Dynamic UserOptimal Departure TimeRoute Choice Model - PowerPoint PPT Presentation

Title: Dynamic UserOptimal Departure TimeRoute Choice Model


1

Trail course Dynamic Travel Choice Models

• Chapter 6
• Dynamic User-Optimal Departure Time/Route Choice
  Model
• Karel Lindveld
• Delft University of Technology
• Faculty of Civil Engineering and Geo Sciences
• Transportation Planning and Traffic Engineering
  Section

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Contents

• Conceptual background
• Follow the book
• Combined choice of route and departure time
• (without schedule delay)
• Departure time choice in the time/space network
• Formulation as a VI problem
• Equivalence theorem
  
• Extension to Chen
• Critique of Chens concept of departure time
  choice
• Schedule delay penalties
• Reformulation
• Summary, errata, discussion

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Conceptual background (I) 1-person case

• Think of 1 individual (on holiday, so no time
  constraints), who
• Chooses a time of day k for an activity
• Morning (early morning (6-8), morning peak
  (8-10), late morning (10-12))
• Afternoon (3 periods)
• Evening (3 periods)
• Given a time period k, choose least-cost route p
• Minimises travel cost C(p,k) depending on t.o.d.
  k and route p

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Conceptual background (II) critique

• Only 1 person has a choice
• no account of how consequences of peoples
  decisions affect others
• Long (2-hour) periods
• no interaction between route- and departure time
  choice

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Conceptual (III) background solution

• 1-person chooses / many persons simultaneously
  choose is similar to ordinary assignment
• Therefore
• Use formulation of user-optimality conditions and
  work out result for simultaneous choices
• As dynamic assignment requires VIP formulation,
  need to incorporate departure time choice into
  VIP framework
• Tea-kettle principle
• we already know how to deal with route choice
  within assignment
• therefore formulate combined dep. time / route
  choice as a route choice problem
• Will now follow the book

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The least-cost departure time problem
Each individual traveller seeks to minimise the
total path cost
If departure time is prescribed, path cost is
minimised over routes p
If departure time is free, path cost is minimised
over routes p and departure times k
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Departure time choice in the time/space network
(1)

• Recapitulate
• DUO (Chen 4) deals only with route choice
• Problem
• how to incorporate departure time choice into a
  route-choice framework ?
• Tea-kettle principle
• formulate new problem in terms of known problem
  (DUO)
• Application
• extend the network with special zero cost links
  for departure times and arrival times
• departure time choice to be modelled as a form of
  route choice

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Departure time choice in the time/space network
(2)
Physical network
STEN (Space/Time Extended Network)
r(4)
s(4)
r(3)
s(3)
s
r
r(2)
s(2)
r(1)
s(1)
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Departure time choice in the time/space network
(3)
Physical network
STEN (Space/Time Extended Network)
r(4)
s(4)
r(3)
s(3)
s
r
r(2)
s(2)
r(1)
s(1)
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Departure time choice in the time/space network
(4)

• Constraints for the DUO problem (see (3.1))
• where is the total flow between r and s
  leaving in period k (given)
• Constraints for the DUO departure time choice
  model
• where is the total flow between r and s
  (given)
• (see (3.2), (3.3))
• Note by limiting the summation in (3.3) to
  we can ensure that all travellers
  leave in this time period

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Formulation as a VI problem
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The equivalence theorem relationships (1)
Consistency
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The equivalence theorem structure (2)

• The equivalence theorems (4.2) and (6.2) have the
  following structure
• with the set of all path flows that
  correspond to the definitional constraints (4.11)
  -(4.12) / (6.9)-(6.10)

(6.17)
(4.18)-(4.22)
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The equivalence theorem proof outline (3)

• Sum equivalence conditions over r,s,p,k,
  eliminate term with minimum path cost
• Assume feasible solution satisfies VIP, retrieve
  optimality conditions by considering 2 possible
  cases for any 2 routes
• substitute definitional constraint
• and use additivity of link cost
• note that derivation of is reversible

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Done are we?
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Critique of Chens concept of departure time
choice

• In real life, people have schedules filled with
  activities
• activities often require people to be in the same
  location at the same time
• this imposes constraints on the starting time
  (and duration) of activities
• Travel is used to get from one activity to
  another
• therefore travel must allow one to be in time
  at a certain location
• if people want to be in time, they will be
  willing to pay a price for this (cost, time,
  discomfort)
• For these reasons Chens concept of free
  departure time choice is unrealistic
• Remedy
• account for penalty cost of being late (or early)
• add this cost to travel cost
• then find minimum cost path

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Extension schedule delay penalties (1)

• Problem
• Chens modelling framework only considers travel
  cost.
• how to incorporate preferred arrival time (
  )?
• How to incorporate preferred departure time (
  )?
• Solution
• add schedule delay penalty on arrival to path
  cost
• define schedule delay penalty (for arrival) as

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Extension schedule delay penalties (2)
STEN (Space/Time Extended Network
r(4)
s(4)
r(3)
s(3)
s
r
r(2)
s(2)
r(1)
s(1)
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Extension schedule delay penalties (3)

• Split cost function into time and others
• Add schedule delay to cost function
  (deterministic)
• 1 PAT and 1 PDT per user-class

(route travel time)
(route travel cost)
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Extension integration into framework (4)

• Use VIP in theorem (6.2) proof still holds with
  extended cost functions
• upper part in proof of (6.2) continues to hold
  because only deals with path costs
• additivity of link cost remains intact, so
  equivalence in lower part of proof of (6.2)/(4.2)
  between path-based and link-based VIP continues
  to hold
• Existence
• Extended cost function c(u) is continuous, so at
  least 1 solution to VIP exists

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Extension schedule delay penalties (5)

• Uniqueness depends on the situation
• Example marginal traveller on pre-loaded network
• Time-dependent travel time
• Schedule delay penalty
• resulting total cost for marginal traveller as a
  function of departure time (
  )

r
s
(assumed fixed)
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Algorithm

• Same as in 4, except minimum cost problem solved
  w.r.t. extended network

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Examples

• Comparison of tables (6.7) and (6.8) shows that
  people will decrease their travel time by
  spreading out.
• This continues until free-flow conditions are
  reached throughout.
• After that, multiple solutions for u are
  guaranteed.

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Summary

• Modelling framework
• Original modelling framework does not include
  departure time choice, only route choice
• Departure time choice can be accommodated as
  route choice when network is extended to a STEN
• Indifference w.r.t. arrival times presented in
  Chen is very unrealistic remedy by including
  schedule delay penalties
• Typos
• (6.3.1) t?k
• add transpose sign to (6.3.2) after last
  bracket