The Northside Bank is working to develop an efficient work schedule for full$-$time and part$-$time tellers. The schedule must provide for efficient operation of the bank, including adequate customer service, employee breaks, and so on$.$ On Fridays, the bank is open from $9$:$00$ A$.$M$.$ to $7$:$00$ P$.$M$.$ The number of tellers necessary to provide adequate customer service during each hour of operation is summarized here.

Time No$.$ of Tellers Time No$.$ of Tellers

$9$:$00$ a$.$m$.-10$:$00$ a$.$m$.62$:$00$ p$.$m$.-3$:$00$ p$.$m$.8$

$10$:$00$ a$.$m$.-11$:$00$ a$.$m$.43$:$00$ p$.$m$.-4$:$00$ p$.$m$.6$

$11$:$00$ a$.$m$.-$Noon $64$:$00$ p$.$m$.-5$:$00$ p$.$m$.6$

Noon$-1$:$00$ p$.$m$.105$:$00$ p$.$m$.-6$:$00$ p$.$m$.7$

$1$:$00$ p$.$m$.-2$:$00$ p$.$m$.116$:$00$ p$.$m$.-7$:$00$ p$.$m$.4$

Each full$-$time employee starts on the hour and works a $4-$hour shift, followed by a $1-$hour break and then a $3-$hour shift. Part$-$time employees work one $4-$hour shift beginning on the hour. Considering salary and fringe benefits, full$-$time employees cost the bank $$15$ per hour $($$$105$ a day$),$ and part$-$time employees cost the bank $$8$ per hour $($$$32$ per day$).$

$($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$12+$

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$3$

$-$ Select your answer $-$

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$3$

$-$ Select your answer $-$

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$3$

$-$ Select your answer $-$

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$3$

$-$ Select your answer $-$

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$3$

$-$ Select your answer $-$

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$3$

$-$ Select your answer $-$

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$3$

$-$ Select your answer $-$

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$3$

$-$ Select your answer $-$

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$3$

$-$ Select your answer $-$

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$3$

$-$ Select your answer $-$

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 six 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: $