Traffic Assignment Part II

1
Traffic Assignment Part II

• CE 573 Transportation Planning
• Lecture 17

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Objectives

• Understand user equilibrium
• Execute a link cost update
• Communicate capacity restrained traffic
  assignment concept
• Execute incremental assignment

3
User Equilibrium Principles

• Principle A Under equilibrium conditions
  traffic arranges itself in congested networks in
  such a way that no individual trip maker can
  reduce his path costs by switching routes.
• Principle B Under equilibrium conditions
  traffic arranges itself in congested networks
  such that all used routes between an O-D pair
  have equal and minimum costs while all unused
  routes have greater or equal costs (to the used
  path costs).

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User Equilibrium Mathematical Form
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Updating Link Costs

• Traffic assignment assign trips to links
• Gives volume on links
• Input link volumes into link cost functions
• Update link costs

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Updating Link Costs (Example)

• Using equations of this form for link cost for
  each link (travel time)
• Where
• ta link travel cost per unit distance on link a,
• toa free flow travel time per unit distance on
  link a,
• Va flow on link a,
• Qpa practical capacity of link a, and
• a and ß calibration parameters for link cost
  (a1.0 and ß3).
• Given the volume estimated from All-Or-Nothing
  (AON) assignment (last class example)
• Update the link costs

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Updating Link Costs (Example)
Links Links Links Links Links Links Links
Link 1-2 2-3 1-4 2-4 2-5 3-5 4-5
Link Volume 2700 1000 0 3100 0 1400 2100
Directional Volume (50/50) 1350 500 0 1550 0 700 1050
to 5 10 15 9 16 12 6
a 1 1 1 1 1 1 1
ß 3 3 3 3 3 3 3
Qp 1000 1000 1000 1000 1000 1000 1000
t (updated link cost) 17.3 11.3 15.0 42.5 16.0 16.1 12.9
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Update P-Matrix and Compare to Previous Matrix

• Which links are, and are not, used?
• If you were to assign traffic with the updated
  trees, would it be the same as it was before?
• Which traffic assignment is correct?
• Should you combine the two traffic assignments
  (before and after the update)? If so then how?

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Capacity Restrained Traffic Assignment Algorithms

• Incremental Assignment
• Method of Successive Averages
• Frank-Wolfe Algorithm

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General Traffic Assignment Approach

• n n 1
• Get previous solution (tan and Van-1)
• Update minimum cost path trees
• Do AON to get Fan?link volumes given previous
  link costs
• Search direction (Fan - Van-1)
• Step size (F)
• Update link volumes (Van)
• Update link costs (tan)

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Incremental Assignment Concept

• The trip matrix T is assigned in proportions.
• Properties of the proportions z
• Each proportion 0 zn 1
• All proportions
• After a proportion of T is assigned to the
  network
• link volumes are updated
• link costs are updated
• Assignment is complete when all proportions (1
  through N) are assigned

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Incremental Assignment Algorithm

• Initialization of network database
• 1) get set of link costs (most likely freeflow)
• 2) set all link volumes to 0,
• 3) select set of proportions
• 4) begin at first proportions(n 0).
• Set n n 1
• Update the minimum cost trees
• All-or-nothing assignment (Fan)
• Update the link flows?Van Van-1 f Fan, where
  f zn.
• Update link costs (tan) based on Van and go to
  step 2.

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Incremental Assignment Application

• Proportions Say there are two
• z1 0.6
• z2 0.4
• Use previous example network
• See Excel file Incremental Assignment
  Example.xls
• What differences do you see between the AON file
  and the Incremental file?
• Why did these differences occur?
• Did we reach equilibrium?

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Method of Successive Averages Concept

• Find previous solution
• Find current AON solution
• Establish a search direction
• Establish a step size 1/(number of iterations)
• Calculate the current solution

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Method of Successive Averages Algorithm

• Initialization of network database
• 1) get set of link costs (most likely freeflow)
• 2) set all link volumes to 0,
• 3) set iteration number (n 0).
• Set n n 1
• Update the minimum cost trees
• All-or-nothing assignment (Fan)
• Update the link flows?, where f 1/n
• Update link costs (tan) given Van.
• Check for convergence if no convergence then go
  to step 2.

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Method of Successive Averages Application

• Use previous example network
• See Excel file MSA Assignment Example.xls
• What differences do you see between the AON file
  and the MSA file?
• Why did these differences occur?
• Did you reach equilibrium?