$($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 shifts.

There are $7$ surplus emplovees scheduled from $2$:$00-3$:$00$ p$.$m$.$ and $4$ from

Score: $0\mathrm{.}64$ out of $0\mathrm{.}64$

Comment:

$($c$)$ Assume that part$-$time employees can be assigned either a $3-$hour or a $4-$hour shift. Develop a minimum$-$cost schedule for the part$-$time employees. $($Let $x_i$ be the same variables from part a$.$ Let $y_i=$ the numbel

of part$-$time employees who start work beginning at hour i where $i=1=11$:$00$ a$.$m$.,i=2=$ Noon, etc$).$

Min$30\mathrm{.}4*(x_1+x_2+x_3+x_4+x_5+x_6+x_7+x_8)+20\mathrm{.}8*(y_1+y_2+y_3+y_4+y_5+y_6+y_7+y_8+y_9)$

$s.t.$

$11$:$00a.m.$

Noon

$1$:$00p.m.x_1+x_2+x_3+y_1+y_2+y_{3}7$

$2$:$00$ p$.$m$.x_1+x_2+x_3+x_4+y_1+y_2+y_3+y_{4}1$

$3$:$00p.m.x_2+x_3+x_4+x_5+y_2+y_3+y_4+y_{5}2$

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

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

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

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

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

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

Enter inequalities using less than greater than $(>),$ less than or equal $(),$ or greater than or equal $().$

$x_{1^{\text{'}}}{x}_{2^{\text{'}}}{x}_{3^{\text{'}}}{x}_{4^{\text{'}}}{x}_{5^{\text{'}}}{x}_{6^{\text{'}}}{x}_{7^{\text{'}}}{x}_{8^{\text{'}}}{y}_{1^{\text{'}}}{y}_{2^{\text{'}}}{y}_{3^{\text{'}}}{y}_{4^{\text{'}}}{y}_{5^{\text{'}}}{y}_{6^{\text{'}}}{y}_{7^{\text{'}}}{y}_{8^{\text{'}}}{y}_{9}0$

Find the optimal solution.

How many part$-$time shifts are needed, and what is the cost savings $($in $ $)$ compared to the previous schedule?

part$-$time shifts needed

cost savings

customer service. The average wage rate for the part$-$time employees is $$7\mathrm{.}60$ per hour. The total number of full$-$time and part$-$time employees needed varies with the time of day as shown.

$($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 $i=1=11$:$00$ a$.$m$.,i=2=$ Noon, etc$).$

Min$30\mathrm{.}4x_1+30\mathrm{.}4x_2+30\mathrm{.}4x_3+30\mathrm{.}4x_4+30\mathrm{.}4x_5+30\mathrm{.}4x_6+30\mathrm{.}4x_7+30\mathrm{.}4x_8$

$s.t.$

$11$:$00a.m.$

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

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

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

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

$6$:$00$ p$.$m$.x_5+x_6+x_7+x_{8}10$

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

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

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

$x_{1^{\text{'}}},x_2,x_{3^{\text{'}}}{x}_{4^{\text{'}}},x_{5^{\text{'}}}{x}_{6^{\text{'}}}{x}_{7^{\text{'}}}{x}_{8}0$