Danny Steel, Inc., fabricates various products from two basic inputs, bar stock and sheet stock.$$ Bar stock is used at a steady rate of $1,000$ units per year and costs $$200$ per bar.$$ Sheet stock is used at a rate of $500$ units per year and costs $$150$ per sheet.$$ The company uses a $20\%$ annual holding cost rate, and the fixed cost to place an order is $$50,$ of which $$10$ is the cost of placing the purchase order and $$40$ is the fixed cost of a truck delivery.$$ The variable $($i$.$e$.,$ per unit charge$)$ trucking cost is included in the unit price.$$ The plant runs $365$ days per year.

$($a$)$ Use the EOQ formula with the full fixed order cost of $$50$ to compute the optimal order quantities, order intervals, and annual cost for bar stock and sheet stock.$$ What fraction of the total annual $($holding plus order$)$ cost consists of fixed trucking cost?

$($b$)$ Using a week $($seven days$)$ as the base interval, round the order intervals for bar stock and sheet stock to the nearest power of $2.$ If you charge the fixed trucking fee only once for deliveries that coincide, what is the annual cost now?