Activity networks
Represent a project as an activity network (activity-on-node) from a precedence table; identify immediate predecessors and successors; recognise when a dummy activity is required.
Worked examples
Reading a precedence table
Straightforward
Problem
A small construction project has the following activities and their immediate predecessors.
| Activity | Description | Duration (days) | Immediate predecessors |
|---|---|---|---|
| A | Site preparation | 3 | — |
| B | Order materials | 2 | — |
| C | Lay foundations | 4 | A |
| D | Deliver materials | 3 | B |
| E | Build frame | 5 | C, D |
| F | Fit roof | 2 | E |
Identify: (a) which activities can start immediately, (b) what E depends on, and (c) which activity leads to the finish.
1
Part (a): find which activities can start immediately.
Activities with no predecessors can start at time zero. Reading the table: A and B both have a dash in the predecessors column — they can start immediately.
2
Part (b): find what activity E depends on.
Activity E lists C and D as immediate predecessors. E cannot start until both the foundations are laid (C) and the materials have been delivered (D).
3
Part (c): find which activity leads to the finish.
Activity F is not listed as a predecessor of any other activity, so F leads directly to the finish of the project.
Answer
(a) A and B can start immediately. (b) E depends on C and D. (c) F leads to the finish.
Identifying a dummy activity
Moderate
Problem
Two activities in a project share a complicated dependency: Activity C depends on A only; Activity D depends on both A and B. Explain why a dummy activity is needed and what it represents.
1
Consider how activity-on-arrow diagrams represent multiple predecessors.
In an activity-on-arrow network, each activity is drawn as an arrow between two nodes. To show that D requires both A and B, the tail of D must connect to the head-nodes of both A and B.
2
Identify the problem with drawing C and D sharing end-nodes.
If we draw C and D sharing the same pair of end-nodes (because both involve A), the diagram incorrectly shows that C also requires B.
3
Introduce a dummy activity to fix the diagram.
The solution is to draw A ending at node 2, then draw a dummy activity (dashed arrow, 0 days) from node 2 to node 3, and have B also end at node 3. D then starts from node 3. C starts from node 2.
4
Explain what the dummy activity represents.
The dummy arrow carries no duration. It simply says: "before D can start, you must wait for both A (via node 2 → node 3 dummy) and B (directly into node 3)." C correctly depends only on A (starting from node 2, before the dummy).
Answer
A dummy activity (zero duration) is needed to correctly show that D depends on both A and B while C depends only on A, without falsely linking C to B.
Finding activities with no successors
Moderate
Problem
For the project below, identify which activities lead directly to the finish node.
| Activity | Immediate predecessors |
|---|---|
| A | — |
| B | — |
| C | A |
| D | A, B |
| E | C, D |
| F | B |
| G | E, F |
1
State the rule for identifying finish activities.
An activity leads to the finish if no other activity lists it as a predecessor.
2
Check each activity in turn.
A: listed as predecessor of C and D — not a finish activity.
B: listed as predecessor of D and F — not a finish activity.
C: listed as predecessor of E — not a finish activity.
D: listed as predecessor of E — not a finish activity.
E: listed as predecessor of G — not a finish activity.
F: listed as predecessor of G — not a finish activity.
G: not listed as a predecessor of any activity — G leads to the finish.
B: listed as predecessor of D and F — not a finish activity.
C: listed as predecessor of E — not a finish activity.
D: listed as predecessor of E — not a finish activity.
E: listed as predecessor of G — not a finish activity.
F: listed as predecessor of G — not a finish activity.
G: not listed as a predecessor of any activity — G leads to the finish.
3
State the conclusion.
Only activity G leads directly to the finish node. The project ends when G is complete.
Answer
Only activity G leads directly to the finish node.
Practise
Q1·Straightforward
A project has four activities with the following immediate predecessors:
| Activity | Immediate predecessors |
|---|---|
| A | — |
| B | A |
| C | A |
| D | B, C |
How many immediate predecessors does activity D have?
Explanation
The table shows that D depends on both B and C. Therefore activity D has **2** immediate predecessors.
