Choice
CEE Route 320 Fall 2008
CEE 320 Steve Muench
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)
Desired output from the traffic forecasting process: how many vehicles at any time on a roadway
CEE 320 Fall 2008
Complexity
Route choice decisions are primarily a function of travel times, which are determined by traffic flow
Traffic flow Travel time Relationship captured by highway performance function
CEE 320 Fall 2008
Outline
1. 2. 3. 4. General HPF Functional Forms Basic Assumptions Route Choice Theories
a. User Equilibrium b. System Optimization c. Comparison
CEE 320 Fall 2008
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
1. 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
CEE 320 Fall 2008
HPF Functional Forms
Travel Time
Linear
Common Non-linear HPF
v T = T0 1 + c
Free Flow
Non-Linear
Capacity
from the Bureau of Public Roads (BPR)
Traffic Flow (veh/hr)
CEE 320 Fall 2008
Speed vs. Flow
2 u q = k j u - uf
uf Free Flow Speed
Speed (mph)
Uncongested Flow um
Congested Flow
CEE 320 Fall 2008
Highest flow, Flow (veh/hr) capacity, qm
qm is bottleneck discharge rate
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
CEE 320 Fall 2008
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"
CEE 320 Fall 2008
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.
CEE 320 Fall 2008
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
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 w = Flow
CEE 320 Fall 2008
xn 0 for all routes
Formulating the UE Problem
min
n xn
t ( w) dw =
n 0 n n
min t n ( w) dw
0
xn
=
n
xn
min t ( w) dw
0
n = Route between given O-D pair tn(w)dw = HPF for a specific route as a function of flow w = Flow
CEE 320 Fall 2008
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..
Route 1
CEE 320 Fall 2008
City
Route 2
Suburb
Example: Compute UE travel times on the two routes
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..
Determine HPFs
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. TT1 = 6 + 4x1 TT2 = 4 + x22 Flow constraint: x1 + x2 = 4.5
CEE 320 Fall 2008
Example: Compute UE travel times on the two routes
Route use check (will both routes be used?)
All or nothing assignment on Route 1
TT1 = 6 + 4( 4.5) = 24 minutes
TT2 = 4 + ( 0 ) = 4 minutes
2
If all the traffic is on Route 1 then Route 2 is the desirable or choice
All nothing assignment on Route 2
TT1 = 6 + 4( 0 ) = 6 minutes
TT2 = 4 + ( 4.5) = 24.25 minutes
2
If all the traffic is on Route 2 then Route 1 is the desirable choice
Therefore, both routes will be used
CEE 320 Fall 2008
Example: Solution
Apply Wardrop's 1st principle requirements. All routes used will have equal times. 6 + 4x1 = 4 + x22 x1 + x2 = 4.5 Substituting and solving: 6 + 4x1 = 4 + (4.5 x1)2 6 + 4x1 = 4 + 20.25 9x1 + x12 x12 13x1 + 18.25 = 0 x1 = 1.6 or 11.4 (total is 4.5 so x1 = 1.6 or 1,600 vehicles) x2 = 4.5 1.6 = 2.9 or 2,900 vehicles Check answer: TT1 = 6 + 4(1.6) = 12.4 minutes TT2 = 4 + (2.9)2 = 12.4 minutes
CEE 320 Fall 2008
Example: Mathematical Solution
minmize S ( x ) =
n xn
t n ( w) dw
0
S ( x ) = (6 + 4w)dw + (4 + w 2 )dw
0 0 3 x2 2 S ( x ) = 6 x1 + 2 x1 + 4 x 2 + 3
x1
x2
S ( x ) = 6w + 2w
2 x1 0
w + 4w + 3
3 x2
0
x1 + x2 = 4.5
x13 4.5 3 2 2 S ( x ) = 6 x1 + 2 x + 18 - 4 x1 + - 4.5 x1 + 4.5 x1 - 3 3
2 1
for a minimum :
CEE 320 Fall 2008
dS = 6 + 4 x1 - 4 - 20.25 + 9 x1 - x12 = 0 dx
simplifying : x12 - 13 x1 + 18.25 = 0 Same equation as before
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.
CEE 320 Fall 2008
Not stable because individuals will be tempted to choose different route.
Formulating the SO Problem
Finding the set of flows that minimizes the following function:
min xnt n ( xn ) = min xnt n ( xn )
n n
n = Route between given O-D pair tn(xn) = travel time for a specific route xn = Flow on a specific route
CEE 320 Fall 2008
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
CEE 320 Fall 2008
City
Route 2
Suburb
Example: Solution
1. Determine HPFs as before
HPF1 = 6 + 4x1 HPF2 = 4 + x22 Flow constraint: x1 + x2 = 4.5
1. Formulate the SO equation
2 S ( x ) = t i xi = ( 6 + 4 x1 ) x1 + 4 + x 2 x 2 i =1 n
(
)
Use the flow constraint(s) to get the equation into o...