Strategic Cost Management · Network Analysis - PERT, CPM

Float and Slack in PERT and CPM: Total, Free, Independent

Updated 11 October 2026 · Fact-checked

Float is the spare time an activity has; slack is the spare time at an event. Total float = LFT − EST − duration. Free float = head event's earliest time − tail event's earliest time − duration. Independent float = head earliest − tail latest − duration. Critical activities have zero total float.

Understand Float and Slack: Total, Free, Independent

Every activity in a project network has a time window. It cannot start before its earliest start time (EST) and must finish by its latest finish time (LFT). If the window is longer than the activity's duration, the extra time is float. Float tells you how much an activity can be delayed or stretched.

Slack is the same idea applied to an event (a node), not an activity. Event slack = latest event time (L) − earliest event time (E). Events on the critical path have zero slack. You use slack in PERT networks and float in CPM, but exam questions often mix the two terms.

There are three kinds of float. Total float is the maximum delay of an activity that does not delay the project end date. It may use up the float of later activities. Free float is the delay possible without affecting the earliest start of any following activity. Independent float is the delay possible even when the tail event occurs at its latest time and the head event at its earliest time. It does not depend on other activities.

The three are always ordered: total float ≥ free float ≥ independent float (when independent float is negative, take it as zero). The gap between total and free float is the interfering float, which equals the head event's slack. The gap between free and independent float is the tail event's slack.

An activity with zero total float is critical. Any delay in it delays the project. Free and independent floats are used when you plan resources, because they show how far you can shift one activity without disturbing the others.

Key rules to remember

Event slack: Slack = L − E; L = latest event time, E = earliest event time. Zero for events on the critical path.
Total float: TF = L(j) − E(i) − D = LST − EST = LFT − EFT; Activity i–j with duration D. i is the tail event and j the head event.
Free float: FF = E(j) − E(i) − D = TF − slack of head event j; Never more than total float.
Independent float: IF = E(j) − L(i) − D = FF − slack of tail event i; If the result is negative, take it as zero.
Interfering float: Interfering float = TF − FF = slack of head event; Part of total float that is shared with the following activities.
Activity times: EFT = EST + D; LST = LFT − D; EST of an activity = E of its tail event. LFT = L of its head event.
Critical activity: TF = 0 (with FF = IF = 0); All floats of a critical activity are zero.

How to solve Float and Slack: Total, Free, Independent questions

Use this order for any float question. Get the event times right first. The floats then follow from the formulas.

1List all activities with tail event, head event and duration. Draw the network if it is not given.
2Forward pass: E of the first event = 0. For each later event, E = the highest of (E of tail + duration) across all activities entering it.
3Backward pass: L of the last event = E of the last event. For each earlier event, L = the lowest of (L of head − duration) across all activities leaving it.
4Find the critical path: events where E = L, joined by activities where L(j) − E(i) = D. Project duration = E of the last event.
5For each activity, compute total float = L(j) − E(i) − D.
6Compute free float = E(j) − E(i) − D, and independent float = E(j) − L(i) − D (use 0 if negative).
7Tabulate EST, EFT, LST, LFT, TF, FF, IF for each activity. Check TF ≥ FF ≥ IF and that critical activities show zero.
8State the interpretation asked: which activities are critical, how much an activity can be delayed, and what happens to the project date.

Quickest way: Event-table shortcut

When to use it: Use it when the network is given and you must find all three floats for many activities in limited time.

Make a small table of events with E and L, and compute slack = L − E for each event.
For each activity compute one base figure: head E − tail E − D. This is the free float.
Total float = free float + head slack.
Independent float = free float − tail slack. If it is negative, write 0.
Activities with zero total float are critical. Confirm they join zero-slack events and add up to the project duration.

Common mistakes in Float and Slack: Total, Free, Independent

Using the same event time for both ends, such as L(j) − L(i) − D for total float.
Students mix up which event time goes with which end of the activity.
Fix: Total float always uses L of the head and E of the tail: L(j) − E(i) − D. Write it down before substituting.
Taking the maximum in the backward pass instead of the minimum.
The forward pass uses the maximum, so students repeat it.
Fix: Forward pass takes the maximum of the incoming values. Backward pass takes the minimum of the outgoing values.
Reporting a negative independent float.
The formula E(j) − L(i) − D can give a negative number when tail slack is large.
Fix: A negative independent float means there is none. Write it as zero.
Calling an activity critical because its free float is zero.
Students confuse free float with total float.
Fix: An activity can have zero free float and positive total float. Only zero total float makes it critical.
Confusing slack and float.
The words are used loosely in textbooks.
Fix: Slack belongs to events (L − E). Float belongs to activities. If a question says 'slack' for an activity, give total float unless told otherwise.
Assuming the total float of an activity can be used without any effect on other activities.
Total float is read as 'free time' for the activity alone.
Fix: Total float is shared along the path. Using all of it can remove float from later activities. Only independent float is fully safe.

