1. **Problem Statement:** Calculate the Expected Time (Te), Variance (Var), Earliest Start (ES), Earliest Finish (EF), Latest Start (LS), Latest Finish (LF), Slack, Critical Path, and Total Project Duration for the Mini Solar Lamp Project using PERT-CPM. 2. **Formulas:** - Expected Time: $$Te = \frac{a + 4m + b}{6}$$ where $a$ = optimistic, $m$ = most likely, $b$ = pessimistic. - Variance: $$Var = \left(\frac{b - a}{6}\right)^2$$ - Slack: $$Slack = LS - ES = LF - EF$$ 3. **Calculate Te and Var for each activity:** - A: $$Te_A = \frac{2 + 4(3) + 4}{6} = \frac{2 + 12 + 4}{6} = \frac{18}{6} = 3$$ $$Var_A = \left(\frac{4 - 2}{6}\right)^2 = \left(\frac{2}{6}\right)^2 = \frac{1}{9} \approx 0.111$$ - B: $$Te_B = \frac{1 + 4(2) + 3}{6} = \frac{1 + 8 + 3}{6} = \frac{12}{6} = 2$$ $$Var_B = \left(\frac{3 - 1}{6}\right)^2 = \left(\frac{2}{6}\right)^2 = 0.111$$ - C: $$Te_C = \frac{1 + 4(2) + 5}{6} = \frac{1 + 8 + 5}{6} = \frac{14}{6} \approx 2.33$$ $$Var_C = \left(\frac{5 - 1}{6}\right)^2 = \left(\frac{4}{6}\right)^2 = \frac{4}{9} \approx 0.444$$ - D: $$Te_D = \frac{2 + 4(3) + 4}{6} = 3$$ $$Var_D = 0.111$$ (same as A and B) - E: $$Te_E = \frac{3 + 4(4) + 7}{6} = \frac{3 + 16 + 7}{6} = \frac{26}{6} \approx 4.33$$ $$Var_E = \left(\frac{7 - 3}{6}\right)^2 = \left(\frac{4}{6}\right)^2 = 0.444$$ - F: $$Te_F = \frac{1 + 4(2) + 3}{6} = 2$$ $$Var_F = 0.111$$ - G: $$Te_G = \frac{1 + 4(2) + 3}{6} = 2$$ $$Var_G = 0.111$$ 4. **Forward Pass (Calculate ES and EF):** - Activity A: ES=0, EF=ES+Te=0+3=3 - B: ES=EF_A=3, EF=3+2=5 - C: ES=EF_A=3, EF=3+2.33=5.33 - D: ES=max(EF_B, EF_C)=max(5,5.33)=5.33, EF=5.33+3=8.33 - E: ES=EF_D=8.33, EF=8.33+4.33=12.66 - F: ES=EF_D=8.33, EF=8.33+2=10.33 - G: ES=max(EF_E, EF_F)=max(12.66,10.33)=12.66, EF=12.66+2=14.66 5. **Backward Pass (Calculate LF and LS):** - Activity G: LF=EF_G=14.66, LS=LF-Te=14.66-2=12.66 - E: LF=LS_G=12.66, LS=LF-Te=12.66-4.33=8.33 - F: LF=LS_G=12.66, LS=12.66-2=10.66 - D: LF=min(LS_E, LS_F)=min(8.33,10.66)=8.33, LS=8.33-3=5.33 - B: LF=LS_D=5.33, LS=5.33-2=3.33 - C: LF=LS_D=5.33, LS=5.33-2.33=3 - A: LF=min(LS_B, LS_C)=min(3.33,3)=3, LS=3-3=0 6. **Calculate Slack:** - A: Slack=LS-ES=0-0=0 - B: 3.33-3=0.33 - C: 3-3=0 - D: 5.33-5.33=0 - E: 8.33-8.33=0 - F: 10.66-8.33=2.33 - G: 12.66-12.66=0 7. **Critical Path:** Activities with zero slack: A → C → D → E → G 8. **Total Project Duration:** EF of last activity G = 14.66 days **Summary Table:** | Activity | ES | EF | LS | LF | Slack | Critical | |----------|-----|------|------|------|-------|----------| | A | 0 | 3 | 0 | 3 | 0 | Yes | | B | 3 | 5 | 3.33 | 5.33 | 0.33 | No | | C | 3 | 5.33 | 3 | 5.33 | 0 | Yes | | D | 5.33| 8.33 | 5.33 | 8.33 | 0 | Yes | | E | 8.33| 12.66| 8.33 | 12.66| 0 | Yes | | F | 8.33| 10.33| 10.66| 12.66| 2.33 | No | | G | 12.66|14.66| 12.66| 14.66| 0 | Yes |