Mathematical programming Engineering & Materials Science 21%
User experience Engineering & Materials Science 18%
Partial differential equations Engineering & Materials Science 18%
Dynamic models Engineering & Materials Science 14%

T1 - Traffic assignment

T2 - A survey of mathematical models and techniques

AU - Kachroo, Pushkin

AU - Özbay, Kaan M.A.

N1 - Publisher Copyright: © Springer International Publishing AG, part of Springer Nature 2018.

N2 - This chapter presents the fundamentals of the theory and techniques of traffic assignment problem. It first presents the steady-state traffic assignment problem formulation which is also called static assignment, followed by Dynamic Traffic Assignment (DTA), where the traffic demand on the network is time varying. The static assignment problem is shown in a mathematical programming setting for two different objectives to be satisfied. The first one where all users experience same travel times in alternate used routes is called user-equilibrium and another setting called system optimum in which the assignment attempts to minimize the total travel time. The alternate formulation uses variational inequality method which is also presented. Dynamic travel routing problem is also reviewed in the variational inequality setting. DTA problem is shown in discrete and continuous time in terms of lumped parameters as well as in a macroscopic setting, where partial differential equations are used for the link traffic dynamics. A Hamilton–Jacobi- based travel time dynamics model is also presented for the links and routes, which is integrated with the macroscopic traffic dynamics. Simulation-based DTA method is also very briefly reviewed. This chapter is taken from the following Springer publication and is reproduced here, with permission and with minor changes: Pushkin Kachroo, and Neveen Shlayan, “Dynamic traffic assignment: A survey of mathematical models and technique,” Advances in Dynamic Network Modeling in Complex Transportation Systems (Editor: Satish V. Ukkusuri and Kaan Özbay) Springer New York, 2013. 1-25.

AB - This chapter presents the fundamentals of the theory and techniques of traffic assignment problem. It first presents the steady-state traffic assignment problem formulation which is also called static assignment, followed by Dynamic Traffic Assignment (DTA), where the traffic demand on the network is time varying. The static assignment problem is shown in a mathematical programming setting for two different objectives to be satisfied. The first one where all users experience same travel times in alternate used routes is called user-equilibrium and another setting called system optimum in which the assignment attempts to minimize the total travel time. The alternate formulation uses variational inequality method which is also presented. Dynamic travel routing problem is also reviewed in the variational inequality setting. DTA problem is shown in discrete and continuous time in terms of lumped parameters as well as in a macroscopic setting, where partial differential equations are used for the link traffic dynamics. A Hamilton–Jacobi- based travel time dynamics model is also presented for the links and routes, which is integrated with the macroscopic traffic dynamics. Simulation-based DTA method is also very briefly reviewed. This chapter is taken from the following Springer publication and is reproduced here, with permission and with minor changes: Pushkin Kachroo, and Neveen Shlayan, “Dynamic traffic assignment: A survey of mathematical models and technique,” Advances in Dynamic Network Modeling in Complex Transportation Systems (Editor: Satish V. Ukkusuri and Kaan Özbay) Springer New York, 2013. 1-25.

UR - http://www.scopus.com/inward/record.url?scp=85047219067&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85047219067&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-69231-9_2

DO - 10.1007/978-3-319-69231-9_2

M3 - Chapter

AN - SCOPUS:85047219067

T3 - Advances in Industrial Control

BT - Advances in Industrial Control

PB - Springer International Publishing

Search code, repositories, users, issues, pull requests...

Provide feedback.

We read every piece of feedback, and take your input very seriously.

Saved searches

Use saved searches to filter your results more quickly.

To see all available qualifiers, see our documentation .

Notifications

Algorithms for Routing and Solving the Traffic Assignment Problem

vlarmet/cppRouting

Folders and files, repository files navigation, cpprouting v3 : algorithms for routing and solving the traffic assignment problem.

Vincent LARMET November 24, 2022

Package presentation

Stable version from cran, or from github, what are we talking about , set the number of threads used by cpprouting, instantiate the graph, main functions, compute isochrones, compute possible detours within a fixed additional cost, contraction hierarchies, work with dual weighted network, network simplification, choosing the best routing algorithm, link-based algorithms, bush-based algorithms, performance comparison, algorithm compatibility, parallel implementation, application 1 : calculate two step floating catchment areas (2sfca) of general practitioners in france, application 2 : calculate the minimum travel time to the closest maternity ward in france, application 3 : calculate average commuting time to go to job in france, application 4 : calculate the flow of people crossing each municipality in the context of commuting in bourgogne-franche-comte region, benchmark with other r packages.

cppRouting is an R package which provide routing algorithms (shortest paths/distances, isochrones) and traffic assignment solvers on non-negative weighted graphs. cppRouting is characterized by :

its ability to work on large road graphs (country/continent scale)
its large choice of one-to-one shortest path algorithms
its implementation of contraction hierarchies and associated routing algorithms
its large choice of algorithms for solving the traffic assignment problem (TAP)
its high performance through memory usage and parallel programming

cppRouting is therefore particularly adapted for geographer, or whoever who need to calculate accessibility indicators at large scale. All algorithms are written in C++ and mainly use containers from the Standard Template Library (STL). This package have been made with Rcpp and RcppParallel packages.

Installation

cppRouting implements algorithms belonging to graph theory , so let’s define what a graph is. A graph is commonly used to represent a network, which is composed of vertices connected by edges.

In cppRouting , an edge has at least three attributes : vertice’s ID from which it start, vertice’s ID from which it end and a weight (length, flow, travel time …).

Readme Data

**This README file and all time measurements were made on a Windows 10 computer, with 11th generation i5 (6 cores) processor and 32GB of memory. ** The data presented here is the official french road network describing over 500000 km of roads. All data used in this README are free and can be downloaded here :

roads : http://professionnels.ign.fr/route500
general practitioners location : https://www.insee.fr/fr/statistiques/3568614?sommaire=3568656#consulter
maternity wards location : https://www.insee.fr/fr/statistiques/3568611?sommaire=3568656#dictionnaire
shapefile of the ~36000 communes in France : http://professionnels.ign.fr/adminexpress
commuting to work from the French national census : https://www.insee.fr/fr/statistiques/3566477#consulter

Graph data have been preprocessed for more readability (see data_preparation.R).

The final graph is composed of 234615 nodes and 685118 edges. Data has to be a 3 columns data.frame or matrix containing from, to and a cost/distance column. Here the cost is the time needed to travel in each edges (in minutes). From and to are vertices IDs (character or numeric).

Head of road network data

Head of coordinates data.

Graph object have some useful attributes for the user :

graph$nbnode : total number of vertices,
graph$dict$ref : vertices ids.

