OPTIMAL INVENTORY TO MEET
FLUCTUATING DEMAND
A firm sells a commodity that is used only during the winter. Because the commodity de-
teriorates with age, units of the commodity that remain unsold by the end of the season
cannot be carried over to the following winter. To allow time for manufacture, the firm
must place its order for the commodity before July 1. Thus, the firm must decide how
many units to stock. For simplicity, the firm orders only in multiples of 10, and it assumes
that the number of units demanded by its customers is also a multiple of 10.
A study of past records reveals that the number of units demanded per season ranges
from 150 to 200, and the probabilities are as shown in Table 28. The cost of the commod-
ity, including purchase price and allowance for handling, storage, and insurance, is $50
per unit; the selling price is $75 per unit. Units that are not sold can be disposed of as
scrap for $6 each. If the firm is unable to satisfy the demand, it suffers a loss of goodwill
because there is some possibility of permanently losing customers to a competitor; this
loss of goodwill is assigned the value of $4 per unsold unit. How many units of this com-
modity should the firm order?
Calculation Procedure:
- Set up the equations for profit
A firm that sells a perishable commodity with a widely fluctuating demand runs a risk at
each end of the spectrum. If its stock is excessive, it suffers a loss on the unsold units; if
its stock is inadequate, it forfeits potential profits and suffers a loss of goodwill. So it
must determine how large a stock to maintain to maximize profits in the long run, apply-
ing past demand as a guide.
Let X = number of units ordered; Y= number of units demanded; P = profit (exclusive
of fixed costs), $. IfX= 7, then P = (JS- 5O)X, or P = 25X, Eq. a. IfX > 7, then P = 757
- 5OJT+ 6(X- 7), or P = -44X+ 697, Eq. b. lfX< 7, then P = (75 - SQ)X-4(Y-X), or
P = 29X- 47, Eq. c.
- Construct the profit matrix
In Table 29, list all possible values of X in the column at the left and all possible values of
7 in the row across the top. Compute the value of P for every possible combination of X
and 7, and record the value in the table. Thus, assume X= Y= 160; by Eq. a, P = 25 x 160
= $4000. Now assume^= 180 and Y= 160; by Eq. b, P = -44X 180 + 69 x 160 = $3120.
Finally, assume X= 160 and Y= 200; by Eq. c, P = 29 x 160 - 4 x 200 - $3840. Table 29
shows that P can range from $1550 (when the stock is highest and the demand is lowest)
to $5000 (when the demand is highest and the stock is adequate for the demand).
Alternatively, find the values of P thus: In Table 29, insert all values lying on the
TABLE 28. Demand Probabilities
Number of units
demanded 150 160 170 180 190 200
Probability, percent 8 13 20 32 18 9