# Fundamentals of Transportation/Mode Choice/Solution

Problem:

You are given the following mode choice model.

${\displaystyle U_{ijm}=-1C_{ijm}+5D_{T}\,\!}$

Where:

• ${\displaystyle C_{ijm}}$ = travel cost between ${\displaystyle i}$ and ${\displaystyle j}$ by mode ${\displaystyle m}$
• ${\displaystyle D_{T}}$ = dummy variable (alternative specific constant) for transit
Auto Travel Times
Origin\Destination Dakotopolis New Fargo
Dakotopolis 5 7
New Fargo 7 5
Transit Travel Times
Origin\Destination Dakotopolis New Fargo
Dakotopolis 10 15
New Fargo 15 8
Solution:

Part A

A. Using a logit model, determine the probability of a traveler driving.

Solution Steps

1. Compute Utility for Each Mode for Each Cell
2. Compute Exponentiated Utilities for Each Cell
3. Sum Exponentiated Utilities
4. Compute Probability for Each Mode for Each Cell
5. Multiply Probability in Each Cell by Number of Trips in Each Cell
Auto Utility: ${\displaystyle U_{auto}}$
Origin\Destination Dakotopolis New Fargo
Dakotopolis -5 -7
New Fargo -7 -5
Transit Utility: ${\displaystyle U_{transit}}$
Origin\Destination Dakotopolis New Fargo
Dakotopolis -5 -10
New Fargo -10 -3
${\displaystyle e^{U_{auto}}}$
Origin\Destination Dakotopolis New Fargo
Dakotopolis 0.0067 0.0009
New Fargo 0.0009 0.0067
${\displaystyle e^{U_{transit}}}$
Origin\Destination Dakotopolis New Fargo
Dakotopolis 0.0067 0.0000454
New Fargo 0.0000454 0.0565
Sum: ${\displaystyle e^{U_{transit}}+e^{U_{auto}}}$
Origin\Destination Dakotopolis New Fargo
Dakotopolis 0.0134 0.0009454
New Fargo 0.0009454 0.0498
P(Auto) = ${\displaystyle e^{U_{auto}}/(e^{U_{auto}}+e^{U_{transit}})}$
Origin\Destination Dakotopolis New Fargo
Dakotopolis 0.5 0.953
New Fargo 0.953 0.12
P(Transit) = ${\displaystyle e^{U_{transit}}/(e^{U_{auto}}+e^{U_{transit}})}$
Origin\Destination Dakotopolis New Fargo
Dakotopolis 0.5 0.047
New Fargo 0.047 0.88

Part B

B. Using the results from the previous problem (#2), how many car trips will there be?

Recall

Total Trips
Origin\Destination Dakotopolis New Fargo
Dakotopolis 9395 5606
New Fargo 6385 15665
Total Trips by Auto = ${\displaystyle T_{ij}*P(Auto)}$
Origin\Destination Dakotopolis New Fargo
Dakotopolis 4697 5339
New Fargo 6511 1867