Critical path & float, 4 ways

What it is

A project is a set of tasks, some of which must wait for others to finish. The critical path is the longest path of tasks through the project. It sets the shortest possible time the whole project can take. Tasks on it have zero float: no spare time, so any delay to them delays the whole project.

The two ideas

critical path = the longest path = the minimum project time

float = spare time a task has = latest start − earliest start

Critical tasks always have a float of zero. Only the tasks off the critical path have any float.

Worked example

Task A (3 wks) must finish before both B (4 wks) and C (2 wks). Both B and C must finish before D (5 wks).

Critical path = the longest path: A-B-D = 3 + 4 + 5 = 12, A-C-D = 3 + 2 + 5 = 10. So critical path = A → B → D, project time = 12 weeks
Forward scan → earliest each task can start (EST): A at 0, then B and C at 3 (after A), D at 7 (waits for both B and C)
Backward scan → latest each can start without delaying the finish (LST): work back from week 12
Float = LST − EST. Zero float means it is on the critical path

Task	Dur	EST	LST	Float = LST − EST
A	3	0	0	0 ← critical
B	4	3	3	0 ← critical
C	2	3	5	2
D	5	7	7	0 ← critical

Only C has float: it could start as late as week 5 instead of week 3, so it can slip up to 2 weeks with no effect on the finish.

Watch out

• Critical path is the longest path, not the shortest. Longest = the bottleneck that decides the finish.
• Critical tasks have zero float, not maximum.
• Float lives on the tasks off the critical path.

The project network

The bold path A → B → D is the critical path (12 weeks). C sits on the lighter path with 2 weeks of float.

Two paths, longest wins

Path	Add the durations	Length
A → B → D	3 + 4 + 5	12 wks ← critical
A → C → D	3 + 2 + 5	10 wks

Longest = critical path = 12 weeks. Then scan for float: C can start as early as week 3 (EST) or as late as week 5 (LST), so float = 5 − 3 = 2 weeks.

Warm up first

Don't read yet, just have a go in your head:

Is the critical path the longest or the shortest path?

The longest. It is the chain that decides the earliest the project can finish.

How much float do tasks on the critical path have?

Zero. No spare time, so any delay to them delays everything.

Faded example

Rung 1 · watch one done fully

Critical path = longest = A-B-D = 12 weeks. Scan the odd one out, C: earliest start 3, latest start 5, so float = 5 − 3 = 2 weeks.

Rung 2 · you fill the gaps

A task has earliest start (EST) = 4 and latest start (LST) = 9. Float = ? − ? = ?

Check my gaps

Float = LST − EST = 9 − 4 = 5 weeks.

Rung 3 · all you

A (2) → B (3) → D (4), and A (2) → C (6) → D (4). Find the critical path and project time, then the float of B (EST then LST). Then check.

Check my answer

Longest path: A-C-D = 2+6+4 = 12 (A-B-D = 9), so critical path = A-C-D, 12 weeks. For B: EST = 2, LST = 5, so float = 5 − 2 = 3 weeks.

Say it back

In one sentence, out loud: why is the longest path the one that sets the project time? If you can say it, you've got it.

⚡ Critical path & float, one look

Critical paththe longest path through the network

It givesthe minimum time the whole project takes

Floatspare time = latest start − earliest start

Critical tasksalways have zero float

ExampleC: EST 3, LST 5 → float = 5 − 3 = 2

Traplongest not shortest · critical float = 0

Methodlongest path = critical & project time · forward scan for EST, backward scan for LST · float = LST − EST