CPM network, 12 activities (A=10 to L=8)

Forward pass and backward pass

• Forward pass: the calculation from the start event to the end event to find EST and EFT of each activity ($EST$ = largest EFT of the predecessors, $EFT = EST + D$). The largest EFT is the project duration.
• Backward pass: the calculation from the end event back to the start event to find LFT and LST ($LFT$ = smallest LST of the successors, $LST = LFT - D$), taking the project duration as LFT of the last activity.

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)
(1) --B(9)--> (3)
(2) --D(9)--> (4)
(2) --C(7)--> (6)
(3) --E(8)--> (4)
(3) --F(5)--> (7)
(4) --G(11)--> (5)
(5) - - dummy - -> (6)
(5) - - dummy - -> (7)
(5) --J(12)--> (10)
(6) --H(6)--> (8)
(7) --K(10)--> (9)
(8) --I(9)--> (10)
(9) --L(8)--> (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 + 10 = 10
• B: EST = 0; EFT = 0 + 9 = 9
• E: EST = EFT of B = 9; EFT = 9 + 8 = 17
• F: EST = EFT of B = 9; EFT = 9 + 5 = 14
• C: EST = EFT of A = 10; EFT = 10 + 7 = 17
• D: EST = EFT of A = 10; EFT = 10 + 9 = 19
• G: EST = max(D:19, E:17) = 19; EFT = 19 + 11 = 30
• H: EST = max(C:17, G:30) = 30; EFT = 30 + 6 = 36
• J: EST = EFT of G = 30; EFT = 30 + 12 = 42
• K: EST = max(G:30, F:14) = 30; EFT = 30 + 10 = 40
• I: EST = EFT of H = 36; EFT = 36 + 9 = 45
• L: EST = EFT of K = 40; EFT = 40 + 8 = 48

Project duration = largest EFT = 48 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 = 48; LST = 48 - 8 = 40
• I: LFT = project duration = 48; LST = 48 - 9 = 39
• K: LFT = LST of L = 40; LST = 40 - 10 = 30
• J: LFT = project duration = 48; LST = 48 - 12 = 36
• H: LFT = LST of I = 39; LST = 39 - 6 = 33
• G: LFT = min(H:33, J:36, K:30) = 30; LST = 30 - 11 = 19
• D: LFT = LST of G = 19; LST = 19 - 9 = 10
• C: LFT = LST of H = 33; LST = 33 - 7 = 26
• F: LFT = LST of K = 30; LST = 30 - 5 = 25
• E: LFT = LST of G = 19; LST = 19 - 8 = 11
• B: LFT = min(E:11, F:25) = 11; LST = 11 - 9 = 2
• A: LFT = min(C:26, D:10) = 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, D, G, K, L
• Critical path: A - D - G - K - L (= 10+9+11+10+8 = 48 days)

Answer: project duration = 48 days; critical path = A - D - G - K - L