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MAT133Y1
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Abe Igelfeld
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Lecture

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Mathematics

MAT133Y1

Abe Igelfeld

Fall

Description

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.
x
x/2
t
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
purchasing costs.
Storage Costs. Suppose the cost in dollars per annum of storing one unit is A . The average
x
inventory level (see Figure 1) is , so we shall take the annual storage cost to be
2
Ax
2 :
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
x
BV
x :
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 :
x
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 :
2 x
Mathematically we can use this model to minimize annual inventory costs.
C (x) = A ▯ BV
2 x2
= 0
, x2 = 2BV
A
r
2BV
) x = A
2BV
This value of x minimizes C since C (x) = is positive (since we can assume x is
x3
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
50(500)
C(100) = 1500 + = 1750 :
100
For a general lot size x , annual cost would be, in dollars,
25;000
C(x) = 15x + :
x
A minimum is reached if
r
x = 2(500)(50)
30
r
25;000
= 15
:
= 40:8
2 Since x must be an integer, we calculate
25;000
C(41) = 615 + 41 = 615 + 609:76 = $1224:76
C(40) = 600 +25;000 = 600 + 625 = $1225 :
40
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

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