Suppose a retailer sells a product at a steady rate all year. The managers want to be sure never to run out of stock, but they don't want to order too many products at a time because storing a large number of items becomes costly. They also don't want to place a large number of orders throughout the year because each order adds associated costs (processing paperwork, shipping, handling). The goal is to find the optimal order size and frequency.

If the retailer sells A items per year at a constant rate and places equally sized orders of x items during the year, then A/x orders will be placed annually. Let's assume perfect timing of the orders, in the sense that a new order arrives just as the previous stock of items is depleted. Because the items are sold at a steady rate, the retailer is storing on average half the number of products in each oder: 1/2x. (Half the time they have more than this, half the time they have less).

**In this explanation, I don't understand why it is A/x and why is is 1/2x. Could you please explain why in greater detail than the book used here. Thank you