Critical Path Calculator

How to Use the Critical Path Calculator

1. 1. List the activities. One per line: an ID, a name, a duration and the IDs of the activities that must finish first, separated by commas. Leave the predecessors blank, or write a dash, for an activity that can start straight away.
2. 2. Choose the kind of duration. One duration per activity gives the classic critical path method (CPM). Three-point estimates — optimistic, most likely, pessimistic — switch to PERT, which turns each into an expected time and a standard deviation.
3. 3. Set the time unit (days, weeks, hours). It is only a label; use one unit for every duration.
4. 4. Read the table. Every activity gets its earliest start and finish (the forward pass), its latest start and finish (the backward pass), its total and free slack, and a critical flag. Critical rows are highlighted.
5. 5. Check the chart and the path. The schedule chart shows each activity on its earliest dates with its slack beside it; the critical path is listed above the table.

How the Critical Path Method Works

A project is a network of activities linked by finish-to-start dependencies. The forward pass goes from the start: an activity's earliest start (ES) is the largest earliest finish (EF) among its predecessors, and EF = ES + duration. The largest EF in the network is the project duration. The backward pass goes from the end: an activity's latest finish (LF) is the smallest latest start (LS) among its successors, with the project duration for activities at the end, and LS = LF − duration.

• Total slack (total float) = LS − ES = LF − EF. It is how long an activity can slip without delaying the project, if nothing else slips.
• Free slack is how long it can slip without delaying the earliest start of any successor: the smallest successor ES minus EF. It is never more than total slack.
• Critical activities have zero total slack. A critical path is an unbroken chain of critical activities from start to finish; its length is the project duration. A network can have more than one, and the calculator lists them all.
• Slack on a chain of non-critical activities is shared: using it up on one activity reduces what is left for the next.

Worked Example

A building project has nine activities. Foundation (B) needs site preparation (A); framing (C), plumbing (D) and electrical rough-in (E) all follow the foundation; roofing (F) follows framing; insulation (G) needs D, E and F; drywall (H) follows G and finishing (I) follows H.

The project takes 24 days and the critical path is A → B → C → F → G → H → I. Insulation cannot start until roofing ends on day 15, so plumbing (5 days of slack) and electrical work (4 days) have room to slip without moving the finish. Shorten any critical activity and the project shortens, until another path becomes critical.

PERT: When Durations Are Uncertain

PERT replaces each duration with a three-point estimate. The expected time is (a + 4m + b) ÷ 6 and the standard deviation is (b − a) ÷ 6, where a, m and b are the optimistic, most likely and pessimistic times. The critical path is found from the expected times. Because independent variances add, the path's variance is the sum of its activities' variances, and its length is treated as approximately normal.

With the PERT estimates in the calculator, the same network has an expected duration of 24.33 days along the critical path. Its variance is 1.444, a standard deviation of 1.202 days. A target of 26 days is (26 − 24.33) ÷ 1.202 = 1.387 standard deviations above the mean, a 91.7% chance of finishing on time; the duration with 95% confidence is 26.31 days. When several paths tie, the one with the largest variance is used. Other near-critical paths also have a chance of running late, so this probability is an approximation that tends to be optimistic.

Frequently Asked Questions

What is the critical path?

The longest chain of dependent activities from start to finish. Its length is the shortest time the project can take, and any delay to an activity on it delays the whole project. Activities on it have zero slack.

How do you calculate slack or float?

Total slack = latest start − earliest start, which is the same as latest finish − earliest finish. Find the earliest times with a forward pass and the latest times with a backward pass from the project's end date.

What is the difference between total slack and free slack?

Total slack is how long an activity can be delayed without delaying the project. Free slack is how long it can be delayed without delaying the earliest start of the next activity. Free slack is never larger than total slack, and it is zero whenever a successor starts the moment the activity finishes.

Can a project have more than one critical path?

Yes. When two or more chains have the same total length, each is critical. The project is then more fragile, because a delay on any of them delays the finish. The calculator lists every critical path it finds.

What is the PERT formula?

Expected time = (optimistic + 4 × most likely + pessimistic) ÷ 6, which weights the most likely estimate four times as heavily as either extreme. The standard deviation is (pessimistic − optimistic) ÷ 6. For an optimistic 2, most likely 5 and pessimistic 14, the expected time is 6 and the standard deviation is 2.

What is the difference between CPM and PERT?

CPM uses a single, deterministic duration for each activity and is used when times are well known. PERT uses three-point estimates and a probability model for uncertain durations. Both find the critical path with the same forward and backward passes.