The SCM department has four professors, Saba, Geoff, Tang and Daves. To schedule their teaching for the next semester, the department chair collected their prep time $($in weeks$)$ for each of the six courses $($shown in table below$).$ Each professor has to teach at least three courses, but no more than four. To meet students' need, the department needs to offer at least eight sections for Intro to SCM $($business core course$).$ The department also needs to offer one section for all the other courses.

Intro to SCM

Project Mgt

Quality Mgt

Control

Logistics

Mgt Science

Saba

$3$

$10$

$12$

$16$

$12$

$7$

Geoff

$4$

$19$

$2$

$10$

$8$

$18$

Tang

$5$

$11$

$4$

$14$

$14$

$3$

Daves

$4$

$11$

$4$

$15$

$17$

$15$

The department chair wants to find the least time$-$consuming schedule for all professors. Let ij $=$ number of sections of course j that Professor i is assigned to teach, where i $=$ S $($Saba$),$ G $($Geoff$),$ T $($Tang$),$ D $($Daves$),$ and j $=$ I $($Intro to SCM$),$ P $($Project Mgt$),$ Q $($Quality Mgt$),$ C $($Control$),$ L $($Logistics$),$ M $($Mgt Science$).$

Sub$-$question $1$: What is an appropriate objective function?$\_\_\_\_$

$\_\_\_\_$

A$)$ Max Z $=3$SI $+10$SP $+12$SQ $+16$SC $+12$SL $+7$SM $+4$GI $+19$GP $+2$GQ $+10$GC $+8$GL $+18$GM $+$

$5$TI $+11$TP $+4$TQ $+14$TC $+14$TL $+3$TM $+4$DI $+11$DP $+4$DQ $+15$DC $+17$DL $+15$DM

B$)$ Min Z $=3$SI $+10$SP $+12$SQ $+16$SC $+12$SL $+7$SM $+4$GI $+19$GP $+2$GQ $+10$GC $+8$GL $+18$GM $+$

$5$TI $+11$TP $+4$TQ $+14$TC $+14$TL $+3$TM $+4$DI $+11$DP $+4$DQ $+15$DC $+17$DL $+15$DM

C$)$ Min Z $=$ SI $+$ SP $+$ SQ $+$ SC $+$ SL $+$ SM $+$ GI $+$ GP $+$ GQ $+$ GC $+$ GL $+$ GM $+$

TI $+$ TP $+$ TQ $+$ TC $+$ TL $+$ TM $+$ DI $+$ DP $+$ DQ $+$ DC $+$ DL $+$ DM

D$)$ Max Z $=$ SI $+$ SP $+$ SQ $+$ SC $+$ SL $+$ SM $+$ GI $+$ GP $+$ GQ $+$ GC $+$ GL $+$ GM $+$

TI $+$ TP $+$ TQ $+$ TC $+$ TL $+$ TM $+$ DI $+$ DP $+$ DQ $+$ DC $+$ DL $+$ DM