The critical path calculation shows B$-$D$-$E$-$F$-$H is the critical path, and the expected project completion time is $21$ weeks. After viewing this information, management requested overtime be used to complete the project in $16$ weeks. Thus, crashing of the project is necessary. The following information is relevant.

Time $($weeks$)$ Cost $($$$)$

Activity Normal Crash Normal Crash

A $319001,700$

B $632,0004,100$

C $215001,000$

D $531,8002,400$

E $431,5001,850$

F $313,0003,900$

G $948,0009,800$

H $321,0002,000$

$($a$)$

Formulate a linear programming model that can be used to make the crashing decisions for this project. $($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$,$ G$,$ and H$.)$