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

Activity Time

A $3$

B $6$

C $2$

D $5$

E $4$

F $3$

G $9$

H $3$

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

Start to A and B

A to C

B to D and G

C to D and G

D to E

E to F

F and G to H

H to Finish

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 $321,0002,000$

$($b$)$ Solve the linear programming model and make the minimum cost crashing decisions. What is the amount of crash time $($in weeks$)$ for each activity?

$($c$)$ Develop a complete activity schedule based on the crashed activity times.