$(30$ points$)$ A punch press takes in coils of sheet metal and can make five different electrical breaker boxes, denoted by B$1,$ B$2,$ B$3,$ B$4,$ and B$5.$ Each box takes exactly $1$ minute to produce. To switch the process from one type of box to another takes $4$ hours. There is demand of $1,800,1,000,600,350,$ and $200$ units per month for boxes B$1,$ B$2,$ B$3,$ B$4,$ and B$5,$ respectively. The plant works one shift, $5$ days per week. After lunch, breaks, and so on$,$ there is $7$ hours available per shift. Assume $52$ weeks per year.

$($a$)$ What is $r_a$ in boxes per hour?

$($b$)$ What would utilization be if there were no setups? $($Note that utilization will approach this as batch sizes approach infinity.$)$

$($c$)$ Suppose the SCV of the press is $0\mathrm{.}2$ no matter what the batch sizes are. What is the average cycle time when the batch sizes are all equal to $1,000($assume $c_a^2=1)?$

$($d$)$ Use trial and error to find a set of batch sizes that minimizes cycle time.