A linear programming computer package is needed. $x_1,x_2,x_3,x_4,x_5,x_{6^{\text{'}}},x_7,x_{B^{\text{'}}}{y}_1,y_2,y_3,y_4,y_5,y_{6^{\text{'}}},y_7,y_{8^{\text{'}}},y_{9}0$

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

Western Family Steakhouse offers a variety of low$-$cost meals and quick service. Other than management, the steakhouse operates with two full$-$time employees who work $8$ hours per day. The rest of the employees are part$-$time employees who are scheduled for $4-$hour shifts during peak meal times. On Saturdays the steakhouse is open from $11$:$00$ a$.$m$.$ to $10$:$00$ p$.$m$.$ Management wants to develop a schedule for part$-$ime employees that will minimize labor costs and still provide excellent customer service. The average wage rate for the part$-$ime 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.

same $4-$hours$-$on$,1-$hour$-$off, $4-$hours$-$on pattern.

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$.$

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

$1$:$00p.m.x_1+x_2+x_{3}7$

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

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

$4$:$00p.m.x_3+x_4+x_5+x_{6}1$

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

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

$7$:$00p.m.x_6+x_7+x_{8}10$

Find the optimal solution.

$(x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8)=1$

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

$

How many part$-$time shifts are needed?

$20$

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

Seven surplus employees are scheduled for $2pm-3pm,$ and four for $8pm-9pm,$

suggesting that employees should be rotated off earlier

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

Comment:

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

Your answer$($s$)$ should be in the form of expression$($s$)$

s$.$t$.$

$11$:$00a.m.$

a$.$m$.$

Noon

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

$2$:$00$ 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$8,$ y$1,$ y$2,$ y$3,$ y$4,$ y$5,$ y$6,$ y$7,$ y$8,$ y$9>=0$

Find the optimal solution.

$($x$1,$ x$2,$ x$3,$ x$4,$ x$5,$ x$6,$ x$7,$ x$8,$ y$1,$ y$2,$ y$3,$ y$4,$ y$5,$ y$6,$ y$7,$ y$8,$ y$9)=$

Incorrect: Your answer is incorrect.

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

part$-$time shifts needed

$14$

Incorrect: Your answer is incorrect.

cost savings