The critical path
Identify the critical path as the longest path through the activity network; determine the minimum project completion time; interpret the effect of delays to critical and non-critical activities.
Worked examples
Identifying the critical path
Moderate
Problem
A project has the following activities. Find the minimum completion time and identify the critical path.
| Activity | Predecessors | Duration (days) |
|---|---|---|
| A | — | 5 |
| B | — | 3 |
| C | A | 4 |
| D | A, B | 6 |
| E | C, D | 3 |
1
Perform a forward scan to find the EST and EFT of A and B, which have no predecessors.
2
Find the EST and EFT of C and D, which follow A (and B for D).
3
Minimum project completion time = 14 days.
Find the EST and EFT of E, which requires both C and D to finish, giving the minimum completion time.
Minimum project completion time = 14 days.
4
Perform a backward scan, setting the LST of the last activity so the project finishes exactly on time.
5
Calculate the float (Float = LST − EST) for each activity and identify the critical path (float = 0).
| Activity | EST | LST | Float |
|---|---|---|---|
| A | 0 | 0 | 0 ← critical |
| B | 0 | 2 | 2 |
| C | 5 | 7 | 2 |
| D | 5 | 5 | 0 ← critical |
| E | 11 | 11 | 0 ← critical |
The critical path is A → D → E, with duration days.
Answer
The minimum completion time is 14 days, and the critical path is A → D → E.
Effect of a delay on a critical activity
Moderate
Problem
Using the project above (critical path A → D → E, minimum completion time 14 days), activity D is delayed by 4 days. What is the new minimum completion time?
1
Consider the effect of a delay to an activity with zero float.
Activity D is on the critical path, so it has zero float. Any delay to D adds directly to the project duration.
2
Add the delay to the original minimum completion time.
3
State the conclusion.
The project is now expected to take 18 days minimum. If Robert wants to recover the original 14-day finish, another activity on the critical path must be shortened by 4 days.
Answer
The new minimum completion time is 18 days.
Effect of a delay on a non-critical activity
Challenging
Problem
Activity B (float = 2 days) is delayed by 5 days. By how many days is the project delayed?
1
Consider how much delay the activity's float can absorb.
Activity B has 2 days of float — it can absorb up to 2 days of delay without affecting the project finish.
2
Find the excess delay beyond the float.
The delay of 5 days exceeds the float by days. The excess delay flows through to the project finish.
3
The new minimum completion time would be days.
State the project delay and new completion time.
The new minimum completion time would be days.
4
Note the effect on B's status.
Once activity B is delayed by more than 2 days, it becomes critical — it now has zero float and joins the critical path.
Answer
The project is delayed by 3 days, giving a new minimum completion time of 17 days.
Practise
Q1·Straightforward
An activity has an earliest start time (EST) of 4 days and a latest start time (LST) of 4 days. What is the float of this activity?
Float = LST − EST.
Float = LST − EST.
Explanation
A float of 0 means this activity is on the critical path — any delay to it will delay the entire project.
Q2·Straightforward
A project has the following activities with durations (in days):
| Activity | Predecessors | Duration |
|---|---|---|
| A | — | 5 |
| B | A | 3 |
| C | A | 6 |
| D | B, C | 4 |
The minimum completion time of the project is the EFT of the last activity. Perform a forward scan and find the minimum completion time.
Explanation
Forward scan:
The minimum project completion time is **15 days**.
The minimum project completion time is **15 days**.
Q3·Straightforward
Which statement correctly describes the critical path in a project network?
Explanation
The critical path is the **longest path** from the start to the finish of the network. Its length equals the **minimum possible project duration**. All activities on the critical path have zero float — any delay to any one of them delays the whole project.
Q4·Moderate
A project has the following activities:
| Activity | Predecessors | Duration (days) |
|---|---|---|
| A | — | 3 |
| B | — | 7 |
| C | A | 5 |
| D | A, B | 2 |
| E | C, D | 4 |
Perform a forward scan and find the minimum project completion time.
