Logit Models

Transportation · FE Reference Handbook section

Transportation

9 formulas

10 exam-style examples

~60 min

All Transportation lectures

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

If two modes, auto (A) and transit (T), are being considered, the probability of selecting the auto Mode A can be written as
If n modes of travel are being considered, the probability of selecting Mode x can be written as:

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1

Logit Models (mode choice probability) — solve for probability of choosing mode i — Logit Models

logit models predicting transit versus auto mode choice Given utility of mode i (U_i) = -1.6000; utility of mode j (U_j) = -1.8000, determine the probability of choosing mode i (P_i).

Given

utility of mode i (U_i) = -1.6000
$utility of mode j (U_j) = -1.8000$

Find

probability of choosing mode i (P_i)

Start with the thinking

The governing relation printed in this handbook section is Logit Models (mode choice probability).
Everything except P_i is given, so isolate P_i symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
Logit models predict the probability a traveler chooses a mode based on relative utilities of the alternatives.

Figure 1 — schematic for Logit Models (mode choice probability) — solve for probability of choosing mode i — Logit Models

Step-by-step solution

Step 1 — State the governing relation:
$P_i = \dfrac{e^{U_i}}{e^{U_i} + e^{U_j}}$
Step 2 — Rearrange the relation so that P_i stands alone on the left-hand side.
Step 3 — List the givens: utility of mode i (U_i) = -1.6000, utility of mode j (U_j) = -1.8000.
Step 4 — Substitute the given values into the rearranged relation.
Step 5 — Evaluate:
$P_{i} = 0.5498$
Step 6 — Check: returning P_i = 0.5498 to
$P_i = \dfrac{e^{U_i}}{e^{U_i} + e^{U_j}}$
reproduces the given quantities, and both sides carry the same units.

Answer:

P_{i} = 0.5498

Why the other options are there

1.0997 — kept a factor of two that cancels in the correct rearrangement.
0.2749 — dropped that same factor in the other direction.
0.6048 — rounded an intermediate value before the final step.

Reference: FE Handbook — Logit Models

Example 2

Logit Models (mode choice probability) — solve for utility of mode i — Logit Models (2)

logit models used in a travel-demand mode-split forecast Given utility of mode j (U_j) = -1.2000; probability of choosing mode i (P_i) = 0.6000, determine the utility of mode i (U_i).

Given

$utility of mode j (U_j) = -1.2000$
probability of choosing mode i (P_i) = 0.6000

Find

utility of mode i (U_i)

Start with the thinking

The governing relation printed in this handbook section is Logit Models (mode choice probability).
Everything except U_i is given, so isolate U_i symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
Logit models predict the probability a traveler chooses a mode based on relative utilities of the alternatives.

Figure 2 — schematic for Logit Models (mode choice probability) — solve for utility of mode i — Logit Models (2)

Step-by-step solution

Step 1 — State the governing relation:
$P_i = \dfrac{e^{U_i}}{e^{U_i} + e^{U_j}}$
Step 2 — Rearrange the relation so that U_i stands alone on the left-hand side.
Step 3 — List the givens: utility of mode j (U_j) = -1.2000, probability of choosing mode i (P_i) = 0.6000.
Step 4 — Substitute the given values into the rearranged relation.
Step 5 — Evaluate:
$U_{i} = -0.7945$
Step 6 — Check: returning U_i = -0.7945 to
$P_i = \dfrac{e^{U_i}}{e^{U_i} + e^{U_j}}$
reproduces the given quantities, and both sides carry the same units.

Answer:

U_{i} = -0.7945

Why the other options are there

-1.5891 — kept a factor of two that cancels in the correct rearrangement.
-0.3973 — dropped that same factor in the other direction.
-0.8740 — rounded an intermediate value before the final step.

Reference: FE Handbook — Logit Models

Example 3

Logit Models (mode choice probability) — solve for probability of choosing mode i (case 2) — Logit Models (3)

logit models estimating probability of route choice Given utility of mode i (U_i) = 2.1000; utility of mode j (U_j) = -1.2000, determine the probability of choosing mode i (P_i).

Given

utility of mode i (U_i) = 2.1000
$utility of mode j (U_j) = -1.2000$

Find

probability of choosing mode i (P_i)

Start with the thinking

The governing relation printed in this handbook section is Logit Models (mode choice probability).
Everything except P_i is given, so isolate P_i symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
Logit models predict the probability a traveler chooses a mode based on relative utilities of the alternatives.

