Suppose backorders are allowed in the EOQ model that was discussed in Section 5.4.3. Each item backordered results in some cost of b per item per year backordered. Derive the new EOQ formula.

Economic Order Quantity (EOQ):

The economic order quantity (EOQ) model is used when annual demand

(d) is constant, the replenishment lead time is zero, and stockouts are not allowed.
We assume that the cost per item is

c, the cost to place an order is A, and the cost to hold an item for one year is h. The holding cost arises from several sources: the cost (interest) of tying up money in inventory, space costs, taxes, and theft. It is common to apply a rate i (in %), so that h = ic. Common values for i range from 15% to 33%. We can determine the optimal order quantity Q, but considering the components of total cost (TC(Q)): annual order cost, annual holding cost, and total purchase cost.  
It is easy to extend the EOQ formula for several cases such as allowing backorders if there are stockouts, adding a lead time, and so on.