) A project schedule has the following characteristics:

Question

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.

Answer 1

To compute the earliest start time (Es) and latest finish time (Lf) for each event, we can use a forward and backward pass method.

1) Forward Pass:
Start with activity 1-2, its Es is 0 (since it's the first activity). We add the activity's time to the Es to get the earliest start time for the next activity.
Activity 1-2: Es = 0, Lf = 4
Activity 1-3: Es = 0 + 4 = 4, Lf = 5
Activity 2-4: Es = 4 + 1 = 5, Lf = 6
Activity 3-4: Es = 5 + 1 = 6, Lf = 7
Activity 3-5: Es = 7 + 1 = 8, Lf = 13
Activity 4-9: Es = 6 + 1 = 7, Lf = 12
Activity 5-6: Es = 4 + 4 = 8, Lf = 12
Activity 5-7: Es = 8 + 4 = 12, Lf = 20
Activity 6-8: Es = 8 + 4 = 12, Lf = 13
Activity 7-8: Es = 20 + 2 = 22, Lf = 24
Activity 8-10: Es = 24 + 5 = 29, Lf = 34
Activity 9-10: Es = 12 + 7 = 19, Lf = 26

2) Backward Pass:
Start with last activity 9-10, its Lf is the same as its Es. We subtract the activity's time from the Lf to get the latest finish time for the previous activity.
Activity 8-10: Es = 29, Lf = 34
Activity 7-8: Es = 22, Lf = 24
Activity 6-8: Es = 12, Lf = 13
Activity 5-7: Es = 12, Lf = 20
Activity 5-6: Es = 8, Lf = 12
Activity 4-9: Es = 7, Lf = 12
Activity 3-5: Es = 8, Lf = 13
Activity 3-4: Es = 6, Lf = 7
Activity 2-4: Es = 5, Lf = 6
Activity 1-3: Es = 4, Lf = 5
Activity 1-2: Es = 0, Lf = 4

Now, we can compute the Total Float and critical path.

Total Float for each activity = Lf - Es - time
Activity 1-2: 4 - 0 - 4 = 0
Activity 1-3: 5 - 4 - 1 = 0
Activity 2-4: 6 - 5 - 1 = 0
Activity 3-4: 7 - 6 - 1 = 0
Activity 3-5: 13 - 7 - 6 = 0
Activity 4-9: 12 - 7 - 5 = 0
Activity 5-6: 12 - 4 - 4 = 4
Activity 5-7: 20 - 12 - 8 = 0
Activity 6-8: 13 - 12 - 1 = 0
Activity 7-8: 24 - 22 - 2 = 0
Activity 8-10: 34 - 24 - 5 = 5
Activity 9-10: 26 - 12 - 7 = 7

Critical path: The activities with 0 Total Float are part of the critical path. Therefore, the critical path consists of activities 1-2, 1-3, 2-4, 3-4, 3-5, 4-9, 5-7, 6-8, 7-8, and 9-10.