$3\mathrm{.}17$ On average, $10$ new batches of parts are started through a certain plant each week. Average batch quantity is $75$ parts that are processed through a sequence of ten machines. Setup time per machine per batch averages $5hr,$ and average operation time per machine for each part is $10$min. Nonoperation time per batch at each machine averages $10hr.$ There are $30$ machines in the factory. Any machine can be set up for any type of batch processed in the plant. The plant operates $40h\frac{r}{w}k.$ The plant manager complains that almost $65hr$ of overtime must be authorized each week to keep up with the production schedule. Determine $($a$)$ manufacturing lead time for an average order, $($b$)$ weekly production capacity of the plant if all machines were operated at $100\%$ utilization, $($c$)$ current utilization of

$87$

the plant, and $($d$)$ average work$-$in$-$process $($number of parts$)$ in the plant. $($e$)$ Why must overtime be authorized to achieve the desired output?