Logit Models

Transportation · FE Reference Handbook section

Transportation

9 formulas

10 exam-style examples

~60 min

All Transportation lectures

Learning objectives

What you must be able to do before leaving this section.

This chapter section covers Logit Models within Transportation. Read it the way you would read a textbook chapter: the theory first so the relations mean something, then every equation with its use and its trap, then 10 fully worked examples with the arithmetic shown line by line, and finally a self-check you should be able to answer without notes.

Explain, in your own words, what logit models describes physically and when it applies.
State every one of the 9 relations the handbook lists here and name each symbol with its unit.
Select the correct relation from the wording of an exam stem within 20 seconds.
Carry a complete solution from givens to a "most nearly" answer with the correct unit.
Recognise the distractors generated by the unit trap: grades as decimals in curve formulas, percent in the stem.

Lecture

Why this section exists. Logit Models is the part of Transportation that lets you connect a vertical or horizontal alignment, or a traffic stream to a number you can defend. Before any equation is useful you must be able to picture the physical situation it describes; the schematic below is that picture.

How the theory is built. The handbook prints results, not derivations. Each relation in this section comes from one governing principle applied to the idealised system: state the principle, impose the stated assumptions, and the printed equation follows. Knowing which assumption each relation rests on is what lets you reject a wrong answer choice in seconds.

How it is examined. Items from this page are written as a curve geometry element or a capacity/flow relationship. Roughly two thirds are direct substitution, one third require one intermediate quantity from a neighbouring relation, and a small number are conceptual — testing whether you know the assumption, not the arithmetic.

The habit that earns the points. Unit discipline. grades as decimals in curve formulas, percent in the stem. Every relation below is dimensionally consistent only when that rule is honoured, and the distractor set is deliberately built from candidates who ignored it. Write the unit next to every number you substitute, every time.

How to study this page. Read the theory, then cover the formula cards and try to reproduce each relation from its description. Then work the examples with the solution hidden, revealing one line at a time. Finish with the self-check questions; if you cannot answer one, return to the matching formula card.

Dense peak-hour traffic queued on an urban arterial at dusk. — Photo 1. Where this shows up in practice: logit models.
Wikimedia Commons, CC BY 2.0

Transportation — Logit Models: reference schematic for orienting the symbols used in this section.

Theory, developed

Read this before the equations — it is what makes them memorable.

The physical situation

Every item from this section describes a vertical or horizontal alignment, or a traffic stream. Sketch it before you compute — a labelled sketch with the givens on it converts a wordy stem into a solvable problem and exposes the quantity the examiner left out on purpose.

The governing principle

The 9 relations on this page are consequences of one principle applied to that idealised system. Identify which quantity is conserved, balanced, or defined, and the correct equation follows without memorisation.

Assumptions and limits of validity

Each printed relation carries silent assumptions — linearity, steady state, uniformity, small deformation, or standard conditions, depending on the subject. Conceptual exam items are written by violating exactly one of these, so read the sentence above the equation as carefully as the equation itself.

Solution procedure you should automate

1) Read the last sentence of the stem to identify the requested quantity. 2) Locate the relation on this page whose left-hand side is that quantity. 3) Tabulate the givens with units and mark the missing symbol. 4) If a symbol is missing, find the one relation that produces it. 5) Rearrange symbolically, substitute once, evaluate, and round only at the end.

Notation used in this section

Ux	Quantity produced by "Ux = / aiXi" — read its definition and unit from the handbook line directly above the equation.
i	Quantity produced by "i=1" — read its definition and unit from the handbook line directly above the equation.
n	Quantity produced by "n = number of attributes" — read its definition and unit from the handbook line directly above the equation.
Xi	Quantity produced by "Xi = attribute value (time, cost, and so forth)" — read its definition and unit from the handbook line directly above the equation.
ai	Quantity produced by "ai = coefficient value for attributes i (negative, since the values are disutilities)" — read its definition and unit from the handbook line directly above the equation.
x	Quantity produced by "x=1" — read its definition and unit from the handbook line directly above the equation.

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

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

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

Trip distribution between two zones with the gravity model — Logit Models

Zone i produces 3,416 trips. Zone 1 has 2,189 attractions with a friction factor of 0.85; zone 2 has 1,349 attractions with a friction factor of 0.85. Use the gravity model to distribute the trips.

Given

P_i = 3,416 trips
A₁ = 2,189, F₁ = 0.85
A₂ = 1,349, F₂ = 0.85

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$2189 (0.85) = 1, 861$
Numerator (zone 2)
$1349 (0.85) = 1, 147$
Denominator
$1, 861 + 1, 147 = 3, 007$
Substituting — T_i1 = 3416(1,861/3,007) = 2,114 trips
Balance — T_i2 = 3416 − 2,114 = 1,302 trips

Answer: Zone 1 receives 2,114 trips; zone 2 receives 1,302 trips

Why the other options are there

1,708 trips each (attractions ignored)
2,114 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Logit Models

Example 2

Trip distribution between two zones with the gravity model — Logit Models (2)

Zone i produces 1,108 trips. Zone 1 has 1,468 attractions with a friction factor of 0.65; zone 2 has 1,406 attractions with a friction factor of 0.70. Use the gravity model to distribute the trips.

