Norton Industries is installing a new computer system. The activities, the activity times, and the project network are as follows.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 $321,0002,000$

$($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$.)$

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

maximum crashing for H

$($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?

Activity Crash Time

A

B

C

D

E

F

G

H

What is the added cost $($in dollars$)$ of meeting the $16-$week completion time?

$

$($c$)$

Develop a complete activity schedule based on the crashed activity times.

Activity Earliest

Start Latest

Start Earliest

Finish Latest

Finish Slack Critical

Activity

A

$---$Select$---$

B

$---$Select$---$

C

$---$Select$---$

D

$---$Select$---$

E

$---$Select$---$

F

$---$Select$---$

G

$---$Select$---$

H

$---$Select$---$