Overview

 General procedure for model application
 Basic assumptions in Random Utility Model
 Uncertainty in choice
 Utility & Logit model
 Numerical example
 Application issues in four step model
 Summary

Individual & Travel Data
Choice
Model
Formulation
Estimate
Dissagregate
Choice
Model(s)
Predict Exogenous
Explanatory & Policy
Variables
Apply
Prediction
Procedure
Aggregate
(TAZ) Travel
Prediction
Insert Predicted
Proportions for Each
Mode in the Four Step
Sequence

 We will skip the more theoretical description of principles, theorems, lemas
 Emphasize practical aspects
 Look at examples
 Note: Dan McFadden is Professor of Economics and Nobel Laureate in Economics http://emlab.berkeley.edu/users/mcfadden/
 A site that contains a very good bibliography on
Random Utility Models

 Suppose a trip maker i faces J options (choices or alternatives) with index j=1,2,3…J.
 Assume that each trip maker associates with each choice j=1,2,...,J a function called
UTILITY representing the "convenience" of choosing mode j.
 j=1,2,..., J is called the choice set. This is the set from which a decision maker chooses an option.
 Note: Let’s assume that choice and consideration sets are the same.

– A person, i, needs to go to work from home to the downtown area.
–
Suppose this person has three possible modes to choose from: Car (j=1), Bus (j=2), and her Bike
(j=3). Total number of options (J=3).
–
One possible form of the person’s convenience function (called utility) is:
U car
=F (car attributes, person characteristics, trip attributes)
U bus
=F (bus attributes, person characteristics, trip attributes)
U bike
=F (bike attributes, person characteristics, trip attributes)

 Variables describing the individual --> this is an attempt to represent
"taste variation" from person to person.
In our example if young persons have systematically differing preferences from the older individuals, then, age would be one of the variables.
 Variables describing the choice characteristics (called choice attributes) in the choice set. For example, some travel modes are less expensive than others. Cost of the trip for each available mode would be another variable in the utility. Travel time is another key variable.
 Variables describing the context such as the trip type, time of day, budget constraints.

 Travelers (decision makers) formulate for each option a utility and they calculate its value.
 Then, they choose the option with the most advantageous utility (maximum utility).
 Example: U
(car,bus,bike)
=-0.5*cost-2*waiting time
 Cost by bus=$1, Waiting time=5 minutes
 Cost by car=$2.5, Waiting time=1 minute
 Cost by bike=$0.2, Waiting time=0 minutes
Which mode is the most desirable, second less desirable, etc?
Utility is actually an Indirect Conditional Utility

 We (analysts) do not know all the factors that influence choice behavior
 Travelers (decision makers) do not always make choices consistently
 We are not interested in including all possible variables that affect behavior in our models
 We are interested in policy variables (taxes, fares, gasoline costs, waiting times) that we can “manipulate” to find travelers reaction
 We are also interested in social, demographic, and economic traveler characteristics because these variables allow us to link models to TAZs

 The example becomes: U
(bus,car,bike)
=-0.5*cost-
2*waiting time + SOMETHING ELSE

The “something else” is an indicator of “general mode attractiveness” AND a random component

Let’s look at the details:

 U ij
= a j
-0.5*cost j
-2*waiting time j
+ b j
* age i
+ e ij
Utility of person i for mode j

A constant for each mode j.
Captures desirability of j for unknown reasons
 U ij time
= a j
-0.5*cost j j
+ b j
* age i
-2*waiting
+ e ij

Cost is different for each mode j
 U ij time
= a j
-0.5*cost j j
+ b j
* age i
-2*waiting
+ e ij
Waiting time is different for each mode j

 U ij time
= a j
-0.5*cost j j
+ b j
* age i
-2*waiting
+ e ij
The effect of the age variable is different for each alternate mode
( Class: Let’s talk about behavioral meaning - bikes?)

 U ij time
= a j
-0.5*cost j j
+ b j
* age i
-2*waiting
+ e ij
The key indicator of uncertainty = our ignorance & traveler variability for unknown reasons

 In a similar way as for age we can include other traveler and trip characteristics (explanatory)
 U ij
= a time j j
-0.5*cost
+ b j
* age i j
-2*waiting
+ e ij
In applications: These are parameters we estimate from data using regression methods

 U ij time
= j a j
-0.5*cost j
+ b j
* age i
-2*waiting
+ e ij
 Can write as: U ij
= V jj
+ e ij
Systematic & measurable part
Random

 U icar
=
6
- 0.5*cost - 2*waiting time +
0.15
* age i
+ e icar
 U ibus
= 5 - 0.5*cost - 2*waiting time + 0.25 * age i
+ e ibus
 U ibike
= 12 - 0.5*cost - 2*waiting time - 0.3 * age i
+ e ibike
Note: Different age coefficients - why?