Given

P_i = 1,108 trips
A₁ = 1,468, F₁ = 0.65
A₂ = 1,406, F₂ = 0.70

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$1468 (0.65) = 954.2$
Numerator (zone 2)
$1406 (0.70) = 984.2$
Denominator
$954.2 + 984.2 = 1, 938$
Substituting — T_i1 = 1108(954.2/1,938) = 545.4 trips
Balance — T_i2 = 1108 − 545.4 = 562.6 trips

Answer: Zone 1 receives 545.4 trips; zone 2 receives 562.6 trips

Why the other options are there

554.0 trips each (attractions ignored)
566.0 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Logit Models

Example 3

Trip distribution between two zones with the gravity model — Logit Models (3)

Zone i produces 2,602 trips. Zone 1 has 2,319 attractions with a friction factor of 0.45; zone 2 has 1,437 attractions with a friction factor of 0.85. Use the gravity model to distribute the trips.

Given

P_i = 2,602 trips
A₁ = 2,319, F₁ = 0.45
A₂ = 1,437, F₂ = 0.85

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$2319 (0.45) = 1, 044$
Numerator (zone 2)
$1437 (0.85) = 1, 221$
Denominator
$1, 044 + 1, 221 = 2, 265$
Substituting — T_i1 = 2602(1,044/2,265) = 1,199 trips
Balance — T_i2 = 2602 − 1,199 = 1,403 trips

Answer: Zone 1 receives 1,199 trips; zone 2 receives 1,403 trips

Why the other options are there

1,301 trips each (attractions ignored)
1,607 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Logit Models

Example 4

Trip distribution between two zones with the gravity model — Logit Models (4)

Zone i produces 3,051 trips. Zone 1 has 1,336 attractions with a friction factor of 0.55; zone 2 has 1,541 attractions with a friction factor of 0.85. Use the gravity model to distribute the trips.

Given

P_i = 3,051 trips
A₁ = 1,336, F₁ = 0.55
A₂ = 1,541, F₂ = 0.85

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$1336 (0.55) = 734.8$
Numerator (zone 2)
$1541 (0.85) = 1, 310$
Denominator
$734.8 + 1, 310 = 2, 045$
Substituting — T_i1 = 3051(734.8/2,045) = 1,096 trips
Balance — T_i2 = 3051 − 1,096 = 1,955 trips

Answer: Zone 1 receives 1,096 trips; zone 2 receives 1,955 trips

Why the other options are there

1,526 trips each (attractions ignored)
1,417 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Logit Models

Example 5

Trip distribution between two zones with the gravity model — Logit Models (5)

Zone i produces 3,335 trips. Zone 1 has 1,273 attractions with a friction factor of 0.65; zone 2 has 505 attractions with a friction factor of 0.85. Use the gravity model to distribute the trips.

Given

P_i = 3,335 trips
A₁ = 1,273, F₁ = 0.65
A₂ = 505, F₂ = 0.85

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$1273 (0.65) = 827.5$
Numerator (zone 2)
$505 (0.85) = 429.3$
Denominator
$827.5 + 429.3 = 1, 257$
Substituting — T_i1 = 3335(827.5/1,257) = 2,196 trips
Balance — T_i2 = 3335 − 2,196 = 1,139 trips

Answer: Zone 1 receives 2,196 trips; zone 2 receives 1,139 trips

Why the other options are there

1,668 trips each (attractions ignored)
2,388 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Logit Models

Example 6

Trip distribution between two zones with the gravity model — Logit Models (6)

Zone i produces 3,663 trips. Zone 1 has 1,774 attractions with a friction factor of 0.95; zone 2 has 1,221 attractions with a friction factor of 0.35. Use the gravity model to distribute the trips.

Given

P_i = 3,663 trips
A₁ = 1,774, F₁ = 0.95
A₂ = 1,221, F₂ = 0.35

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$1774 (0.95) = 1, 685$
Numerator (zone 2)
$1221 (0.35) = 427.3$
Denominator
$1, 685 + 427.3 = 2, 113$
Substituting — T_i1 = 3663(1,685/2,113) = 2,922 trips
Balance — T_i2 = 3663 − 2,922 = 741.0 trips

Answer: Zone 1 receives 2,922 trips; zone 2 receives 741.0 trips

Why the other options are there

1,832 trips each (attractions ignored)
2,170 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Logit Models

Example 7

Trip distribution between two zones with the gravity model — Logit Models (7)

Zone i produces 2,371 trips. Zone 1 has 2,481 attractions with a friction factor of 0.85; zone 2 has 2,565 attractions with a friction factor of 0.95. Use the gravity model to distribute the trips.