Other attributes are internals data and have no interest for the user. All graph attributes should never be modified by the user.

cppRouting package provide these functions :

get_distance_matrix : compute distance matrix (between all combinations origin-destination nodes - one-to-many ),
get_distance_pair : compute distances between origin and destination by pair ( one-to-one ),
get_path_pair : compute shortest paths between origin and destination by pair ( one-to-one ),
get_multi_paths : compute shortest paths between all origin nodes and all destination nodes ( one-to-many ),
get_isochrone : compute isochrones/isodistances with one or multiple breaks.
get_detour : return nodes that are reachable within a fixed additional cost around shortest paths. This function can be useful in producing accessibility indicators.
cpp_simplify : remove non-intersection nodes, duplicated edges and isolated loops in the graph. Graph topology is preserved so distance calculation is faster and remains true. This function can be applied to very large graphs (several millions of nodes).
cpp_contract : contract the graph by applying contraction hierarchies algorithm.
get_aon : given an origin-destination matrix, compute All-or-Nothing assignment.
assign_traffic : given an origin-destination matrix, estimate the traffic flows on the network.

As the package name suggest, cppRouting is initially aimed to provide efficient algorithms for finding shortest paths.

Path algorithms proposed by the package are :

1 uni-directional Dijkstra algorithm,
2 bi-directional Dijkstra algorithm,
3 uni-directional A* algorithm
4 New bi-directional A* algorithm (Piljs & Post, 2009 : see http://repub.eur.nl/pub/16100/ei2009-10.pdf )
5 one-to-one bi-directional Dijkstra adapted to contraction hierarchies (Geisberger & al., 2008)
6 many-to-many bi-directional Dijkstra adapted to contraction hierarchies (Geisberger & al., 2008)
7 PHAST algorithm (Hardware-accelerated shortest path trees), one-to-all algorithm adapted to contraction hierarchies (Delling & al., 2011)

1 , 2 , 3 and 4 are available for one-to-one calculation in get_distance_pair and get_path_pair functions on a non-contracted graph. In these functions, uni-directional Dijkstra algorithm is stopped when the destination node is reached. A* and NBA* are relevant if geographic coordinates of all nodes are provided. Note that coordinates should be expressed in a projection system . To be accurate and efficient, A* and NBA* algorithms should use an admissible heuristic function (here the Euclidean distance), i.e cost and heuristic function must be expressed in the same unit. In cppRouting , heuristic function h for a node (n) is defined such that : h(n,d) = ED(n,d) / k with h the heuristic, ED the Euclidean distance, d the destination node and a constant k . So in the case where coordinates are expressed in meters and cost is expressed in time, k is the maximum speed allowed on the road. By default, constant is 1 and is designed for graphs with cost expressed in the same unit than coordinates (for example in meters). If coordinates cannot be provided, bi-directional Dijkstra algorithm is the best option in terms of performance.

5 is used for one-to-one calculation in get_distance_pair and get_path_pair functions on a contracted graph.

1 is used for one-to-many calculation in get_distance_matrix function on a non-contracted graph.

6 and 7 are available for one-to-many calculation in get_distance_matrix function on a contracted graph.

Let’s compare different path algorithms in terms of performance. For A* and NBA algorithms, coordinates are defined in meters and max speed is 110km/h; so for the heuristic function to be admissible, we have to convert meters to minutes by setting constant to 110/0.06 :

So, how to choose the algorithm ? It’s simple, the faster, the better. If coordinates are provided, go for NBA , else go for bidirectional Dijkstra. Uni-directional Dijkstra and A* algorithms should be used if main memory is (almost) full because they require only one graph instead of two (forward and backward).

An isochrone is a set of nodes reachable from a node within a fixed limit. Let’s compute isochrones around Dijon city

get_detour function returns all reachable nodes within a fixed detour time around the shortest path between origin and destination nodes. Returned nodes (n) meet the following condition : SP(o,n) + SP(n,d) < SP(o,d) + t with SP shortest distance/time, o the origin node, d the destination node and t the extra cost. The algorithm used is a slightly modified bidirectional Dijkstra. Let’s see an example for the path between Dijon and Lyon city :

Contraction hierarchies is a speed-up technique for finding shortest path on a network. It was proposed by Geisberger & al.(2008). Initially created for one-to-one queries, it has been extended to many-to-many and distance matrix calculation. This technique is composed of two phases:

preprocessing phase called contraction with cpp_contract function
query phase : a slightly modified version of bidirectional search for one-to-one query, available in get_distance_pair and get_path_pair ; PHAST algorithm and a many-to-many algorithm using buckets available in get_distance_matrix function.

Contraction phase consists of iteratively removing a vertex v from the graph and creating a shortcut for each pair (u,w) of v ’s neighborhood if the shortest path from u to w contains v . To be efficient and avoid creating too much shortcuts, vertices have to be ordered according to several heuristics. The two heuristics used by cppRouting are :

edge difference (number of shortcuts potentially created by removing v - number of incoming edges - number of outcoming edges)
deleted neighbors (number of already contracted neighbors)

The nodes are initially ordered using only edge difference , then importance of v is lazily updated during contraction phase with the combined two heuristics. To see more detailed explanations, see these ressources :

quick review
authors article
detailed author thesis

Compare outputs

Distance by pair.

Here are the measurements of contraction time and query time (in second) of contraction hierarchies on different graphs :

As we can see on the plot, the larger is the graph, the higher is the benefit of using contraction hierarchies. For OSM Europe, query time can be faster by a factor of 5000 compared to bidirectional Dijkstra and 2500 to NBA.

Distance matrix

Here are the measurements of query time (in second) of contraction hierarchies on different graphs. We compare PHAST and many to many CH to Dijkstra algorithm on square matrix (i.e the sets of source and target nodes are of equal length).

Here are the plots (in log-log) of query time improvement factor of PHAST and many to many CH compared to Dijkstra algorithm :

Benefits are less important than one-to-one queries but still interesting. For OSM Europe, query time can be faster by a factor of * 100 . PHAST’s improvement is constant since it iteratively perform an *one-to-all* search, just like original Dijkstra. many to many CH is well adapted for **square matrix**.

Here are the plots of query time of PHAST and many to many CH on assymetric matrix (i.e. number of source and number of target are unequal) with |S| / |T| the number of sources divided by the number of targets :

Note that this plot is the same for |T| / |S| . PHAST algorithm is much faster for rectangular matrix. The rate |S| / |T| where many to many CH is better varies according the graph size. For example, if we have to calculate a distance matrix between 10000 sources and 10 targets (or 10 sources and 10000 targets) on OSM France, we must use PHAST . On the other hand, if we want a matrix of 10000 sources and 8000 targets, we use many to many CH algorithm.

Sometimes it can be useful to sum up additional weights along the shortest path. For a use-case example, let’s say we want to compute the distance along the shortest time path or the time needed to travel the shortest distance path. It is now possible to set an auxiliary set of edge weights during graph construction in makegraph() function and set aggregate_aux to TRUE in get_distance_* functions. Let’s see an example where we would like to compute the number of edges within each shortest path :

Note that this functionality work for contracted graphs as well.

cpp_simplify ’s internal function performs two major steps :

removing non-intersection nodes between two intersection nodes then calculate cost of the new edges,
removing duplicate edges that are potentially created in the first step.

In order to remove maximum number of nodes, some iterations are needed until only intersection nodes are remained.

