Critical Path Method (CPM): Steps, Example and Formulas

What is the critical path method?

The critical path method (CPM) is a way of calculating a project schedule from its activities, durations and dependencies. It identifies the longest sequence of dependent activities from start to finish, called the critical path. Because every activity on that path must happen one after another, its total length is the shortest possible project duration.

CPM gives you three things a simple task list cannot:

• Dates: the earliest and latest each activity can start and finish without delaying the project.
• Float: how much each activity can slip before it affects the finish date (see float and slack).
• Focus: a short list of activities where any delay moves the end date, so you know where to spend management attention.

The method was developed in the late 1950s by Morgan R. Walker of DuPont and James E. Kelley Jr. of Remington Rand to plan large industrial projects. PERT, a closely related technique that uses three duration estimates, was developed at around the same time for the US Navy. Today the same calculation runs inside almost every scheduling tool, and it remains a core topic in the PMP certification exam.

CPM is one technique within project scheduling. It answers "how long will this take and what drives the end date?" once you know the activities and their logic.

Key CPM terms and formulas

The formulas below use the day 0 convention, where the project starts at time 0 and an activity's early finish is the moment it ends. It is the convention most textbooks and the PMP exam use; the alternative day 1 convention is explained further down.

• Forward pass: ES = the largest EF of all predecessors (0 for the first activity); EF = ES + D.
• Backward pass: LF = the smallest LS of all successors (the project finish for the last activity); LS = LF − D.
• Total float = LS − ES = LF − EF.
• Free float = the smallest ES of the successors − EF.

How to find the critical path in 6 steps

1. List the activities and durations

Start from the work breakdown structure and list every activity with a duration estimate in the same unit, usually working days. Include milestones as zero-duration activities.

2. Identify the dependencies

For each activity, record which activities must finish before it can start. These are its predecessors and successors. Most links are finish-to-start; our guide to task dependencies covers the other types.

3. Draw the network diagram

Draw each activity as a box (the activity-on-node or precedence diagramming method) and connect boxes with arrows that follow the dependencies. Every activity except the first should have a predecessor, and every activity except the last should have a successor.

4. Do the forward pass

Work from left to right. The first activity starts at 0. For every other activity, the early start is the largest early finish among its predecessors, because it cannot start until all of them are done. Add the duration to get the early finish. The largest early finish in the network is the project duration.

5. Do the backward pass

Work from right to left. Set the late finish of the last activity equal to the project duration. For every other activity, the late finish is the smallest late start among its successors, because it must finish in time for the earliest of them. Subtract the duration to get the late start.

6. Calculate float and identify the critical path

Subtract ES from LS for each activity. Activities with zero total float are critical, and joining them from start to finish gives the critical path. Highlight it on your Gantt chart so the whole team can see it.

Critical path method example

A small team is launching a new website. The project manager lists nine activities, with durations in working days:

The paths through the network

Listing every path is a useful cross-check on small networks:

The longest path is A-B-E-G-H-I at 21 days, so that is the critical path. On real projects with hundreds of activities you cannot list every path, which is why the forward and backward passes matter.

Forward pass

• A starts at 0 and finishes at 3.
• B, C and D all follow A, so each starts at 3. B finishes at 8, C at 5 and D at 9.
• E follows B: starts at 8, finishes at 15.
• F follows B and C. It must wait for the later of the two (B at 8), so it starts at 8 and finishes at 14.
• G follows D (9) and E (15), so it starts at 15 and finishes at 17.
• H follows F (14) and G (17), so it starts at 17 and finishes at 20.
• I starts at 20 and finishes at 21. The project takes 21 days.

Backward pass

• I must finish by 21, so it must start by 20. H must finish by 20 and start by 17.
• G must finish by 17 (H's late start) and start by 15. F must also finish by 17, so it can start as late as 11.
• E must finish by 15 (G's late start) and start by 8. D must also finish by 15, so it can start as late as 9.
• C must finish by 11 (F's late start), so it can start as late as 9.
• B has two successors: E (late start 8) and F (late start 11). It takes the smaller, so B must finish by 8 and start by 3.
• A has three successors with late starts of 3, 9 and 9. It must finish by 3 and start by 0.

Float and the critical path

Activities A, B, E, G, H and I have zero total float and form the critical path. Three points are worth noticing:

• C has 6 days of total float but only 3 days of free float. It can slip 3 days without affecting anything. Beyond that, it starts to delay F, eating into F's float, although the project still finishes on time until all 6 days are used.
• F is near-critical. With only 3 days of float, a modest overrun in the back-end build would make path A-B-F-H-I critical.
• D's float is entirely its own. Content writing can slip up to 6 days without affecting any other activity, so it is a good candidate to lend people to critical work if needed.

