A chemical plant produces sodium bisulfate in $140$ kg bags. Demand for this product is $28$ tonnes per day. The capacity for producing this product is $50$ tonnes per day. Setup cost is $$420,$ and holding cost is $$200$ per tonne per year. The company operates $270$ days a year. $($Note: $1$ tonne $=1,000$ kg$).$

a$.$ What is the optimal number of bags per production run? $($Round your intermediate calculations to $2$ decimal places and the final answer to the nearest whole number.$)$

Optimal number of bags

b$.$ What would the average inventory level be for this lot size? $($Round your intermediate calculations to $2$ decimal places and the final answer to the nearest whole number.$)$

Average inventory bags

c$.$ Determine the approximate length of a production run, in days. $($Round the intermediate calculations and final answer to $2$ decimal places.$)$

Run length days

d$.$ About how many production runs per year would there be$?($Round your intermediate calculations to $2$ decimal places and the final answer to $1$ decimal place.$)$

Runs per year

e$.$ How much could the company save annually in inventory control cost if the setup cost could be reduced to $$210$ per production run and the optimal production quantity is recalculated and used? $($Round the intermediate calculations and final answer to $2$ decimal places.$)$

Annual Savings $