Let’s see a small example :

Here, junction nodes are e , h , d , k , l , i and m . So b , c , f and n have been contracted in the first step of the function. By contracting n , an edge with cost of 2 has been created between m and l nodes. The second step of the function has removed this edge which is greater than the original one (i.e 1), and the whole process now need a second iteration to remove m and l that aren’t intersection nodes anymore. Let’s try with iterate argument set to TRUE :

French road network simplification

Running time.

Here are running times in second on graphs of different sizes :

data presented here,
OpenStreetMap french road network from Geofabrik ,
an assembly of several european countries from OSM (France, Italy, Spain, Germany, Portugal, Switzerland, Belgium and Netherlands).

OpenStreetMap data have been extracted with osm2po tool, which I highly recommend.

Traffic assignment

Traffic assignment models are used to estimate the traffic flows on a network. It take as input origin-destinations matrix describing volume between each OD pair. For this part, we will use standard networks massively used in traffic modelling studies : Siouxfalls and Chicago networks ( source ). Let’s import Siouxfalls. It is composed of 24 nodes and 76 links.

We also have OD matrix as well :

All-or-Nothing (AON)

All-or-Nothing assignment (AON) is the most simplistic (and fastest) method to load flow on a network, since it assume there is no congestion effects. The assignment algorithm itself is the procedure that loads the origin-destination matrix to the shortest path trees and produces the flows. In cppRouting , OD matrix is represented as 3 vectors of equal length :

from : origin node,
to : destination node,
demand : volume.

Let’s load flows on the network using get_aon() function :

get_aon work also for contracted networks .

Computing time is directly linked to shortest paths calculations. The user has the choice between two kind of routing algorithms, depending the sparsity of OD matrix. By default, algorithm argument is set to bidirectional Dijkstra. Just like get_*_pair functions, it run length(from) times routing algorithm for finding each OD pair’s shortest path. If OD matrix is dense, recursive one-to-many methods like Dijkstra algorithm would be preferred. Given an origin node, bidirectional Dijkstra et NBA* algorithms are on average 2 and 5 times faster than Dijkstra algorithm, respectively.

This rule-of-thumb apply also when the network have been contracted using cpp_contract function. phast algorithm is for matrix-like shortest paths calculation, and bi is pairwise. In that case, bi algorithm is 200 times faster than phast for a given node.

For resuming :

matrix-like calculation : optimal for dense matrix, run one-to-many search for min(length(unique(from)), length(unique(to))) times,
pairwise calculation : optimal for highly sparse matrix, run one-to-one search for length(from) times.

User Equilibrium (UE)

The term “User Equilibrium” is used to describe a route choice assumption formally proposed by Wardrop : “The journey times on all the routes actually used are equal and less than those which would be experienced by a single vehicle on any unused routes” . Note that this principle follows directly from the assumptions that drivers choose minimum time paths, and are well-informed about network conditions.

Unlike AON assignment, this more realistic way to assign flows on a network take into account congestion effect . In this paradigm, the cost of a given link is dependent of the flow on it.

As an example, let’s assume 3000 commuters going from one node to another connected by two links. The User equilibrium is illustrated on this figure ( source ):

The relation between cost and flow is called volume decay function and is written as :

$t = t_{0}.(1+\alpha.(\frac{v}{c})^{\beta})$

Traffic Assignment Problem (TAP) is a convex optimization problem solved by iterative algorithms. The relative gap is the metric to minimize and is written as :

$gap=|(\frac{TSTT}{SPTT}) - 1|$

Link-based algorithms implemented in cppRouting are, in increasing order of complexity :

$\frac{1}{it}$

By going down through that list, we slightly increase computing time in each iteration but we will need less iterations to reach a given gap. Let’s equilibrate traffic within our small network, first we need to construct the network with important parameters : capacity , alpha and beta .

For these algorithms, 99% of computation time is done within AON assignment.

Returned data contains the equilibrated network with the following edge attributes :

ftt : free-travel time i.e. the initial cost set during graph construction,
cost : actual travel time at equilibrium,
flow : equilibrated flow.

Now, we can plot the result with line size varying with road capacity and the color with flow.

cppRouting also propose a bush-based algorithm called Algorithm B from R.B. Dial (2006) . The problem is decomposed into sub-problems, corresponding to each origin of the OD matrix, that operate on acyclic sub-networks of the original transportation network, called bushes . Link flows are shifted from the longest path to the shortest path recursively within each bush using Newton method. Unlike link-based algorithm, Algorithm B can achieve very precise solution by minimizing relative gap down to 1e-16.

The main steps of the procedure are :

Initialization

bushes initialization : for each unique origin node of the OD matrix, an acyclic sub-network is created by computing shortest path tree from root node to all other nodes.

For each bush :

Topological ordering of the nodes
for each node, shortest path and longest path is computed and stored.
given an OD matrix, flows are loaded on each bush using shortest path labels.

Iteration For each bush :

shortest and longest paths are updated
bush optimization : the bush is updated by removing unused links (no flow on it) and adding links leading to shorter paths. Network costs are updated in the same time.
bush equilibration : the flow is shifted from the shortest to the longest path using Newton method.

Evaluation We compute AON assignment and relative gap.

For detailed explanations of Algorithm B, please see this course ( part 1 , part 2 ).

Important note : computation time for algorithm-B is depending of the number of origin node AND AON assignment.

Algorithm B is used by setting dial to algorithm argument :

Now, let’s measure performance of Traffic assignment algorithms on a larger road network. We use Chicago network, composed of 12982 nodes and 39018 edges.

OD matrix contains 2,297,945 trips.

Since All-or-Nothing algorithm will be called multiple times, we have to choose the fastest AON assignment algorithm.

By setting matrix-like AON calculation, we speed up computation time by a factor of 50 for link-based methods.

Algorithm B could achieve very small gap within an reasonable amount of time, while link-based methods seems to converge much more slowly. For achieving highly precise solutions, algorithm B should be implemented. However, algorithm B generally use more memory than link-based methods, depending network size and OD matrix. On the other hand, link-based method may have a higher benefit through parallel computing since computation time is mainly due to AON assignment.

Note : network can be contracted “on-the-fly” at each iteration to speed-up AON calculation, by setting cphast or cbi in aon_method argument.

Here is a table summarizing cppRouting functions compatibility with network nature (contracted, simplified, normal).

Now a table summarizing cppRouting functions compatibility with parallel computing.

Applications

Except application 4, all indicators are calculated at the country scale but for the limited R ’s ability to plot large shapefile, only one region is mapped.

2SFCA method is explained here : https://en.wikipedia.org/wiki/Two-step_floating_catchment_area_method

First step Isochrones are calculated with the cppRouting function get_isochrone

Second step

Plot the map for Bourgogne-Franche-Comte region

Shortest travel time matrix The shortest travel time is computed with the cppRouting function get_distance_matrix . In order to compute multiple distances from one source, original uni-directional Dijkstra algorithm is ran without early stopping. We compute travel time from all commune nodes to all maternity ward nodes (i.e ~36000*400 distances).

