Example $5$

A final assembly plant for a certain automobile model is to have a capacity of $225,000$ units annually. The plant will operate $50$ weeks$/$yr$,$ two shifts$/$day$,5$ days$/$week$,$ and $7\mathrm{.}5$ hours$/$shift$.$ It will be divided into three departments: $(1)$ body shop, $(2)$ paint shop, $(3)$ general assembly department. The body shop welds the car bodies using robots, and the paint shop coats the bodies. Both of these departments are highly automated. General assembly has no automation. There are $15\mathrm{.}0$ hours of work content time on each car in this third department, where cars are moved by a continuous conveyor. Determine $($a$)$ hourly production rate of the plant, $($b$)$ number of workers and workstations required in trim chassis final if no automated stations are used, the average manning level is $2\mathrm{.}5,$ balancing efficiency $=90\%,$ proportion uptime $=95\%,$ and a repositioning time of $0\mathrm{.}15$min is allowed for each worker.