Forward and backward scanning
Apply forward scanning to find earliest start times (EST) and backward scanning to find latest start times (LST); calculate float (slack) time for each activity.
Worked examples
Forward scan — finding earliest start times
Straightforward
Problem
A project has the following activities. Perform a forward scan to find the EST and EFT of each activity.
| Activity | Predecessors | Duration (days) |
|---|---|---|
| A | — | 4 |
| B | — | 6 |
| C | A | 3 |
| D | A, B | 5 |
| E | C, D | 2 |
1
State the forward scan rule.
For activities with no predecessors, EST = 0. For all others, EST = maximum of the EFTs of all immediate predecessors. EFT = EST + duration.
2
Find the EST and EFT of A and B, which have no predecessors.
3
Find the EST and EFT of C, which follows A only.
4
Find the EST and EFT of D, which follows both A and B.
D must wait for the later finish of its predecessors.
5
Find the EST and EFT of E, which follows both C and D.
Answer
The minimum project completion time is the EFT of the last activity: 13 days.
Backward scan — finding latest start times
Moderate
Problem
Using the same project from Example 1 (minimum duration = 13 days), perform a backward scan to find the LST of each activity.
1
State the backward scan rule.
For the last activity, LST = project duration − duration of that activity. For all others, LST = minimum of the LSTs of all immediate successors, minus this activity's duration.
2
Find the LST of E, the last activity.
3
Find the LST of D, for which only E follows.
4
Find the LST of C, for which only E follows.
5
Find the LST of B, for which only D follows.
6
Find the LST of A, for which both C and D follow — take the minimum.
Answer
LST values: A = 2, B = 0, C = 8, D = 6, E = 11.
Calculating float
Challenging
Problem
Using the ESTs from Example 1 and LSTs from Example 2, calculate the float of each activity and identify which have zero float. Float = LST − EST. An activity with float = 0 is on the critical path.
1
Tabulate EST and LST for each activity and compute the float.
| Activity | EST | LST | Float = LST − EST |
|---|---|---|---|
| A | 0 | 2 | 2 |
| B | 0 | 0 | 0 |
| C | 4 | 8 | 4 |
| D | 6 | 6 | 0 |
| E | 11 | 11 | 0 |
2
Identify the activities with zero float.
Activities B, D and E all have float = 0. These are on the critical path.
3
Note the float of the non-critical activities.
Activity A has 2 days of float — it can start up to 2 days late without delaying the project. Activity C has 4 days of float.
4
State the critical path and confirm its total duration.
The critical path is B → D → E, with total duration days, confirming the minimum project time.
Answer
The critical path is B → D → E (float = 0), with total duration 13 days.
Practise
Q1·Straightforward
Activity A has no predecessors and a duration of 6 days. Using a forward scan (earliest start time = 0 for activities with no predecessors), what is the earliest finish time (EFT) of activity A?
Earliest finish time = EST + duration.
Earliest finish time = EST + duration.
Explanation
Activity A can finish as early as day 6.
Q2·Straightforward
Activity B has an earliest start time (EST) of 8 days and a latest start time (LST) of 11 days. What is the float (slack) of activity B?
Float = LST − EST.
Float = LST − EST.
Explanation
Activity B can be delayed by up to 3 days without delaying the project.
Q3·Straightforward
A project must be completed in 20 days. Activity D is the last activity and has a duration of 4 days. Using a backward scan, what is the latest start time (LST) of activity D?
For the last activity: LST = project duration − duration of the activity.
For the last activity: LST = project duration − duration of the activity.
Explanation
Activity D must start no later than day 16 to ensure the project finishes on time.
Q4·Moderate
Activity A (duration 3 days) and Activity B (duration 5 days) both have no predecessors, so each has EST = 0. Activity C cannot start until both A and B are finished. What is the earliest start time (EST) of activity C?
Explanation
The earliest C can start is the latest of the finish times of A and B:
C cannot start until day 5.