Given

P_i = 2,371 trips
A₁ = 2,481, F₁ = 0.85
A₂ = 2,565, F₂ = 0.95

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$2481 (0.85) = 2, 109$
Numerator (zone 2)
$2565 (0.95) = 2, 437$
Denominator
$2, 109 + 2, 437 = 4, 546$
Substituting — T_i1 = 2371(2,109/4,546) = 1,100 trips
Balance — T_i2 = 2371 − 1,100 = 1,271 trips

Answer: Zone 1 receives 1,100 trips; zone 2 receives 1,271 trips

Why the other options are there

1,186 trips each (attractions ignored)
1,166 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Logit Models

Example 8

Trip distribution between two zones with the gravity model — Logit Models (8)

Zone i produces 1,895 trips. Zone 1 has 2,890 attractions with a friction factor of 0.65; zone 2 has 780 attractions with a friction factor of 0.75. Use the gravity model to distribute the trips.

Given

P_i = 1,895 trips
A₁ = 2,890, F₁ = 0.65
A₂ = 780, F₂ = 0.75

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$2890 (0.65) = 1, 879$
Numerator (zone 2)
$780 (0.75) = 585.0$
Denominator
$1, 879 + 585.0 = 2, 464$
Substituting — T_i1 = 1895(1,879/2,464) = 1,445 trips
Balance — T_i2 = 1895 − 1,445 = 450.0 trips

Answer: Zone 1 receives 1,445 trips; zone 2 receives 450.0 trips

Why the other options are there

947.5 trips each (attractions ignored)
1,492 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Logit Models

Example 9

Trip distribution between two zones with the gravity model — Logit Models (9)

Zone i produces 1,103 trips. Zone 1 has 1,141 attractions with a friction factor of 0.60; zone 2 has 726 attractions with a friction factor of 0.95. Use the gravity model to distribute the trips.

Given

P_i = 1,103 trips
A₁ = 1,141, F₁ = 0.60
A₂ = 726, F₂ = 0.95

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$1141 (0.60) = 684.6$
Numerator (zone 2)
$726 (0.95) = 689.7$
Denominator
$684.6 + 689.7 = 1, 374$
Substituting — T_i1 = 1103(684.6/1,374) = 549.5 trips
Balance — T_i2 = 1103 − 549.5 = 553.5 trips

Answer: Zone 1 receives 549.5 trips; zone 2 receives 553.5 trips

Why the other options are there

551.5 trips each (attractions ignored)
674.1 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Logit Models

Example 10

Trip distribution between two zones with the gravity model — Logit Models (10)

Zone i produces 2,580 trips. Zone 1 has 1,726 attractions with a friction factor of 0.70; zone 2 has 2,647 attractions with a friction factor of 0.65. Use the gravity model to distribute the trips.

Given

P_i = 2,580 trips
A₁ = 1,726, F₁ = 0.70
A₂ = 2,647, F₂ = 0.65

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$1726 (0.70) = 1, 208$
Numerator (zone 2)
$2647 (0.65) = 1, 721$
Denominator
$1, 208 + 1, 721 = 2, 929$
Substituting — T_i1 = 2580(1,208/2,929) = 1,064 trips
Balance — T_i2 = 2580 − 1,064 = 1,516 trips

Answer: Zone 1 receives 1,064 trips; zone 2 receives 1,516 trips

Why the other options are there

1,290 trips each (attractions ignored)
1,018 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Logit Models

Self-check

Answer these without notes before moving on.

Without looking, state the relation on this page whose left-hand side is the quantity most often requested, and name every symbol in it.
Which assumption, if violated, makes the main relation of this section invalid?
Given a vertical or horizontal alignment, or a traffic stream, what is the first quantity you would compute, and why that one first?
Which unit conversion in this subject most often produces a wrong answer choice, and what is its numerical factor?
Rework Example 1 above from the givens alone, without reading the solution lines.

Chapter summary

Logit Models contains 9 relations; you must be able to find this page in under 15 seconds.
Exam style: a curve geometry element or a capacity/flow relationship.
Unit rule: grades as decimals in curve formulas, percent in the stem.
Work the 10 examples until the solution path, not the answer, is automatic.

Common traps in this section

grades as decimals in curve formulas, percent in the stem
Answering the intermediate quantity instead of the quantity requested.
Rounding intermediate values before the final step.
Using a relation from an adjacent handbook section that shares a symbol.
Skipping the sketch — most lost points on this page start with a misread geometry.

Logit Models

Learning objectives

Lecture

Theory, developed

Notation used in this section

Handbook notes for this section

Core formulas for this FE topic

Ux=/aiXi

i=1

Ux=utilityofModex

n=numberofattributes

Xi=attributevalue(time,cost,andsoforth)

ai=coefficientvalueforattributesi(negative,sincethevaluesaredisutilities)

P_ Ai =Needs technical review

P_ x i = nNeeds technical review

x=1

Worked exam-style examples

Self-check

Chapter summary

Common traps in this section