Understand the idea.
Minimum under a model
The forward scan gives the earliest completion under the stated dependencies and durations. It assumes every ready activity can obtain the resources it needs immediately. A single shared machine may prevent the parallel starts that this basic model allows.
The display tasks contain days of activity time, but overlap permits a 13-day elapsed schedule. Total work and elapsed duration answer different questions; the network minimum is conditional on its timing and resource assumptions.
| Activity | Duration | Immediate predecessors |
|---|---|---|
| A | 4 | None |
| B | 3 | None |
| C | 5 | A |
| D | 4 | A |
| E | 2 | B and D |
| F | 3 | C and E |
Shortening the project
To shorten completion, change a controlling activity or a dependency that can legitimately be changed, then rescan. The saving on an individual task need not equal the saving for the whole project: another path can become the longest.
Reducing D from 4 to 3 days makes A–D–E–F take 12 days, tying A–C–F. Further reductions to D alone leave A–C–F at 12, so they cannot reduce the overall finish below that limit.
See the method in action.
Work versus elapsed time
Tasks A (5 hours) and B (8 hours) start together. C (2 hours) follows both. Find the total work and minimum elapsed duration, assuming independent resources.
Worked solution
- 1
Add every activity to measure total work.
Total work counts both parallel activities as well as the following activity.
- 2
Find when C can start.
A and B start together, and C requires completion of both.
- 3
Add the final sequential activity.
Only C's two hours follow the parallel stage, so elapsed time is less than total work.
15 task-hours of work; 10 hours elapsed.
Check a proposed reduction
A project has path durations 18 and 16 days. A task only on the 18-day path can be shortened by 4 days. What is the new earliest completion?
Worked solution
- 1
Reduce the affected route.
The shortened task occurs only on the original 18-day path.
- 2
Retain the other route's duration.
None of the changed work lies on this route.
- 3
Recalculate project completion.
The unchanged route becomes critical and prevents a 14-day project finish.
- 4
Measure the overall saving.
Only two of the four task-days saved reduce elapsed project duration.
16 days: the four-day task reduction saves only two days overall.
Try it yourself.
Check your understanding
The longest route is 30 days and the next longest is 28. A task appearing only on the longest is shortened by 5 days. What finish time results?
Show answer & reasoning
28 days.
The changed route takes days, while the other remains at 28. Their maximum is 28 days, so the project saves two days rather than five.
Keep practising
6 worksheet questions · 16 marks
Read the questions online
- Find the display project's total activity duration and its earliest elapsed completion.3 marks
A school display project. Times are in days. Activity Duration Immediate predecessors A 4 None B 3 None C 5 A D 4 A E 2 B and D F 3 C and E - Reduce D in the display project from 4 to 3 days. Find the new completion and all critical real-activity paths.3 marks
A school display project. Times are in days. Activity Duration Immediate predecessors A 4 None B 3 None C 5 A D 4 A E 2 B and D F 3 C and E - Instead reduce D from 4 to 1 day. Find the new project completion.3 marks
A school display project. Times are in days. Activity Duration Immediate predecessors A 4 None B 3 None C 5 A D 4 A E 2 B and D F 3 C and E - Two independent tasks each last 6 hours but need the same one available operator. A final 1-hour task follows both. Compare the dependency-only minimum with a schedule respecting the operator limit.3 marks
- A project has a 22-day critical path and a 21-day alternative. A one-day reduction is available on a task shared by both routes. What revised completion results?2 marks
- A contractor promises the exact network minimum with no allowance for uncertainty. Give two assumptions that should be checked.2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Find the display project's total activity duration and its earliest elapsed completion.3 marks
A school display project. Times are in days. Activity Duration Immediate predecessors A 4 None B 3 None C 5 A D 4 A E 2 B and D F 3 C and E Sum .
The controlling route A–D–E–F takes 13.
21 task-days; 13 elapsed days. - Reduce D in the display project from 4 to 3 days. Find the new completion and all critical real-activity paths.3 marks
A school display project. Times are in days. Activity Duration Immediate predecessors A 4 None B 3 None C 5 A D 4 A E 2 B and D F 3 C and E The changed D route is .
The C route remains .
12 days; A–D–E–F and A–C–F. - Instead reduce D from 4 to 1 day. Find the new project completion.3 marks
A school display project. Times are in days. Activity Duration Immediate predecessors A 4 None B 3 None C 5 A D 4 A E 2 B and D F 3 C and E The D route becomes .
The unchanged A–C–F route still needs 12.
12 days. - Two independent tasks each last 6 hours but need the same one available operator. A final 1-hour task follows both. Compare the dependency-only minimum with a schedule respecting the operator limit.3 marks
Parallel model: .
One operator must do the two tasks sequentially: .
Dependency-only minimum 7 hours; with one operator, 13 hours. - A project has a 22-day critical path and a 21-day alternative. A one-day reduction is available on a task shared by both routes. What revised completion results?2 marks
Both path durations fall by one.
They become 21 and 20; the maximum is 21.
21 days. - A contractor promises the exact network minimum with no allowance for uncertainty. Give two assumptions that should be checked.2 marks
The calculation uses fixed durations and immediate starts.
Supply delays, resource conflicts or rework can invalidate these assumptions.
For example: the activity durations are reliable, and enough independent resources exist for parallel tasks.
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 ↗