route choice - uw courses web servercourses.washington.edu/cee320ag/lecture/route_choice.pdf ·...
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![Page 1: Route Choice - UW Courses Web Servercourses.washington.edu/cee320ag/Lecture/Route_Choice.pdf · Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are](https://reader034.vdocuments.net/reader034/viewer/2022050208/5f5b6f331fafa546fd2b3c67/html5/thumbnails/1.jpg)
CE
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Route
Choice
![Page 2: Route Choice - UW Courses Web Servercourses.washington.edu/cee320ag/Lecture/Route_Choice.pdf · Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are](https://reader034.vdocuments.net/reader034/viewer/2022050208/5f5b6f331fafa546fd2b3c67/html5/thumbnails/2.jpg)
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Route Choice
• Final step in sequential approach– Trip generation (number of trips)
– Trip distribution (origins and destinations)
– Mode choice (bus, train, etc.)
– Route choice (specific roadways used for each origin and destination)
• Desired output from the traffic forecasting process: how many vehicles at any time on a roadway
![Page 3: Route Choice - UW Courses Web Servercourses.washington.edu/cee320ag/Lecture/Route_Choice.pdf · Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are](https://reader034.vdocuments.net/reader034/viewer/2022050208/5f5b6f331fafa546fd2b3c67/html5/thumbnails/3.jpg)
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Complexity
• Route choice decisions are primarily a
function of travel times, which are
determined by traffic flow
Travel time
Traffic flow
Relationship can be
captured in a variety of
ways, including by
highway performance
function
![Page 4: Route Choice - UW Courses Web Servercourses.washington.edu/cee320ag/Lecture/Route_Choice.pdf · Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are](https://reader034.vdocuments.net/reader034/viewer/2022050208/5f5b6f331fafa546fd2b3c67/html5/thumbnails/4.jpg)
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Outline
1. General
2. HPF Functional Forms
3. Basic Assumptions
4. Route Choice Theories
a. User Equilibrium
b. System Optimization
c. Comparison
![Page 5: Route Choice - UW Courses Web Servercourses.washington.edu/cee320ag/Lecture/Route_Choice.pdf · Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are](https://reader034.vdocuments.net/reader034/viewer/2022050208/5f5b6f331fafa546fd2b3c67/html5/thumbnails/5.jpg)
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Basic Assumptions
1. Travelers select routes on the basis of route travel times only– People select the path with the shortest TT
– Premise: TT is the major criterion, quality factors such as “scenery” do not count
– Generally, this is reasonable
2. Travelers know travel times on all available routes between their origin and destination – Strong assumption: Travelers may not use all available
routes, and may base TTs on perception
3. Travelers all make this choice at the same time
![Page 6: Route Choice - UW Courses Web Servercourses.washington.edu/cee320ag/Lecture/Route_Choice.pdf · Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are](https://reader034.vdocuments.net/reader034/viewer/2022050208/5f5b6f331fafa546fd2b3c67/html5/thumbnails/6.jpg)
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HPF Functional FormsT
ravel T
ime
Traffic Flow (veh/hr)Capacity
Linear
Non-LinearFree
Flow
c
vTT 10
Common Non-linear HPF
from the Bureau of Public Roads (BPR)
![Page 7: Route Choice - UW Courses Web Servercourses.washington.edu/cee320ag/Lecture/Route_Choice.pdf · Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are](https://reader034.vdocuments.net/reader034/viewer/2022050208/5f5b6f331fafa546fd2b3c67/html5/thumbnails/7.jpg)
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Speed vs. Flow
f
ju
uukq
2
Flow (veh/hr)
Spee
d (
mph)
uf
Free Flow Speed
Highest flow,
capacity, qm
Uncongested Flow
Congested Flow
um
qm is bottleneck discharge rate
![Page 8: Route Choice - UW Courses Web Servercourses.washington.edu/cee320ag/Lecture/Route_Choice.pdf · Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are](https://reader034.vdocuments.net/reader034/viewer/2022050208/5f5b6f331fafa546fd2b3c67/html5/thumbnails/8.jpg)
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Theory of User Equilibrium
Travelers will select a route so as to minimize
their personal travel time between their origin
and destination. User equilibrium (UE) is said to
exist when travelers at the individual level
cannot unilaterally improve their travel times by
changing routes.
