CPM network, hydropower project (months)

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

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

The network has 7 events and 1 dummy activity. Start event is (1) and the end event is (7).

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 + 3 = 3
• B: EST = 0; EFT = 0 + 5 = 5
• C: EST = 0; EFT = 0 + 8 = 8
• D: EST = EFT of A = 3; EFT = 3 + 4 = 7
• E: EST = EFT of A = 3; EFT = 3 + 7 = 10
• H: EST = EFT of A = 3; EFT = 3 + 7 = 10
• F: EST = EFT of B = 5; EFT = 5 + 2 = 7
• G: EST = EFT of C = 8; EFT = 8 + 1 = 9
• J: EST = EFT of C = 8; EFT = 8 + 9 = 17
• I: EST = max(E:10, F:7, G:9) = 10; EFT = 10 + 4 = 14

Project duration = largest EFT = 17 months.

Backward pass (LFT and LST)

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

• I: LFT = project duration = 17; LST = 17 - 4 = 13
• J: LFT = project duration = 17; LST = 17 - 9 = 8
• G: LFT = LST of I = 13; LST = 13 - 1 = 12
• F: LFT = LST of I = 13; LST = 13 - 2 = 11
• H: LFT = project duration = 17; LST = 17 - 7 = 10
• E: LFT = LST of I = 13; LST = 13 - 7 = 6
• D: LFT = project duration = 17; LST = 17 - 4 = 13
• C: LFT = min(G:12, J:8) = 8; LST = 8 - 8 = 0
• B: LFT = LST of F = 11; LST = 11 - 5 = 6
• A: LFT = min(D:13, E:6, H:10) = 6; LST = 6 - 3 = 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): C, J
• Critical path: C - J (= 8+9 = 17 months)

Answer: project duration = 17 months; critical path = C - J

Analysis of the project situation

• Project completion time = 17 months.
• Critical path = C - J (8 + 9 = 17 months); critical activities = C and J (zero float).
• All other activities (A, B, D, E, F, G, H, I) are non-critical; they can be delayed by their floats (see table) without delaying the project, but a delay of C or J delays the hydropower project month for month.
• Management should give priority in resources, funds and supervision to C and J, and use the float of the other activities for levelling the resources.