A project involves eight activities with the precedence relationship and duration as shown in the table below. The slack for the activity D is _____ hours (answer in integer).

Activity	Immediate predecessor	Duration (hours)
A	-	4
B	A	8
C	A	5
D	B	2
E	B	7
F	C	6
G	D	3
H	E, F, G	9

The correct answer is 2.

To find the slack for activity D, we perform a Critical Path Method (CPM) analysis, calculating the Earliest Start (ES), Earliest Finish (EF), Latest Start (LS), and Latest Finish (LF) times for each activity.

Step 1: Forward Pass (Earliest Times)
We calculate the earliest start and finish times for each activity starting from the beginning of the project (Time = 0).
The earliest finish time for any activity is given by:
$$EF=ES+\text{Duration}$$
For activities with multiple predecessors, the earliest start time is the maximum of the earliest finish times of all its immediate predecessors:
$$ES=max(E{F}_{\text{predecessors}}$$
Let us calculate these values:
1. Activity A (No predecessor):
ES(A) = 0
EF(A) = 0 + 4 = 4

2. Activity B (Predecessor: A):
ES(B) = EF(A) = 4
EF(B) = 4 + 8 = 12

3. Activity C (Predecessor: A):
ES(C) = EF(A) = 4
EF(C) = 4 + 5 = 9

4. Activity D (Predecessor: B):
ES(D) = EF(B) = 12
EF(D) = 12 + 2 = 14

5. Activity E (Predecessor: B):
ES(E) = EF(B) = 12
EF(E) = 12 + 7 = 19

6. Activity F (Predecessor: C):
ES(F) = EF(C) = 9
EF(F) = 9 + 6 = 15

7. Activity G (Predecessor: D):
ES(G) = EF(D) = 14
EF(G) = 14 + 3 = 17

8. Activity H (Predecessors: E, F, G):
ES(H) = max(EF(E), EF(F), EF(G)) = max(19, 15, 17) = 19
EF(H) = 19 + 9 = 28

The total project completion time is 28 hours.

Step 2: Backward Pass (Latest Times)
We calculate the latest start and finish times starting from the project end (Time = 28).
For the final activity H:
LF(H) = 28
LS(H) = 28 - 9 = 19

For other activities, the latest finish time is the minimum of the latest start times of its successor activities:
$$LF=min(L{S}_{\text{successors}}$$
The latest start time is:
$$LS=LF-\text{Duration}$$
Let us calculate these values backward:
1. Activity E (Successor: H):
LF(E) = LS(H) = 19
LS(E) = 19 - 7 = 12

2. Activity F (Successor: H):
LF(F) = LS(H) = 19
LS(F) = 19 - 6 = 13

3. Activity G (Successor: H):
LF(G) = LS(H) = 19
LS(G) = 19 - 3 = 16

4. Activity D (Successor: G):
LF(D) = LS(G) = 16
LS(D) = 16 - 2 = 14

Step 3: Calculating Slack for Activity D
The slack (or total float) for an activity is the amount of time that it can be delayed without delaying the project completion date. It is calculated as:
$$\text{Slack}=LS-ES=LF-EF$$
For activity D:
Slack(D) = LS(D) - ES(D) = 14 - 12 = 2 hours.
Alternatively, using finish times:
Slack(D) = LF(D) - EF(D) = 16 - 14 = 2 hours.