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?