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CPM network, 12 activities (A=1 to L=4)

Short answer

Network analysis gives project duration 21 weeks. Critical activities: B, D, H, I, K, L; path B-D-H-I-K-L.

Draw a network diagram from the following data. (i) Find critical path and critical activities. (ii) Find ES, EF, LS, LF, TF, FF, Ind.F and Int.F.

ActivityDuration (weeks)Predecessor
A1None
B3None
C2A
D4A, B
E3B
F5C, D
G1D
H2D
I6E, H
J3E, H
K2F, G, I
L4K, J

Network diagram (activity on node)

Each node is: activity (duration). Arrows show precedence.

A(1) --+--> C(2) -----------------> F(5) --------+
       |                             ^           |
       +--> D(4) --+-----------------+           |
            ^      |                             v
B(3) --+----+      +--> G(1) -----------------> K(2) ---> L(4)
       |           |                             ^         ^
       |           +--> H(2) --+--> I(6) --------+         |
       |                       |                           |
       +--> E(3) --------------+--> J(3) ------------------+

Links: A and B start; C after A; D after A and B; E after B; F after C and D; G and H after D; I and J after E and H; K after F, G and I; L after K and J.

Forward and backward passes

ESTEST = largest EFTEFT of predecessors; LFTLFT = smallest LSTLST of successors. Project duration T=21T = 21 weeks. TF=LST−ESTTF = LST - EST; FF=min⁡(ESTsucc)−EFTFF = \min(EST_{succ}) - EFT; Int.F=TF−FFInt.F = TF - FF; Ind.F=max⁡{0,min⁡(ESTsucc)−max⁡(LFTpred)−D}Ind.F = \max\{0, \min(EST_{succ}) - \max(LFT_{pred}) - D\}.

ActDESEFLSLFTFFFInt.FInd.FCritical
A101232020No
B303030000Yes
C2138107432No
D437370000Yes
E336693303No
F571210153300No
G17814157707No
H279790000Yes
I69159150000Yes
J391214175505No
K2151715170000Yes
L4172117210000Yes

Sample check, activity C: ES=EFA=1ES = EF_A = 1; EF=3EF = 3; LF=LSF=10LF = LS_F = 10; LS=8LS = 8; TF=8−1=7TF = 8-1 = 7; FF=ESF−EFC=7−3=4FF = ES_F - EF_C = 7-3 = 4; Int.F=3Int.F = 3; Ind.F=7−3−2=2Ind.F = 7 - 3 - 2 = 2 (using LFA=3LF_A = 3).

(i) Critical path

The activities with TF=0TF = 0 form the critical path B - D - H - I - K - L: 3+4+2+6+2+4=213+4+2+6+2+4 = 21 weeks.

Answer: Critical activities: B, D, H, I, K, L. Project duration = 21 weeks.

✓ Answer checked by Claude Opus 5.5 (high effort)

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