CPM network, building project (A to K)

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

(1) --A(10)--> (2)
(1) --B(12)--> (3)
(1) --C(9)--> (4)
(2) --D(8)--> (5)
(2) --E(5)--> (6)
(3) --F(13)--> (6)
(4) --G(6)--> (6)
(4) --H(4)--> (7)
(5) --I(15)--> (8)
(6) --J(7)--> (8)
(7) --K(9)--> (8)

The network has 8 events and no dummy activities. The start event is (1) and the end event is (8).

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 + 12 = 12
• C: EST = 0; EFT = 0 + 9 = 9
• G: EST = EFT of C = 9; EFT = 9 + 6 = 15
• H: EST = EFT of C = 9; EFT = 9 + 4 = 13
• D: EST = EFT of A = 10; EFT = 10 + 8 = 18
• E: EST = EFT of A = 10; EFT = 10 + 5 = 15
• F: EST = EFT of B = 12; EFT = 12 + 13 = 25
• K: EST = EFT of H = 13; EFT = 13 + 9 = 22
• I: EST = EFT of D = 18; EFT = 18 + 15 = 33
• J: EST = max(E:15, F:25, G:15) = 25; EFT = 25 + 7 = 32

Project duration = largest EFT = 33 weeks.

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 = 33; LST = 33 - 7 = 26
• I: LFT = project duration = 33; LST = 33 - 15 = 18
• K: LFT = project duration = 33; LST = 33 - 9 = 24
• F: LFT = LST of J = 26; LST = 26 - 13 = 13
• E: LFT = LST of J = 26; LST = 26 - 5 = 21
• D: LFT = LST of I = 18; LST = 18 - 8 = 10
• H: LFT = LST of K = 24; LST = 24 - 4 = 20
• G: LFT = LST of J = 26; LST = 26 - 6 = 20
• C: LFT = min(G:20, H:20) = 20; LST = 20 - 9 = 11
• B: LFT = LST of F = 13; LST = 13 - 12 = 1
• A: LFT = min(D:10, E:21) = 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, I
• Critical path: A - D - I (= 10+8+15 = 33 weeks)

Answer: the building project is completed in 33 weeks; critical path = A - D - I

Significance of the critical path

• It gives the minimum time in which the project can be completed (33 weeks here).
• Critical activities (A, D, I) have zero float; a delay in any of them delays the building by the same time, so they need close supervision and first priority for labour, material and equipment.
• To shorten the project (crashing), only critical activities need to be speeded up; shortening non-critical activities wastes money.
• Non-critical activities (e.g. B, F, H, K) have float and can be rescheduled to level the resources.
• It is the basis for monitoring and updating the schedule; the critical path can change when non-critical activities use up their float.