CPM network, 13 activities (A=2 to M=7)

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(2)--> (2)
(1) --B(4)--> (3)
(2) --C(3)--> (4)
(2) - - dummy - -> (6)
(3) --E(5)--> (5)
(3) - - dummy - -> (6)
(4) --F(1)--> (10)
(5) --I(7)--> (9)
(6) --D(5)--> (7)
(7) --G(4)--> (8)
(7) --H(5)--> (9)
(7) - - dummy - -> (10)
(8) --M(7)--> (12)
(9) --K(3)--> (11)
(10) --J(2)--> (12)
(11) --L(6)--> (12)

The network has 12 events and 3 dummy activities. Start event is (1) and the end event is (12).

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 + 2 = 2
• B: EST = 0; EFT = 0 + 4 = 4
• C: EST = EFT of A = 2; EFT = 2 + 3 = 5
• D: EST = max(A:2, B:4) = 4; EFT = 4 + 5 = 9
• E: EST = EFT of B = 4; EFT = 4 + 5 = 9
• F: EST = EFT of C = 5; EFT = 5 + 1 = 6
• G: EST = EFT of D = 9; EFT = 9 + 4 = 13
• H: EST = EFT of D = 9; EFT = 9 + 5 = 14
• I: EST = EFT of E = 9; EFT = 9 + 7 = 16
• J: EST = max(D:9, F:6) = 9; EFT = 9 + 2 = 11
• M: EST = EFT of G = 13; EFT = 13 + 7 = 20
• K: EST = max(H:14, I:16) = 16; EFT = 16 + 3 = 19
• L: EST = EFT of K = 19; EFT = 19 + 6 = 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 - 6 = 19
• K: LFT = LST of L = 19; LST = 19 - 3 = 16
• M: LFT = project duration = 25; LST = 25 - 7 = 18
• J: LFT = project duration = 25; LST = 25 - 2 = 23
• I: LFT = LST of K = 16; LST = 16 - 7 = 9
• H: LFT = LST of K = 16; LST = 16 - 5 = 11
• G: LFT = LST of M = 18; LST = 18 - 4 = 14
• F: LFT = LST of J = 23; LST = 23 - 1 = 22
• E: LFT = LST of I = 9; LST = 9 - 5 = 4
• D: LFT = min(G:14, H:11, J:23) = 11; LST = 11 - 5 = 6
• C: LFT = LST of F = 22; LST = 22 - 3 = 19
• B: LFT = min(D:6, E:4) = 4; LST = 4 - 4 = 0
• A: LFT = min(C:19, D:6) = 6; LST = 6 - 2 = 4

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, E, I, K, L
• Critical path: B - E - I - K - L (= 4+5+7+3+6 = 25 days)

Answer: project duration = 25 days; critical path = B - E - I - K - L