Critical Path Analysis
Learn how to schedule projects efficiently using precedence tables, forward and backward passes, and float time calculations.
Precedence Tables and Network Diagrams
A precedence table lists each activity in a project, its duration, and which activities must be completed before it can begin (its predecessors). This information is used to draw a network diagram.
Example Precedence Table
| Activity | Duration | Predecessors |
|---|---|---|
| A | 3 | — |
| B | 4 | — |
| C | 2 | A |
| D | 5 | A, B |
| E | 3 | C, D |
Activities without predecessors can start immediately. Activities with predecessors cannot start until all their predecessors are complete.
Forward and Backward Pass
The forward pass calculates the Earliest Start Time (EST) and Earliest Finish Time (EFT) for each activity by working left to right through the network:
EFT = EST + Duration
When an activity has multiple predecessors, its EST is the maximum EFT of all predecessors.
The backward pass calculates the Latest Finish Time (LFT) and Latest Start Time (LST) by working right to left:
LST = LFT − Duration
When an activity has multiple successors, its LFT is the minimum LST of all successors.
Float Time and the Critical Path
Float (or slack) is the amount of time an activity can be delayed without affecting the overall project completion time:
Float = LST − EST = LFT − EFT
Activities with zero float are critical activities. The critical path is the longest path through the network, connecting all critical activities. It determines the minimum project duration.
Any delay to a critical activity will delay the entire project. Non-critical activities have flexibility (float) within which they can be rescheduled.
Key Vocabulary
Critical Path
The longest path through the network; its length equals the minimum project duration.
Float (Slack)
The maximum time an activity can be delayed without delaying the project.
Forward Pass
The process of calculating earliest start and finish times from left to right.
Backward Pass
The process of calculating latest start and finish times from right to left.
Worked Examples
Using the table above, find the EST and EFT for each activity.
A: EST = 0, EFT = 0 + 3 = 3.
B: EST = 0, EFT = 0 + 4 = 4.
C: EST = 3 (after A), EFT = 3 + 2 = 5.
D: EST = max(3, 4) = 4 (after A and B), EFT = 4 + 5 = 9.
E: EST = max(5, 9) = 9 (after C and D), EFT = 9 + 3 = 12.
Find the LST and LFT for each activity (project duration = 12).
E: LFT = 12, LST = 12 − 3 = 9.
D: LFT = 9 (before E), LST = 9 − 5 = 4.
C: LFT = 9 (before E), LST = 9 − 2 = 7.
B: LFT = 4 (before D), LST = 4 − 4 = 0.
A: LFT = min(7, 4) = 4 (before C and D), LST = 4 − 3 = 1.
Find the float for each activity and identify the critical path.
A: Float = 1 − 0 = 1. B: Float = 0 − 0 = 0.
C: Float = 7 − 3 = 4. D: Float = 4 − 4 = 0.
E: Float = 9 − 9 = 0.
Critical path: B → D → E (all activities with float = 0). Minimum duration = 12.
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What does the critical path determine?
Question 2
An activity has EST = 3 and LST = 7. What is its float?
Question 3
Activities on the critical path have a float of:
Question 4
Activity X has predecessors P (EFT = 5) and Q (EFT = 8). What is the EST of X?
Question 5
In the backward pass, if activity Y has successors with LST values of 10 and 14, what is the LFT of Y?
Key Concepts Summary
- ● A precedence table lists activities, durations, and prerequisites.
- ● The forward pass finds EST and EFT; use the maximum when multiple predecessors merge.
- ● The backward pass finds LST and LFT; use the minimum when multiple successors diverge.
- ● Float = LST − EST. Activities with zero float are on the critical path.
- ● The critical path is the longest path and determines the minimum project duration.