Plot the map of minimum travel time in Bourgogne-Franche-Comte region

Commuting data from national census is composed of 968794 unique pairs of origin - destination locations (home municipality, job municipality). Using other R packages like igraph or dodgr , we would have to calculate the whole matrix between all communes (36000 x 36000). We would end with a 10Gb matrix whereas we only need 0.075% of the result (968794/(36000 x 36000)). So this example illustrate the advantage of calculating distance by pair .

Plot the map of average travel time of Bourgogne-Franche-Comte inhabitants

First, we must determine in which commune each node is located.

If a commuter crosses multiple nodes in a municipality, we count it only once.

**Plot the flow of people crossing Bourgogne-Franche-Comte’s communes **

To show the efficiency of cppRouting , we can make some benchmarking with the famous R package igraph , and the dodgr package. Estimation of the User Equilibrium cannot be compared because of the absence of this kind of algorithm within R landscape (to my knowledge).

Distance matrix : one core

Even if we add the preprocessing time (i.e. 14s) to the query time, the whole process of contraction hierarchies is still faster.

All-or-Nothing assignment : sparse OD matrix

All-or-Nothing assignment : dense OD matrix

Please don’t forget to cite cppRouting package in your work !

Dynamic Traffic Assignment

Early Experiences

Current Practices

Research Needs

Page categories

Activity Based Models

Network Assignment

Topic Circles

More pages in this category:

(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.

← Network assignment Land Use-Transport Modeling →

This site uses cookies to learn which topics interest our readers.

We've detected unusual activity from your computer network

To continue, please click the box below to let us know you're not a robot.

Why did this happen?

Please make sure your browser supports JavaScript and cookies and that you are not blocking them from loading. For more information you can review our Terms of Service and Cookie Policy .

For inquiries related to this message please contact our support team and provide the reference ID below.

North Jersey traffic - Construction, road work, road maintance on GWB, 80, 1&9, 9W, 46

At the Prudential Center, the Devils have a home game on Saturday, and Nicki Minaj performs on Thursday.

Soccer is at Red Bull Arena with the New York Red Bulls vs. Inter Miami CF on Saturday and Ecuador National Football Team vs. Italy on Sunday.

Scheduled construction, road work, road maintenance, drainage improvements, utility work, water main work, bridge inspection, signal repair, bridge work and CCTV repairs will cause daytime lane closures on the George Washington Bridge; Interstate 80; U.S. highways 1&9, 9W and 46; and state routes 4, 5, 10, 15, 17, 208 and 505.

The New Jersey Department of Transportation is repairing potholes statewide. Motorists can expect to see crews repairing roadways at any time during the day.

Heavy rain is in the forecast for Saturday, which can cause dangerous conditions on the roads.

Visit 511nj.org for up-to-date traffic information and real-time traffic cameras from throughout the state, or call 511.

Bergen County traffic

Construction on E. Palisade Avenue will close all lanes in both directions between Route 9W and Summit Street in Englewood Cliffs, 7 a.m. to 3 p.m. March 22.
Construction on the George Washington Bridge eastbound will close one lane on each level, 8:30 a.m. to 4:30 p.m. March 22; all lanes on the lower level, 11 p.m. March 22 to 8 a.m. March 23; and three lanes on the upper level, 11 p.m. March 23 to 8 a.m. March 24.
Drainage improvements on Route 17 southbound between Allendale and Sheridan avenues in Saddle River, 9 a.m. to 1 p.m. March 22.
Road maintenance operations on Bergen Boulevard will close the northbound left lane near Fletcher Avenue in Fort Lee, 9 a.m. to 3 p.m. March 22.
Signal repair on Route 4 will intermittently close all eastbound lanes near Midland Avenue in Fair Lawn, 9 a.m. to 3 p.m. March 22.
Drainage improvements on Route 5 will close the right lane in both directions between Bergen Boulevard in Fort Lee and River Road in Edgewater, 9 a.m. to 3 p.m. March 22.
On Route 9W southbound, drainage improvements will close the right lane between Hillside Avenue in Alpine and Hollywood Avenue in Englewood Cliffs, 9 a.m. to 3 p.m. March 22; and utility work will close the right lane near Charlotte Place in Englewood Cliffs, 9 a.m. to 3 p.m. through March 24.
Utility work will close portions of local roads in Tenafly, Bergenfield and Englewood, through April 12.
Guard rail repairs closes the northbound entrance ramp to Route 208 from Maple Avenue in Fair Lawn, 9 a.m. to 3 p.m. March 22.
Roadwork on I-80 westbound closes the left lane between exits 64 and 63 in Lodi, 10 a.m. to 3 p.m. March 22.

Are you a collector? American Pickers' are coming to NJ, maybe you have their next pick

Essex County traffic

Construction on Route 46 closes the left lane in both directions near Plymouth Road in Fairfield, 9 a.m. to 3 p.m. March 22.
On the New Jersey Turnpike southbound between exit 14 in Newark and exit 6 in Mansfield, paving operations and utility work will close the outer roadway, 9 p.m. March 22 to 5 a.m. March 23; and construction and utility work will close the inner roadway, 10 p.m. March 23 to 6 a.m. March 24.
Roadwork on I-280 westbound will close the left lane between exits 15 and 13 in Newark, 11 p.m. March 22 to 6 a.m. March 23.
The New Jersey Devils are home at the Prudential Center to face the Ottawa Senators at 7 p.m. March 23. Then Nicki Minaj brings her Pink Friday 2 World Tour to the arena at 8 p.m. March 28.

Hudson County traffic

On Tonnelle Avenue , water main work will close the right lane in both directions near Route 3 in North Bergen, 10 a.m. to 2 p.m., March 22, March 26, March 27 and March 28; and all northbound lanes intermittently, 10 p.m. March 22 to 6 a.m. March 23.
Soccer season continues at Red Bull Arena in Harrison, with Ecuador National Football Team vs. Guatemala National Team at 8 p.m. March 21, the New York Red Bulls vs. Inter Miami CF at 2 p.m. March 23, and Ecuador National Football Team vs. Italy at 4 p.m. March 24.
Construction on the Lincoln Tunnel eastbound will close the south tube, 11 p.m. to 5 a.m., through March 22.
Construction on the Holland Tunnel south tube will close one eastbound lane, 11 p.m. to 5 a.m. through March 22.
Drainage improvements on Route 3 westbound will close the right lane near Harmon Cove in Secaucus, 8 p.m. March 22 to 8 a.m. March 23.

Morris County traffic

Construction on Route 46 will close the one lane in both directions between Main Street in Rockaway and Route 53 in Denville, and near Beverwyck Road in Parsippany, 8 a.m. to 3 p.m. March 22.
On Route 15 , bridge work and CCTV repairs will close one lane near Espanong Road in Jefferson, 9 a.m. to 3 p.m. March 22.
Construction on Route 10 will close the eastbound right lane near Dover Chester Road in Randolph, 9 a.m. to 3 p.m. April 2.