Figure 3 — schematic for Logit Models (mode choice probability) — solve for probability of choosing mode i (case 2) — Logit Models (3)

Step-by-step solution

Step 1 — State the governing relation:
$P_i = \dfrac{e^{U_i}}{e^{U_i} + e^{U_j}}$
Step 2 — Rearrange the relation so that P_i stands alone on the left-hand side.
Step 3 — List the givens: utility of mode i (U_i) = 2.1000, utility of mode j (U_j) = -1.2000.
Step 4 — Substitute the given values into the rearranged relation.
Step 5 — Evaluate:
$P_{i} = 0.9644$
Step 6 — Check: returning P_i = 0.9644 to
$P_i = \dfrac{e^{U_i}}{e^{U_i} + e^{U_j}}$
reproduces the given quantities, and both sides carry the same units.

Answer:

P_{i} = 0.9644

Why the other options are there

1.9289 — kept a factor of two that cancels in the correct rearrangement.
0.4822 — dropped that same factor in the other direction.
1.0609 — rounded an intermediate value before the final step.

Reference: FE Handbook — Logit Models

Example 4

Logit Models (mode choice probability) — solve for utility of mode i (case 2) — Logit Models (4)

logit models predicting transit versus auto mode choice Given utility of mode j (U_j) = -0.2000; probability of choosing mode i (P_i) = 0.6400, determine the utility of mode i (U_i).

Given

$utility of mode j (U_j) = -0.2000$
probability of choosing mode i (P_i) = 0.6400

Find

utility of mode i (U_i)

Start with the thinking

The governing relation printed in this handbook section is Logit Models (mode choice probability).
Everything except U_i is given, so isolate U_i symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
Logit models predict the probability a traveler chooses a mode based on relative utilities of the alternatives.

Figure 4 — schematic for Logit Models (mode choice probability) — solve for utility of mode i (case 2) — Logit Models (4)

Step-by-step solution

Step 1 — State the governing relation:
$P_i = \dfrac{e^{U_i}}{e^{U_i} + e^{U_j}}$
Step 2 — Rearrange the relation so that U_i stands alone on the left-hand side.
Step 3 — List the givens: utility of mode j (U_j) = -0.2000, probability of choosing mode i (P_i) = 0.6400.
Step 4 — Substitute the given values into the rearranged relation.
Step 5 — Evaluate:
$U_{i} = 0.3754$
Step 6 — Check: returning U_i = 0.3754 to
$P_i = \dfrac{e^{U_i}}{e^{U_i} + e^{U_j}}$
reproduces the given quantities, and both sides carry the same units.

Answer:

U_{i} = 0.3754

Why the other options are there

0.7507 — kept a factor of two that cancels in the correct rearrangement.
0.1877 — dropped that same factor in the other direction.
0.4129 — rounded an intermediate value before the final step.

Reference: FE Handbook — Logit Models

Example 5

Logit Models (mode choice probability) — solve for probability of choosing mode i (case 3) — Logit Models (5)

logit models used in a travel-demand mode-split forecast Given utility of mode i (U_i) = -1.2000; utility of mode j (U_j) = 1.8000, determine the probability of choosing mode i (P_i).

Given

utility of mode i (U_i) = -1.2000
$utility of mode j (U_j) = 1.8000$

Find

probability of choosing mode i (P_i)

Start with the thinking

The governing relation printed in this handbook section is Logit Models (mode choice probability).
Everything except P_i is given, so isolate P_i symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
Logit models predict the probability a traveler chooses a mode based on relative utilities of the alternatives.

Figure 5 — schematic for Logit Models (mode choice probability) — solve for probability of choosing mode i (case 3) — Logit Models (5)

Step-by-step solution

Step 1 — State the governing relation:
$P_i = \dfrac{e^{U_i}}{e^{U_i} + e^{U_j}}$
Step 2 — Rearrange the relation so that P_i stands alone on the left-hand side.
Step 3 — List the givens: utility of mode i (U_i) = -1.2000, utility of mode j (U_j) = 1.8000.
Step 4 — Substitute the given values into the rearranged relation.
Step 5 — Evaluate:
$P_{i} = 0.0474$
Step 6 — Check: returning P_i = 0.0474 to
$P_i = \dfrac{e^{U_i}}{e^{U_i} + e^{U_j}}$
reproduces the given quantities, and both sides carry the same units.

Answer:

P_{i} = 0.0474

Why the other options are there

0.0949 — kept a factor of two that cancels in the correct rearrangement.
0.0237 — dropped that same factor in the other direction.
0.0522 — rounded an intermediate value before the final step.

Reference: FE Handbook — Logit Models

Example 6

Logit Models (mode choice probability) — solve for utility of mode i (case 3) — Logit Models (6)

logit models estimating probability of route choice Given utility of mode j (U_j) = -1.7000; probability of choosing mode i (P_i) = 0.4100, determine the utility of mode i (U_i).