C cannot start until day 5.
Q5·Moderate
A project has the following activities (duration in days):
| Activity | Predecessors | Duration |
|---|---|---|
| A | — | 4 |
| B | — | 7 |
| C | A | 3 |
| D | B | 2 |
| E | C, D | 5 |
Perform a forward scan and find the EST of activity E.
Explanation
Forward scan:
Q6·Moderate
A project has the following activities (duration in days):
| Activity | Predecessors | Duration |
|---|---|---|
| A | — | 3 |
| B | — | 6 |
| C | A, B | 4 |
| D | A | 5 |
| E | C, D | 2 |
Perform a forward scan and find the EST of activity C.
Explanation
C can start no earlier than day 6.
Q7·Moderate
A project has a minimum completion time of 18 days. Activity F has a duration of 4 days and is the last activity before the finish. Using a backward scan, what is the latest start time (LST) of activity F?
Explanation
If F starts any later than day 14, the project will not finish on time.
Q8·Moderate
A project has the following activities:
| Activity | Predecessors | Duration (days) |
|---|---|---|
| A | — | 5 |
| B | A | 3 |
| C | A | 4 |
| D | B, C | 2 |
| E | D | 6 |
After performing a complete forward and backward scan, the project duration is 17 days. The LST of D is 9 days and its EST is 9 days. What is the float of activity B?
(Hint: EFT of A = 5. EST of B = 5. EFT of B = 8. LST of D = 9, so LST of B = 9 − 3 = 6.)
(Hint: EFT of A = 5. EST of B = 5. EFT of B = 8. LST of D = 9, so LST of B = 9 − 3 = 6.)
Explanation
Forward scan:
Backward scan:
Float of B:
Backward scan:
Float of B:
Q9·Challenging
A project has the following activities:
| Activity | Predecessors | Duration (days) |
|---|---|---|
| A | — | 3 |
| B | — | 5 |
| C | A | 4 |
| D | A, B | 2 |
| E | A | 6 |
| F | C, D, E | 3 |
Perform a forward scan. What is the EST of activity D?
Explanation
Forward scan:
Q10·Challenging
Using the same project as the previous question:
| Activity | Predecessors | Duration (days) |
|---|---|---|
| A | — | 3 |
| B | — | 5 |
| C | A | 4 |
| D | A, B | 2 |
| E | A | 6 |
| F | C, D, E | 3 |
Complete the forward scan and find the minimum project completion time.
Explanation
Forward scan:
The minimum project completion time is **12 days**.
The minimum project completion time is **12 days**.
Q11·Challenging
A project has the following activities:
| Activity | Predecessors | Duration (days) |
|---|---|---|
| A | — | 3 |
| B | — | 5 |
| C | A | 4 |
| D | A, B | 2 |
| E | A | 6 |
| F | C, D, E | 3 |
The minimum project completion time is 12 days. Perform a backward scan and find the float of activity D.
Explanation
From the forward scan: .
Backward scan:
Float of D:
Activity D has 2 days of float — it can be delayed by up to 2 days without affecting the project completion time.
Backward scan:
Float of D:
Activity D has 2 days of float — it can be delayed by up to 2 days without affecting the project completion time.
Q12·Challenging
A project has the following activities:
| Activity | Predecessors | Duration (days) |
|---|---|---|
| A | — | 3 |
| B | — | 5 |
| C | A | 4 |
| D | A, B | 2 |
| E | A | 6 |
| F | C, D, E | 3 |
The minimum project completion time is 12 days. Perform a complete backward scan and find the latest start time (LST) of activity B.
Explanation
Backward scan (project duration = 12):
For B: only D follows B, so:
Activity B must start by day 2 at the latest.
For B: only D follows B, so:
Activity B must start by day 2 at the latest.
