) 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 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.