A chemical firm produces sodium bisulfate in $100-$pound bags. Demand for this product is $20$ tons per day. The capacity for producing the product is $50$ tons per day. Setup costs $$100,$ and storage and handling costs are $$5$ per ton a year. The firm operates $200$ days a year. $($Note: $1$ ton $=2,000$ pounds.$)$a$.$ How many bags per run are optimal?b$.$ What would the average inventory be for this lot size?c$.$ Determine the approximate length of a production run, in days.d$.$ About how many runs per year would there be$?$e$.$ How much could the company save annually if the setup cost could be reduced to $$25$ per run