To solve this problem, we need to analyze the given activity network diagram and answer the questions related to critical path analysis. The steps involved include finding the earliest start time for an activity, determining the minimum completion time, identifying the critical path, and determining which activity has a float time of two days.

Step 1: Identify the Activities and Durations

Let's first list the activities and their respective durations based on the diagram:

• Activity A: Duration = 2 days
• Activity B: Duration = 3 days
• Activity C: Duration = 4 days
• Activity D: Duration = 3 days
• Activity E: Duration = 6 days
• Activity F: Duration = 2 days

Step 2: Determine the Earliest Start Time for Activity E

To determine the earliest start time for Activity E, we need to consider the paths leading to Activity E.

Paths to Activity E:

• Start → A → D → E
• Start → B → C → E

We calculate the earliest time each path reaches Activity E:

• Path 1: Start (0) → A (2 days) → D (3 days) → E → Total: 2 + 3 = 5 days
• Path 2: Start (0) → B (3 days) → C (4 days) → E → Total: 3 + 4 = 7 days

The earliest start time for Activity E is the maximum of these two times, which is 7 days.

Step 3: Find the Minimum Completion Time for the Project

The minimum completion time for the project is determined by the longest path through the network, which is the critical path.

Possible paths from Start to Finish:

• Start → A → D → E → Finish
• Start → B → C → E → Finish
• Start → A → F → Finish
• Start → B → C → F → Finish

We calculate the total duration for each path:

• Path 1: Start → A → D → E → Finish = 2 + 3 + 6 = 11 days
• Path 2: Start → B → C → E → Finish = 3 + 4 + 6 = 13 days
• Path 3: Start → A → F → Finish = 2 + 2 = 4 days
• Path 4: Start → B → C → F → Finish = 3 + 4 + 2 = 9 days

The minimum completion time for the project is determined by the longest path, which is 13 days.

Step 4: Identify the Critical Path

The critical path is the path that has the longest duration, which means it determines the project duration.

From the calculations in Step 3, the critical path is: Start → B → C → E → Finish, with a total duration of 13 days.

Step 5: Determine Which Activity Has a Float Time of Two Days

Float time (or slack) is the amount of time that an activity can be delayed without delaying the project.

We can check the float for each non-critical path activity:

• For Activity A (Start → A → D → E → Finish): The duration is 11 days, so the float is $13 - 11 = 2$ days. Activity A has a float of 2 days.
• For Activities B, C, E, and F: They are on the critical path or do not have slack.

Activity A has a float time of 2 days.

Summary of Answers:

1. Earliest Start Time for Activity E: 7 days
2. Minimum Completion Time for the Project: 13 days
3. Critical Path: Start → B → C → E → Finish (13 days)
4. Activity with a Float Time of Two Days: Activity A

Would you like more details or have any questions? Here are some related questions to expand your understanding:

1. How do you calculate the latest start and finish times for each activity?
2. What is the significance of the float time in project management?
3. Can you describe how to determine the critical path in a more complex network?
4. How does resource allocation affect the critical path and project duration?
5. What methods can be used to shorten the project duration if necessary?

Tip: Always identify the critical path in project management, as it directly impacts the project timeline and helps in prioritizing tasks.