Enter your project activities, their durations and predecessor links. The calculator runs a forward and backward pass to compute earliest/latest start and finish times, slack for every activity, the critical path, and total project duration. Switch to PERT mode to enter optimistic / most-likely / pessimistic estimates and get the expected duration plus project variance and standard deviation along the critical path.
| ID | Name | Opt (a) | Likely (m) | Pess (b) | Predecessors | |
|---|---|---|---|---|---|---|
| Activity | te | ΟΒ² | ES | EF | LS | LF | Slack | Critical |
|---|---|---|---|---|---|---|---|---|
| ARequirements | 3 | 0.111 | 0 | 3 | 0 | 3 | 0 | β yes |
| BDesign | 5.17 | 0.694 | 3 | 8.17 | 3 | 8.17 | 0 | β yes |
| CProcurement | 4 | 0.444 | 3 | 7 | 4.17 | 8.17 | 1.17 | no |
| DBuild | 8.5 | 2.25 | 8.17 | 16.67 | 8.17 | 16.67 | 0 | β yes |
| ETest & Deploy | 3.17 | 0.25 | 16.67 | 19.83 | 16.67 | 19.83 | 0 | β yes |
The Critical Path Method (CPM) and the Program Evaluation and Review Technique (PERT) are the two foundational techniques of project schedule analysis. Given a network of activities and their dependencies, this tool performs a forward pass and a backward pass to find the earliest and latest each activity can start and finish, the slack (float) available to each, the critical path of zero-slack activities that determines the project's minimum duration, and β in PERT mode β the expected duration and statistical uncertainty of the schedule.
Forward pass (earliest times): start at activities with no predecessors at time 0. Each activity's Earliest Start (ES) is the maximum Earliest Finish (EF) of its predecessors; its EF = ES + duration. The largest EF across all activities is the project duration.
Backward pass (latest times): start from the project duration. Each activity's Latest Finish (LF) is the minimum Latest Start (LS) of its successors (or the project duration if it has none); its LS = LF β duration.
Slack (total float) = LS β ES = LF β EF. It is how long an activity can slip without delaying the project. Activities with zero slack lie on the critical path β delay any of them and the whole project slips.
CPM uses a single deterministic duration per activity. PERT acknowledges uncertainty by asking for three estimates: optimistic (a), most likely (m), and pessimistic (b). It models each duration as a Beta distribution and computes the expected duration te = (a + 4m + b) / 6 β a weighted average that gives the most-likely value four times the weight of the extremes.
Each activity's variance is ((b β a) / 6)Β², on the rule of thumb that the range a-to-b spans roughly six standard deviations. The expected te values are then fed through the same forward/backward pass as CPM to find the critical path.
A key PERT result is that the project's schedule variance equals the SUM of the variances of the activities ON the critical path (treating them as independent). The project standard deviation Ο is the square root of that sum.
With the expected project duration and Ο, you can apply the Central Limit Theorem and a normal approximation to answer questions like "what is the probability of finishing within X days?" using a z-score: z = (target β expected) Γ· Ο. This is why PERT reports Ο β it turns a single deadline into a risk statement. Note the standard caveat: a near-critical path with high variance can sometimes become the true bottleneck once randomness is considered.
The results table highlights critical activities (slack = 0). To shorten the project you must shorten the critical path β adding resources to non-critical activities only increases their slack and changes nothing. This is "crashing": compress critical activities, but watch for the path shifting, because once a critical activity is shortened enough, a previously non-critical path can become the new critical path.
Non-critical activities have positive slack β schedule flexibility you can exploit to level resources, smooth workload, or absorb minor delays without risking the deadline.
The critical path is the longest chain of dependent activities through the project network, and therefore the shortest possible project duration. Every activity on it has zero slack β any delay to a critical activity delays the entire project. The calculator identifies it as the set of zero-slack activities found from the forward and backward passes.
CPM uses a single, known duration for each activity and is deterministic β ideal when durations are well understood (e.g. construction). PERT uses three estimates (optimistic, most likely, pessimistic) per activity to model uncertainty, computing an expected duration te = (a + 4m + b)/6 and a variance per activity. PERT then reports a project standard deviation so you can assess the probability of meeting a deadline. This tool does both.
Slack, or total float, is the amount of time an activity can be delayed without delaying the project finish. It equals LS β ES (equivalently LF β EF). Critical activities have zero slack; non-critical activities have positive slack that gives you scheduling flexibility for resource leveling or absorbing minor delays.
te = (a + 4m + b) / 6, where a is optimistic, m is most likely, and b is pessimistic. It is a Beta-distribution weighted average that weights the most-likely estimate four times more than the two extremes. Each activity's variance is ((b β a) / 6)Β², and the project variance is the sum of variances along the critical path.
Because the project duration is set entirely by the critical path. Speeding up an activity that already has slack just increases its slack β the project still finishes when the critical path finishes. To shorten the schedule you must compress (crash) critical activities, while watching for the critical path to shift to another chain once you do.