Aproject consists of six activities. The immediate predecessor of each activity and the estimated duration is also provided in the table below:

Activity	Immediate predecessor	Estimated duration (weeks)
P		5
Q		1
R	Q	2
S	P,R	4
T	P	6
U	S,T	3

If all the activities other than S take the estimated amount of time, the maximum duration (in weeks) of the activity S without delaying the completion of the project is ___________.

The correct answer is 6.

To find the maximum duration of activity S without delaying the completion of the project, we must first analyze the project network and determine the critical path and the minimum project completion time when all other activities take their estimated durations.

Let us list all the activities along with their estimated durations:
- Activity P: 5 weeks (Predecessor: None)
- Activity Q: 1 week (Predecessor: None)
- Activity R: 2 weeks (Predecessor: Q)
- Activity S: Predecessors are P and R. Its duration is variable.
- Activity T: 6 weeks (Predecessor: P)
- Activity U: 3 weeks (Predecessors: S and T)

The project is completed when the final activity, U, is completed. Let us trace the different paths through the network that lead to the completion of U:

Path 1: P → T → U
This path does not involve activity S. Its duration is constant and calculated as:
$\text{Duration}=\text{Duration}\left(\text{P}\right)+\text{Duration}\left(\text{T}\right)+\text{Duration}\left(\text{U}\right)$
$\text{Duration}=5+6+3=14\text{weeks}$

Since the duration of Path 1 is 14 weeks and does not depend on S, the overall project completion time cannot be less than 14 weeks. Therefore, the scheduled project duration is 14 weeks.

Now, let us examine the paths that include activity S:

Path 2: P → S → U
The duration of this path is:
$\text{Duration}=\text{Duration}\left(\text{P}\right)+\text{Duration}\left(\text{S}\right)+\text{Duration}\left(\text{U}\right)$
$\text{Duration}=5+\text{Duration}\left(\text{S}\right)+3=8+\text{Duration}\left(\text{S}\right)$

To ensure that this path does not delay the project completion beyond 14 weeks, its duration must be less than or equal to 14 weeks:
$8+\text{Duration}\left(\text{S}\right)\le 14$
$\text{Duration}\left(\text{S}\right)\le 6\text{weeks}$

Path 3: Q → R → S → U
The duration of this path is:
$\text{Duration}=\text{Duration}\left(\text{Q}\right)+\text{Duration}\left(\text{R}\right)+\text{Duration}\left(\text{S}\right)+\text{Duration}\left(\text{U}\right)$
$\text{Duration}=1+2+\text{Duration}\left(\text{S}\right)+3=6+\text{Duration}\left(\text{S}\right)$

To ensure that this path does not delay the project completion beyond 14 weeks:
$6+\text{Duration}\left(\text{S}\right)\le 14$
$\text{Duration}\left(\text{S}\right)\le 8\text{weeks}$

Comparing the two constraints on the duration of S:
- From Path 2: Duration(S) ≤ 6 weeks
- From Path 3: Duration(S) ≤ 8 weeks

To satisfy both conditions and avoid delaying the project, the maximum allowable duration for activity S is the minimum of these limits, which is 6 weeks.