Norton Industries is installing a new computer system. The activities, the activity times, and the project network are as follows.

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.

$($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 $x_i=$ the finish time

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

Min${|}_{{?}_{?}}$

s$.$t$.$

constraint A

constraint B

$C$ depends on $A$

D depends on $C$

D depends on $B$

E depends on $D$

$F$ depends on $E$

$G$ depends on $C$

$G$ depends on $B$

$H$ depends on $G$

$H$ depends on $F$

finish time for $H$

maximum crashing for $A$

maximum crashing for $B$

maximum crashing for $C$

maximum crashing for $D$

maximum crashing for $E$

maximum crashing for $F$

maximum crashing for $G$