A chemical firm produces sodium bisulfate in $20-$pound bags. Demand for this product is $20$ tons per day. The capacity for producing the product is $502$ tons per day. Set up costs $$100,$ and storage and handling costs are $$5$ per ton a year. The firm operates $200$ days a year. $($Note: $1$ ton $=2000$ pounds$).($a$)$ How many bags per run are optimal?$($b$)$ What would average inventory be for this lot size?$($c$)$ Determine the approximate length of a production run?$($d$)$ About how many runs per year would there be$?($e$)$ How much could the company save annually if the set up cost could be reduced to $$25$ per run?