Hello there! In this video on project scheduling, I will
be calculating expected times and variances for uncertain activity times. I’ll also
be calculating probabilities of completing a project at specified times. And lastly,
I’ll be calculating the completion time of a project at a stated probability.
Here we have a table showing activity predecessors and their uncertain time estimates. Optimistic,
Most Likely, and Pessimistic times are represented by a, m, and b respectively.
To calculate the expected time t we apply this formula: t = a plus 4m plus b over 6.
So the expected time for A is calculated as 4 + 4 times 6 + 14, divided by 6. And that
gives 7. We do the same for Activity B, C, D, E, F,
and G. And using these expected activity times and
predecessors, I completed the project network from scratch in an earlier video. You will
find the link provided in description. The critical path is BDEF and the project
completion time is 32 weeks. Note that the project completion time can
also be found by adding up the expected activity times for the critical activities.
Now, the formula for calculating the variance for each activity is b minus a divided by
6 squared. Just like the project completion time can
be found by adding up the expected activity times for critical activities, the project
variance is also found by adding up the variances for critical activities. So we only need to
calculate the variances for the critical activities. Thus, for B, the variance is 2.78, for D,
1.78, for E, 2.78 and for F, the variance is 1.
And adding these, we have the project variance of 8.33. Then taking the square root we have
the project standard deviation of 2.8867. Although the uncertain time estimates are
beta distributed, we can approximate the project completion time by a normal distribution.
In this case, with mean equals 32 and standard deviation 2.8867.
So we can apply the z formula to answer these questions.
First, we find the probability that the project will be completed within 35 weeks. That is,
x equals 35 and that gives z = 1.04. And looking it up in tables we have a less-than area of
.8508. For x greater than 31, we have z greater than
-0.35. The area less than -0.35 is 0.3632. And subtracting it from 1 we obtain the more-than
area of 0.6368. Between 31 and 35, we find the area between
their z values of -0.35 and 1.04 by subtracting the smaller area from the larger one. And
that gives 0.4876. For d), less than 31 or more than 35 is the
complement of our result in c). So we just subtract the result from 1 to obtain 0.5124.
Next, if we want a 90% probability of completing the project on time, we do a reverse lookup
for a less-than area of 0.90. That will correspond to a z-score of 1.28.
And substituting that into the z formula we obtain a completion time of about 35.7. That
is, the project should begin about 36 weeks earlier in order to be completed on time with
a 90% probability. And that’s it for this video.
Thanks for watching.