CPM network, 12 activities (A=8 to L=3)

IF is taken as independent float.

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 (time units). Events are circled numbers; dashed arrows are dummy activities (zero duration) that show dependencies only.

(1) --A(8)--> (2)
(1) --B(2)--> (3)
(2) --C(1)--> (4)
(3) --D(9)--> (4)
(3) --E(4)--> (5)
(4) --F(5)--> (7)
(5) --H(3)--> (6)
(5) - - dummy - -> (7)
(5) --G(6)--> (8)
(6) - - dummy - -> (8)
(6) --J(5)--> (9)
(7) --L(3)--> (10)
(8) --I(3)--> (9)
(9) --K(2)--> (10)

The network has 10 events and 2 dummy activities. Start event is (1) and the end event is (10).

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 + 8 = 8
• B: EST = 0; EFT = 0 + 2 = 2
• D: EST = EFT of B = 2; EFT = 2 + 9 = 11
• E: EST = EFT of B = 2; EFT = 2 + 4 = 6
• G: EST = EFT of E = 6; EFT = 6 + 6 = 12
• H: EST = EFT of E = 6; EFT = 6 + 3 = 9
• C: EST = EFT of A = 8; EFT = 8 + 1 = 9
• J: EST = EFT of H = 9; EFT = 9 + 5 = 14
• F: EST = max(C:9, D:11) = 11; EFT = 11 + 5 = 16
• I: EST = max(G:12, H:9) = 12; EFT = 12 + 3 = 15
• K: EST = max(I:15, J:14) = 15; EFT = 15 + 2 = 17
• L: EST = max(E:6, F:16) = 16; EFT = 16 + 3 = 19

Project duration = largest EFT = 19 time units.

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 = 19; LST = 19 - 3 = 16
• K: LFT = project duration = 19; LST = 19 - 2 = 17
• I: LFT = LST of K = 17; LST = 17 - 3 = 14
• F: LFT = LST of L = 16; LST = 16 - 5 = 11
• J: LFT = LST of K = 17; LST = 17 - 5 = 12
• C: LFT = LST of F = 11; LST = 11 - 1 = 10
• H: LFT = min(I:14, J:12) = 12; LST = 12 - 3 = 9
• G: LFT = LST of I = 14; LST = 14 - 6 = 8
• E: LFT = min(G:8, H:9, L:16) = 8; LST = 8 - 4 = 4
• D: LFT = LST of F = 11; LST = 11 - 9 = 2
• B: LFT = min(D:2, E:4) = 2; LST = 2 - 2 = 0
• A: LFT = LST of C = 10; LST = 10 - 8 = 2

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): B, D, F, L
• Critical path: B - D - F - L (= 2+9+5+3 = 19 time units)

Answer: project duration = 19 time units; critical path = B - D - F - L