Earliest/latest times | Year 12

01 / LEARN

Understand the idea.

Four times for each activity

For an activity from event $i$ to event $j$ , use the tail's earliest time for $E S$ and add the duration for $E F$ . Use the head's latest time for $L F$ and subtract the duration for $L S$ . Keep these four labels distinct.

The activity's earliest finish need not equal the head event's earliest time. Another incoming activity may finish later and hold up the event, even though this particular activity has already finished.

\begin{aligned} E S & = E_{i} \\ E F & = E_{i} + d \\ L F & = L_{j} \\ L S & = L_{j} - d \end{aligned}

Forward and backward scans for the display network, with completion at day 13.
Event	Earliest time	Latest time
1	0	0
2	4	4
3	3	8
4	8	8
5	10	10
6	13	13

Read a time window

For B, $E S = 0$ , $E F = 3$ , $L F = 8$ and $L S = 5$ . Its possible start window is day 0 through day 5 if the day-13 project finish is protected and the remaining work is scheduled consistently.

For C, starting at day 4 gives $E F = 4 + 5 = 9$ , while the merge event waits until day 10 for E. Describe a window as actual start and finish times, and retain the model's assumptions about reliable durations and sufficient parallel resources.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Use event values

An activity takes 6 hours. Its tail has earliest time 3 and its head has latest time 14. Find ES, EF, LS and LF.

Worked solution

1
Use the tail event for earliest start.
$E S = 3 hours$
The activity can start as soon as its tail event occurs.
2
Calculate its earliest finish.
$E F = 3 + 6 = 9 hours$
Finishing requires its full six-hour duration after starting.
3
Use the head event for latest finish.
$L F = 14 hours$
The activity must be complete by the latest permitted occurrence of its head event.
4
Work backwards to latest start.
$L S = 14 - 6 = 8 hours$
Starting at hour 8 leaves exactly six hours before the latest finish.

Answer

ES=3, EF=9, LS=8, LF=14 hours.

EXAMPLE 02

Do not copy the head's earliest time

Activities P and Q finish at the same event. P starts at 2 and lasts 4 hours; Q starts at 5 and lasts 3 hours. Find P's earliest finish and the event's earliest time.

Worked solution

1
Calculate P's own finish.
$E F_{P} = 2 + 4 = 6 hours$
P's finish uses P's start and duration, independently of Q.
2
Calculate Q's finish.
$E F_{Q} = 5 + 3 = 8 hours$
Q reaches the shared event two hours later than P.
3
Find the shared event time.
$E = max (6, 8) = 8 hours$
The event must wait for Q even though P has already finished at hour 6.

Answer

P has EF=6 hours; the event occurs at 8 hours.

03 / YOUR TURN

Try it yourself.

Check your understanding

An activity has ES=7, duration=4 and LF=13. What is its latest start?

Show answer & reasoning

Use the latest finish, not the earliest start: $L S = L F - d = 13 - 4 = 9$ . A start at time 9 completes the activity exactly at its latest finish of 13.

Keep practising

6 worksheet questions · 15 marks

Open worksheet Worked solutions

Read the questions online

Use the display project to find $ES$ , $EF$ , $LS$ and $LF$ for B.
3 marks
Forward and backward scans for the display network, with completion at day 13.
Event Earliest time Latest time
1 0 0
2 4 4
3 3 8
4 8 8
5 10 10
6 13 13
Find all four activity times for C in the display project.
3 marks
Forward and backward scans for the display network, with completion at day 13.
Event Earliest time Latest time
1 0 0
2 4 4
3 3 8
4 8 8
5 10 10
6 13 13
Find all four times for E in the display project.
2 marks
Forward and backward scans for the display network, with completion at day 13.
Event Earliest time Latest time
1 0 0
2 4 4
3 3 8
4 8 8
5 10 10
6 13 13
A task lasts 8 minutes. Its $ES$ is 12 and $LS$ is 15. Find $EF$ and $LF$ .
2 marks
An activity runs 2→5 for 3 days. $E_{2} = 6$ , $E_{5} = 12$ and $L_{5} = 14$ . Find its earliest finish and explain why it differs from $E_{5}$ .
3 marks
A claimed schedule gives $ES = 5$ , $EF = 11$ , $LS = 9$ and $LF = 14$ for one fixed-duration activity. Identify the inconsistency.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Use the display project to find $ES$ , $EF$ , $LS$ and $LF$ for B.
3 marks
Forward and backward scans for the display network, with completion at day 13.
Event Earliest time Latest time
1 0 0
2 4 4
3 3 8
4 8 8
5 10 10
6 13 13
B is 1→3 of duration 3.
$ES = E_{1} = 0$ ; $EF = 3$ ; $LF = L_{3} = 8$ ; $LS = 8 - 3 = 5$ .
0, 3, 5 and 8 days respectively.
Find all four activity times for C in the display project.
3 marks
Forward and backward scans for the display network, with completion at day 13.
Event Earliest time Latest time
1 0 0
2 4 4
3 3 8
4 8 8
5 10 10
6 13 13
C is 2→5 of duration 5.
$ES = 4$ ; $EF = 4 + 5 = 9$ ; $LF = 10$ ; $LS = 10 - 5 = 5$ .
$ES = 4$ , $EF = 9$ , $LS = 5$ , $LF = 10$ days.
Find all four times for E in the display project.
2 marks
Forward and backward scans for the display network, with completion at day 13.
Event Earliest time Latest time
1 0 0
2 4 4
3 3 8
4 8 8
5 10 10
6 13 13
E is 4→5 of duration 2.
Use $E_{4} = 8$ and $L_{5} = 10$ .
$ES = 8$ , $EF = 10$ , $LS = 8$ , $LF = 10$ days.
A task lasts 8 minutes. Its $ES$ is 12 and $LS$ is 15. Find $EF$ and $LF$ .
2 marks
Add the same fixed duration to each start.
$12 + 8 = 20$ ; $15 + 8 = 23$ .
$EF = 20$ and $LF = 23$ minutes.
An activity runs 2→5 for 3 days. $E_{2} = 6$ , $E_{5} = 12$ and $L_{5} = 14$ . Find its earliest finish and explain why it differs from $E_{5}$ .
3 marks
The task itself finishes at $6 + 3 = 9$ .
An event combines all incoming requirements.
$EF = 9$ days; event 5 waits for another incoming activity until day 12.
A claimed schedule gives $ES = 5$ , $EF = 11$ , $LS = 9$ and $LF = 14$ for one fixed-duration activity. Identify the inconsistency.
2 marks
$11 - 5 = 6$ .
$14 - 9 = 5$ ; the same activity cannot have both durations in this model.
The duration would be 6 from $ES$ / $EF$ but 5 from $LS$ / $LF$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 General Mathematics: discrete models, project scheduling and assignment. Activity-on-arrow conventions are used for project networks; technology may verify calculations but does not replace interpreting the model.

Official curriculum source ↗

In this lesson

Understand the idea.

Four times for each activity

Read a time window

See the method in action.

Use event values

Use the tail event for earliest start.

Calculate its earliest finish.

Use the head event for latest finish.

Work backwards to latest start.

Do not copy the head's earliest time

Calculate P's own finish.

Calculate Q's finish.

Find the shared event time.

Try it yourself.

Check your understanding

Keep practising