Given

$utility of mode j (U_j) = -1.7000$
probability of choosing mode i (P_i) = 0.4100

Find

utility of mode i (U_i)

Start with the thinking

The governing relation printed in this handbook section is Logit Models (mode choice probability).
Everything except U_i is given, so isolate U_i symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
Logit models predict the probability a traveler chooses a mode based on relative utilities of the alternatives.

Figure 6 — schematic for Logit Models (mode choice probability) — solve for utility of mode i (case 3) — Logit Models (6)

Step-by-step solution

Step 1 — State the governing relation:
$P_i = \dfrac{e^{U_i}}{e^{U_i} + e^{U_j}}$
Step 2 — Rearrange the relation so that U_i stands alone on the left-hand side.
Step 3 — List the givens: utility of mode j (U_j) = -1.7000, probability of choosing mode i (P_i) = 0.4100.
Step 4 — Substitute the given values into the rearranged relation.
Step 5 — Evaluate:
$U_{i} = -2.0640$
Step 6 — Check: returning U_i = -2.0640 to
$P_i = \dfrac{e^{U_i}}{e^{U_i} + e^{U_j}}$
reproduces the given quantities, and both sides carry the same units.

Answer:

U_{i} = -2.0640

Why the other options are there

-4.1279 — kept a factor of two that cancels in the correct rearrangement.
-1.0320 — dropped that same factor in the other direction.
-2.2704 — rounded an intermediate value before the final step.

Reference: FE Handbook — Logit Models

Example 7

Logit Models (mode choice probability) — solve for probability of choosing mode i (case 4) — Logit Models (7)

logit models predicting transit versus auto mode choice Given utility of mode i (U_i) = 0.9000; utility of mode j (U_j) = 1.7000, determine the probability of choosing mode i (P_i).

Given

utility of mode i (U_i) = 0.9000
$utility of mode j (U_j) = 1.7000$

Find

probability of choosing mode i (P_i)

Start with the thinking

The governing relation printed in this handbook section is Logit Models (mode choice probability).
Everything except P_i is given, so isolate P_i symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
Logit models predict the probability a traveler chooses a mode based on relative utilities of the alternatives.

Figure 7 — schematic for Logit Models (mode choice probability) — solve for probability of choosing mode i (case 4) — Logit Models (7)

Step-by-step solution

Step 1 — State the governing relation:
$P_i = \dfrac{e^{U_i}}{e^{U_i} + e^{U_j}}$
Step 2 — Rearrange the relation so that P_i stands alone on the left-hand side.
Step 3 — List the givens: utility of mode i (U_i) = 0.9000, utility of mode j (U_j) = 1.7000.
Step 4 — Substitute the given values into the rearranged relation.
Step 5 — Evaluate:
$P_{i} = 0.3100$
Step 6 — Check: returning P_i = 0.3100 to
$P_i = \dfrac{e^{U_i}}{e^{U_i} + e^{U_j}}$
reproduces the given quantities, and both sides carry the same units.

Answer:

P_{i} = 0.3100

Why the other options are there

0.6201 — kept a factor of two that cancels in the correct rearrangement.
0.1550 — dropped that same factor in the other direction.
0.3410 — rounded an intermediate value before the final step.

Reference: FE Handbook — Logit Models

Example 8

Logit Models (mode choice probability) — solve for utility of mode i (case 4) — Logit Models (8)

logit models used in a travel-demand mode-split forecast Given utility of mode j (U_j) = -1.9000; probability of choosing mode i (P_i) = 0.9600, determine the utility of mode i (U_i).

Given

$utility of mode j (U_j) = -1.9000$
probability of choosing mode i (P_i) = 0.9600

Find

utility of mode i (U_i)

Start with the thinking

The governing relation printed in this handbook section is Logit Models (mode choice probability).
Everything except U_i is given, so isolate U_i symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
Logit models predict the probability a traveler chooses a mode based on relative utilities of the alternatives.

Figure 8 — schematic for Logit Models (mode choice probability) — solve for utility of mode i (case 4) — Logit Models (8)

Step-by-step solution

Step 1 — State the governing relation:
$P_i = \dfrac{e^{U_i}}{e^{U_i} + e^{U_j}}$
Step 2 — Rearrange the relation so that U_i stands alone on the left-hand side.
Step 3 — List the givens: utility of mode j (U_j) = -1.9000, probability of choosing mode i (P_i) = 0.9600.
Step 4 — Substitute the given values into the rearranged relation.
Step 5 — Evaluate:
$U_{i} = 1.2781$
Step 6 — Check: returning U_i = 1.2781 to
$P_i = \dfrac{e^{U_i}}{e^{U_i} + e^{U_j}}$
reproduces the given quantities, and both sides carry the same units.