Explanation
Forward scan:
Minimum completion time = **13 days**.
Minimum completion time = **13 days**.
Q5·Moderate
Using the same project as the previous question (minimum completion time = 13 days):
| Activity | Predecessors | Duration (days) |
|---|---|---|
| A | — | 3 |
| B | — | 7 |
| C | A | 5 |
| D | A, B | 2 |
| E | C, D | 4 |
After completing a forward and backward scan, the float values are:
- A: float = 4, B: float = 0, C: float = 4, D: float = 0, E: float = 0
How many activities are on the critical path?
- A: float = 4, B: float = 0, C: float = 4, D: float = 0, E: float = 0
How many activities are on the critical path?
Explanation
Activities B, D and E all have float = 0, so they lie on the critical path. There are **3** activities on the critical path.
The critical path is B → D → E, with total duration days.
The critical path is B → D → E, with total duration days.
Q6·Moderate
A project has minimum completion time of 20 days. Activity G is on the critical path and has a duration of 3 days. If activity G is delayed by 2 days, what is the new minimum project completion time (in days)?
Explanation
Because G is on the critical path (float = 0), a 2-day delay to G causes a 2-day delay to the entire project.
Q7·Moderate
Activity H has float of 5 days. If activity H is delayed by 3 days, by how many days is the project completion time extended?
Explanation
Activity H has 5 days of float. A delay of 3 days is less than the float, so the project finish date is unaffected.
The project completion time is extended by **0 days**.
The project completion time is extended by **0 days**.
Q8·Moderate
A project has the following activities:
| Activity | Predecessors | Duration (days) |
|---|---|---|
| A | — | 4 |
| B | — | 2 |
| C | A | 3 |
| D | B | 6 |
| E | C, D | 5 |
The critical path runs through A → C → E or B → D → E. Determine the minimum project completion time.
Explanation
Path A → C → E: days
Path B → D → E: days
The longest path is B → D → E with **13 days**, so the minimum project completion time is 13 days.
Path B → D → E: days
The longest path is B → D → E with **13 days**, so the minimum project completion time is 13 days.
Q9·Challenging
A project has the following activities:
| Activity | Predecessors | Duration (days) |
|---|---|---|
| A | — | 2 |
| B | — | 5 |
| C | A | 4 |
| D | A, B | 3 |
| E | C | 6 |
| F | D, E | 2 |
Perform a complete forward scan and find the minimum project completion time.
Explanation
Forward scan:
Minimum completion time = **14 days**.
Minimum completion time = **14 days**.
Q10·Challenging
Using the same project as the previous question (minimum completion time = 14 days):
| Activity | Predecessors | Duration (days) |
|---|---|---|
| A | — | 2 |
| B | — | 5 |
| C | A | 4 |
| D | A, B | 3 |
| E | C | 6 |
| F | D, E | 2 |
Perform a backward scan. What is the float of activity B?
Explanation
Backward scan (project duration = 14):
For B (only D follows B):
From the forward scan: .
For B (only D follows B):
From the forward scan: .
Q11·Challenging
Using the same project (completion time = 14 days):
| Activity | Predecessors | Duration (days) |
|---|---|---|
| A | — | 2 |
| B | — | 5 |
| C | A | 4 |
| D | A, B | 3 |
| E | C | 6 |
| F | D, E | 2 |
Activity B has float of 4 days. If activity B is delayed by 5 days, by how many days is the minimum project completion time extended?
Explanation
Activity B has 4 days of float. A delay of 5 days exceeds the float by day.
The excess delay propagates to the finish of the project, extending the minimum completion time by **1 day** (from 14 days to 15 days).
The excess delay propagates to the finish of the project, extending the minimum completion time by **1 day** (from 14 days to 15 days).
Q12·Challenging
A project has the following activities:
| Activity | Predecessors | Duration (days) |
|---|---|---|
| A | — | 6 |
| B | — | 4 |
| C | A | 3 |
| D | A, B | 5 |
| E | B | 8 |
| F | C, D | 2 |
| G | E, F | 1 |
Find the minimum project completion time.
