CE 326: Transportation Planning

Mode Choice

The Four Step Model

• Trip Generation: determine the number of productions from and attractions to individual zones

• Trip Distribution: determine the origin-destination matrix between zones

• Mode Choice: allocate trip shares to individual modes

• Trip Assignment: allocate trip shares to individual routes

Model Types

• Trip-End Models

• Only “captive” riders will choose transit

• Estimate market share of captive riders using socioeconomic variables

• Can be performed before distribution in 4-Step Model

• Trip Interchange Models

• Modes “compete” for ridership

• Modal preference is measured by utilities

• Must be performed after distribution in 4-Step Model

Utility Functions

Trip Interchange Model: Discrete Choice Model

• Discrete set of alternatives

• Universal Choice Set

• Individual Choice Set

• Each alternative is described by a utility function

• Mode choice models generally measure disutility

Utility Theory

• If the utility of mode i is greater than the utility of mode j for user t, then user t will choose mode i.

• Utility is a function both of the attributes of each mode and of the attributes of the individual user

Utility Function

• Utility functions are used to estimate the utility of a mode as a function of trip and user characteristics

• The observable utility of mode i for user t, Vit, is estimated as a function of three different variable types: mode attributes, individual characteristics, and interaction terms

Modeling Variable Types

• For service variables and interaction terms, we estimate a single parameter value because the impact of a change in these variables is equal across all modes

• For individual characteristics we have to estimate mode-specific parameters because the impacts of these variables on modal utility vary across modes

• Mode-specific parameters measure the relative impact of an individual characteristic compared to a defined base mode

Variable Type Parameter Value Variable

Modal Service Variables Constant Variable

Individual Characteristics Variable Constant

Attributes of Modal Alternatives (1)

• Service variables

• Travel time – total, in-vehicle, out-of-vehicle

• Travel cost

• Number of transfers

• Walk distance

• Reliability of on-time arrival

• Service variable values will vary across different across modes

• E.g. Bus travel time ≠ drive alone travel time, subway # transfers ≠ bus # transfers

• The impact on utility of a unit change in a service variable is the same across all modes; therefore, we can estimate one parameter value that will apply to all modes.

• “People perceive services indirectly in terms of their attributes”

Attributes of Modal Alternatives (2)

• Since not all variables can be observed, modal bias parameters (constants) are included in modal utility functions to capture the observed impacts on modal utilities that are not captured by the variables included in the model

• Modal bias parameters are constant for all decision-makers (regardless of chosen mode)

• Modal bias parameters measure the expected impact on mode utility due to unobserved variables relative to a base mode

• The parameter value for a base mode relative to itself will be zero; therefore, no modal bias parameter should be included in the utility function for the base mode

Characteristics of the Decision Maker

• Individual Characteristics

• Sex

• Age

• Household income

• Household size

• Household number of automobiles

• Household number of workers

• Individual characteristics remain constant, regardless of the chosen mode

• As a result, we cannot simply include individual characteristics in the model; in order to measure their impacts we must create mode-specific variables, for which we can estimate different parameter values

Mode-Specific Individual Characteristics

• To estimate a mode-specific variable for an individual characteristic, we multiply the dummy variable for each chosen mode by the variable value

• For modes that are not chosen, the value of this new variable will be zero, and the variable will have no impact on the model

Mode-Specific Parameter Values

• Parameter values for mode-specific variables vary across modes

• Parameter values for mode-specific variables measure the expected impact on mode utility of a unit change in the variable relative to a base mode

• The parameter value for a base mode relative to itself will be zero; therefore, mode-specific variables should not be included in the utility function for the base mode

Final Utility Function

𝑉𝑖𝑡 = 𝛼 𝑖 + 𝛾𝑗𝑋𝑖𝑗𝑡𝑗

+ 𝛽𝑖𝑗𝑆𝑖𝑗𝑡𝑗

Deterministic Model

• Utility for each mode and individual is evaluated as a function of observed variables

• Each user will choose the mode for which their utility is maximized

Weaknesses of Deterministic Model

• We cannot observe all possible variables that may impact utility

• Sources of error

• Incorrect or incomplete information about attributes of alternatives

• Incorrect or incomplete information about attributes of individuals

• Lack of knowledge of special circumstances

• We can represent this error in the utility function as a random variable

Probabilistic Model • The mathematical form of a probabilistic model is

determined by the distribution of the error term

• According to the Central Limit Theorem, the error term should be normally distributed

• A model with a normally distributed error term is called a Multinomial Probit Model

• However, this model is mathematically complex and difficult to evaluate

• A more commonly applied, and less complex to solve, model is the Multinomial Logit Model

Multinomial Logit Model

Gumbel Distribution (1)

• The Multinomial Logit Model assumes that the error term is independently and identically distributed for all modes according to a Gumbel Distribution

• The Gumbel Distribution is given by:

Gumbel Distribution (2)

• The Gumbel Distribution closely approximates the Normal Distribution

Source: Koppelman and Bhat

Multinomial Logit Model

• Assumption of the Gumbel Distribution results in the closed-form Multinomial Logit Model

Maximum Likelihood Estimation

• Parameter values are estimated using maximum likelihood estimation

• Maximize the likelihood function or the log likelihood function

Properties of the Multinomial Logit Model

Equivalent Differences Property

• Choice probabilities depend only on the differences in the observed utilities of different modes

Independence of Irrelevant Alternatives

• The ratio of the likelihoods of choosing two alternatives is not affected by the utility of any other alternatives

IIA Example

Mode 1 Mode 2

IIA Example

• MNL does not recognize any relationship between alternative modes

Mode 2

Mode 1

Mode 3

Derivatives and Elasticities

Measures of Response to Changes in Attributes of Alternatives - Derivatives

• Assuming variables are continuous

• Direct: change in probability of an alternative with respect to the change in attributes of that alternative

• Cross: change in probability of one alternative with respect to the change in attributes of another alternative

• Note:

• the sum of the derivatives across all alternatives should equal zero

Measures of Response to Changes in Attributes of Alternatives – Elasticities (1)

• Unlike derivatives, elasticities are normalized

• There is debate over whether elasticities should be normalized to the original values, the new values, or the midpoint

• To avoid this, we can estimate point elasticities, which measure elasticities for infinitesimally small changes

Measures of Response to Changes in Attributes of Alternatives – Elasticities (2)

• Assuming variables are continuous

• Direct: percentage change in the probability of one alternative with respect to a one percent change in an attribute of that alternative

• Cross: percentage change in the probability of one alternative with respect to a one percent change in an attribute of another alternative

Nested Logit Model

Nested Logit Model

• Hierarchical model

Mode Category Choice

Passenger

Car

Specific Mode Choice

Drive Alone

Shared Ride

Transit

Specific Mode Choice

Bus

Rail

Two decision levels

Conditional Probabilities

Mode Category Choice

Passenger

Car Transit

Drive Alone Shared Ride Bus Rail

Specific Mode Choice

Specific Mode Choice

Sources

• Koppelman and Bhat (http://www.ce.utexas.edu/prof/bhat/COURSES/LM_Draft_060131Final-060630.pdf)