The same calculation, with a six-activity network, is laid out cell by cell in our forward and backward pass guide. You can also download our free critical path template, which calculates float and flags critical activities for you.

Day 0 or day 1? Two calculation conventions

You will see two ways of numbering the forward and backward pass, and they give answers that differ by one. Neither is wrong, but you must not mix them.

In the day 0 convention, "finishes at 3" means at the end of day 3. In the day 1 convention, activity A in our example runs from day 1 to day 3, B from day 4 to day 8, and so on, which matches how people read a calendar. Float and the critical path are identical in both. If an exam answer or a colleague's figures are one day out on every activity, a different convention is usually the reason.

Dependencies, lags and constraints in CPM

The basic method assumes every link is finish-to-start with no gap. Real schedules are more varied, and the passes adjust as follows:

• Lags add waiting time to a link. A finish-to-start link with a 2-day lag means the successor's ES is the predecessor's EF + 2. A lead is a negative lag that lets work overlap. See leads and lags.
• Start-to-start (SS) links tie a successor's start to the predecessor's start, for example "start testing 3 days after development starts".
• Finish-to-finish (FF) links tie the two finishes together, for example "documentation finishes when the build finishes".
• Date constraints such as "must finish on" or "start no earlier than" override the calculated dates. Use them sparingly, because they stop the schedule recalculating naturally.
• Negative float appears when a constraint or required finish date is earlier than the calculated finish. The amount of negative float tells you how much time you need to recover.

Most scheduling software handles all of these automatically. When you check a critical path by hand, look for lags and constraints first, because they are the usual reason a manual calculation does not match the tool.

CPM vs PERT vs critical chain

CPM is often taught alongside two related techniques. They answer slightly different questions.

In practice they combine well: you can use PERT estimates as the durations in a CPM calculation, then run a Monte Carlo simulation to see the likely range of finish dates.

Using the critical path on a live project

Finding the critical path at the planning stage is only half the value. During delivery:

1. Update progress and recalculate at a fixed cadence, usually weekly. Record actual dates and remaining durations, then let the tool redo the passes.
2. Check whether the path has moved. If a non-critical activity slips by more than its float, its path becomes critical. If a critical activity finishes early, another path may take over.
3. Watch near-critical paths. Report activities with, say, five days of float or less alongside the critical ones. In our example, F needs as much attention as the critical activities.
4. Be wary of several critical paths. When two or more paths have zero float, a delay on any one of them delays the project, so the overall risk of finishing late is higher.
5. Check resources. CPM assumes people are available whenever the logic allows. Resource levelling to remove overloads can lengthen the critical path, or create a new one.

How to shorten the critical path

If the calculated finish is later than the business needs, you have four main options: crash critical activities by adding resources, fast track by overlapping activities that were planned in sequence, reduce scope, or change the logic. Our guides to schedule compression and fast tracking vs crashing explain the trade-offs.

Shortening has a limit. In the example, cutting the front-end build (E) from 7 to 4 days saves 3 days, bringing the project to 18 days. Cutting E further saves nothing, because path A-B-F-H-I (18 days) becomes critical. Always recalculate after each change to see where the new critical path lies.

Advantages and limitations of the critical path method

For uncertain durations, add PERT estimates or simulation. For resource-heavy projects, level the schedule or consider critical chain project management. For Agile work, use the critical path at the level of releases and milestones rather than individual stories.

Common critical path mistakes

• Missing dependencies. An activity with no successor ends the calculation early and can hide the real critical path.
• Hard-coded dates. Typing start dates instead of linking activities stops the passes from working.
• Mixing conventions. Using day 0 for the forward pass and day 1 for the backward pass produces float that is out by one.
• Ignoring calendars. Weekends, holidays and part-time resources change the real dates even when the logic is correct.
• Calculating once and never again. The critical path can move every week; a path from the planning stage may be irrelevant two months later.
• Treating float as spare time for everyone. Total float is shared along a path. If one activity uses it, later activities on the same path have less.

Critical path tools and templates

For a small network, a spreadsheet works well and helps you understand the calculation. Our free critical path template includes columns for duration, predecessors, early and late dates, total float and a critical flag, and the project schedule template is useful once you want to turn the result into dates and owners.

For anything larger, use software that recalculates the passes automatically whenever a duration or dependency changes. Most project management tools with dependency-aware Gantt charts can show which tasks drive the finish date. Wrike, our top-ranked tool, suits teams managing several projects with interactive Gantt charts and dependencies, while Smartsheet suits planners who prefer a spreadsheet layout. Check which plan includes the scheduling features you need before you buy.

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