Route%20Choice

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Route Choice
CEE 320Steve Muench
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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)
• Desired output from the traffic forecasting
  process how many vehicles at any time on a
  roadway

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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
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Outline

• General
• HPF Functional Forms
• Basic Assumptions
• Route Choice Theories
• User Equilibrium
• System Optimization
• Comparison

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Basic Assumptions

• 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
• 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

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HPF Functional Forms
Common Non-linear HPF
Linear
Travel Time
FreeFlow
Non-Linear
from the Bureau of Public Roads (BPR)
Capacity
Traffic Flow (veh/hr)
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Speed vs. Flow
ufFree Flow Speed
Uncongested Flow
um
Speed (mph)
Highest flow, capacity, qm
Congested Flow
Flow (veh/hr)
qm is bottleneck discharge rate
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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
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Wardrop
Wardrops 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
Wardrops 2nd principle At equilibrium the
average journey time is minimum
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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.
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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
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
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Formulating the UE Problem
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
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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
Suburb
Route 2
City
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Example Compute UE travel times on the two routes

• 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

• 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..

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Example Compute UE travel times on the two routes

• Route use check (will both routes be used?)
• All or nothing assignment on Route 1
• All or nothing assignment on Route 2
• Therefore, both routes will be used

If all the traffic is on Route 1 then Route 2 is
the desirable choice
If all the traffic is on Route 2 then Route 1 is
the desirable choice
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Example Solution

• Apply Wardrops 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

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Example Mathematical Solution
?
?
?
? Same equation as before
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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.
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Formulating the SO Problem
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
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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
Suburb
Route 2
City
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Example Solution

• Determine HPFs as before
• HPF1 6 4x1
• HPF2 4 x22
• Flow constraint x1 x2 4.5
• Formulate the SO equation
• Use the flow constraint(s) to get the equation
  into one variable

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Example Solution

• Minimize the SO function
• Solve the minimized function

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Example Solution

• Solve the minimized function
• Find travel times
• Find the total vehicular delay

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Compare UE and SO Solutions

• User equilibrium
• t1 12.4 minutes
• t2 12.4 minutes
• x1 1,600 vehicles
• x2 2,900 vehicles
• tixi 55,800 veh-min

• System optimization
• t1 14.3 minutes
• t2 10.08 minutes
• x1 2,033 vehicles
• x2 2,467 vehicles
• tixi 53,592 veh-min

Route 1
Suburb
Route 2
City
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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?

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UE
SO
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Total Travel time is Minimum at SO
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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 X11600 is 55829
• Total travel time if X11601 is 55819