Central has grown from a small two$-$person operation to a $$75-$ million per year business in a span of $25$ years. From its inception Central has been a profitable business in sound financial condition. Despite the continued growth of profits in absolute terms, however, Central found that profits as a percentage of sales have declined. When management became aware of the seriousness of the problem, it was decided to undertake a thorough review of policies and procedures in the area of inventory management.

One of the first items the company reviewed was the cost of preparing and processing a requisition, preparing a purchase order, and making necessary record changes. This was estimated to be $$50$ per order.

One of the typical items of inventory analyzed in detail was a small hydraulic fitting. Central sells about $21,840$ of these fittings per year $($each fitting is purchased for $$14$ and is sold for $$19).$ The manufacturer from whom Central buys the fitting does not offer any quantity discount. The manufacturer is located about $1,500$ miles away and the fittings are shipped to Central by truck. They all arrive at once.

The annual per unit inventory holding cost is $20\%$ of the item's value.

A$)$ What is the order quantity that minimizes total annual cost of inventory $($TAC$)?$ What is the minimum TAC?

B$)$ Suppose that the supplier of the fittings only has the capacity to deliver the fittings at the rate of $100$ per day. Recalculate the optimum order quantity and the minimum TAC.

C$)$ Suppose the supplier were to offer the following quantity discounts:

$\backslash$table$[[$Quantity Ordered,Price per fitting$$14\mathrm{.}00$