The Northshore 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.

$$ Time$$ Number of Tellers$$ Time$$ Number of Tellers

$9$:$00$ a$.$m$.10$:$00$ a$.$m$.62$:$00$ p$.$m$.3$:$00$ p$.$m$.6$

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

$11$:$00$ a$.$m$.$Noon$84$:$00$ p$.$m$.5$:$00$ p$.$m$.7$

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

$1$:$00$ p$.$m$.2$:$00$ p$.$m$.96$:$00$ p$.$m$.7$:$00$ p$.$m$.6$

Each full$-$time employee starts on the hour and works a $4-$hour shift, followed by $1$ hour for lunch 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 parttime employees cost the bank $$8$ per hour $($$$32$ per day$).$

$1.$ Formulate an integer programming model that can be used to develop a schedule that willsatisfy customer service needs at a minimum employee cost. $($Hint: Let xi $=$ number of fulltime employees coming on duty at the beginning of hour i and yi $=$ number of part$-$time employees coming on duty at the beginning of hour i$.)$

$2.$ Solve the LP Relaxation of your model in part $($a$).$

$3.$ Solve for the optimal schedule of tellers. Comment on the solution.

After reviewing the solution to part $($c$),$ the bank manager realized that some additionalrequirements must be specified. Specifically, she wants to ensure that one fulltime 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.