CPM network, 13 activities (A to M)

Notation: EST/EFT = earliest start/finish, LST/LFT = latest start/finish, TF = total float, FF = free float, Int.F = interfering float, Ind.F = independent float.

Network diagram (activity-on-arrow)

Each activity is an arrow with its duration in brackets (days). Events are circled numbers; dashed arrows are dummy activities (zero duration) that show dependencies only.

(1) --A(9)--> (2)
(1) --D(9)--> (3)
(1) --B(8)--> (4)
(1) --C(7)--> (5)
(2) --F(4)--> (4)
(2) --E(5)--> (6)
(3) --G(3)--> (5)
(3) --H(6)--> (8)
(4) - - dummy - -> (6)
(4) --I(7)--> (7)
(5) --J(8)--> (7)
(5) - - dummy - -> (8)
(6) --K(4)--> (9)
(7) --L(5)--> (9)
(8) --M(3)--> (9)

The network has 9 events and 2 dummy activities. Start event is (1) and the end event is (9).

Forward pass (EST and EFT)

Rule: EST = largest EFT of the preceding activities (0 for start activities); EFT = EST + D.

• A: EST = 0; EFT = 0 + 9 = 9
• B: EST = 0; EFT = 0 + 8 = 8
• C: EST = 0; EFT = 0 + 7 = 7
• D: EST = 0; EFT = 0 + 9 = 9
• E: EST = EFT of A = 9; EFT = 9 + 5 = 14
• F: EST = EFT of A = 9; EFT = 9 + 4 = 13
• G: EST = EFT of D = 9; EFT = 9 + 3 = 12
• H: EST = EFT of D = 9; EFT = 9 + 6 = 15
• J: EST = max(C:7, G:12) = 12; EFT = 12 + 8 = 20
• I: EST = max(B:8, F:13) = 13; EFT = 13 + 7 = 20
• K: EST = max(B:8, E:14, F:13) = 14; EFT = 14 + 4 = 18
• M: EST = max(C:7, G:12, H:15) = 15; EFT = 15 + 3 = 18
• L: EST = max(I:20, J:20) = 20; EFT = 20 + 5 = 25

Project duration = largest EFT = 25 days.

Backward pass (LFT and LST)

Rule: LFT = smallest LST of the succeeding activities (project duration for end activities); LST = LFT - D.

• L: LFT = project duration = 25; LST = 25 - 5 = 20
• M: LFT = project duration = 25; LST = 25 - 3 = 22
• K: LFT = project duration = 25; LST = 25 - 4 = 21
• I: LFT = LST of L = 20; LST = 20 - 7 = 13
• J: LFT = LST of L = 20; LST = 20 - 8 = 12
• H: LFT = LST of M = 22; LST = 22 - 6 = 16
• G: LFT = min(J:12, M:22) = 12; LST = 12 - 3 = 9
• F: LFT = min(I:13, K:21) = 13; LST = 13 - 4 = 9
• E: LFT = LST of K = 21; LST = 21 - 5 = 16
• D: LFT = min(G:9, H:16) = 9; LST = 9 - 9 = 0
• C: LFT = min(J:12, M:22) = 12; LST = 12 - 7 = 5
• B: LFT = min(I:13, K:21) = 13; LST = 13 - 8 = 5
• A: LFT = min(E:16, F:9) = 9; LST = 9 - 9 = 0

Floats

• Total float $TF = LST - EST = LFT - EFT$
• Free float $FF = \min(\text{EST of successors}) - EFT$
• Interfering float $IntF = TF - FF$
• Independent float $IndF = \max\left(0,\ \min(\text{EST of successors}) - \max(\text{LFT of predecessors}) - D\right)$

Result table

Event times (TE from the forward pass, TL from the backward pass):

Critical path and project duration

• Critical activities (TF = 0): A, D, F, G, I, J, L
• Critical path: A - F - I - L (= 9+4+7+5 = 25 days)
• Critical path: D - G - J - L (= 9+3+8+5 = 25 days)
• There are two critical paths of the same length, so a delay on either one delays the project.

Answer: project duration = 25 days; critical paths = A - F - I - L and D - G - J - L