Q2·Straightforward
A project has five activities with the following immediate predecessors:
| Activity | Immediate predecessors |
|---|---|
| A | — |
| B | — |
| C | — |
| D | A, B |
| E | B, C |
How many activities have no immediate predecessor (i.e., can start at the beginning of the project)?
Explanation
Activities A, B and C all have no immediate predecessor — they can all begin at the start of the project. That is **3** activities.
Q3·Straightforward
In an activity network, a dummy activity is sometimes added to correctly represent dependencies without creating a false link. What is the duration (in days) of a dummy activity?
Explanation
A dummy activity has a duration of **0** days. It is a dashed arrow used purely to show a dependency relationship; it does not represent any real work in the project.
Q4·Moderate
A project has six activities:
| Activity | Immediate predecessors |
|---|---|
| A | — |
| B | — |
| C | A |
| D | A, B |
| E | C |
| F | D, E |
How many immediate predecessors does activity D have?
Explanation
Activity D has immediate predecessors A and B — so it has **2** immediate predecessors. D cannot start until both A and B are complete.
Q5·Moderate
Using the same project as the previous question:
| Activity | Immediate predecessors |
|---|---|
| A | — |
| B | — |
| C | A |
| D | A, B |
| E | C |
| F | D, E |
How many activities have B as an immediate predecessor?
Explanation
Only activity D lists B as an immediate predecessor. Therefore B is an immediate predecessor of **1** activity.
Q6·Moderate
A project has six activities:
| Activity | Immediate predecessors |
|---|---|
| A | — |
| B | — |
| C | A, B |
| D | B |
| E | C, D |
| F | C |
How many immediate predecessors does activity E have?
Explanation
Activity E depends on C and D, so it has **2** immediate predecessors.
Q7·Moderate
A project has seven activities:
| Activity | Immediate predecessors |
|---|---|
| A | — |
| B | A |
| C | A |
| D | B, C |
| E | D |
| F | B |
| G | E, F |
How many activities lead directly to the finish of the project (i.e., have no successors — nothing depends on them)?
Explanation
Check each activity: B is a predecessor of D and F; C is a predecessor of D; D is a predecessor of E; E is a predecessor of G; F is a predecessor of G. Only **G** appears in no other activity's predecessors column, so 1 activity leads directly to the finish.
Q8·Moderate
In a project network, dummy activities are used to keep the diagram logically correct. Each dummy activity has a duration of 0 days. If a project requires 3 dummy activities, what is the total number of days contributed by all dummy activities combined?
Explanation
Each dummy activity has duration 0 days, so days in total. Dummy activities never add to the project duration — they only show logical dependencies.
Q9·Challenging
A project has seven activities:
| Activity | Immediate predecessors |
|---|---|
| A | — |
| B | — |
| C | A, B |
| D | A |
| E | C, D |
| F | C |
| G | E, F |
How many immediate predecessors does activity G have?
Explanation
Activity G depends on E and F, giving it **2** immediate predecessors.
Q10·Challenging
A project has eight activities:
| Activity | Immediate predecessors |
|---|---|
| A | — |
| B | — |
| C | A |
| D | A, B |
| E | C |
| F | D, E |
| G | B |
| H | F, G |
How many activities have exactly 2 immediate predecessors?
Explanation
Checking each activity:
- A: 0 predecessors
- B: 0 predecessors
- C: 1 predecessor (A)
- D: 2 predecessors (A, B) ✓
- E: 1 predecessor (C)
- F: 2 predecessors (D, E) ✓
- G: 1 predecessor (B)
- H: 2 predecessors (F, G) ✓
Activities D, F and H each have exactly 2 immediate predecessors — so the answer is **3**.
- A: 0 predecessors
- B: 0 predecessors
- C: 1 predecessor (A)
- D: 2 predecessors (A, B) ✓
- E: 1 predecessor (C)
- F: 2 predecessors (D, E) ✓
- G: 1 predecessor (B)
- H: 2 predecessors (F, G) ✓
Activities D, F and H each have exactly 2 immediate predecessors — so the answer is **3**.
