The expected time (P days) and standard deviation (Q days) of the different activities (A to F) in a project are indicated in the network diagram as (P, Q). The standard deviation of the project completion time, in days, is . (Rounded off to one decimal place)

Question

The primary purpose of mine ventilation is to:

Answer 1

A PERT network gives two numbers for every activity: the expected time P and its standard deviation Q. To find the spread (standard deviation) of the whole project's finish time, we first have to decide which chain of activities actually controls the schedule, the critical path, and then combine only that chain's variability.

Trace the two routes from start to finish. Node 1 splits into two branches. One branch runs 1-2-4-5-6 using activities A, C, D, F. The other runs 1-3-6 using activities B, E.
Total the expected times P for each route. Route A-C-D-F: $12+15+18+6=51$ days. Route B-E: $23+27=50$ days.
Pick the longer route as critical. A project finishes only when every activity, on every route, is done. The route that takes longer, here 51 days on A-C-D-F, is the bottleneck and is called the critical path. The other route (50 days) has slack and does not control the finish date.
Convert each critical activity's Q into a variance. The variance of an activity is $Q^2$, not $Q$ itself, because variances, not standard deviations, add up for independent activities in series. For A, C, D, F, the Q values are $0, 1, 2, 2$, giving variances $0, 1, 4, 4$.
Sum the variances and take the square root. Total variance $= 0+1+4+4 = 9$. The standard deviation of the whole project is $\sqrt{9} = 3$ days.

Notice that we never use activities B and E in this calculation, even though the question gives their Q values too. Only the critical path's variability affects the completion time's spread, because the non-critical route (B-E) has 1 day of slack and is not the limiting chain.

Let's summarize:

The critical path is the longest route by expected time: A-C-D-F at 51 days.
Only critical-path activities contribute to the project's variance.
Variance adds as $Q^2$, giving total variance 9 and standard deviation 3 days.

So the standard deviation of the project completion time is 3.0 days.

Answer 2

A PERT network gives two numbers for every activity: the expected time P and its standard deviation Q. To find the spread (standard deviation) of the whole project's finish time, we first have to decide which chain of activities actually controls the schedule, the critical path, and then combine only that chain's variability.

Trace the two routes from start to finish. Node 1 splits into two branches. One branch runs 1-2-4-5-6 using activities A, C, D, F. The other runs 1-3-6 using activities B, E.
Total the expected times P for each route. Route A-C-D-F: $12+15+18+6=51$ days. Route B-E: $23+27=50$ days.
Pick the longer route as critical. A project finishes only when every activity, on every route, is done. The route that takes longer, here 51 days on A-C-D-F, is the bottleneck and is called the critical path. The other route (50 days) has slack and does not control the finish date.
Convert each critical activity's Q into a variance. The variance of an activity is $Q^2$, not $Q$ itself, because variances, not standard deviations, add up for independent activities in series. For A, C, D, F, the Q values are $0, 1, 2, 2$, giving variances $0, 1, 4, 4$.
Sum the variances and take the square root. Total variance $= 0+1+4+4 = 9$. The standard deviation of the whole project is $\sqrt{9} = 3$ days.

Notice that we never use activities B and E in this calculation, even though the question gives their Q values too. Only the critical path's variability affects the completion time's spread, because the non-critical route (B-E) has 1 day of slack and is not the limiting chain.

Let's summarize:

The critical path is the longest route by expected time: A-C-D-F at 51 days.
Only critical-path activities contribute to the project's variance.
Variance adds as $Q^2$, giving total variance 9 and standard deviation 3 days.

So the standard deviation of the project completion time is 3.0 days.

The expected time (P days) and standard deviation (Q days) of the different activities (A to F) in a project are indicated in the network diagram as (P, Q). The standard deviation of the project completion time, in days, is . (Rounded off to one decimal place)

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Correct Answer: 3

Solution and Explanation

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