A project schedule has the following characteristics:
Activity Time (Weeks) Activity Time (Weeks)
1-2 4 5-6 4
1-3 1 5-7 8
2-4 1 6-8 1
3-4 1 7-8 2
3-5 6 8-10 5
4-9 5 9-10 7
1) Compute earliest start time (Es) and latest finish time (Lf) for each event.
2) Compute Total float and critical path.
To compute the earliest start time (Es) and latest finish time (Lf) for each event, we can use the following steps:
Step 1: Create a table with all the activities and their corresponding time (weeks) values.
Activity Time (Weeks)
1-2 4
1-3 1
2-4 1
3-4 1
3-5 6
4-9 5
5-6 4
5-7 8
6-8 1
7-8 2
8-10 5
9-10 7
Step 2: Compute the Es and Lf values for each activity.
Start with the first activity, 1-2:
- Es: The earliest start time for activity 1-2 is 0 (since it is the first activity).
- Lf: The latest finish time for activity 1-2 is 4 (Es + Time = 0 + 4 = 4).
Continue with the remaining activities:
- Activity 1-3: Es = 0 (since it is dependent on activity 1-2), Lf = 1 + 4 = 5.
- Activity 2-4: Es = 4 (since it is dependent on activity 1-2), Lf = 4 + 1 = 5.
- Activity 3-4: Es = 5 (since it is dependent on activity 1-3), Lf = 5 + 1 = 6.
- Activity 3-5: Es = 5 (since it is dependent on activity 1-3), Lf = 5 + 6 = 11.
- Activity 4-9: Es = 6 (since it is dependent on activity 3-4), Lf = 6 + 5 = 11.
- Activity 5-6: Es = 11 (since it is dependent on activity 3-5), Lf = 11 + 4 = 15.
- Activity 5-7: Es = 11 (since it is dependent on activity 3-5), Lf = 11 + 8 = 19.
- Activity 6-8: Es = 15 (since it is dependent on activity 5-6), Lf = 15 + 1 = 16.
- Activity 7-8: Es = 19 (since it is dependent on activity 5-7), Lf = 19 + 2 = 21.
- Activity 8-10: Es = 16 (since it is dependent on activity 6-8), Lf = 16 + 5 = 21.
- Activity 9-10: Es = 11 (since it is dependent on activity 5-7), Lf = 11 + 7 = 18.
The computed Es and Lf values for each activity are as follows:
Activity Es Lf
1-2 0 4
1-3 0 5
2-4 4 5
3-4 5 6
3-5 5 11
4-9 6 11
5-6 11 15
5-7 11 19
6-8 15 16
7-8 19 21
8-10 16 21
9-10 11 18
To compute the total float and critical path, we can use the following steps:
Step 1: Calculate the total float for each activity using the formula:
Total Float = Lf - Es - Time
Step 2: Identify the critical path, which is the longest path with zero total float.
The calculated total float and critical path are as follows:
Activity Total Float Critical Path
1-2 0 Yes
1-3 4 No
2-4 0 Yes
3-4 0 Yes
3-5 0 Yes
4-9 0 Yes
5-6 0 Yes
5-7 5 No
6-8 0 Yes
7-8 2 No
8-10 0 Yes
9-10 7 No
The critical path consists of activities 1-2, 2-4, 3-4, 3-5, 4-9, 5-6, 6-8, and 8-10.