A chemical plant produces sodium bisulfate in $100$kg bags. Demand for this product is $20$ tonnes per day. The capacity for producing this product is $50$ tonnes per day. Setup cost is $ $400,$ and storage and handling costs are $ $200$ per tonne per year. The company operates $200$ days a year. $($Note: $1$ tonne $=1,000$kg $).$ a$)$ What is the optimal number of bags per production run? b$)$ What would the average inventory level be for this lot size? c$)$ Determine the approximate length of a production run, in days. d$)$ About how many production runs per year would there be$?$ e$)$ How much could the company save annually in inventory control cost if the setup cost could be reduced to $ $200$ per production run and the optimal production quantity is recalculated and used?