A typical part must be processed through $5$ machines on average in a certain factory. The

factory operates on batch bases; average batch size $=25$ parts. Part average processing time

$=3$ min., average part handling time $=2$ min, average tool change time per part $=1$ min.,

average setup time for a machine $=4$ hours, and average nonoperation time for a machine

per batch $=10$ hr$/$machine$.$ There are $15$ machines in the plant working in parallel. Each of

the machines can be set up for any type of job processed in the plant. The plant operates an

average of $80$ production hours per week. Determine $($a$)$ manufacturing lead time for an

average part, $($b$)$ plant capacity,