A linear programming computer package is needed.

$\backslash$table$[[$Time$,\backslash$table$[[$Number of$],[$Employes$],[$Needed$]]],[11$:$00$ a$.$m$.-$Noon,$9],[$Noon$-1$:$00$ p$.$m$.,9],[1$:$00$ p$.$m$.-2$:$00$ p$.$m$.,9],[2$:$00$ p$.$m$.-3$:$00$ p$.$m$.,3],[3$:$00$ p$.$m$.-4$:$00$ p$.$m$.,3],[4$:$00$ p$.$m$.-5$:$00$ p$.$m$.,3],[5$:$00$ p$.$m$.-6$:$00$ p$.$m$.,6],[6$:$00$ p$.$m$.-7$:$00$ p$.$m$.,12],[7$:$00$ p$.$m$.-8$:$00$ p$.$m$.,12],[8$:$00$ p$.$m$.-9$:$00$ p$.$m$.,7],[9$:$00$ p$.$m$.-10$:$00$ p$.$m$.,7]]$

$($a$)$ Develop a minimum$-$cost schedule for part$-$time employees. $($Let $x_i=$ the number of part$-$time employees who start work beginning at hour I where $l=1=11$:$00$ a$.$m$.,I=2=$ Noon, etc$).$

Min $$

s$.$t$.$

$11$:$00$ a$.$m$.$

Noon $$

$1$:$00$ p$.$m$.$

$2$:$00$ p$.$m$.$

$3$:$00$ p$.$m$.$

$4$:$00$ p$.$m$.$

$5$:$00$ p$.$m$.$

$6$:$00$ p$.$m$.$

$7$:$00$ p$.$m$.$

$8$:$00$ p$.$m$.$

$9$:$00$ p$.$m$.$

$x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_{8}0$

Find the optimal solution.

$($b$)$ What is the total payroll $($in $ $)$ for the part$-$time employees?

$$

How many part$-$time shifts are needed?

$$ shifts

Use the surplus variables to comment on the desirability of scheduling at least some of the part$-$time employees for $3-$hour shifs.