Answer:
Explanation:
(a) Constructing the project network:
The project network is typically represented using the Program Evaluation and Review Technique (PERT) method, which considers the optimistic, most probable, and pessimistic times to estimate the expected duration and critical path. Let's create the network:
- Activity A (Report generation): Optimistic = 2 weeks, Most Probable = 9 weeks, Pessimistic = 13 weeks.
- Activity B (Web scraping): Optimistic = 6 weeks, Most Probable = 10 weeks, Pessimistic = 12 weeks.
- Activity C (Testing): Dependencies on both A and B.
- For C, we use the following formula to calculate the Expected Time (TE) using the PERT method:
\[TE = (O + 4M + P) / 6\]
- For activity C, TE = (1 + 4 * 1 + 1) / 6 = 1 week.
(b) Based on the critical path, estimate the probability that the project will be complete within 12 weeks:
The critical path is the longest path through the network. We calculate the expected duration for each path and identify the critical path:
- Path A-C: Expected Time (TE) = 2 (A) + 1 (C) = 3 weeks.
- Path B-C: Expected Time (TE) = 6 (B) + 1 (C) = 7 weeks.
The critical path is B-C with an expected duration of 7 weeks.
To estimate the probability that the project will be complete within 12 weeks, you can use the z-score formula and the Central Limit Theorem. First, find the standard deviation (SD) for the critical path:
- Variance (Var) = [(P - O) / 6]^2 = [(12 - 6) / 6]^2 = 1.
- Standard Deviation (SD) = √Var = √1 = 1.
Now, calculate the z-score:
\[z = (T - TE) / SD\]
Where T is the desired completion time (12 weeks).
\[z = (12 - 7) / 1 = 5\]
Consult a standard normal distribution table or calculator to find the probability associated with a z-score of 5. Typically, z-scores this extreme are very close to 1, indicating that the probability of completing the project within 12 weeks is very high (almost certain).
(c) Using all paths through the project network, estimate the probability that the project will be complete within 12 weeks:
To estimate the probability using all paths, you need to consider both paths A-C and B-C. We've already calculated the expected durations for both paths:
- Path A-C: Expected Time (TE) = 3 weeks.
- Path B-C: Expected Time (TE) = 7 weeks.
The project is complete when either path A-C or path B-C is complete. Therefore, the probability of completing the project within 12 weeks is determined by the probability of both paths being less than or equal to 12 weeks.
To find this combined probability, you'd calculate it using a joint probability approach. However, to perform this calculation, you would need to know the correlation between the durations of paths A-C and B-C, which is typically not provided in the PERT analysis.
Without information on the correlation, it's challenging to provide a specific probability estimate for the entire project. You might assume that paths A-C and B-C are independent and use a conservative approach, but that may not accurately reflect the real-world scenario.