The bank now employs $12$ full$-$time tellers, but many people are on its roster of available part$-$time employees. A part$-$time employee must put in exactly $4$ hours per day but can start anytime between $9$ a$.$m$.$ and $1$ p$.$m$.$ Part$-$timers are fairly inexpensive labor pool, since no retirement or lunch benefits are provided for them. Full$-$timers, on the other hand, work from $9$ a$.$m$.$ to $5$ p$.$m$.$ but are allowed $1$ hour for lunch. $($Half of the full$-$timers eat at $11$ a$.$m$.,$ the other half at noon.$)$ Full$-$timers thus provide $35$ hours per week of productive labor time.

By corporate policy, the bank limits part$-$time hours to a maximum of $50\%$ of the day's total requirement. Part$-$timers earn $$4$ per hour $($or $$16$ per day$)$ on average, and full$-$timers earn $$50$ per day in salary and benefits, on average. The bank would like to set a schedule that would minimize the total personnel costs. It will release one or more of its full$-$time tellers if it is possible to do so$.$ It is your job to help the bank determine the schedule.