CPM network, 10 activities (A=10 to J=4)

Here IF is taken as the independent float; interfering float = TF - FF.

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(10)--> (2)
(2) --B(12)--> (3)
(2) --C(8)--> (4)
(2) --D(6)--> (5)
(3) --E(10)--> (6)
(4) --G(4)--> (7)
(5) --F(8)--> (8)
(6) --I(6)--> (8)
(7) --H(10)--> (8)
(8) --J(4)--> (9)

The network has 9 events and 0 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 + 10 = 10
• B: EST = EFT of A = 10; EFT = 10 + 12 = 22
• C: EST = EFT of A = 10; EFT = 10 + 8 = 18
• D: EST = EFT of A = 10; EFT = 10 + 6 = 16
• F: EST = EFT of D = 16; EFT = 16 + 8 = 24
• G: EST = EFT of C = 18; EFT = 18 + 4 = 22
• E: EST = EFT of B = 22; EFT = 22 + 10 = 32
• H: EST = EFT of G = 22; EFT = 22 + 10 = 32
• I: EST = EFT of E = 32; EFT = 32 + 6 = 38
• J: EST = max(F:24, H:32, I:38) = 38; EFT = 38 + 4 = 42

Project duration = largest EFT = 42 days.

Backward pass (LFT and LST)

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

• J: LFT = project duration = 42; LST = 42 - 4 = 38
• I: LFT = LST of J = 38; LST = 38 - 6 = 32
• H: LFT = LST of J = 38; LST = 38 - 10 = 28
• E: LFT = LST of I = 32; LST = 32 - 10 = 22
• G: LFT = LST of H = 28; LST = 28 - 4 = 24
• F: LFT = LST of J = 38; LST = 38 - 8 = 30
• D: LFT = LST of F = 30; LST = 30 - 6 = 24
• C: LFT = LST of G = 24; LST = 24 - 8 = 16
• B: LFT = LST of E = 22; LST = 22 - 12 = 10
• A: LFT = min(B:10, C:16, D:24) = 10; LST = 10 - 10 = 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, B, E, I, J
• Critical path: A - B - E - I - J (= 10+12+10+6+4 = 42 days)

Answer: project duration = 42 days; critical path = A - B - E - I - J