Passaic County traffic

Utility work on Route 4 will close the right lane in both directions between the Passaic River and Route 20 in Paterson, 9 a.m. to 3 p.m. March 22.
On Route 20 in both directions, drainage improvements and utility work will close the right lane between Route 46 in Clifton to E. 24th Street in Paterson, 9 a.m. to 3 p.m. March 22. Construction will close two lanes between 21st and 17th avenues in Paterson, 8 p.m. March 22 to 7 a.m. March 23.
Roadwork on I-80 eastbound will close the right lane near exit 55 in Totowa, 8 p.m. March 22 through 8 a.m. March 23.
Construction on Route 21 southbound will close all lanes near exit 11 in Passaic, 9 p.m. to 5 a.m. March 22.
On Route 46, bridge inspection will close the left lane in both directions near the Garden State Parkway in Clifton and Elmwood Park, 10 a.m. to 3 p.m., April 4 and April 5.

M25 closure: Everything you need to know as drivers face delays on first morning of 'nightmare' diversion

The UK's busiest motorway is shut in both directions this weekend across a five-mile stretch. Here's what you need to know about how bad traffic could be, a map of the diversion route and why it's happening.

News reporter

Sunday 17 March 2024 05:25, UK

A section of the M25 is closed while a bridge is demolished. Pic: PA

Drivers are facing significant delays as a section of the M25 shuts in both directions this weekend in an unprecedented move.

Motorists saw two miles of congestion on the approach to the closure in Surrey this morning, as traffic builds along the main diversion route through Byfleet, West Byfleet, Woking and Ottershaw.

National Highways estimated that the average journey time along the diversion route on Saturday morning was 25 minutes.

The closure of a five-mile stretch between junctions 10 and 11 on the UK's busiest motorway is already causing travel chaos - just hours into a two-and-a-half day closure.

Daryl Jordan, of Woking Borough Council, said residents will be "affected massively" by the closure, adding: "It's going to be a nightmare."

South East Coast Ambulance Service, which covers Surrey, also urged drivers to clear the way for ambulances ahead of a "challenging weekend" for crews.

But how long will the section be shut, where are the diversion routes and why is it happening?

More on Travel Chaos

Aerial pictures show the M25 on Friday ahead of this weekend's closure.

M25 closure: Aerial pictures show calm amid weekend gridlock fears with section of motorway shut

Air traffic chaos after engineers were working from home

Passengers 'bounced off plane roof' after mid-air flight incident from Sydney to Auckland

Balancing fairness and efficiency in traffic routing via interpolated traffic assignment

Published: 02 August 2023
Volume 37 , article number 32 , ( 2023 )

Cite this article

Devansh Jalota 1 ,
Kiril Solovey 2 ,
Matthew Tsao 3 ,
Stephen Zoepf 4 &
Marco Pavone 5

312 Accesses

3 Citations

Explore all metrics

System optimum (SO) routing, wherein the total travel time of all users is minimized, is a holy grail for transportation authorities. However, SO routing may discriminate against users who incur much larger travel times than others to achieve high system efficiency, i.e., low total travel times. To address the inherent unfairness of SO routing, we study the ${\beta }$ -fair SO problem whose goal is to minimize the total travel time while guaranteeing a ${\beta \ge 1}$ level of unfairness, which specifies the maximum possible ratio between the travel times of different users with shared origins and destinations. To obtain feasible solutions to the ${\beta }$ -fair SO problem while achieving high system efficiency, we develop a new convex program, the interpolated traffic assignment problem (I-TAP), which interpolates between a fairness-promoting and an efficiency-promoting traffic-assignment objective. We evaluate the efficacy of I-TAP through theoretical bounds on the total system travel time and level of unfairness in terms of its interpolation parameter, as well as present a numerical comparison between I-TAP and a state-of-the-art algorithm on a range of transportation networks. The numerical results indicate that our approach is faster by several orders of magnitude as compared to the benchmark algorithm, while achieving higher system efficiency for all desirable levels of unfairness. We further leverage the structure of I-TAP to develop two pricing mechanisms to collectively enforce the I-TAP solution in the presence of selfish homogeneous and heterogeneous users, respectively, that independently choose routes to minimize their own travel costs. We mention that this is the first study of pricing in the context of fair routing for general road networks (as opposed to, e.g., parallel road networks).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price includes VAT (Russian Federation)

Instant access to the full article PDF.

Rent this article via DeepDyve

Institutional subscriptions

Availability of data and materials

The data-sets used for our numerical experiments are publicly available at the following Github links: frank-wolfe-traffic , fair-routing .

Code availability

Our code for our numerical experiments is publicly available at the following Github links: frank-wolfe-traffic , fair-routing .

Fleming, S. (2019). Traffic congestion cost the US economy nearly \$87 billion in 2018. WEF. Retrieved February 19, 2021 from https://www.weforum.org/agenda/2019/03/traffic-congestion-cost-the-us-economy-nearly-87-billion-in-2018/ .

Roughgarden, T., & Tardos, E. (2002). How bad is selfish routing? Journal of the ACM, 49 (2), 236–259. https://doi.org/10.1145/506147.506153

Article MathSciNet Google Scholar

Roughgarden, T., & Tardos, Éva. (2004). Bounding the inefficiency of equilibria in nonatomic congestion games. Games and Economic Behavior, 47 (2), 389–403. https://doi.org/10.1016/j.geb.2003.06.004

Sheffi, Y. (1985). Urban transportation networks: Equilibrium analysis with mathematical programming methods (1st ed.). Prentice-Hall.

Google Scholar

Zhang, J., Pourazarm, S., Cassandras, C., & Paschalidis, I. (2018). The price of anarchy in transportation networks: Data-driven evaluation and reduction strategies. Proceedings of the IEEE, 106 (4), 538–553. https://doi.org/10.1109/JPROC.2018.2790405

Article Google Scholar

Sharon, G., Albert, M., Rambha, T., Boyles, S., & Stone, P. (2018). Traffic optimization for a mixture of self-interested and compliant agents. In International symposium on artificial intelligence and mathematics .

Pigou, A. (1912). Wealth and Welfare (1st ed.). Macmillan and Co.

Wardrop, J. G. (1952). Some theoretical aspects of road traffic research. Proceedings of the Institution of Civil Engineers, 1 (3), 325–362. https://doi.org/10.1680/ipeds.1952.11259

Yang, H., & Huang, H.-J. (2005). Mathematical and economic theory of road pricing (1st ed.). Emerald Publishing.

Book Google Scholar

Jahn, O., Möhring, R., Schulz, A., & Stier-Moses, N. (2005). System-optimal routing of traffic flows with user constraints in networks with congestion. Operations Research, 53 (4), 600–616. https://doi.org/10.1287/opre.1040.0197

Roughgarden, T. (2002). How unfair is optimal routing? In ACM-SIAM symposium on discrete algorithms .

