Consider the following project network and activity times $($in days$).$ Assume that management desires a $12-$day project completion time.

A project network is given with the starting point on the left and the ending finish point on the right with activities A through G$.$ The arcs between the activities are listed as follows.

Start to A and B

A to C

B to D

C to E and F

D to E and F

F to G

E and G to Finish

Activity A B C D E F G

Time $3255622$

The crashing data for this project are as follows.

Time $($days$)$ Cost $($$$)$

Activity Normal Crash Normal Crash

A $328001,400$

B $211,2001,900$

C $532,0002,800$

D $531,5002,300$

E $641,8002,800$

F $216001,000$

G $215001,000$

$($a$)$

Formulate a linear programming model that can be used to assist with the crashing decisions. $($The objective function should be the additional cost of crashing. Let

xi $=$

the finish time for activity i and

yi $=$

the amount of time activity i is crashed where i $=$ A$,$ B$,$ C$,$ D$,$ E$,$ F$,$ and G$.$ Let

xFIN $=$

the finish time for the entire project.$)$