Q11·Challenging
A project has the following activities:
| Activity | Immediate predecessors |
|---|---|
| A | — |
| B | — |
| C | A, B |
| D | B |
| E | C, D |
| F | A |
| G | E, F |
Activity B is an immediate predecessor of how many activities?
Explanation
Activity C has predecessors A and B — so B is a predecessor of C. Activity D has predecessor B — so B is a predecessor of D. No other activity lists B. Therefore B is an immediate predecessor of **2** activities (C and D).
Q12·Challenging
A project manager creates the following precedence table:
| Activity | Immediate predecessors |
|---|---|
| A | — |
| B | — |
| C | A |
| D | A, B |
| E | C |
| F | D, E, B |
| G | F |
How many immediate predecessors does activity F have?
Explanation
Activity F lists D, E and B as immediate predecessors — so it has **3** immediate predecessors. F cannot begin until all three (D, E and B) are complete.
Open Math
Activity networks
Networks · MS-N3
Name:
Date:
Q1Straightforward
A project has four activities with the following immediate predecessors:
| Activity | Immediate predecessors |
|---|---|
| A | — |
| B | A |
| C | A |
| D | B, C |
How many immediate predecessors does activity D have?
Q2Straightforward
A project has five activities with the following immediate predecessors:
| Activity | Immediate predecessors |
|---|---|
| A | — |
| B | — |
| C | — |
| D | A, B |
| E | B, C |
How many activities have no immediate predecessor (i.e., can start at the beginning of the project)?
Q3Straightforward
In an activity network, a dummy activity is sometimes added to correctly represent dependencies without creating a false link. What is the duration (in days) of a dummy activity?
Q4Moderate
A project has six activities:
| Activity | Immediate predecessors |
|---|---|
| A | — |
| B | — |
| C | A |
| D | A, B |
| E | C |
| F | D, E |
How many immediate predecessors does activity D have?
Q5Moderate
Using the same project as the previous question:
| Activity | Immediate predecessors |
|---|---|
| A | — |
| B | — |
| C | A |
| D | A, B |
| E | C |
| F | D, E |
How many activities have B as an immediate predecessor?
Q6Moderate
A project has six activities:
| Activity | Immediate predecessors |
|---|---|
| A | — |
| B | — |
| C | A, B |
| D | B |
| E | C, D |
| F | C |
How many immediate predecessors does activity E have?
Q7Moderate
A project has seven activities:
| Activity | Immediate predecessors |
|---|---|
| A | — |
| B | A |
| C | A |
| D | B, C |
| E | D |
| F | B |
| G | E, F |
How many activities lead directly to the finish of the project (i.e., have no successors — nothing depends on them)?
Q8Moderate
In a project network, dummy activities are used to keep the diagram logically correct. Each dummy activity has a duration of 0 days. If a project requires 3 dummy activities, what is the total number of days contributed by all dummy activities combined?
Q9Challenging
A project has seven activities:
| Activity | Immediate predecessors |
|---|---|
| A | — |
| B | — |
| C | A, B |
| D | A |
| E | C, D |
| F | C |
| G | E, F |
How many immediate predecessors does activity G have?
Q10Challenging
A project has eight activities:
| Activity | Immediate predecessors |
|---|---|
| A | — |
| B | — |
| C | A |
| D | A, B |
| E | C |
| F | D, E |
| G | B |
| H | F, G |
How many activities have exactly 2 immediate predecessors?
Q11Challenging
A project has the following activities:
| Activity | Immediate predecessors |
|---|---|
| A | — |
| B | — |
| C | A, B |
| D | B |
| E | C, D |
| F | A |
| G | E, F |
Activity B is an immediate predecessor of how many activities?
Q12Challenging
A project manager creates the following precedence table:
| Activity | Immediate predecessors |
|---|---|
| A | — |
| B | — |
| C | A |
| D | A, B |
| E | C |
| F | D, E, B |
| G | F |
How many immediate predecessors does activity F have?
Worked solutions and answers at openmath.au/year-12/standard-2/critical-path-analysis/activity-networks