Choice
CEE Route 320 Spring 2007
CEE 320 Steve Muench
Outline
1. 2. 3. 4. General HPF Functional Forms Basic Assumptions Route Choice Theories
a. User Equilibrium b. System Optimization c. Comparison
CEE 320 Spring 2007
Route Choice
Equilibrium problem for alternate routes Requires relationship between:
Travel time (TT) Traffic flow (TF)
Highway Performance Function (HPF)
Common term for this relationship
CEE 320 Spring 2007
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 Spring 2007
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
CEE 320 Spring 2007
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
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. Wardrops 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
CEE 320 Spring 2007
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 w = Flow
CEE 320 Spring 2007
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.
Route 1
CEE 320 Spring 2007
City
Route 2
Suburb
Example: Solution
1. 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. HPF1 = 6 + 4x1 HPF2 = 4 + x22 Flow constraint: x1 + x2 = 4.5
1. Route use check (will both routes be used?)
All or nothing assignment on Route 1
TT1 = 6 + 4( 4.5) = 24 minutes 2 TT2 = 4 + ( 0) = 4 minutes TT1 = 6 + 4( 0 ) = 6 minutes 2 TT2 = 4 + ( 4.5) = 24.25 minutes
If all the traffic is on Route 1 then Route 2 is the desirable choice
All or nothing assignment on Route 2
If all the traffic is on Route 2 then Route 1 is the desirable choice
CEE 320 Spring 2007
Therefore, routes both will be used
Example: Solution
1. Equate TTs
Apply Wardrops 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.
CEE 320 Spring 2007
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 Spring 2007
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
Wardrops 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.
CEE 320 Spring 2007
Formulating the SO Problem
Finding the set of flows that minimizes the following function:
S ( x ) = xn t n ( xn )
n
n = Route between given O-D pair tn(xn) = travel time for a specific route xn = Flow on a specific route
CEE 320 Spring 2007
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 vehic...