A practical guide to CPM and PERT project scheduling — building the network, the forward and backward pass, the critical path, slack/float, PERT three-point estimates, and crashing to compress schedules.
What Are CPM and PERT?
CPM (Critical Path Method) and PERT (Program Evaluation and Review Technique) are network-based techniques for planning, scheduling, and controlling projects. Both model a project as a network of activities with dependencies, and both identify the critical path — the chain of activities that sets the project's minimum duration. They differ mainly in how they treat time: CPM uses fixed, deterministic durations and excels at time-cost trade-offs; PERT uses probabilistic estimates to handle uncertainty. Modern project management blends them.
Building the Network
A project is broken into activities, each with a duration and a list of predecessors (activities that must finish first). The most common diagram is activity-on-node (AON), where each box is an activity and arrows show dependencies. Consider this small project:
| Activity | Predecessor | Duration (days) |
|---|
| A | — | 3 |
| B | A | 4 |
| C | A | 2 |
| D | B, C | 5 |
| E | C | 6 |
| F | D, E | 2 |
The Forward Pass: Earliest Times
The forward pass moves through the network from start to finish computing each activity's earliest start (ES) and earliest finish (EF = ES + duration). An activity cannot start until all predecessors finish, so its ES equals the largest EF among its predecessors. For the example (starting at day 0):
- A: ES=0, EF=3
- B: ES=3, EF=7; C: ES=3, EF=5
- D: ES=max(7,5)=7, EF=12; E: ES=5, EF=11
- F: ES=max(12,11)=12, EF=14
The project's earliest completion is 14 days.
The Backward Pass: Latest Times
The backward pass moves from finish to start computing each activity's latest finish (LF) and latest start (LS = LF − duration) — the latest times that do not delay the project. Starting from the project finish at 14:
- F: LF=14, LS=12
- D: LF=12, LS=7; E: LF=12, LS=6
- B: LF=7, LS=3; C: LF=min(7,6)=6, LS=4
- A: LF=min(3,4)=3, LS=0
Slack and the Critical Path
Slack (float) is the amount an activity can be delayed without delaying the project: Slack = LS − ES = LF − EF. Activities with zero slack lie on the critical path — delaying any of them delays the whole project.
| Activity | ES | EF | LS | LF | Slack |
|---|
| A | 0 | 3 | 0 | 3 | 0 (critical) |
| B | 3 | 7 | 3 | 7 | 0 (critical) |
| C | 3 | 5 | 4 | 6 | 1 |
| D | 7 | 12 | 7 | 12 | 0 (critical) |
| E | 5 | 11 | 6 | 12 | 1 |
| F | 12 | 14 | 12 | 14 | 0 (critical) |
The critical path is A → B → D → F, totaling 3+4+5+2 = 14 days. You can build and analyze networks like this on the CPM/PERT calculator.
PERT: Handling Uncertainty
Real durations are uncertain. PERT replaces single estimates with a three-point estimate per activity — optimistic (O), most likely (M), and pessimistic (P) — and computes an expected time using a beta-distribution approximation:
tₑ = (O + 4M + P) / 6
and a variance:
σ² = ((P − O) / 6)²
The expected project duration is the sum of tₑ along the critical path, and the project variance is the sum of the activity variances on that path. With the central limit theorem, you can then estimate the probability of finishing by a target date using the normal distribution: z = (target − expected) / σpath.
Example. If an activity has O=2, M=5, P=14, then tₑ = (2 + 20 + 14)/6 = 6 days, and σ² = ((14−2)/6)² = 4.
Crashing: Compressing the Schedule
Crashing shortens the project by adding resources — overtime, extra crews, expedited materials — to reduce critical-activity durations. Each activity has a normal time/cost and a crash time/cost, and the key figure is the crash cost per day saved:
crash cost slope = (crash cost − normal cost) / (normal time − crash time)
Rules of thumb for crashing:
- Only crashing critical-path activities shortens the project.
- Crash the critical activity with the lowest cost slope first.
- Re-check the critical path after each crash — shortening one path can make a parallel path become critical, after which you may need to crash activities on both paths simultaneously.
- Stop when the target is met, when no critical activity can be crashed further, or when the marginal crash cost exceeds the value of time saved.
CPM vs. PERT at a Glance
| CPM | PERT |
|---|
| Time estimates | Single, deterministic | Three-point, probabilistic |
| Focus | Time-cost trade-off (crashing) | Uncertainty and probability of completion |
| Typical use | Construction, repetitive projects | R&D, novel projects |
In practice, planners use the same network and pass logic for both, applying PERT estimates to capture risk and CPM crashing to manage the time-cost trade-off.