Angelelli, E., Arsik, I., Morandi, V., Savelsbergh, M., & Speranza, M. G. (2016). Proactive route guidance to avoid congestion. Transportation Research Part B: Methodological, 94 , 1–21. https://doi.org/10.1016/j.trb.2016.08.015

Angelelli, E., Morandi, V., Savelsbergh, M., & Speranza, M. G. (2020). System optimal routing of traffic flows with user constraints using linear programming. European Journal of Operational Research . https://doi.org/10.1016/j.ejor.2020.12.043

Angelelli, E., Morandi, V., & Speranza, M. G. (2018). Congestion avoiding heuristic path generation for the proactive route guidance. Computers & Operations Research, 99 , 234–248. https://doi.org/10.1016/j.cor.2018.07.009

Angelelli, E., Morandi, V., & Speranza, M. G. (2020). Minimizing the total travel time with limited unfairness in traffic networks. Computers & Operations Research, 123 , 105016. https://doi.org/10.1016/j.cor.2020.105016

Fleischer, L., Jain, K., & Mahdian, M. (2004). Tolls for heterogeneous selfish users in multicommodity networks and generalized congestion games. In IEEE symposium on foundations of computer science . https://doi.org/10.1109/FOCS.2004.69 .

Jalota, D., Solovey, K., Tsao, M., Zoepf, S., & Pavone, M. (2022). Balancing fairness and efficiency in traffic routing via interpolated traffic assignment. In: Proceedings of the 21st international conference on autonomous agents and multiagent systems. AAMAS ’22 (pp. 678–686). International Foundation for Autonomous Agents and Multiagent Systems, Richland, SC.

Rahmattalabi, A., Jabbari, S., Lakkaraju, H., Vayanos, P., Rice, E., & Tambe, M. (2020). Fair influence maximization: A welfare optimization approach. CoRR arXiv:2006.07906 .

Bertsimas, D., Farias, V., & Trichakis, N. (2012). On the efficiency-fairness trade-off. Management Science, 58 (12), 2234–2250. https://doi.org/10.1287/mnsc.1120.1549

Stoica, A.-A., Han, J.X., & Chaintreau, A. (2020). Seeding network influence in biased networks and the benefits of diversity. In Proceedings of the web conference 2020. WWW ’20 (pp. 2089–2098). Association for Computing Machinery, New York, NY, USA. https://doi.org/10.1145/3366423.3380275 .

Hu, L., & Chen, Y. (2020). Fair classification and social welfare. In Proceedings of the 2020 conference on fairness, accountability, and transparency. FAT* ’20 (pp. 535–545). Association for Computing Machinery, New York, NY, USA. https://doi.org/10.1145/3351095.3372857 .

Tsang, A., Wilder, B., Rice, E., Tambe, M., & Zick, Y. (2019). Group-fairness in influence maximization. CoRR arxiv:1903.00967 .

Dial, R. (1996). Bicriterion traffic assignment: Basic theory and elementary algorithms. Transportation Science, 30 (2), 93–111. https://doi.org/10.1287/trsc.30.2.93

Dial, R. (1997). Bicriterion traffic assignment: Efficient algorithms plus examples. Transportation Research Part B: Methodological, 31 (5), 357–379. https://doi.org/10.1016/S0191-2615(96)00034-3

Schulz, A., & Stier-Moses, N. (2006). Efficiency and fairness of system-optimal routing with user constraints. Networks, 48 (4), 223–234. https://doi.org/10.1002/net.20133

Bayram, V., Tansel, B., & Yaman, H. (2015). Compromising system and user interests in shelter location and evacuation planning. Transportation Research Part B: Methodological, 72 , 146–163. https://doi.org/10.1016/j.trb.2014.11.010

Ferguson, B., & Marden, J. (2021). The impact of fairness on performance in congestion networks. In 2021 American control conference (ACC) (pp. 4521–4526). https://doi.org/10.23919/ACC50511.2021.9483197 .

Lujak, M., Giordani, S., & Ossowski, S. (2014). Route guidance: Bridging system and user optimization in traffic assignment. Neurocomputing . https://doi.org/10.1016/j.neucom.2014.08.071

Koutsoupias, E., & Papadimitriou, C. (1999). Worst-case equilibria. In Theoretical aspects of computer science .

Basu, S., Yang, G., Lianeas, T., Nikolova, E., & Chen, Y. (2017). Reconciling selfish routing with social good. CoRR arXiv:1707.00208 .

Cole, R., Dodis, Y., & Roughgarden, T. (2003). Pricing network edges for heterogeneous selfish users. In Proceedings of the thirty-fifth annual ACM symposium on theory of computing. STOC ’03 (pp. 521–530). Association for Computing Machinery, New York, NY, USA. https://doi.org/10.1145/780542.780618 .

Solodov, M. (2007). An explicit descent method for bilevel convex optimization. Journal of Convex Analysis, 14 (2), 227.

MathSciNet Google Scholar

Caltrans. (2020). US 101 South, Corridor system management plan, 2010. Retrieved July 01.

Still, G. (2018). Lectures on parametric optimization: An introduction. Optimization Online.

Yang, H., & Wang, X. (2011). Managing network mobility with tradable credits. Transportation Research Part B: Methodological, 45 (3), 580–594. https://doi.org/10.1016/j.trb.2010.10.002

Mitradjieva, M., & Lindberg, P. O. (2013). The stiff is moving—Conjugate direction Frank-Wolfe methods with applications to traffic assignment. Transportation Science, 47 (2), 280–293.

Buchhold, V., Sanders, P., & Wagner, D. (2019). Real-time traffic assignment using engineered customizable contraction hierarchies. ACM Journal of Experimental Algorithmics . https://doi.org/10.1145/3362693

Bar-Gera, H. (2002). Origin-based algorithm for the traffic assignment problem. Transportation Science, 36 (4), 398–417.

Bar-Gera, H. (2010). Traffic assignment by paired alternative segments. Transportation Research Part B: Methodological, 44 (8), 1022–1046. https://doi.org/10.1016/j.trb.2009.11.004

Patriksson, M. (2015). The traffic assignment problem: Models and methods. Courier Dover Publications, 2015.

Szeto, W., & Wong, S. (2012). Dynamic traffic assignment: Model classifications and recent advances in travel choice principles. Open Engineering, 2 (1), 1–18. https://doi.org/10.2478/s13531-011-0057-y

Wei, W., Hu, L., Wu, Q., & Ding, T. (2019). Efficient computation of user optimal traffic assignment via second-order cone and linear programming techniques. IEEE Access, 7 , 137010–137019. https://doi.org/10.1109/ACCESS.2019.2942497

Dezső, B., Jüttner, A., & Kovács, P. (2011). Lemon-an open source c++ graph template library. Electronic Notes in Theoretical Computer Science, 264 (5), 23–45.

Boost C++ Libraries. (2010). www.boost.org .

Sedgewick, R., & Wayne, K. (2011). Algorithms (4th ed., p. 1955). Addison-Wesley: Princeton University.

Transportation Networks for Research Core Team. (2016). Transportation networks for research. Retrieved Accessed Jan 20, 2021, from github.com/bstabler/TransportationNetworks.

Geisberger, R., Sanders, P., Schultes, D., & Vetter, C. (2012). Exact routing in large road networks using contraction hierarchies. Transportation Science, 46 (3), 388–404.

Correa, J., Schulz, A., & Stier-Moses, N. (2007). Fast, fair, and efficient flows in networks. Operations Research, 55 (2), 215–225. https://doi.org/10.1287/opre.1070.0383

Farris, F. (2010). The Gini index and measures of inequality. The American Mathematical Monthly, 117 (10), 851–864. https://doi.org/10.4169/000298910X523344

Wu, D., Yin, Y., Lawphongpanich, S., & Yang, H. (2012). Design of more equitable congestion pricing and tradable credit schemes for multimodal transportation networks. Transportation Research Part B: Methodological, 46 (9), 1273–1287.

