The $21$st Century Studios is about to begin the production of its most important

$($and most expensive$)$ movie of the year. The movie$$s producer, Dusty Hoffmer, has

decided to use CPM to help plan and control this key project. He has identified the

eight major activities $($labeled A$,$ B$,...,$ H$)$ required to produce the movie. Their

precedence relationships are shown in the project network below.J

I

A

B D

C

E

F

G

HStart Finish

Dusty now has learned that another studio also will be coming out with a blockbuster

movie during the middle of the upcoming summer, just when his movie was to be re$-$

leased. This would be very unfortunate timing. Therefore, he and the top management

of $21$st Century Studios have concluded that they must accelerate production of their

movie and bring it out at the beginning of the summer $(16$ weeks from now$)$ to estab$-$

lish it as THE movie of the year. Although this will require substantially increasing an

already huge budget, management feels that this will pay off in much larger box office

earnings both nationally and internationally. Dusty now wants to determine the least

costly way of meeting the new deadline $16$ weeks hence. Using the CPM method of

time$-$cost trade$-$offs, he has obtained the cost and timing information as given in the

table below for activities the activities, where J and I can not be crashed.

Activity Normal Time Crash Time Normal Cost Crash Cost

A $5$ weeks $3$ weeks $$20$ million $$30$ million

B $3$ weeks $2$ weeks $$10$ million $$20$ million

C $4$ weeks $2$ weeks $$16$ million $$24$ million

D $6$ weeks $3$ week $$25$ million $$43$ million

E $5$ weeks $4$ weeks $$22$ million $$30$ million

F $7$ weeks $4$ weeks $$30$ million $$48$ million

G $9$ weeks $5$ weeks $$25$ million $$45$ million

H $8$ weeks $6$ weeks $$30$ million $$44$ million

I $3$ weeks N$/$A N$/$A N$/$A

J $3$ weeks N$/$A N$/$A N$/$A

$($a$)$ Run the critical path method algorithm to determine which activities should be

crashed and by how much to achieve the deadline. Give the final xj for each

activity $($by filling in the Times Crashed column$)$ and calculate the final Z $($total

amount spent crashing$)$ value.

$($b$)$ Formulate a linear programming model for the problem.