Mathematics Standard • Year 12 • Module 6 • Lesson 1
Introduction to Project Networks, Skill Drill
Build fluency in the language of project networks: activities, events, precedence, dummies, AOA vs AON, one term, one diagram and one path at a time.
1. Quick recall
Answer each question in the space provided. 1 mark each
Q1.1 Complete each definition with one short phrase.
An activity is ____________________________________________.
An event is ____________________________________________.
An immediate predecessor of activity B is ____________________________________________.
Q1.2 In AOA notation, activities are drawn as ____________ and events are drawn as ____________.
A dummy activity is drawn as a ____________ arrow and has duration ____________.
Q1.3 State the one-line rule that forces us to use a dummy activity in some networks. Rule: __________________________________________________________________.
2. Worked example, precedence table and minimum project time
Follow each line of working. Every step has a reason on the right.
Problem. A small project has activities A(4), B(3), C(2), D(5). A has no predecessor. B and C both need A. D needs both B and C. Write the precedence table and find the minimum project time.
Step 1, Write the precedence table.
A(4) B(3, A) C(2, A) D(5, B and C)
Reason: list activity, duration, then only the immediate predecessors.
Step 2, List every path from start to finish.
Path 1: A → B → D Path 2: A → C → D
Reason: D needs BOTH B and C, so it starts only after the slower of the two paths finishes.
Step 3, Add the durations on each path.
Path 1 = 4 + 3 + 5 = 12 Path 2 = 4 + 2 + 5 = 11
Reason: project time on any path is the sum of its activity durations.
Step 4, Take the maximum path length.
Minimum project time = max(12, 11) = 12 days
Reason: D cannot start until B AND C finish, so the longer path wins.
Conclusion. The project takes a minimum of 12 days, controlled by the path A → B → D.
3. Faded example, fill in the missing steps
A project has activities P(2), Q(4), R(3), S(2), T(5). P has no predecessor. Q and R need P. S needs Q. T needs R and S. Find the minimum project time. 4 marks
Step 1, Precedence table:
P(__, __) Q(__, __) R(__, __) S(__, __) T(__, __ and __)
Step 2, List every path from start to finish.
Path 1: P → Q → S → T Path 2: P → R → T
Step 3, Add the durations on each path.
Path 1 = __ + __ + __ + __ = ____ Path 2 = __ + __ + __ = ____
Step 4, Take the maximum:
Min project time = max( __, __ ) = ____ days
Conclusion. Minimum project time = ____ days, controlled by path ____________.
4. Graduated practice, vocabulary, tables and minimum times
Show your working in the space below each part. For every minimum-time question, write at least one path-sum line.
Foundation, single-idea recall (4 questions)
| Q | Problem | Answer |
|---|---|---|
| 4.1 1 | True or false: an event has zero duration. | |
| 4.2 1 | True or false: a dummy activity is drawn as a solid arrow. | |
| 4.3 1 | In a precedence table, do you list all predecessors or only the immediate predecessors? | |
| 4.4 1 | What letter normally labels the start activity (the one with no predecessor) in our worked examples? |
Standard, typical HSC difficulty (6 questions)
For network questions, write the path-sums clearly before stating the minimum time.
4.5 A project has A(3), B(2,A), C(4,A), D(1,B), E(2,C,D). Write the precedence table in (activity, duration, immediate predecessors) form. 2 marks
4.6 For the project in Q4.5, list every path from start to finish. 2 marks
4.7 For the project in Q4.5, find the minimum project time. 2 marks
4.8 Activities: M(5), N(3,M), O(4,M), P(2,N,O). Find the minimum project time. 2 marks
4.9 In the AOA network of Q4.8, can N and O run in parallel? In one short sentence, explain why or why not. 2 marks
4.10 A project has A(2), B(3), C(4,A), D(5,B), E(1,C,D). Find the minimum project time. 2 marks
Extension, combine ideas (2 questions)
4.11 A house-build project has: Site prep S(2), Foundation F(4,S), Walls W(6,F), Roof R(3,W), Electrical E(4,W), Plumbing P(3,W), Drywall D(5,E,P), Finish N(2,D,R). Write the precedence table and find the minimum project time, showing every path. 3 marks
4.12 A small software project has: Design U(3), Backend B(6,U), Frontend F(5,U), Test T(2,B,F), Deploy D(1,T). The team wants the project finished in 11 working days, is this possible without changing any activity? Show the path-sum that decides this. 3 marks
5. Self-check the easy 3
Tick the first three once you've checked your method works.