Jalota, D., Solovey, K., Gopalakrishnan, K., Zoepf, S., Balakrishnan, H., & Pavone, M. (2021). When efficiency meets equity in congestion pricing and revenue refunding schemes. CoRR arXiv:2106.10407 .

Kleinberg, J. (2018). Inherent trade-offs in algorithmic fairness. SIGMETRICS Performance Evaluation Review, 46 (1), 40. https://doi.org/10.1145/3292040.3219634

Download references

Acknowledgements

This work was supported in part by the National Science Foundation (NSF) CAREER Award CMMI1454737, NSF Award 1830554, Toyota Research Institute, and the Ravitz Foundation.

Author information

Authors and affiliations.

Institute for Computational and Mathematical Engineering, Stanford University, 450 Serra Mall, Stanford, CA, 94305, USA

Devansh Jalota

Faculty of Electrical and Computer Engineering, Technion - Israel Institute of Technology, 3200003, Haifa, Israel

Kiril Solovey

Data Science Division, Lyft Inc, 185 Berry St, San Francisco, CA, 94107, USA

Matthew Tsao

MIT Mobility Initiative, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA, 02139, USA

Stephen Zoepf

Department of Aeronautics and Astronautics, Stanford University, 450 Serra Mall, Stanford, CA, 94305, USA

Marco Pavone

You can also search for this author in PubMed Google Scholar

Contributions

DJ contributed to all the sections of this paper, including algorithm design, theoretical analysis, and numerical experiments, wrote most sections in the initial draft of the paper, and revised it critically. KS contributed to the algorithm design, implemented the Frank Wolfe algorithm for the numerical experiments, wrote several sections in the initial draft of the paper, and revised the paper critically. MT contributed to several theoretical results and proofread the paper. SZ and MP initiated the overarching idea behind the paper and revised it critically.

Corresponding author

Correspondence to Devansh Jalota .

Ethics declarations

Conflict of interest.

The authors have no competing interests to declare that are relevant to the content of this article.

Ethical approval

Not applicable.

Consent to participate

Consent for publication, additional information, publisher's note.

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Jalota, D., Solovey, K., Tsao, M. et al. Balancing fairness and efficiency in traffic routing via interpolated traffic assignment. Auton Agent Multi-Agent Syst 37 , 32 (2023). https://doi.org/10.1007/s10458-023-09616-7

Download citation

Accepted : 09 July 2023

Published : 02 August 2023

DOI : https://doi.org/10.1007/s10458-023-09616-7

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Traffic assignment
Congestion games
Nash equilibria

Find a journal
Publish with us
Track your research

Plan to enjoy the weather this weekend? Here are the road closures to watch out for

As the sun peeks out of the clouds again this weekend, residents in the Valley are expected to get behind the wheel to enjoy the brief period of perfect weather before the summer heat arrives. If you plan on traveling, be aware of these notable closures along Interstate 10 and State Route 143.

The Arizona Department of Transportation advised drivers to prepare to spend extra time on the roads and plan alternate routes if necessary. Drivers were also asked to be ready to slow down and to merge safely when maneuvering through work zones.

Metro Phoenix drivers can check real-time travel and road conditions online at the Arizona Department of Transportations' az511.gov website.

Here's where the freeways will be closed or restricted and how to avoid traffic in those areas.

Eastbound I-10 closed between Loop 202 and I-17

Details: Eastbound I-10 closed between Loop 202 (South Mountain Freeway) and I-17 (Stack Interchange) for pavement improvement work.

Northbound Loop 202 ramps to eastbound I-10 closed.

When: From 9 p.m. Friday, March 22, to 5 a.m. Monday, March 25.

Alternate Routes: Drivers heading eastbound on I-10 can detour to southbound or eastbound Loop 202 and reconnect with the I-10 near Chandler Boulevard in the Ahwatukee area. Drivers can also use eastbound Van Buren Street or Buckeye Road to reach southbound I-17 or exit ahead of the closure and use eastbound McDowell Road or Thomas Road to reach northbound I-17.

Southbound SR 143 closed between Loop 202 and I-10

Details: Southbound SR 143 (Hohokam Expressway) closed between Loop 202 (Red Mountain Freeway) and I-10 for work on the I-10 Broadway Curve Improvement Project.

The westbound Loop 202 exit to Sky Harbor Airport closed.
The eastbound Sky Harbor Boulevard ramp to southbound SR 143 (airport exit ramp).
The eastbound I-10 off-ramp at 48th Street will be limited to right turns only allowing no access to northbound SR 143.

When: From 10 p.m. Friday, March 22, to 4 p.m. Monday, March 25.

Alternate Routes: Drivers can take the westbound Loop 202 (Red Mountain Freeway) to eastbound I-10 to reach destinations including Sky Harbor. Loop 101 drivers can also take southbound 44th Street to reach the airport.

Eastbound Loop 101 narrowed to two lanes between Seventh Street and 16th Street

Details: Eastbound Loop 101 (Pima Freeway) narrowed to two lanes between Seventh Street and 16th Street in north Phoenix for freeway wall project.

Plan for 16th Street to be closed in both directions beneath the Loop 101 overpass.

Alternate Routes: Drivers can travel around the 16th Street closure by using the Loop 101/Beardsley frontage roads and turning at either Cave Creek Road or Seventh Street.

IMAGES

Traffic assignment classification, components, and traffic assignment
Internet Routing and Traffic Engineering
NEW Azure Traffic Manager Tutorial
Traffic Assignment Process.
Azure Traffic Manager
Inputs and Outputs of Traffic Assignment