Frank Knight, 1924
![Page 9: Route Choice - UW Courses Web Servercourses.washington.edu/cee320ag/Lecture/Route_Choice.pdf · Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are](https://reader034.vdocuments.net/reader034/viewer/2022050208/5f5b6f331fafa546fd2b3c67/html5/thumbnails/9.jpg)
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Wardrop
Wardrop’s 1st principle
“The journey times in all routes actually used are equal and less
than those which would be experienced by a single vehicle on any
unused route ”
Wardrop’s 2nd principle
“At equilibrium the average journey time is minimum”
![Page 10: Route Choice - UW Courses Web Servercourses.washington.edu/cee320ag/Lecture/Route_Choice.pdf · Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are](https://reader034.vdocuments.net/reader034/viewer/2022050208/5f5b6f331fafa546fd2b3c67/html5/thumbnails/10.jpg)
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Theory of System-Optimal Route Choice
Preferred routes are those, which minimize total
system travel time. With System-Optimal (SO)
route choices, no traveler can switch to a
different route without increasing total system
travel time. Realistically, travelers will likely
switch to non-System-Optimal routes to improve
their own TTs.
![Page 11: Route Choice - UW Courses Web Servercourses.washington.edu/cee320ag/Lecture/Route_Choice.pdf · Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are](https://reader034.vdocuments.net/reader034/viewer/2022050208/5f5b6f331fafa546fd2b3c67/html5/thumbnails/11.jpg)
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Formulating the UE Problem
dwwtnx
n
n 0
min
Finding the set of flows that equates TTs on all
used routes can be cumbersome.
Alternatively, one can minimize the following
function:
n = Route between given O-D pair
tn(w)dw = HPF for a specific route as a function of flow
w = Flow
xn ≥ 0 for all routes
Minimize travel times
![Page 12: Route Choice - UW Courses Web Servercourses.washington.edu/cee320ag/Lecture/Route_Choice.pdf · Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are](https://reader034.vdocuments.net/reader034/viewer/2022050208/5f5b6f331fafa546fd2b3c67/html5/thumbnails/12.jpg)
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Formulating the UE Problem
dwwt
dwwtdwwt
n
nn
x
n
n
x
n
n
x
n
n
0
00
min
minmin
n = Route between given O-D pair
tn(w)dw = HPF for a specific route as a function of flow
w = Flow
xn ≥ 0 for all routes
![Page 13: Route Choice - UW Courses Web Servercourses.washington.edu/cee320ag/Lecture/Route_Choice.pdf · Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are](https://reader034.vdocuments.net/reader034/viewer/2022050208/5f5b6f331fafa546fd2b3c67/html5/thumbnails/13.jpg)
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Example (UE)
Two routes connect a city and a suburb. During the peak-hour morning
commute, a total of 4,500 vehicles travel from the suburb to the city.
Route 1 has a 60-mph speed limit and is 6 miles long. Route 2 is half as
long with a 45-mph speed limit. The HPFs for the route 1 & 2 are as
follows:
•Route 1 HPF increases at the rate of 4 minutes for every additional
1,000 vehicles per hour.
•Route 2 HPF increases as the square of volume of vehicles in
thousands per hour..
Route 1
Route 2City Suburb
![Page 14: Route Choice - UW Courses Web Servercourses.washington.edu/cee320ag/Lecture/Route_Choice.pdf · Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are](https://reader034.vdocuments.net/reader034/viewer/2022050208/5f5b6f331fafa546fd2b3c67/html5/thumbnails/14.jpg)
CE
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Example: Compute UE travel times on
the two routes
![Page 15: Route Choice - UW Courses Web Servercourses.washington.edu/cee320ag/Lecture/Route_Choice.pdf · Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are](https://reader034.vdocuments.net/reader034/viewer/2022050208/5f5b6f331fafa546fd2b3c67/html5/thumbnails/15.jpg)
CE
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Example: Compute UE travel times on
the two routes
![Page 16: Route Choice - UW Courses Web Servercourses.washington.edu/cee320ag/Lecture/Route_Choice.pdf · Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are](https://reader034.vdocuments.net/reader034/viewer/2022050208/5f5b6f331fafa546fd2b3c67/html5/thumbnails/16.jpg)
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Example: Compute UE travel times on
the two routes
![Page 17: Route Choice - UW Courses Web Servercourses.washington.edu/cee320ag/Lecture/Route_Choice.pdf · Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are](https://reader034.vdocuments.net/reader034/viewer/2022050208/5f5b6f331fafa546fd2b3c67/html5/thumbnails/17.jpg)
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Example: Compute UE travel times on
the two routes
![Page 18: Route Choice - UW Courses Web Servercourses.washington.edu/cee320ag/Lecture/Route_Choice.pdf · Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are](https://reader034.vdocuments.net/reader034/viewer/2022050208/5f5b6f331fafa546fd2b3c67/html5/thumbnails/18.jpg)
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Example: Compute UE travel times on
the two routes
![Page 19: Route Choice - UW Courses Web Servercourses.washington.edu/cee320ag/Lecture/Route_Choice.pdf · Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are](https://reader034.vdocuments.net/reader034/viewer/2022050208/5f5b6f331fafa546fd2b3c67/html5/thumbnails/19.jpg)
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Theory of System-Optimal Route Choice
Preferred routes are those, which minimize total
system travel time. With System-Optimal (SO)
route choices, no traveler can switch to a
different route without increasing total system
travel time. Travelers can switch to routes
decreasing their TTs but only if System-Optimal
flows are maintained. Realistically, travelers will
likely switch to non-System-Optimal routes to
improve their own TTs.
