CHAPTER 4 Dynamic UserOptimal Route Choice Model

1

Trail course Variational Inequalities

• CHAPTER 4Dynamic User-Optimal Route Choice
  Model
• Francesco Viti
• Delft University of Technology
• Faculty of Civil Engineering and Geo Sciences
• Transportation Planning and Traffic Engineering
  Section

2
Topics of the chapter

• Recapitulation of the DUO definition
• DUO formulated using Variational Inequality
  approach
• Equivalence analysis
• Introduction to time-space network (STEN)
• Numerical examples and results
• Questions raised

3
Purpose of Chapter 4

• Static User Optimal assignment is applicable
  when
• A simple approximation to the true system
  characteristics is sufficient for the purpose of
  the study.
• Dynamic User Optimal assignment is needed when
• Transport supply and/or demand change
  dynamically
• A more detailed model is needed
• It is absolutely indispensible when
• User make their route choice decision based on
  predictive traffic information, such as via tv,
  radio, internet etc.
• When we want to take account of en-route path
  changes
• When we want to take account of dynamic
  propagation of flows.

4
Data on which route choice should be based

• Whats the difference between perfect information
  and perfect knowledge from experience and
  historic data?
• Route choice is based on route travel costs they
  could be
• Istantaneous / actual
• Real / predicted
• HP. Route choice decision for travellers is based
  on ACTUAL route travel times rather than reactive
• They choose routes that are ACTUALLY the
  shortest when arriving at their destination.

5
Dynamic User-Optimal Conditions (1)

• No Departure time choice
• O-D demand is fixed and time-dependent

6
Dynamic User-Optimal Conditions (2)
Users are divided in departure time intervals in
which the cost of all used routes are equal and
minimal
Where, with the hypothesis of additive costs
7
Variational Inequality formulation (assumpitions)

• Traffic information available to travellers is
  perfect at instant and in the future
• Travel time considered as a deterministic
  variable
• Travellers are homogeneous
• Single class model
• Route choice is pre-trip made
• After depart user wont change his route anymore
• No changes in signal timing plan
• Network capacity is considered time independent
• No hard link capacity restraint
• Flow can exceed capacity and approach infinity
• Model is discrete in time
• Demand is aggregated into fixed time-slices no
  individual users are distinguished within time
  periods.
• FIFO not considered during this chapter

8
THEOREM
The DUO route choice problem is equivalent to
finding a solution u?? such that
That is
is a union of all ? defined by a particular
choice of
When the set of feasible solutions depends on the
solution itself the problem is called Quasi-VI
problem.
9
CONSTRAINTS (1)
Flow conservation
o
d
Flow propagation
10
CONSTRAINTS (2)
Relationship between flow propagation and route
choice
Other travel costs
Travel time
Flow Propagation
Route choice
Link flows
11
CONSTRAINTS (3)
Nonnegativity constraint
Definitional constraints
12
Problem of non-uniqueness
The feasible region of the DUO Route Choice model
delineated by VIP is not convex implying multiple
solutions.
Two cases in which we can define feasaible set
13
A parenthesis (1)

• Which are route costs?
• Route costs are dominated by route travel time
  attributes
• Travel times and other attributes are less
  predictable, this implies the dealing with
  stochastic decision process and EXPECTED travel
  times
• Costs are generally stochastic dued to the weak
  information about the attributes or to the
  stochasticity of the flows propagation.
• Is not always easy to assign a monetary value to
  attributes.

14
A parenthesis (2)

• Which are link costs?
• Link travel time is assumed equal for all users
  entering that link at the same interval, so in
  discrete link travel time is a step function
• for all a,t within a period k

How can they be related?
15
Dynamic Traffic Assignment
Decision level
Performance level
16
Equivalence Analysis
Theorem under a certain flow propagation
relationship
equilibrium conditions ltgt VIP
17
Proof of necessity (1)
user-optimal conditions can be written
Knowing that and
Subtracting the two inequations
Summing over r,s,p,k and considering the
constraint
18
Proof of necessity (2)
Route based DUO route choice model
To prove the asserted we have to convert this
model in a link based formulation Knowing that
19
Proof of sufficiency (1)

• Imagine a solution equal to except for two
  routes,
• then we could find two equilibriums
• a)

Switching some amount ?1 frome one route to the
other with
Using the route based DUO model and the previous
The same can be proved using a certain ? 2 so
20
Time-Space Network
Network is usually represented by one dimension
(e.g. distance) or by cartesian coordinates.
In the dynamic case the network needs to be
represented by two dimensions (space and time) by
treating time as a space parameter.
21
STEN (Space-Time Extended Network)
Time-extended origin
time
t4
Phisical origin
t3
Phisical destination
t2
Time-extended destination
t1
Slope depends on link TT
space
22
CASE STUDY
A first case study proposed by Chen is useful for
comparing the static model with the dynamic one.
2
4
5
3
1
Inclusion of propagation constraints
Major difference
23
RESULTS
Dynamic case
Static case
24
Questions raised

• How about the lenght of interval k?
• Should depend on strategy applied?
• Should be function of some travel times?
• Should be function of graph size?
• We only know that
• Larger k coarser approximation
• Shorter k more computational effort, lack of
  inputs.
• How about route alternatives?
• Should be exaustive?
• Should we count efficient routes?