Open Math
Forward and backward scanning
Networks · MS-N3
Name:
Date:
Q1Straightforward
Activity A has no predecessors and a duration of 6 days. Using a forward scan (earliest start time = 0 for activities with no predecessors), what is the earliest finish time (EFT) of activity A?
Earliest finish time = EST + duration.
Earliest finish time = EST + duration.
Q2Straightforward
Activity B has an earliest start time (EST) of 8 days and a latest start time (LST) of 11 days. What is the float (slack) of activity B?
Float = LST − EST.
Float = LST − EST.
Q3Straightforward
A project must be completed in 20 days. Activity D is the last activity and has a duration of 4 days. Using a backward scan, what is the latest start time (LST) of activity D?
For the last activity: LST = project duration − duration of the activity.
For the last activity: LST = project duration − duration of the activity.
Q4Moderate
Activity A (duration 3 days) and Activity B (duration 5 days) both have no predecessors, so each has EST = 0. Activity C cannot start until both A and B are finished. What is the earliest start time (EST) of activity C?
Q5Moderate
A project has the following activities (duration in days):
| Activity | Predecessors | Duration |
|---|---|---|
| A | — | 4 |
| B | — | 7 |
| C | A | 3 |
| D | B | 2 |
| E | C, D | 5 |
Perform a forward scan and find the EST of activity E.
Q6Moderate
A project has the following activities (duration in days):
| Activity | Predecessors | Duration |
|---|---|---|
| A | — | 3 |
| B | — | 6 |
| C | A, B | 4 |
| D | A | 5 |
| E | C, D | 2 |
Perform a forward scan and find the EST of activity C.
Q7Moderate
A project has a minimum completion time of 18 days. Activity F has a duration of 4 days and is the last activity before the finish. Using a backward scan, what is the latest start time (LST) of activity F?
Q8Moderate
A project has the following activities:
| Activity | Predecessors | Duration (days) |
|---|---|---|
| A | — | 5 |
| B | A | 3 |
| C | A | 4 |
| D | B, C | 2 |
| E | D | 6 |
After performing a complete forward and backward scan, the project duration is 17 days. The LST of D is 9 days and its EST is 9 days. What is the float of activity B?
(Hint: EFT of A = 5. EST of B = 5. EFT of B = 8. LST of D = 9, so LST of B = 9 − 3 = 6.)
(Hint: EFT of A = 5. EST of B = 5. EFT of B = 8. LST of D = 9, so LST of B = 9 − 3 = 6.)
Q9Challenging
A project has the following activities:
| Activity | Predecessors | Duration (days) |
|---|---|---|
| A | — | 3 |
| B | — | 5 |
| C | A | 4 |
| D | A, B | 2 |
| E | A | 6 |
| F | C, D, E | 3 |
Perform a forward scan. What is the EST of activity D?
Q10Challenging
Using the same project as the previous question:
| Activity | Predecessors | Duration (days) |
|---|---|---|
| A | — | 3 |
| B | — | 5 |
| C | A | 4 |
| D | A, B | 2 |
| E | A | 6 |
| F | C, D, E | 3 |
Complete the forward scan and find the minimum project completion time.
Q11Challenging
A project has the following activities:
| Activity | Predecessors | Duration (days) |
|---|---|---|
| A | — | 3 |
| B | — | 5 |
| C | A | 4 |
| D | A, B | 2 |
| E | A | 6 |
| F | C, D, E | 3 |
The minimum project completion time is 12 days. Perform a backward scan and find the float of activity D.
Q12Challenging
A project has the following activities:
| Activity | Predecessors | Duration (days) |
|---|---|---|
| A | — | 3 |
| B | — | 5 |
| C | A | 4 |
| D | A, B | 2 |
| E | A | 6 |
| F | C, D, E | 3 |
The minimum project completion time is 12 days. Perform a complete backward scan and find the latest start time (LST) of activity B.
Worked solutions and answers at openmath.au/year-12/standard-2/critical-path-analysis/forward-and-backward-scanning