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 as follows:

$\backslash$table$[[$Time$,$No$.$ of Tellers$],[9$:$00$ a$.$m$.-10$:$00$ a$.$m$.,6],[10$:$00$ a$.$m$.-11$:$00$ a$.$m$.,4],[11$:$00$ a$.$m$.-$ Noon,$8],[$Noon $-1$:$00$ p$.$m$.,10],[1$:$00$ p$.$m$.-2$:$00$ p$.$m$.,9],[2$:$00$ p$.$m$.-3$:$00$ p$.$m$.,6],[3$:$00$ p$.$m$.-4$:$00$ p$.$m$.,4],[4$:$00$ p$.$m$.-5$:$00$ p$.$m$.,7],[5$:$00$ p$.$m$.-6$:$00$ p$.$m$.,6],[6$:$00$ p$.$m$.-7$:$00$ p$.$m$.,6]]$

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$).$

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:

$x_i=$ number of full$-$time employees coming on duty at the beginningof hour i and

$y_j=$ number of part$-$time employees coming on duty at the beginningof hour $j.)$

For example, $x_{9}9$ denotes the number of FT employees coming at $9$:$00$ a$.$m$.$ and $y_3$ denotes the number of PT employees coming at $3$:$00$ p$.$m$.(30\%)$

Solve the LP relaxation of your model in MS Excel to get the optimal schedule for both FT and PT employees. For daily operation, how many FT tellers should be hired? How many PT tellers should be hired? How much is the total cost per day? $(30\%)$

Please comment on the optimal schedule of tellers. Any surplus of tellers at certain hours? $(20\%)$

After reviewing the solution in the above, 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. $(20\%)$