Question

a) Calculate mean duration and standard deviation for all the activities using the beta distribution. [4pts]
  
b) Construct a network diagram for this problem using the mean durations calculated in part (a), calculate the LS(Foll.), ES(Prec.) and total float for all the activities, and hence identify the critical path . What is the mean completion time for the project? What is the standard deviation of the critical path? [30 pts ]
  
c) What is the 92% confidence interval for the length of the critical path? [4 pts]
  
d) Assuming the probability distribution of the length of the critical path can be approximated by a normal distribution with the mean and standard deviation calculated in part (b), calculate the probability of completing the project within 42 weeks. [4 pts]

e) Calculate the probability of completing the project between 35 and 40 weeks? [4 pts]
  
f) Answer the project manager’s question: “I want to tell the client that there is a 10.03% chance the project will take longer than X weeks _{- what figure should I give them (i.e. find X)?” [4]}

(please show work step by step and excel file)

activity	follows	optimistic duration	most likely direction	pessimistic duration
A	-	4	6	14
B	A,C	3	4	5
C	-	3	5	13
D	A,E	12	18	24
E	-	8	10	18
F	A,E	4	6	8
G	B,F	7	8	9
H	G	10	12	14
I	G	5	6	7
J	D,I	5	7	9

Answer 1

Answer:

Given that,

(a).

Calculate mean duration and standard deviation for all the activities using the beta distribution:

Mean duration and standard deviation for all activities are as follows:

Activity	Optimistic Duration (t0)	Most likely duration (tm)	Pessimistic Duration (tp)	Expected Duration te=(t0+4tm+tp)/6	Variance=(tp-t0)2/36
A	4	6	14	7	2.7778
B	3	4	5	4	0.1111
C	3	5	13	6	2.7778
D	12	18	24	18	4.0000
E	8	10	18	11	2.7778
F	4	6	8	6	0.4444
G	7	8	9	8	0.1111
H	10	12	14	12	0.4444
I	5	6	7	6	0.1111
J	5	7	9	7	0.4444

(b).

Construct a network diagram for this problem using the mean durations calculated in part (a), calculate the LS(Foll.), ES(Prec.) and total float for all the activities, and hence identify the critical path:

Latest start time (LS), Earliest Start (ES) and Total Float (TF) of all activities are as follows:

Activity	LS=Latest Finish-Duration	ES	TF=LS-ES
A	11-7=4	0	4
B	17-4=13	7	6
C	13-6=7	0	7
D	31-18=13	11	2
E	11-11=0	0	0
F	17-6=11	11	0
G	25-8=17	17	0
H	38-12=26	25	1
I	31-6=25	25	0
J	38-7=31	31	0

So, Critical path is given by,

With critical activities E, F, G, I and J.

Mean completion time of the project is 38 weeks.

Variance of critical path = 2.7778+0.4444+0.1111+0.1111+0.4444

= 3.8888

So, standard deviation of critical path = (3.8888) 0.5

= 1.9720

(c).

What is the 92% confidence interval for the length of the critical path:

Length of critical path X N(38,3.8888)

We know,

So, 92% confidence interval is (34.5476,41.4524).

(d).

Calculate the probability of completing the project within 42 weeks:

Assuming the probability distribution of the length of the critical path can be approximated by a normal distribution with the mean and standard deviation calculated in part (b).

Required probability is given by.

=0.9787388 (Using R-code 'pnorm(2.02837))

=0.9787

(e).

Calculate the probability of completing the project between 35 and 40 weeks:

Required probability is given by,

=0.7806577 [Using R-code 'pnorm(1.0412)-pnorm(-1.5213)']

=0.7807

(f).

Answer the project manager’s question:

“I want to tell the client that there is a 10.03% chance the project will take longer than X weeks - what figure should I give them (i.e. find X)?”

Suppose, required time be k weeks.