A product design project is broken down into $10$ activities, as listed in the following table.

Activity Predecessor Time $($Weeks$)$

A $--4$

B $--2$

C A $4$

D A $3$

E B$,$ C $5$

F C $6$

G D $2$

H E $3$

I F$,$ G $7$

J H $7$

a$)$ Draw a Gantt chart for the project, assuming each activity starts as early as possible.

b$)$ Draw the activity$-$on$-$node $($precedence diagram$)$ of the project.

c$)$ Compute the earliest start, earliest finish, latest start, latest finish, and slack times of the activities using the critical path method.

d$)$ What is the project duration?

e$)$ Which activities are critical?

f$)$ What is$($are$)$ the critical paths of the project?

g$)$ Suppose all activities except A and C can be shortened by $1$ week each. The crash cost is

$$5,000$ per week per activity.

Which activities would you shorten to reduce the project duration by $3$ weeks?

In which sequence should these activities be shortened?

How much crashing cost will be incurred to reduce the project duration by $3$ weeks? What is$($are$)$ the critical path$($s$)$ of the project after crashing?