Refer to Table $1$ below. This table shows your estimates of optimistic, likely and pessimistic durations for activities contained within one of the works' packages.

Table $1-$Estimated activity times for activities included within a work package

Activity

Optimistic duration $($weeks$)$

Likely duration $($weeks$)$

A$-$B

$1$

$3$

Pessimistic duration $($weeks$)$

$5$

B$-$C

$4$

$6$

B$-$D

$1$

$2$

$3$

C$-$B

CF

CG

$4$

$2$

$1$

$1$

$5$

$3$

$2$

$3$

$6$

$4$

$6$

$4$

$3$

D$-$E

$2$

EH

$4$

$8$

$3$

$4$

PH

$1$

$2$

$3$

G$-$H

$1$

$2$

$4$

G

$3$

$6$

HI

$3$

$5$

$3$

$4$

$1$

$2$

$5$

$3$

Time left

a$)$ Using a programme evaluation and review technique $($PERT$)$ approach, draw a project schedule showing all activities, the critical path and project mean duration. $(10)$ marks$)$

$(5$ marks$)$ b$)$ Calculate the probability of the work package being completed by week $20,$

Discuss the main applications and limitations of PERT based project planning and examine the likely suitability of a PERT based approach in the case study. $(10$ marks$)$