Activity Time Predecessor(s)
A 3 None
B 4 A
C 3 B
D 5 B
E 10 C, D
F 3 E
G 6 E
H 5 G, F
Please assist with finding the ES, EF, LS, LF and slack time for ALL activities.
also, the critical path?
and, the project completion time?
To find the ES (Earliest Start), EF (Earliest Finish), LS (Latest Start), LF (Latest Finish), and Slack Time for all activities, we need to calculate these values using the given information.
1. ES (Earliest Start): The earliest possible start time of an activity is determined by the maximum EF (Earliest Finish) of all its preceding activities.
2. EF (Earliest Finish): The earliest possible finish time of an activity is calculated by adding the activity's duration to its ES (Earliest Start) time.
3. LS (Latest Start): The latest possible start time of an activity is determined by subtracting its duration from the LF (Latest Finish) of the last activity in the project.
4. LF (Latest Finish): The latest possible finish time of an activity is determined by the minimum LS (Latest Start) of all its succeeding activities.
5. Slack Time: Slack Time for an activity is the amount of time that an activity can be delayed without affecting the project's completion time. It is calculated by subtracting the EF (Earliest Finish) from the LF (Latest Finish) of an activity.
Now, let's calculate the values for each activity step-by-step:
Activity A:
ES = 0 (no predecessors)
EF = ES + duration = 0 + 3 = 3
Activity B:
ES = EF of predecessor activity A = 3
EF = ES + duration = 3 + 4 = 7
Activity C:
ES = EF of predecessor activity B = 7
EF = ES + duration = 7 + 3 = 10
Activity D:
ES = EF of predecessor activity B = 7
EF = ES + duration = 7 + 5 = 12
Activity E:
ES = maximum EF of predecessor activities C and D = maximum(10, 12) = 12
EF = ES + duration = 12 + 10 = 22
Activity F:
ES = EF of predecessor activity E = 22
EF = ES + duration = 22 + 3 = 25
Activity G:
ES = EF of predecessor activity E = 22
EF = ES + duration = 22 + 6 = 28
Activity H:
ES = EF of predecessor activities G and F = maximum(28, 25) = 28
EF = ES + duration = 28 + 5 = 33
Now, let's calculate the LS, LF, and Slack Time for each activity:
Activity H:
LF = project completion time = 33
LS = LF - duration = 33 - 5 = 28
Slack Time = LF - EF = 33 - 33 = 0
Activity G:
LF = LS of successor activity H = 28
LS = LF - duration = 28 - 6 = 22
Slack Time = LF - EF = 28 - 28 = 0
Activity F:
LF = LS of successor activity H = 28
LS = LF - duration = 28 - 3 = 25
Slack Time = LF - EF = 28 - 25 = 3
Activity E:
LF = LS of successor activities F and G = minimum(25, 28) = 25
LS = LF - duration = 25 - 10 = 15
Slack Time = LF - EF = 25 - 22 = 3
Activity D:
LF = LS of successor activity E = 15
LS = LF - duration = 15 - 5 = 10
Slack Time = LF - EF = 15 - 12 = 3
Activity C:
LF = LS of successor activity E = 15
LS = LF - duration = 15 - 3 = 12
Slack Time = LF - EF = 15 - 10 = 5
Activity B:
LF = LS of successor activities C and D = minimum(12, 15) = 12
LS = LF - duration = 12 - 4 = 8
Slack Time = LF - EF = 12 - 7 = 5
Activity A:
LF = LS of successor activity B = 8
LS = LF - duration = 8 - 3 = 5
Slack Time = LF - EF = 8 - 3 = 5
Critical Path: The critical path is the longest path through the project, which is the sequence of activities that have zero slack time. On this path, each activity's EF is equal to the ES of the next activity.
In this project, the critical path is A -> B -> C -> E -> F -> H.
Project Completion Time: The project completion time is equal to the LF (Latest Finish) of the last activity in the project, which is 33 in this case.