Fundamentals of Transportation/Mode Choice/Solution

Problem:

You are given the following mode choice model.

$U_{ijm} = -1 C_{ijm} + 5 D_T\,\!$

Where:

• C_ijm = travel cost between i and j by mode m
• 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: U_auto

Origin\Destination	Dakotopolis	New Fargo
Dakotopolis	-5	-7
New Fargo	-7	-5

Transit Utility: U_transit

Origin\Destination	Dakotopolis	New Fargo
Dakotopolis	-5	-10
New Fargo	-10	-3

$e^{U_{auto}}$

Origin\Destination	Dakotopolis	New Fargo
Dakotopolis	0.0067	0.0009
New Fargo	0.0009	0.0067

$e^{U_{transit}}$

Origin\Destination	Dakotopolis	New Fargo
Dakotopolis	0.0067	0.0000454
New Fargo	0.0000454	0.0565

Sum: $e^{U_{transit}} + e^{U_{auto}}$

Origin\Destination	Dakotopolis	New Fargo
Dakotopolis	0.0134	0.0009454
New Fargo	0.0009454	0.0498

P(Auto) = $e^{U_{auto}}/(e^{U_{auto}} + e^{U_{transit}})$

Origin\Destination	Dakotopolis	New Fargo
Dakotopolis	0.5	0.953
New Fargo	0.953	0.12

P(Transit) = $e^{U_{transit}}/(e^{U_{auto}} + e^{U_{transit}})$

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 = T_ij * P(Auto)

Origin\Destination	Dakotopolis	New Fargo
Dakotopolis	4697	5339
New Fargo	6511	1867