Explanation
Forward scan:
Minimum completion time = **14 days**.
Minimum completion time = **14 days**.
Open Math
The critical path
Networks · MS-N3
Name:
Date:
Q1Straightforward
An activity has an earliest start time (EST) of 4 days and a latest start time (LST) of 4 days. What is the float of this activity?
Float = LST − EST.
Float = LST − EST.
Q2Straightforward
A project has the following activities with durations (in days):
| Activity | Predecessors | Duration |
|---|---|---|
| A | — | 5 |
| B | A | 3 |
| C | A | 6 |
| D | B, C | 4 |
The minimum completion time of the project is the EFT of the last activity. Perform a forward scan and find the minimum completion time.
Q3Straightforward
Which statement correctly describes the critical path in a project network?
- A.The shortest path from start to finish
- B.The path of activities with the most float
- C.The longest path from start to finish, determining the minimum project duration
- D.The path with the fewest activities
Q4Moderate
A project has the following activities:
| Activity | Predecessors | Duration (days) |
|---|---|---|
| A | — | 3 |
| B | — | 7 |
| C | A | 5 |
| D | A, B | 2 |
| E | C, D | 4 |
Perform a forward scan and find the minimum project completion time.
Q5Moderate
Using the same project as the previous question (minimum completion time = 13 days):
| Activity | Predecessors | Duration (days) |
|---|---|---|
| A | — | 3 |
| B | — | 7 |
| C | A | 5 |
| D | A, B | 2 |
| E | C, D | 4 |
After completing a forward and backward scan, the float values are:
- A: float = 4, B: float = 0, C: float = 4, D: float = 0, E: float = 0
How many activities are on the critical path?
- A: float = 4, B: float = 0, C: float = 4, D: float = 0, E: float = 0
How many activities are on the critical path?
Q6Moderate
A project has minimum completion time of 20 days. Activity G is on the critical path and has a duration of 3 days. If activity G is delayed by 2 days, what is the new minimum project completion time (in days)?
Q7Moderate
Activity H has float of 5 days. If activity H is delayed by 3 days, by how many days is the project completion time extended?
Q8Moderate
A project has the following activities:
| Activity | Predecessors | Duration (days) |
|---|---|---|
| A | — | 4 |
| B | — | 2 |
| C | A | 3 |
| D | B | 6 |
| E | C, D | 5 |
The critical path runs through A → C → E or B → D → E. Determine the minimum project completion time.
Q9Challenging
A project has the following activities:
| Activity | Predecessors | Duration (days) |
|---|---|---|
| A | — | 2 |
| B | — | 5 |
| C | A | 4 |
| D | A, B | 3 |
| E | C | 6 |
| F | D, E | 2 |
Perform a complete forward scan and find the minimum project completion time.
Q10Challenging
Using the same project as the previous question (minimum completion time = 14 days):
| Activity | Predecessors | Duration (days) |
|---|---|---|
| A | — | 2 |
| B | — | 5 |
| C | A | 4 |
| D | A, B | 3 |
| E | C | 6 |
| F | D, E | 2 |
Perform a backward scan. What is the float of activity B?
Q11Challenging
Using the same project (completion time = 14 days):
| Activity | Predecessors | Duration (days) |
|---|---|---|
| A | — | 2 |
| B | — | 5 |
| C | A | 4 |
| D | A, B | 3 |
| E | C | 6 |
| F | D, E | 2 |
Activity B has float of 4 days. If activity B is delayed by 5 days, by how many days is the minimum project completion time extended?
Q12Challenging
A project has the following activities:
| Activity | Predecessors | Duration (days) |
|---|---|---|
| A | — | 6 |
| B | — | 4 |
| C | A | 3 |
| D | A, B | 5 |
| E | B | 8 |
| F | C, D | 2 |
| G | E, F | 1 |
Find the minimum project completion time.
Worked solutions and answers at openmath.au/year-12/standard-2/critical-path-analysis/the-critical-path