$($CLO: $2,$ CLO: $4,$ CLO: $6))$ Consider the following project problem with detailed information in the table below. Times are given in weeks.

Activity

Immediate

Predecessor$($s$)$

Optimistic

Time

Most Likely

Time

Pessimistic

Time

A

$--$

$11$

$13$

$18$

B

A

$6$

$6$

$12$

C

B

$12$

$16$

$20$

D

A

$7$

$9$

$14$

E

B

$8$

$10$

$12$

F

B

$10$

$12$

$14$

G

C$,$ F

$6$

$7$

$8$

H

D

$6$

$6$

$12$

I

H

$10$

$12$

$20$

J

G$,$ I

$8$

$10$

$18$

K

E$,$ J

$6$

$6$

$9$

a$)$ Calculate the expected duration $($D$),$ and variance of the following activities.

Activity

D

Variance

B

Answer $1$ Question $1$

$7$

Answer $2$ Question $1$

$1$

E

Answer $3$ Question $1$

$10$

Answer $4$ Question $1$

$0\mathrm{.}44$

K

Answer $5$ Question $1$

$6\mathrm{.}5$

Answer $6$ Question $1$

$0\mathrm{.}25$

b$)$ Which activities form the critical path? $($N$.$B: The answer to this question should be written in the following format: ABCDEF, no space between letters$)$

Answer $7$ Question $1$

c$)$ What is the estimated time of the critical path?

Answer $8$ Question $1$

weeks

d$)$ If the project is due to be completed in $64$ weeks, what is the probability that the project will be completed on the due date?

Answer $9$ Question $1$

$\%$

e$)$ If the project manager wants at least a $95\%$ probability that the project will be completed on the due date, what is the shortest project due date that will satisfy the manager?

Answer $10$ Question $1$

weeks

f$)$ How many week$($s$)$ can you delay activity F$,$ without impacting the expected project completion date?

Answer $11$ Question $1$

week$($s