Not stable because individuals will be
tempted to choose different route.
![Page 20: Route Choice - UW Courses Web Servercourses.washington.edu/cee320ag/Lecture/Route_Choice.pdf · Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are](https://reader034.vdocuments.net/reader034/viewer/2022050208/5f5b6f331fafa546fd2b3c67/html5/thumbnails/20.jpg)
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Formulating the SO Problem
n
n
nnn
n
nn xtxxtx minmin
Finding the set of flows that minimizes the
following function:
n = Route between given O-D pair
tn(xn) = travel time for a specific route
xn = Flow on a specific route
Minimize travel time times flow
![Page 21: Route Choice - UW Courses Web Servercourses.washington.edu/cee320ag/Lecture/Route_Choice.pdf · Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are](https://reader034.vdocuments.net/reader034/viewer/2022050208/5f5b6f331fafa546fd2b3c67/html5/thumbnails/21.jpg)
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Example (SO)
Two routes connect a city and a suburb. During the peak-hour morning
commute, a total of 4,500 vehicles travel from the suburb to the city.
Route 1 has a 60-mph speed limit and is 6 miles long. Route 2 is half as
long with a 45-mph speed limit. The HPFs for the route 1 & 2 are as
follows:
•Route 1 HPF increases at the rate of 4 minutes for every additional
1,000 vehicles per hour.
•Route 2 HPF increases as the square of volume of vehicles in
thousands per hour. Compute UE travel times on the two routes.
Route 1
Route 2City Suburb
![Page 22: Route Choice - UW Courses Web Servercourses.washington.edu/cee320ag/Lecture/Route_Choice.pdf · Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are](https://reader034.vdocuments.net/reader034/viewer/2022050208/5f5b6f331fafa546fd2b3c67/html5/thumbnails/22.jpg)
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Example: Solution
![Page 23: Route Choice - UW Courses Web Servercourses.washington.edu/cee320ag/Lecture/Route_Choice.pdf · Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are](https://reader034.vdocuments.net/reader034/viewer/2022050208/5f5b6f331fafa546fd2b3c67/html5/thumbnails/23.jpg)
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Example: Solution
![Page 24: Route Choice - UW Courses Web Servercourses.washington.edu/cee320ag/Lecture/Route_Choice.pdf · Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are](https://reader034.vdocuments.net/reader034/viewer/2022050208/5f5b6f331fafa546fd2b3c67/html5/thumbnails/24.jpg)
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Compare UE and SO Solutions
• User equilibrium • System optimization
Route 1
Route 2City Suburb
![Page 25: Route Choice - UW Courses Web Servercourses.washington.edu/cee320ag/Lecture/Route_Choice.pdf · Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are](https://reader034.vdocuments.net/reader034/viewer/2022050208/5f5b6f331fafa546fd2b3c67/html5/thumbnails/25.jpg)
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Why are the solutions different?
• Why is total travel time shorter?
• Notice in SO we expect some drivers to
take a longer route.
• How can we force the SO?
• Why would we want to force the SO?
![Page 26: Route Choice - UW Courses Web Servercourses.washington.edu/cee320ag/Lecture/Route_Choice.pdf · Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are](https://reader034.vdocuments.net/reader034/viewer/2022050208/5f5b6f331fafa546fd2b3c67/html5/thumbnails/26.jpg)
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0
5
10
15
20
25
30
0
300
600
900
1200
1500
1800
2100
2400
2700
3000
3300
3600
3900
4200
4500
flow (link 1)
travel ti
me (
min
ute
s)
TT1
TT2
UE
SO
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Total Travel time is Minimum at SO
0
20000
40000
60000
80000
100000
120000
0 500 1000 1500 2000 2500 3000 3500 4000 4500
Flow (link 1)
To
tal T
ravel ti
me (
min
)
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• By asking one driver to take 3 minutes
longer, I save more than 3 minutes in the
reduced travel time of all drivers (non-
linear)
• Total travel time if X1=1600 is 55829
• Total travel time if X1=1601 is 55819