VIDEO

Email Traffic—Post Office Protocol
Traffic Section Part1

COMMENTS

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.
3.6: 3-6 Route Choice
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.
Traffic Networks: Dynamic Traffic Routing, Assignment, and ...
Alternatively, the system‐optimum traffic assignment is achieved at a volume distribution of 500 veh on routes 1 and 2, respectively. From a traffic engineering point of view, the difference in total travel time between the system and user‐optimum traffic assignment (16,250 versus 16,667 veh‐min) is of interest.
A.8
Cost Minimization and Efficiency Maximization in Route Selection 3. Traffic Assignment. Contemporary transportation networks are intensively used and congested to various degrees, notably road transportation systems in urban areas. Less known is the spatial logic behind the generation, attraction, and distribution of traffic on a network.
Traffic Assignment Optimization Using Flow-Based Multi-maps
In traffic assignment, vehicle trips (demand) are assigned to the traffic network (supply) by strategically performing route choices. It is an NP-hard, convex problem (Traffic Assignment Problem) that requires approximate methods [ 12 , 13 ].
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,
Dynamic Traffic Routing, Assignment, and Assessment of ...
The traffic modeler can either combine traffic modeling with traffic assignment or alternatively utilize the routes computed by the O-D estimator to route traffic. This closed loop optimal control framework can involve a single loop or in most cases may involve an iterative loop to attain equilibrium.
Dynamic traffic assignment: A review of the ...
A semi-dynamic traffic assignment model can be considered a series of connected STA models (e.g., Nakayama and Conors, 2014). Unlike STA, a semi-dynamic traffic assignment model has multiple time periods for route choice and allows the residual traffic of one period to transfer to the following time periods.
Balancing fairness and efficiency in traffic routing via interpolated
To obtain feasible solutions to the β-fair SO problem while achieving high system efficiency, we develop a new convex program, the interpolated traffic assignment problem (I-TAP), which interpolates between a fairness-promoting and an efficiency-promoting traffic-assignment objective. We evaluate the efficacy of I-TAP through theoretical ...
Traffic Networks: Dynamic Traffic Routing, Assignment, and Assessment
The main object of the following paper is to describe and illustrate some approximation methods that can be used to obtain rough estimates of queue lengths, delays, etc., for various queueing problemis, particularly highway traffic intersection problems, which may be too difficult to solve exactly, or if solved exactly give formulas that are more detailed than one needs for the purpose of ...
Traffic Assignment Problem
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.
Traffic assignment: A survey of mathematical models and techniques
It first presents the steady-state traffic assignment problem formulation which is also called static assignment, followed by Dynamic Traffic Assignment (DTA), where the traffic demand on the network is time varying. ... Dynamic travel routing problem is also reviewed in the variational inequality setting. DTA problem is shown in discrete and ...
Graphical solution for system optimum dynamic traffic assignment with
This paper analyzes the dynamic traffic assignment problem on a three-alternative network with day-based incentive routing strategies by using graphical solution method. It is assumed that the cumulative count curve of vehicles is known and that the arrival rate is unimodal.
A scalable vehicle assignment and routing strategy for real-time on
Similarly, Seo and Asakura (2021) proposed a linear programming problem to optimise fleet size, road network design, and parking lot deployment of a SAV system, where traffic congestion is considered via a point-queue dynamic traffic assignment model. However, vehicle routing and static ridesharing are considered at macroscopic level, while ...
cppRouting v3 : Algorithms for Routing and Solving the Traffic ...
cppRouting is an R package which provide routing algorithms (shortest paths/distances, isochrones) and traffic assignment solvers on non-negative weighted graphs. cppRouting is characterized by :. its ability to work on large road graphs (country/continent scale) its large choice of one-to-one shortest path algorithms; its implementation of contraction hierarchies and associated routing algorithms
Application of Traffic Weighted Multi-Maps Based on Disjoint Routing
Classical traffic assignment and routing works consider that all the drivers share the same network view (map), assuming this is a fundamental hypothesis. Thus, algorithms rely on it and focus on other topics, such as edge occupancy and toll costs.
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. ... The simulation models themselves, however, do not predict the routing of traffic, and therefore are unable ...
PDF Network Analysis Module (NM-NAM)
NM-NAM Operating Topologies and IP Address Assignments This section includes the following topics: † Management Traffic—Choose One of the NM-NAM Interfaces, page 6 † Monitored Traffic—Use One or Both of the NM-NAM Interfaces, page 7 † Sample Operating Topologies, page 8 Management Traffic—Choose One of the NM-NAM Interfaces
PDF Unity EdgeConnect SD-WAN Solution
traffic from the data center across the WAN to help ... > Routing: EdgeConnect supports standard Layer 2 and Layer 3 open networking protocols such as VLAN (802.1Q), LAG (802.3ad), IPv4 and IPv6 ... assignment of business intent policies to ensure faster and easier connectivity across multiple
Official MapQuest
Official MapQuest website, find driving directions, maps, live traffic updates and road conditions. Find nearby businesses, restaurants and hotels. Explore!
One-way Traffic Control for Lagunitas Creek Bridge ...
Schedule: Tuesday, March 26 through Thursday, March 28 . MARIN COUNTY — Caltrans is scheduled to start confirmatory drilling for the Lagunitas Creek Bridge Replacement Project near Point Reyes Station from Tuesday, March 26 through Thursday, March 28 from 8 a.m. until 4 p.m. One-way traffic control will be utilized during those hours. No night work is expected for the drilling.
Mississippi River Traffic Threatened by Climate-Driven Drought For a
The Mississippi River is at risk of bottlenecks for a third straight year as warm, dry spring weather and low winter snowpack limit the amount of water feeding into it, according to the National ...
Traffic Advisory Tuolumne County 3/21/24
One-Way Traffic Control on State Route 120. Tuolumne County - The California Department of Transportation (Caltrans) will perform one-way traffic control on State Route 120 (SR-120). Work is Scheduled to occur as follows: State Route 120. One-way traffic control on SR-120 between Sweetwater Campground and Sawmill Mountain Road, east of Groveland, for slope repair/clearing beginning Monday ...
North Jersey traffic
Bergen County traffic. Construction on E. Palisade Avenue will close all lanes in both directions between Route 9W and Summit Street in Englewood Cliffs, 7 a.m. to 3 p.m. March 22.; Construction ...
INTEGRATION Framework for Modeling Eco-routing ...
The paper presents the INTEGRATION microscopic traffic assignment and simulation framework for modeling eco-routing strategies. Two eco-routing algorithms are developed: one based on vehicle sub-populations (ECO-Subpopulation Feedback Assignment or ECO-SFA) and another based on individual agents (ECO-Agent Feedback Assignment or ECO-AFA).
M25 closure: Everything you need to know as drivers face delays on
National Highways estimated that the average journey time along the diversion route on Saturday morning was 25 minutes. The closure of a five-mile stretch between junctions 10 and 11 on the UK's ...
Balancing fairness and efficiency in traffic routing via interpolated
System optimum (SO) routing, wherein the total travel time of all users is minimized, is a holy grail for transportation authorities. However, SO routing may discriminate against users who incur much larger travel times than others to achieve high system efficiency, i.e., low total travel times. To address the inherent unfairness of SO routing, we study the $${\\beta }$$ β -fair SO problem ...
Traffic Advisory Amador County 3/21/24
One-Way Traffic Control State Routes 49 & 88. Amador County - The California Department of Transportation (Caltrans) will perform various lane closures on State Route 49 (SR-49) and SR-88 for maintenance and construction operations. Work to occur as follows; State Route 49. Long-term, one-way traffic control at the Mokelumne River Bridge (Amador/Calaveras County line) for bridge work through ...
Closures along the I-10 and SR 143 this March weekend
When: From 9 p.m. Friday, March 22, to 5 a.m. Monday, March 25. Alternate Routes: Drivers heading eastbound on I-10 can detour to southbound or eastbound Loop 202 and reconnect with the I-10 near ...
PDF Vehicles Involved: MODEL DAKOTA UNIT
On 03/14/2024, at approx. 0355 hours, the Pennsylvania State Police were dispatched to 3928 Route 249 Chatham Township, PA 16950, Tioga County, PA for a report of a motorcycle crash. This crash occurred as Unit 1 was traveling straight on Route 249 when a deer crossed from the right side of the roadway. Unit 1 struck the deer near the 12