$($a$)$ Formulate an integer programming model that can be used to develop a schedule that will satisfy customer service needs at a minimum employee cost. $($Hint: Let xi $=$ number of full$-$time employees coming on duty at the beginning of hour i and yj $=$ number of part$-$time employees coming on duty at the beginning of hour j$.)$

If your answer is zero enter $0$ and if the constant is $"1"$ it must be entered in the box.

Min

x$9+$

x$10+$

x$11+$

y$9+$

y$10+$

y$11+$

y$12+$

$4$

y$1+$

y$2+$

y$3$

s$.$t$.$

x$9+$

x$10+$

x$11+$

y$9+$

y$10+$

y$11+$

y$12+$

y$1+$

y$2+$

y$3$

$>=$

Time $(9$:$00$ a$.$m$.-10$:$00$ a$.$m$.)$

x$9+$

x$10+$

x$11+$

y$9+$

y$10+$

y$11+$

y$12+$

y$1+$

y$2+$

y$3$

$>=$

Time $(10$:$00$ a$.$m$.-11$:$00$ a$.$m$.)$

x$9+$

x$10+$

x$11+$

y$9+$

y$10+$

y$11+$

y$12+$

y$1+$

y$2+$

y$3$

$>=$

Time $(11$:$00$ a$.$m$.-$Noon.$)$

x$9+$

x$10+$

x$11+$

y$9+$

y$10+$

y$11+$

y$12+$

y$1+$

y$2+$

y$3$

$>=$

Time $($Noon$.-1$:$00$ p$.$m$.)$

x$9+$

x$10+$

x$11+$

y$9+$

y$10+$

y$11+$

y$12+$

y$1+$

y$2+$

y$3$

$>=$

Time $(1$:$00$ p$.$m$.-2$:$00$ p$.$m$.)$

x$9+$

x$10+$

x$11+$

y$9+$

y$10+$

y$11+$

y$12+$

y$1+$

y$2+$

y$3$

$>=$

Time $(2$:$00$ p$.$m$.-3$:$00$ p$.$m$.)$

x$9+$

x$10+$

x$11+$

y$9+$

y$10+$

y$11+$

y$12+$

y$1+$

y$2+$

y$3$

$>=$

Time $(3$:$00$ p$.$m$.-4$:$00$ p$.$m$.)$

x$9+$

x$10+$

x$11+$

y$9+$

y$10+$

y$11+$

y$12+$

y$1+$

y$2+$

y$3$

$>=$

Time $(4$:$00$ p$.$m$.-5$:$00$ p$.$m$.)$

x$9+$

x$10+$

x$11+$

y$9+$

y$10+$

y$11+$

y$12+$

y$1+$

y$2+$

y$3$

$>=$

Time $(5$:$00$ p$.$m$.-6$:$00$ p$.$m$.)$

x$9+$

x$10+$

x$11+$

y$9+$

y$10+$

y$11+$

y$12+$

y$1+$

y$2+$

y$3$

$>=$

Time $(6$:$00$ p$.$m$.-7$:$00$ p$.$m$.)$

xi$,$ yj $>=0$ and integer for i $=9,10,11$ and j $=9,10,11,12,1,2,3$

$($b$)$ Solve the LP Relaxation of your model in part $($a$).$ Enter the total cost value under the optimal solution.

Total Cost: $

$($c$)$ Solve for the optimal schedule of tellers. Enter the total number of full$-$time and part$-$time employees required.

If your answer is zero enter $0.$

Total Number of Full$-$Time Employees:

Total Number of Part$-$Time Employees:

$($d$)$ After reviewing the solution to part $($c$),$ the bank manager realized that some additional requirements must be specified. Specifically, she wants to ensure that one full$-$time employee is on duty at all times and that there is a staff of at least five full$-$time employees. Revise your model to incorporate these additional requirements, and solve for the optimal solution. Enter the total number of full$-$time and part$-$time employees required and the total cost under the revised optimal solution.

If required, round your answers to the nearest whole number. If your answer is zero enter $0.$

Total Number of Full$-$Time Employees:

Total Number of Part$-$Time Employees:

Total Cost: $