The four dates on every activity
Each activity in a network diagram carries four dates. This page uses the common zero-based convention (the project starts at time 0).
- ES — Early Start: earliest an activity can begin.
- EF — Early Finish:
EF = ES + Duration. - LS — Late Start: latest it can begin without delaying the project.
- LF — Late Finish: latest it can finish without delaying the project.
Forward pass, then backward pass
| Pass | Rule |
|---|---|
| Forward pass (find ES/EF) | First activity ES = 0. EF = ES + Duration. A successor's ES = max(EF) of all its predecessors. |
| Backward pass (find LS/LF) | Last activity LF = project EF. LS = LF − Duration. A predecessor's LF = min(LS) of all its successors. |
The critical path is the longest path through the network — and equivalently, the chain of activities with zero total float. It sets the shortest possible project duration. Any delay on it delays the whole project.
Total float vs free float
| Quantity | Formula | Meaning |
|---|---|---|
| Total float | LS − ES (= LF − EF) | Delay available without delaying the project |
| Free float | ES(next) − EF(this) | Delay available without delaying the next activity |
Critical-path activities have total float = 0. Free float ≤ total float, always.
Worked example
Path A→B→D takes 3 + 5 + 4 = 12 days; path A→C→D takes 3 + 2 + 4 = 9 days. The longer path (12) is critical.
- For C: forward pass gives
ES = 3, EF = 5. Backward pass (so D can still start at day 8) givesLF = 8, LS = 6. Total float of C = LS − ES = 6 − 3 = 3 days→ C is not critical.- A, B, D all have zero float → they form the critical path.
Mistakes examiners test
- Picking the shortest path. The critical path is the longest duration path, not the shortest.
- Confusing total and free float. Total = vs the project end; free = vs the next activity.
- Forgetting near-critical paths. A path with small float can become critical if it slips.
- Mixing date conventions. Be consistent: zero-based vs one-based shifts every number by one.
Test yourself
4 quick questions — tap an answer to check it instantly. Nothing is sent anywhere.
1. What is the critical path?
2. Total float of an activity equals…
3. Activities on the critical path have a total float of…
4. Free float is the delay possible without delaying…
Sources
- Project Management Institute — pmi.org.
- PMI, A Guide to the Project Management Body of Knowledge (PMBOK® Guide) — schedule management.
Study summary only. "PMP" and "PMBOK" are marks of the Project Management Institute, which does not endorse this material.