Question

Consider a project consisting of four activities A, B, C, and D. The following are constraints within which the project has to be conducted

• A and B, the first activities of the project, can be started simultaneously.

• C can be started only after A is completed.

• D can be started only after B is completed

Suppose the activity times for the activities are A = 4 weeks, B = 3 weeks, C = 2 weeks, D = 3 weeks.

1. How long does the project take to complete?

2. We ask the engineers working on activity A to factor in uncertainty in activity times and to provide better estimates of the activity times. The engineers tell us that activity A takes 2 weeks 25% of the time, 4 weeks 50% of the time and 6 weeks 25% of the time. How long does the project now take to complete (on average)?

3. Suppose the project manager wants to be 60% certain that the project would be completed? What is deadline should they commit to?

4. We now also ask the engineer working on activity B to factor in uncertainty in activity times and to provide better estimates of the activity times. The engineer now tell us that activity B takes 2 weeks 25% of the time, and 3 weeks 50% of the time and 4 weeks 25% of the time time. How long does the project now take to complete (on average)? [ASSUME THAT BOTH ACTIVITY A AND ACTIVITY B ARE UNCERTAIN]

5. For this scenario, what deadline should they commit to that would ensure a 60% probability of completion?

Answer 1

1). A=4 weeks, B=3 weeks, C=2 weeks, D=3 weeks

Assuming A & B start simultaneously, C starts after A and D starts after B

A + C=6 weeks

B + D=6 weeks

Hence all 4 activities are completed in 6 weeks

2). Using the uncertainity probabilities,

when A takes 2 weeks, A+C=4 weeks & B+D=6 weeks, so the project will take 6 weeks(when all activities are done)

when A takes 4 weeks, the project will take 6 weeks (as in problem 1)

when A takes 6 weeks, A+C=8 weeks & B+D=6 weeks, so the project will take 8 weeks

Applying the probabilities, Avg. completion time= 6*.25 + 6*.5 +8*.25 = 6.5 weeks

3). For 60% certainity, the deadlines of 6 weeks from the above problem will combine to give a 75% probability of work completion, which will satisfy the 60% criteria. Hence they should commit to 6 weeks.

4). Considering A+C is one activity, and B+D is another (since these 2 sums determine the project completion)

when B=2 weeks, B+D=5 weeks

when B=3 weeks, B+D=6 weeks

when B=4 weeks, B+D=7 weeks

Now both A and B are uncertain activities, so we have 9 combinations of competion times. In each combination, the task taking longer time is the defining component. And each completion time will be multiplies with respective probabilities of A and B.

For example, when A+B=5 and B+D=4, completion time=5 & probability =.25*.25

when A+B=5 anad B+D=6, completion time=6 & probability =.25*.5

Similarly all probabilities can be completed

So Average completion time = (5*.25*.25+6*.25*.5+....) = 6.625 weeks

5). For 60% probability, we just need to start adding up the probabilties from the shortest project completion times.

P(5 weeks)= .25*.25= .0625

P(6 weeks)= .25*.5 + .25*.5 + .5*.5 = .5

so the total probabiltiy is still less than 6 weeks

Adding the 7 weeks probability as well, it would become .75, which would fulfill the 60% criteris

Hence they should commit for 7 weeks