Critical path method (CPM): a worked example with floats

What CPM is for

The critical path method (CPM) is a way of planning a project made of many activities, some of which must wait for others. It answers three questions every project manager asks:

1. What is the shortest time in which the whole project can be finished?
2. Which activities cannot be delayed without delaying the project? (These are critical.)
3. How much can each of the other activities slip? (This is its float, or slack.)

CPM was developed in 1957 by Morgan Walker of DuPont and James Kelley of Remington Rand to plan plant maintenance. It is used wherever durations are fairly well known: construction, events, product launches, shutdowns.

Rules for drawing the network

This note uses activity-on-arrow (AOA) diagrams, the convention in most Indian B.B.A. syllabi.

• Each activity is an arrow. Each event (the start or end of activities) is a numbered circle.
• An activity cannot start until every activity leading into its start event is finished.
• Number events so that the tail of every arrow has a smaller number than its head (Fulkerson’s rule).
• There is one start event and one end event, and no loops.
• Two activities may not share both the same start and the same end event; use a dummy (a dashed arrow of zero duration) to separate them.

Worked example

A small firm is setting up a new office. The activities are:

Activity	Description	Depends on	Duration (days)
A	Finalise layout	none	3
B	Order furniture	none	4
C	Electrical work	A	2
D	Network cabling	A	5
E	Fit out furniture	B, C	4
F	Install computers and test	D, E	3

Step 1: draw the network

A and B start at event 1. A ends at event 2, where C and D begin. E needs both B and C, so B and C both end at event 3. F needs D and E, so both end at event 4. F runs from event 4 to the final event 5. No dummy is needed.

Step 2: forward pass (earliest times)

Start with E(1) = 0. For every other event, take the largest value of (earliest time of the tail event + activity duration) over the arrows entering it.

Event	Calculation	Earliest time E
1	start	0
2	0 + 3 (A)	3
3	max(0 + 4 via B, 3 + 2 via C)	5
4	max(3 + 5 via D, 5 + 4 via E)	9
5	9 + 3 (F)	12

The project takes 12 days.

Step 3: backward pass (latest times)

Set L(5) = E(5) = 12. Work backwards, taking the smallest value of (latest time of the head event − activity duration) over the arrows leaving each event.

Event	Calculation	Latest time L
5	end	12
4	12 − 3 (F)	9
3	9 − 4 (E)	5
2	min(5 − 2 via C, 9 − 5 via D)	3
1	min(3 − 3 via A, 5 − 4 via B)	0

Step 4: floats

For an activity from event i to event j with duration t:

• Total float = L(j) − E(i) − t: how far it can slip without delaying the project.
• Free float = E(j) − E(i) − t: how far it can slip without delaying any following activity.
• Independent float = E(j) − L(i) − t (taken as zero if negative): slack available even if earlier activities finish late and later ones must start early.

Activity	i → j	t	ES	EF	LS	LF	Total float	Free float	Independent float
A	1 → 2	3	0	3	0	3	0	0	0
B	1 → 3	4	0	4	1	5	1	1	1
C	2 → 3	2	3	5	3	5	0	0	0
D	2 → 4	5	3	8	4	9	1	1	1
E	3 → 4	4	5	9	5	9	0	0	0
F	4 → 5	3	9	12	9	12	0	0	0

Step 5: the critical path

Activities with zero total float are A, C, E and F. The critical path is A → C → E → F, 3 + 2 + 4 + 3 = 12 days. Check it against the other paths: A → D → F takes 11 days and B → E → F takes 11 days, so the critical path is indeed the longest.

What this tells the manager: the layout, electrical work and furniture fitting need daily attention. Ordering furniture (B) and network cabling (D) can each slip by one day without moving the finish date.

Check your work with the simulator

I wrote the Network Diagram Simulator, a free web app, for exactly this kind of problem. Enter the activities, dependencies and durations above, and it draws the network, runs both passes, marks the critical path and plots a Gantt chart. Solve the problem by hand first, then use the app to check.

Common mistakes in exams

• Taking the smallest value in the forward pass. Forward pass: largest. Backward pass: smallest.
• Forgetting that an activity with two predecessors cannot start until both are done.
• Calling the path with the most activities “critical”. Critical means longest in time, not in number of steps.
• Mixing up total and free float. Total float is shared along a path; free float belongs to the activity.

Practice questions

1. Explain the terms event, activity, dummy activity and critical path, with a diagram.
2. A project has activities P (none, 4 days), Q (none, 6), R (P, 3), S (P, 5), T (Q, R, 2), U (S, T, 4). Draw the network, find the critical path and the project duration, and compute total and free float for each activity. (Answer: 13 days, with two critical paths, P → R → T → U and P → S → U.)
3. Distinguish between CPM and PERT.
4. “Activities with zero total float deserve the manager’s closest attention.” Discuss.