PERT and CPM: How to Find the Critical Path (Worked Example)

PERT and CPM: How to Find the Critical Path (Worked Example)

PERT (Program Evaluation and Review Technique) and CPM (Critical Path Method) are the two classic tools for planning and scheduling projects. They break a project into activities, map the dependencies between them, and find the critical path — the sequence that determines the shortest possible project duration. This tutorial shows the full calculation.

Build your project network and find the critical path with the free PERT/CPM Solver.

PERT vs. CPM

• CPM uses a single, deterministic time estimate per activity — best when durations are well known.
• PERT uses three estimates (optimistic, most likely, pessimistic) to account for uncertainty.

Step 1: Expected Time and Variance (PERT)

For each activity, PERT computes:

For example, an activity with a = 4, m = 5, b = 12 has te = (4 + 20 + 12)/6 = 6.

Step 2: Forward Pass (Earliest Times)

Working from the start, compute each activity's:

• Earliest Start (ES) = the maximum Earliest Finish of its predecessors
• Earliest Finish (EF) = ES + duration

The largest EF among the final activities is the project duration.

Step 3: Backward Pass (Latest Times)

Working backward from the project end, compute each activity's:

• Latest Finish (LF) = the minimum Latest Start of its successors
• Latest Start (LS) = LF − duration

Step 4: Slack and the Critical Path

Activities with zero slack lie on the critical path. Any delay to a critical activity delays the whole project. Non-critical activities have slack — they can slip a little without affecting the finish date.

Worked Example

Consider a small project A→C, B→C, B→D, C→E, D→E with durations A = 3, B = 4, C = 2, D = 5, E = 1. The forward and backward passes give:

So the project takes 10 time units, and the critical path is B → D → E.

Why the Critical Path Matters

• Focus management attention on critical activities — they cannot be delayed.
• Allocate slack on non-critical activities to smooth resources.
• Estimate risk — with PERT variances, you can estimate the probability of finishing by a target date.

Common Mistakes

• Missing dependencies — an incomplete network gives the wrong critical path.
• Confusing slack — only zero-slack activities are critical.
• Ignoring parallel paths — the longest path, not the busiest, sets the duration.

Try It Yourself

Enter your activities, dependencies, and time estimates into the PERT/CPM Solver to compute expected times, the forward and backward passes, slack, and the critical path automatically. To compress a schedule at least cost, look at project crashing (time–cost tradeoff) in the solver library.

Key Takeaways

• PERT uses three time estimates; CPM uses one.
• The forward pass gives earliest times; the backward pass gives latest times.
• Zero-slack activities form the critical path, which sets the project duration.
• In the example, the critical path is B → D → E with a duration of 10.