4. An Inventory Model
In this section we shall construct a simple quantitative model to describe the cost of maintaining
an inventory. Suppose you must meet an annual demand of V units of a certain product for
which the rate of demand is constant throughout the year. Suppose further that you replenish
your stock periodically throughout the year by ordering x units of the product just when your
stock is about depleted. In this case a graph of your inventory level versus time would look
something like Figure 1.
The decreasing lines are parallel since the rate of demand is constant throughout the year.
We shall incorporate three costs into our model: storage costs, re-ordering costs, and
Storage Costs. Suppose the cost in dollars per annum of storing one unit is A . The average
inventory level (see Figure 1) is , so we shall take the annual storage cost to be
Re-ordering Costs. Suppose the ▯xed cost in dollars of placing an order is B . If we order
x units at a time, we must order V times per year, so the re-ordering cost is
Purchasing Costs. Let p(x) be the cost in dollars of purchasing x units. In practice this
cost may involve discounts as an incentive to place large orders [see Example 2, Exercises 4, 5],
1 but for this model let us assume that the per unit cost of purchasing x units is ▯xed| that
is, p(x) = kx | so that the total cost of purchasing V units is
V ▯ kx = V k :
This is simply the cost of purchasing V units, and is a constant.
If C(x) is the cost of maintaining the inventory, we then have
C(x) = Ax + BV + kV :
Mathematically we can use this model to minimize annual inventory costs.
C (x) = A ▯ BV
, x2 = 2BV
) x = A
This value of x minimizes C since C (x) = is positive (since we can assume x is
positive), and is called the economic lot size. Note that the economic lot size is independent of
purchasing cost, so that our simple model depends only on storage and re-ordering costs.
Example 1. A department store sells 500 refrigerators per year. The annual storage and
carrying cost per unit is $30 and the ▯xed reorder costs are $50. At present, lots of 100 are
ordered. How much can be saved by an adjustment of the order size?
Solution. Present annual costs are
C(100) = 1500 + = 1750 :
For a general lot size x , annual cost would be, in dollars,
C(x) = 15x + :
A minimum is reached if
x = 2(500)(50)
2 Since x must be an integer, we calculate
C(41) = 615 + 41 = 615 + 609:76 = $1224:76
C(40) = 600 +25;000 = 600 + 625 = $1225 :
Judgment suggests that the round number of 40 units is worth the 26 cents per year of savings
foregone. Hence the saving by a change from 100 to 40 units per order is 1750 ▯ 1225 = 625