Route Choice

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Route Choice
CEE 320Anne Goodchild
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Outline

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

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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

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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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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
  
• Some studies say perception bias is small

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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.
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
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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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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
Suburb
Route 2
City
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Example Solution

• 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
• 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

• 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.

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Example Mathematical Solution
?
?
?
? Same equation as before
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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.
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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
• 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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Primary Reference

• Mannering, F.L. Kilareski, W.P. and Washburn,
  S.S. (2005). Principles of Highway Engineering
  and Traffic Analysis, Third Edition. Chapter 8