CPM network, 10 activities (A=4 to J=9)

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(4)--> (2)
(1) --B(7)--> (3)
(2) - - dummy - -> (4)
(2) --E(2)--> (8)
(3) - - dummy - -> (4)
(3) --D(3)--> (7)
(4) --C(4)--> (5)
(5) --F(1)--> (6)
(6) - - dummy - -> (7)
(6) - - dummy - -> (8)
(7) --H(5)--> (9)
(8) --G(6)--> (9)
(9) --I(8)--> (10)
(10) --J(9)--> (11)

The network has 11 events and 4 dummy activities. Start event is (1) and the end event is (11).

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 + 4 = 4
• B: EST = 0; EFT = 0 + 7 = 7
• E: EST = EFT of A = 4; EFT = 4 + 2 = 6
• C: EST = max(A:4, B:7) = 7; EFT = 7 + 4 = 11
• D: EST = EFT of B = 7; EFT = 7 + 3 = 10
• F: EST = EFT of C = 11; EFT = 11 + 1 = 12
• G: EST = max(E:6, F:12) = 12; EFT = 12 + 6 = 18
• H: EST = max(D:10, F:12) = 12; EFT = 12 + 5 = 17
• I: EST = max(G:18, H:17) = 18; EFT = 18 + 8 = 26
• J: EST = EFT of I = 26; EFT = 26 + 9 = 35

Project duration = largest EFT = 35 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 = 35; LST = 35 - 9 = 26
• I: LFT = LST of J = 26; LST = 26 - 8 = 18
• H: LFT = LST of I = 18; LST = 18 - 5 = 13
• G: LFT = LST of I = 18; LST = 18 - 6 = 12
• F: LFT = min(G:12, H:13) = 12; LST = 12 - 1 = 11
• D: LFT = LST of H = 13; LST = 13 - 3 = 10
• C: LFT = LST of F = 11; LST = 11 - 4 = 7
• E: LFT = LST of G = 12; LST = 12 - 2 = 10
• B: LFT = min(C:7, D:10) = 7; LST = 7 - 7 = 0
• A: LFT = min(C:7, E:10) = 7; LST = 7 - 4 = 3

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, C, F, G, I, J
• Critical path: B - C - F - G - I - J (= 7+4+1+6+8+9 = 35 days)

Answer: project duration = 35 days; critical path = B - C - F - G - I - J