One of the formulas for inventory management says that the average weekly cost of ordering, paying for, and holding merchandise is

A(q) = (km)/q + cm + (hq)/2

Where q is the quantity you order when things run low (shoes, brooms, radios, etc), k is the cost of placing an order (the same, no matter how often you order), c is the cost of one item (a constant), m is the number of items sold each week (a constant), and h is the weekly holding cost per item (a constant that takes into account things such as space, utilities, insurance, and security).

a) Your job, as the inventory manager for your store, is to find the quantity that will minimize A(q). What is it? (The formula you will get for the answer is called the Wilson lot size formula.)

b) Shipping costs sometimes depend on order size. When they do, it is more realistic to replace k by k + bq, the sum of k and a constant multiple of q. What is the most economical quantity to order now?