Worked examples

Example 1

A project has activities A (1–2) 4 days, B (1–3) 3 days, C (2–4) 5 days, D (3–4) 2 days and E (4–5) 3 days. Find the event times, the critical path and the total, free and independent float of activities B and D.

Show the solution

Forward pass: E1 = 0; E2 = 0 + 4 = 4; E3 = 0 + 3 = 3; E4 = max(4 + 5, 3 + 2) = max(9, 5) = 9; E5 = 9 + 3 = 12.
Backward pass: L5 = 12; L4 = 12 − 3 = 9; L3 = 9 − 2 = 7; L2 = 9 − 5 = 4; L1 = min(4 − 4, 7 − 3) = min(0, 4) = 0.
Event slack: events 1, 2, 4 and 5 have zero slack. Event 3 has slack 7 − 3 = 4.
Critical path: 1–2–4–5 (A, C, E) = 4 + 5 + 3 = 12 days.
Activity B (1–3): TF = L3 − E1 − 3 = 7 − 0 − 3 = 4. FF = E3 − E1 − 3 = 3 − 0 − 3 = 0. IF = E3 − L1 − 3 = 3 − 0 − 3 = 0.
Activity D (3–4): TF = L4 − E3 − 2 = 9 − 3 − 2 = 4. FF = E4 − E3 − 2 = 9 − 3 − 2 = 4. IF = E4 − L3 − 2 = 9 − 7 − 2 = 0.
Check: for B, TF − FF = 4 = slack of head event 3. For D, FF − IF = 4 = slack of tail event 3.

Answer: Critical path is 1–2–4–5 with a project duration of 12 days. Activity B: TF = 4, FF = 0, IF = 0. Activity D: TF = 4, FF = 4, IF = 0. B and D share the same 4 days of float. If B uses it, D loses it.

Example 2

For an activity 3–5 with duration 4 days, the event times are E3 = 5, L3 = 9, E5 = 11 and L5 = 14. Calculate its total, free and independent float, and state whether it is critical.

Show the solution

Total float = L5 − E3 − D = 14 − 5 − 4 = 5 days.
Free float = E5 − E3 − D = 11 − 5 − 4 = 2 days.
Independent float = E5 − L3 − D = 11 − 9 − 4 = −2, so independent float = 0.
Check with slack: head slack = 14 − 11 = 3, so FF = TF − 3 = 2. Tail slack = 9 − 5 = 4, so FF − 4 = −2, taken as 0.
Total float is 5, which is more than zero, so the activity is not critical.

Answer: TF = 5 days, FF = 2 days, IF = 0 (calculated −2). The activity is not critical. It can be delayed by 2 days without affecting any following activity's earliest start, and by 5 days without delaying the project.

Exam tips

Show the forward and backward pass values on the network or in a table. Marks are given for event times even if a float is wrong.
Always write the formula with the event labels (for example L5 − E3 − D) before substituting. It prevents head-tail mix-ups.
Tabulate EST, EFT, LST, LFT, TF, FF and IF for each activity in one table. It makes checking easy and fits case-based questions.
MCQs often test ordering and meaning: total float ≥ free float ≥ independent float, and critical means zero total float. Learn these as rules.
Finish with a one-line interpretation, such as which activity has float to spare for resource shifting. Application earns marks.

Practice questions from Network Analysis - PERT, CPM

Float and Slack: Total, Free, Independent: frequently asked questions

What is the difference between total float and free float?

Total float is the delay an activity can take without delaying the project end date. Free float is the delay it can take without delaying the earliest start of the next activities. Free float is never greater than total float. The difference is the slack of the head event.

Can independent float be negative?

The formula E(j) − L(i) − D can give a negative value. A negative value means the activity has no independent float, so you show it as zero.

What is the difference between slack and float?

Slack is the spare time at an event and equals L − E. Float is the spare time of an activity. Event slack is the link between the floats: head slack = TF − FF and tail slack = FF − IF.

Do critical activities have float?

No. Critical activities have zero total float, and so zero free and independent float as well. Any delay in them delays the whole project.

Which float should I use for resource scheduling?

Total float shows the overall room to shift an activity but is shared with later activities. Free float is safe for the following activities. Independent float is safest because it does not depend on when other activities occur.