Question

QUESTION 7    [18 MARKS)

In the following table, optimistic, most likely and pessimistic time estimates have been given for each activity for a project.

Task	Immediate Predecessor(s)		Time (days)
		Optimistic (a)	Most Likely (m)	Pessimistic (b)
A	None	5	7	12
B	None	8	8	8
C	A	2	6	10
D	A,B	12	14	19
E	C,D	6	6	12
F	B,E	3	12	18
G	F	6	8	10

1. Construct an activity on node network diagram for this project.                               
2. If the project is due to be completed in 47 days, what is the probability that this project will be completed on or before the due date?                                                                
3. If the project manager wants at least a 95% probability that the project will be completed on or before the due date, what is the shortest project time due date that will satisfy the manager?

                                                                                                                                                 

Answer 1

Solution:
Expected time for each activity can be calculated as
Expected time = (O+4*M+P)/6
Variance = ((P-O)^2)/36

Task	O	M	P	Expected Time	Variance
A	5	7	12	7.5	1.361111111
B	8	8	8	8	0
C	2	6	10	6	1.777777778
D	12	14	19	14.5	1.361111111
E	6	6	12	7	1
F	3	12	18	11.5	6.25
G	6	8	10	8	0.444444444

Paths through which this task can be completed as
A-C-E-F-G with the completion time (7.5+6+7+11.5+8) = 40
A-D-E-F-G with the completion time (7.5+14.5+7+11.5+8) = 48.5
B-D-E-F-G with the completion time (8+14.5+7+11.5+8) = 49.5
B-F-G with the completion time (8+11.5+8) = 27.5
So critical path is B-D-E-F-G with a critical time 49.5 days and Variance = (0+1.36+1+6.25+0.44) = 9.06
Standard deviation = sqrt(9.06) = 3.01
Solution(b)
We need to calculate the probability that the project is due to complete in 47 days
P(X<47) = ?
Here we will use the standard normal distribution, First, we will calculate Z-score which can be calculated as
Z= (47-49.5)/3.01 = -0.83
So from Z table, we found a p-value
P(X<47) = 0.2033
So there is a 20.33% probability that the project is due to be completed in 47 days.
Solution(c)
Given that P-value = 0.95, So Z score from Z table is 1.645
So Due date can be calculated as
Due date = 49.5 + 1.645*3.01 = 49.5 + 5 = 54.5 days
So the shortest project time due date that will satisfy the manager is 54.5 days.