Transcript Route Choice CEE 320 Steve Muench

CEE 320
Winter 2006
Route
Choice
CEE 320
Steve Muench
Outline
1.
2.
3.
4.
General
HPF Functional Forms
Basic Assumptions
Route Choice Theories
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Winter 2006
a. User Equilibrium
b. System Optimization
c. Comparison
Route Choice
• Equilibrium problem for alternate routes
• Requires relationship between:
– Travel time (TT)
– Traffic flow (TF)
• Highway Performance Function (HPF)
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– Common term for this relationship
Travel Time
HPF Functional Forms
Free
Flow



v


T  T0 1     


c




from the Bureau of Public Roads (BPR)
Non-Linear
Traffic Flow (veh/hr)
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Winter 2006
Common Non-linear HPF
Linear
Capacity
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
–
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Strong assumption: Travelers may not use all available
routes, and may base TTs on perception
Some studies say perception bias is small
Theory of User Equilibrium
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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.
Wadrop definition: A.K.A. Wardrop’s 1st principle
“The travel time between a specified origin & destination
on all used routes is equal, and less than or equal to the
travel time that would be experienced by a traveler on any
unused route”
Formulating the UE Problem
Finding the set of flows that equates TTs on all
used routes can be cumbersome.
Alternatively, one can minimize the following
function:
min S x   
n
xn
 t wdw
n
0
n = Route between given O-D pair
tn(w)dw = HPF for a specific route as a function of flow
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w = Flow
xn ≥ 0 for all routes
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. Compute UE travel times on the two routes.
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Winter 2006
Route 1
City
Route 2
Suburb
Example: Solution
1. Determine HPFs
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–
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Route 1 free-flow TT is 6 minutes, since at 60 mph, 1 mile takes 1 minute.
Route 2 free-flow TT is 4 minutes, since at 45 mph, 1 mile takes 4/3 minutes.
HPF1 = 6 + 4x1
HPF2 = 4 + x22
Flow constraint: x1 + x2 = 4.5
2. Route use check (will both routes be used?)
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All or nothing assignment on Route 1
TT1  6  44.5  24 minutes
2
TT2  4  0  4 minutes
If all the traffic is on Route 1 then
Route 2 is the desirable choice
All or nothing assignment on Route 2
TT1  6  40   6 minutes
2
TT2  4  4.5  24.25 minutes
Therefore, both routes will be used
If all the traffic is on Route 2 then
Route 1 is the desirable choice
Example: Solution
3. Equate TTs
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Apply Wardrop’s 1st principle requirements. All routes used will have equal
times, and ≤ those on unused routes. Hence, if flows are distributed between
Route1 and Route 2, then both must be used on travel time equivalency
bases.
Example: Mathematical Solution
minmize S x   
n
S x   6w  2w
x1  x2  4.5
xn
 t n wdw

0
2 x1
0
3 x2
w
 4w 
3

0
x1
x2
0
0
S x    (6  4w)dw   (4  w 2 )dw

x23 
2
S x   6 x1  2 x1  4 x2  
3

3
3
x
4
.
5
2
2
 S x   6 x1  2 x  18  4 x1 
 4.5 x1  4.5 x1  1
3
3
2
1
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dS
for a minimum :
 6  4 x1  4  20.25  9 x1  x12  0
dx
simplifyin g : x12  13x1  18.25  0  Same equation as before
Theory of System-Optimal Route Choice
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Wardrop’s Second Principle:
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.
Formulating the SO Problem
Finding the set of flows that minimizes the
following function:
S x    xntn xn 
n
n = Route between given O-D pair
tn(xn) = travel time for a specific route
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xn = Flow on a specific route
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.
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Winter 2006
Route 1
City
Route 2
Suburb
Example: Solution
1. Determine HPFs as before
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HPF1 = 6 + 4x1
HPF2 = 4 + x22
Flow constraint: x1 + x2 = 4.5
2. Formulate the SO equation
S  x    t i xi  6  4 x1 x1  4  x 22 x 2
n
i 1
–
Use the flow constraint(s) to get the equation into one variable
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S ( x)  6(4.5  x2 )  4(4.5  x2 ) 2  4 x2  x23
Example: Solution
1. Minimize the SO function
dS
 6  42 4.5  x 2  1  4  3 x 22  0
dx

3x22  8x2  38  0
2. Solve the minimized function
x2  2.467
x1  2.033
3. Find the total vehicular delay
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n
S  x    t i xi  14.132033  10.082467   53,592 vehicle - minutes
i 1
Compare UE and SO Solutions
• User equilibrium
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t1 = 12.4 minutes
t2 = 12.4 minutes
x1 = 1,600 vehicles
x2 = 2,900 vehicles
tixi = 55,800 veh-min
• System optimization
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t1 = 14.3 minutes
t2 = 10.08 minutes
x1 = 2,033 vehicles
x2 = 2,467 vehicles
tixi = 53,592 veh-min
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Route 1
City
Route 2
Suburb
Primary Reference
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Winter 2006
• Mannering, F.L.; Kilareski, W.P. and Washburn, S.S. (2005).
Principles of Highway Engineering and Traffic Analysis, Third
Edition. Chapter 8