 Compute for each person the systematic part of utility for each mode
 Plot all V (syst. utilities) for all persons
 Horizontal = age
 Vertical = V the systematic part of utility of each mode

Modal Utilities
20
10
0
-10
0
-20
20 40 60 80 100
Vcar
Vbus
Vbike
Age

 We need to convert utilities to an estimate of the chance to choose a mode
 The specific equation to use depends on the probability distribution of the random component ( e
) in the utility function
(U=V+ e
)
 Ease of calculations should be considered in selecting a probability function

 Assume the random components ( e i
) of the utility are independent identically Gumbel distributed random variables then:
P i
( car )
 exp( V icar
) bus , bike j

 car exp( V ij
)

P i
( bus )
 exp( V ibus
) bus , bike j

 car exp( V ij
)
P i
( bike )
 exp( V ibike
) bus , bike j

 car exp( V ij
)

Probability
1.2
1
0.8
0.6
0.4
0.2
0
0 20 40 60
Age of Traveler
80 100
Pcar
Pbus
Pbike

 Modal split (type of mode)
 Route choice (link by link or entire path)
 Car ownership (type of car)
 Destination choice (shopping place)
 Activity types (type of activity)
 Residential unit (size and type of home)

 Choice set - consideration set
 Variables to include in utility
 Measurement of mode attributes (e.g.,invehicle-travel-time)
 Need survey data and mode by mode attributes!

Next: TAZ application and “complete” enumeration

Individual & Travel Data
Choice
Model
Formulation
Estimate
Dissagregate
Choice
Model(s)
Predict Exogenous
Explanatory & Policy
Variables
Apply
Prediction
Procedure
Aggregate
(TAZ) Travel
Prediction
Insert in the
Four Step
Sequence

 We need aggregate TAZ proportions by each mode (% of trips by car, % trips by bus, % trips by bike)
 We have a disaggregate (individual) model which tells us the likelihood (chance) of a person to choose each mode
 We need a procedure to go from disaggregate predictions of chance to aggregate predictions of proportions

 (Pa+Pb)/2 is not the same as P ([Va+Vb]/2)
- a and b are value points for V
 When the two are equated we have the
Naïve method of aggregation
 Bias depends on how close the probability function is to a linear function
 Following is an example from Probability to choose bus as an option

Pbus
1
P(V=12)=0.679
0.8
0.6
0.4
P(V=2)=0.034
0.2
0
-5 0
V=2
10
V=12
5 15
Systematic Utility of Bus (Vbus)
20
Pbus

 (P(V=2)+P(V=12))/2
 or
 P((2+12)/2)=P(V=7)

Pbus
The correct value is: [P(V=2)+P(V=12)]/2=0.357
1.2
1
0.8
P(V=12)=0.679
0.6
0.4
P(V=7)=0.223
0.2
P(V=2)=0.034
0
-5 0
V=2
5
V=7
10
V=12
15
Systematic Utility of Bus (Vbus)
20
Pbus

Pbus
0.8
0.6
[P(V=2)+P(V=12)]/2=0.357
0.4
P(V=7)=0.223
0.2
-5
1.2
1
Bias (see page 310 OW)
0
0
V=2
5
V=7
10
V=12
15
Systematic Utility of Bus (Vbus)
20
Pbus

 For each TAZ take the average value of explanatory variables
 Compute average value for each utility function for each mode
 Compute the corresponding probability and use it as the TAZ proportion choosing each mode

 Divide the residents in each TAZ into relatively homogeneous segments

Apply Naïve aggregation to each segment and get proportions for each mode
 Compute the TAZ proportion either as average segment-specific proportion or weighted segment-specific proportion

 Compute for each person and for each mode the probability to choose a mode
 Compute the proportion for each mode as an average of the individual probabilities
 Stochastic microsimulation is a method derived from this - see also Chapter 12 of
Goulias, 2003 (red book)

Segment 1
Segment 2
Segment 2
Segment 3
Average
Exp (U)
Naïve Prob
Age Vcar
45 7.500
21
20
Vbus
5.750
3.900
-0.250
3.750
-0.500
79 12.600
14.250
41.25
6.938
4.813
1030.192
123.039
0.893
0.107
Vbike
-1.600
5.600
5.900
-11.800
-0.475
0.622
0.001
V icar
=
6
- 0.5*cost - 2*waiting time +
0.15
* age i
V ibus
V ibike
= 5 - 0.5*cost - 2*waiting time + 0.25 * age i
= 12 - 0.5*cost - 2*waiting time - 0.3 * age i

Pcar
0.372
Pbus
0.329
Pbike
0.298
Average of Segments
Weigthed Average of Segments 0.305
0.247
0.447
Naïve Aggregation
Complete Enumeration
0.893
0.318
0.107
0.248
0.001
0.434

 Gumbel IID convenient but is it realistic?
 IID components imply unrelated options in the unobserved components - new models account for relations
 Trips are related - different formulations
 See CE 523

 http://www.bts.gov/ntl/DOCS/SICM.html
(Spear’s report on how to apply models)
 http://www.bts.gov/ntl/DOCS/UT.html
(self-instructional overview with examples)
 http://www.tfhrc.gov///////safety/pedbike/vol
2/sec2.5.htm (simple description of most of the key issues)
 All sites accessed September 22, 2003


Rational economic behavior
 Utility linear in systematic and random components
 Choice probability is function of utilities – non linear function!
 Application by enumeration is best weighted average by market segments may be good - depends on application!
 Aggregate models are also available – approximate!
 Surveys must be used for this step

 Ortuzar Willumsen - Chapter 8 (8.1, 8.2,
8.3)
 Ortuzar Willumsen - Chapter 9 (9.1, 9.2,
9.3)