Q$8$: $(12$ pts$)-$

A single$-$station robotic assembly system performs a series of five assembly elements, each of which adds a different component to a base part. Each element takes $3\mathrm{.}5$ sec $.$ In addition, the handling time needed to move the base part into and out of position is $4\mathrm{.}5$ sec $.$ The fraction defect rate is $0\mathrm{.}003$ for all components, and the probability of a jam by a defective component is $0\mathrm{.}7.$ Average downtime per occurrence is $2\mathrm{.}5$ min $.$

Determine:

$($a$)$ Hourly production rate,

$($b$)$ Yield of good product in the output, and

$($c$)$ Uptime efficiency.

$($d$)$ What proportion of the output contains a defective component from the third of the five elements performed in the work cycle?