Answer:

U_{i} = 1.2781

Why the other options are there

2.5561 — kept a factor of two that cancels in the correct rearrangement.
0.6390 — dropped that same factor in the other direction.
1.4059 — rounded an intermediate value before the final step.

Reference: FE Handbook — Logit Models

Example 9

Logit Models (mode choice probability) — solve for probability of choosing mode i (case 5) — Logit Models (9)

logit models estimating probability of route choice Given utility of mode i (U_i) = 0.9000; utility of mode j (U_j) = 1.6000, determine the probability of choosing mode i (P_i).

Given

utility of mode i (U_i) = 0.9000
$utility of mode j (U_j) = 1.6000$

Find

probability of choosing mode i (P_i)

Start with the thinking

The governing relation printed in this handbook section is Logit Models (mode choice probability).
Everything except P_i is given, so isolate P_i symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
Logit models predict the probability a traveler chooses a mode based on relative utilities of the alternatives.

Figure 9 — schematic for Logit Models (mode choice probability) — solve for probability of choosing mode i (case 5) — Logit Models (9)

Step-by-step solution

Step 1 — State the governing relation:
$P_i = \dfrac{e^{U_i}}{e^{U_i} + e^{U_j}}$
Step 2 — Rearrange the relation so that P_i stands alone on the left-hand side.
Step 3 — List the givens: utility of mode i (U_i) = 0.9000, utility of mode j (U_j) = 1.6000.
Step 4 — Substitute the given values into the rearranged relation.
Step 5 — Evaluate:
$P_{i} = 0.3318$
Step 6 — Check: returning P_i = 0.3318 to
$P_i = \dfrac{e^{U_i}}{e^{U_i} + e^{U_j}}$
reproduces the given quantities, and both sides carry the same units.

Answer:

P_{i} = 0.3318

Why the other options are there

0.6636 — kept a factor of two that cancels in the correct rearrangement.
0.1659 — dropped that same factor in the other direction.
0.3650 — rounded an intermediate value before the final step.

Reference: FE Handbook — Logit Models

Example 10

Logit Models (mode choice probability) — solve for utility of mode i (case 5) — Logit Models (10)

logit models predicting transit versus auto mode choice Given utility of mode j (U_j) = -2.2000; probability of choosing mode i (P_i) = 0.9800, determine the utility of mode i (U_i).

Given

$utility of mode j (U_j) = -2.2000$
probability of choosing mode i (P_i) = 0.9800

Find

utility of mode i (U_i)

Start with the thinking

The governing relation printed in this handbook section is Logit Models (mode choice probability).
Everything except U_i is given, so isolate U_i symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
Logit models predict the probability a traveler chooses a mode based on relative utilities of the alternatives.

Figure 10 — schematic for Logit Models (mode choice probability) — solve for utility of mode i (case 5) — Logit Models (10)

Step-by-step solution

Step 1 — State the governing relation:
$P_i = \dfrac{e^{U_i}}{e^{U_i} + e^{U_j}}$
Step 2 — Rearrange the relation so that U_i stands alone on the left-hand side.
Step 3 — List the givens: utility of mode j (U_j) = -2.2000, probability of choosing mode i (P_i) = 0.9800.
Step 4 — Substitute the given values into the rearranged relation.
Step 5 — Evaluate:
$U_{i} = 1.6918$
Step 6 — Check: returning U_i = 1.6918 to
$P_i = \dfrac{e^{U_i}}{e^{U_i} + e^{U_j}}$
reproduces the given quantities, and both sides carry the same units.

Answer:

U_{i} = 1.6918

Why the other options are there

3.3836 — kept a factor of two that cancels in the correct rearrangement.
0.8459 — dropped that same factor in the other direction.
1.8610 — rounded an intermediate value before the final step.

Reference: FE Handbook — Logit Models

Logit Models

Handbook notes for this section

Core formulas for this FE topic

Ux=/aiXiUx = / aiXiUx=/aiXi

i=1i=1i=1

Ux=utilityofModexUx = utility of Mode xUx=utilityofModex

n=numberofattributesn = number of attributesn=numberofattributes

Xi=attributevalue(time,cost,andsoforth)Xi = attribute value (time, cost, and so forth)Xi=attributevalue(time,cost,andsoforth)

ai=coefficientvalueforattributesi(negative,sincethevaluesaredisutilities)ai = coefficient value for attributes i (negative, since the values are disutilities)ai=coefficientvalueforattributesi(negative,sincethevaluesaredisutilities)

P_ Ai =Needs technical review

P_ x i = nNeeds technical review

x=1x=1x=1

Worked exam-style examples