A Simple Inventory Model

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Formulas for Optimal Order Size

Inventory and Cash Models

A number of models have been developed to derive optimal levels of inventory or cash. Here are some well-known examples.

A Simple Inventory Model
Everyman’s Bookstore experiences a steady demand for Principles of Corporate Finance from customers who find that it makes a serviceable bookend. Suppose that the bookstore sells 100 copies of the book a year and that it orders Q books at a time from the publishers. Then it will need to place 100/Q orders per year:

Number of orders per year = [pic][pic]
Just before each delivery, the bookstore has effectively no inventory in Principles of Corporate Finance. Just after each delivery it has an inventory of Q books. Therefore, its average inventory is midway between 0 books and Q books:

Average inventory = Q/2 books
For example, if the store increases its regular order by one book, the average inventory increases by ½ book.
There re two costs to holding this inventory. First, there is the carrying cost. This includes the cost of the capital that is tied up in inventory, the cost of the shelf space, and so on. Let us suppose that these costs work out to a dollar per book per year. The effect of adding one more book to each order is therefore to increase the average inventory by ½ book and the carrying cost by ½ x $1.00 = $0.50. Marginal carrying cost = [pic]

The second type of cost is the order cost. Imagine that each order placed with the publisher involves a fixed clerical and handling expense of $2. The bookstore gets a large reduction in costs when it orders two books at a time rather than one, but thereafter the savings from increases in order size steadily diminish. In fact, the marginal reduction in order cost depends on the square of the order size.[1] Marginal reduction in order cost = [pic]

Here, then, is the kernel of the inventory problem. As the bookstore increases its order size, the number of orders falls but the average inventory rises. Costs that are related to the number of orders decline; those that are related to inventory size increase. It is worth increasing order size as long as the decline in order cost outweighs the increase in carrying cost. The optimal order size is the point at which these two effects exactly offset each other. In our example this occurs when Q = 20:

Marginal reduction in order cost = [pic] = $0.50
Marginal carrying cost = [pic]
The optimal order size is 20 books. Five times a year the bookstore should place an order for 20 books, and it should work off this inventory over the following 10 weeks.
The general formula for optimum order size is found by setting marginal reduction in order cost equal to the marginal carrying cost and solving for Q:
Marginal reduction in order cost = Marginal carrying cost

In our example,


Cash Balances
William Baumol was the first to notice that this simple inventory model can tell us something about the management of cash balances[2]. Suppose that you keep a reservoir of cash that is steadily drawn to pay bills. When it runs out you replenish the cash balances by selling Treasury bills. The main carrying cost of holding this cash is the interest that you are losing. The order cost is the fixed administrative expense of each sale of Treasury bills.

In other words, your cash management problem is exactly analogous to the problem of optimum order size faced by Everyman’s Bookstore. You just have to redefine variables. Instead of books per order, Q becomes the amount of Treasury bills sold each time cash is replenished. Cost per order becomes cost per sale of Treasury bills. Carrying cost is the interest rate foregone by holding cash. Total cash disbursements take the place of books sold. The optimum Q is:

Q = [pic]
Suppose that the interest rate on Treasury bills is 8%, but every sale of bills costs you...
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