$19.$ HAL Ltd$.,$ discussed in Problem $9,$ can produce the disk drive housings in the Hamilton, Ontario, plant at a rate of $150$ housings per month. The housings cost HAL $$85$ each to produce, and the set$-$up cost for beginning a production run is $$700.$ Assume an annual interest rate of $28$ percent for determining the holding cost.

Annual Demand $($D$)=720$ units $/$ yearly

a$.$ What is the optimal number of housings for HAL to produce in each production run?

b$.$ Find the time between initiation of production runs, the time devoted to production, and the downtime each production cycle.