1. **Problem Statement:** Calculate the expected time ($t_e$) and variance ($\sigma^2$) for each activity in a PERT project with given optimistic (O), most likely (M), and pessimistic (P) times. 2. **PERT Formulas:** - Expected time: $$t_e = \frac{O + 4M + P}{6}$$ - Variance: $$\sigma^2 = \left(\frac{P - O}{6}\right)^2$$ 3. **Calculations:** - Activity A: $$t_e = \frac{2 + 4(3) + 7}{6} = \frac{2 + 12 + 7}{6} = \frac{21}{6} = 3.5$$ $$\sigma^2 = \left(\frac{7 - 2}{6}\right)^2 = \left(\frac{5}{6}\right)^2 = 0.6944$$ - Activity B: $$t_e = \frac{3 + 4(5) + 7}{6} = \frac{3 + 20 + 7}{6} = \frac{30}{6} = 5$$ $$\sigma^2 = \left(\frac{7 - 3}{6}\right)^2 = \left(\frac{4}{6}\right)^2 = 0.4444$$ - Activity C: $$t_e = \frac{1 + 4(2) + 10}{6} = \frac{1 + 8 + 10}{6} = \frac{19}{6} \approx 3.167$$ $$\sigma^2 = \left(\frac{10 - 1}{6}\right)^2 = \left(\frac{9}{6}\right)^2 = 2.25$$ - Activity D: $$t_e = \frac{4 + 4(6) + 6}{6} = \frac{4 + 24 + 6}{6} = \frac{34}{6} \approx 5.667$$ $$\sigma^2 = \left(\frac{6 - 4}{6}\right)^2 = \left(\frac{2}{6}\right)^2 = 0.1111$$ - Activity E: $$t_e = \frac{2 + 4(4) + 11}{6} = \frac{2 + 16 + 11}{6} = \frac{29}{6} \approx 4.833$$ $$\sigma^2 = \left(\frac{11 - 2}{6}\right)^2 = \left(\frac{9}{6}\right)^2 = 2.25$$ - Activity F: $$t_e = \frac{3 + 4(5) + 9}{6} = \frac{3 + 20 + 9}{6} = \frac{32}{6} \approx 5.333$$ $$\sigma^2 = \left(\frac{9 - 3}{6}\right)^2 = \left(\frac{6}{6}\right)^2 = 1$$ - Activity G: $$t_e = \frac{2 + 4(3) + 8}{6} = \frac{2 + 12 + 8}{6} = \frac{22}{6} \approx 3.667$$ $$\sigma^2 = \left(\frac{8 - 2}{6}\right)^2 = \left(\frac{6}{6}\right)^2 = 1$$ - Activity H: $$t_e = \frac{1 + 4(3) + 7}{6} = \frac{1 + 12 + 7}{6} = \frac{20}{6} \approx 3.333$$ $$\sigma^2 = \left(\frac{7 - 1}{6}\right)^2 = \left(\frac{6}{6}\right)^2 = 1$$ - Activity I: $$t_e = \frac{3 + 4(4) + 8}{6} = \frac{3 + 16 + 8}{6} = \frac{27}{6} = 4.5$$ $$\sigma^2 = \left(\frac{8 - 3}{6}\right)^2 = \left(\frac{5}{6}\right)^2 = 0.6944$$ 4. **Summary Table:** | Activity | $t_e$ (weeks) | $\sigma^2$ | |----------|---------------|-------------| | A | 3.5 | 0.6944 | | B | 5 | 0.4444 | | C | 3.167 | 2.25 | | D | 5.667 | 0.1111 | | E | 4.833 | 2.25 | | F | 5.333 | 1 | | G | 3.667 | 1 | | H | 3.333 | 1 | | I | 4.5 | 0.6944 | This completes the calculation of expected times and variances for all activities using the PERT formulae.