How did this worksheet feel?
What I'll revisit before next class:
Q1.1, Definitions
Activity: a task that takes time (and resources) to complete, drawn as an arrow in AOA.
Event: a point in time when one or more activities start or finish, drawn as a numbered circle (node), zero duration.
Immediate predecessor of B: an activity that must finish before B can start (no intermediate activities in between).
Q1.2, AOA notation
Activities are drawn as arrows (arcs); events are drawn as circles (nodes). A dummy is drawn as a dashed arrow with duration 0.
Q1.3, The unique-event rule
No two activities can share both the same start event and the same end event. When the precedence table forces this, we insert a dummy to give one of the activities a different start or end event.
Q3, Faded example (P, Q, R, S, T)
Step 1: P(2), Q(4, P), R(3, P), S(2, Q), T(5, R and S).
Step 2: Path 1 P → Q → S → T; Path 2 P → R → T.
Step 3: Path 1 = 2 + 4 + 2 + 5 = 13; Path 2 = 2 + 3 + 5 = 10.
Step 4: Min project time = max(13, 10) = 13.
Conclusion: minimum project time = 13 days, controlled by path P → Q → S → T.
Q4.1, Event duration
True. An event is a point in time, so its duration is 0. Time passes during activities, not at events.
Q4.2, Dummy arrow style
False. A dummy is drawn as a dashed arrow (and labelled with duration 0). Solid arrows are reserved for real activities.
Q4.3, Predecessor scope
Only the immediate predecessors. Listing all ancestors is redundant and can produce incorrect network diagrams.
Q4.4, Start-activity label
In all our worked examples the start activity is the one with the empty predecessor column (typically A). A project may also have more than one start activity.
Q4.5, Precedence table for the small network
A(3), B(2, A), C(4, A), D(1, B), E(2, C and D).
Q4.6, Paths from start to finish
Path 1: A → B → D → E. Path 2: A → C → E.
Q4.7, Minimum project time
Path 1 = 3 + 2 + 1 + 2 = 8. Path 2 = 3 + 4 + 2 = 9. Min time = max(8, 9) = 9 days. Controlled by A → C → E.
Q4.8, M, N, O, P network
Path M → N → P = 5 + 3 + 2 = 10. Path M → O → P = 5 + 4 + 2 = 11. Min time = max(10, 11) = 11 days.
Q4.9, Parallel N and O
Yes. N and O share the same single predecessor (M) and neither one depends on the other, so they can run in parallel after M finishes.
Q4.10, Two start activities A and B
Path A → C → E = 2 + 4 + 1 = 7. Path B → D → E = 3 + 5 + 1 = 9. Min time = max(7, 9) = 9 days. (Two start activities is allowed: the project starts when either can begin.)
Q4.11, House-build network
Table: S(2), F(4, S), W(6, F), R(3, W), E(4, W), P(3, W), D(5, E and P), N(2, D and R).
Paths from S to N:
S → F → W → R → N = 2 + 4 + 6 + 3 + 2 = 17.
S → F → W → E → D → N = 2 + 4 + 6 + 4 + 5 + 2 = 23.
S → F → W → P → D → N = 2 + 4 + 6 + 3 + 5 + 2 = 22.
Min project time = max(17, 23, 22) = 23 days, controlled by S → F → W → E → D → N.
Q4.12-11-day finish for software project
Path U → B → T → D = 3 + 6 + 2 + 1 = 12. Path U → F → T → D = 3 + 5 + 2 + 1 = 11.
Min project time = max(12, 11) = 12 days. No, 11 days is not possible without changing any activity, because the longer (critical) path is 12 days.