A network consists of the following list. Times are given in weeks.
 
Activity

Preceding

Optimistic

Probable

Pessimistic

A

--

5

11

14

B

-

3

3

9

C

--

6

10

14

D

A, B

3

5

7

E

B

4

6

11

F

C

6

8

13

G

D, E

2

4

6

H

F

3

3

9
 
A. Draw the activity network for this problem                      .                                   ***(8 marks)***

B. Determine the Expected Time and Variance for each activity.                             ***(4 marks)***

C. Show the activity schedule (ES, EF, LS, and LF) as well as slack (in table).       ***(12 marks)***
D. Determine and state the critical path for this project. This must be based on your work in
E. What are the expected time and the variance of the project?                                 ***(3 marks)***

F. Calculate the probability that the project will be completed in less than 28 weeks.**(3 marks)**

Let's complete the solution with the calculation for the probability of completing the project in less than 28 weeks.

### F. Probability of Completing the Project in Less Than 28 Weeks

To calculate the probability, we use the expected time and variance for the critical path, which we have already determined:

- **Expected Time (TE) of Project:** 22.5 weeks
- **Variance (V) of Project:** 4.14

First, we calculate the standard deviation (σ) of the project duration:

\[
\sigma = \sqrt{V}
\]

Next, we use the Z-score formula to find the probability:

\[
Z = \frac{X - TE}{\sigma}
\]

Where:
- \(X\) is the target time (28 weeks),
- \(TE\) is the expected time (22.5 weeks),
- \(\sigma\) is the standard deviation.

Then, we find the probability from the Z-score using the standard normal distribution table.

Let's calculate the values.It seems that there was an unexpected issue, and the previous calculations were lost. Let's quickly redo the necessary steps to find the probability of completing the project in less than 28 weeks.

### Summary of Previous Calculations
- **Expected Time (TE) of Project:** 22.5 weeks
- **Variance (V) of Project:** 4.14

We need to calculate the standard deviation (σ), Z-score, and the probability:

1. **Standard Deviation (σ):** \( \sigma = \sqrt{V} \)
2. **Z-score (Z):** \( Z = \frac{X - TE}{\sigma} \)
3. **Probability:** We can find it using the cumulative distribution function (CDF) for the normal distribution, approximated by the error function (erf).

Let me quickly recalculate these values.### F. Probability of Completing the Project in Less Than 28 Weeks

- **Standard Deviation (σ):** \( 2.035 \) weeks
- **Z-score (Z):** \( 2.703 \)

Using the Z-score, the probability that the project will be completed in less than 28 weeks is approximately \( 0.997 \) or 99.66%.

This high probability indicates a strong likelihood that the project will finish within the 28-week timeframe.

---

Would you like further details on any of the calculations or concepts? Here are some additional questions you might consider:

1. How would a change in the variance of activities affect the project's completion probability?
2. What strategies could be implemented if the critical path activities are delayed?
3. How can we manage resources to ensure that non-critical activities do not become critical?
4. What would be the impact on the project if a non-critical activity with high slack becomes delayed?
5. How can the project manager utilize the slack time in activities to improve project efficiency?
6. What methods can be used to compress the schedule if the project needs to be completed sooner?
7. How does the estimation accuracy of the optimistic, probable, and pessimistic times affect project planning?
8. What are some risk management strategies for handling uncertainty in project scheduling?

**Tip:** Regularly update the project schedule and monitor the critical path to ensure timely completion and address any delays proactively.

Learn how to calculate the probability of completing a project in less than 28 weeks using Project Management techniques. Includes step-by-step calculations for expected time, variance, standard deviation, and Z-score. Suitable for postgraduate students and professionals.

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