has undertaken an incremental investment, and the profit that accrues from this incremen-
tal investment is called the incremental profit. Since the objective is to maximize profits
from the sale of this commodity without reference to the rate of return that the firm earns
on invested capital, the incremental investment is justified if the incremental profit has a
positive value.
If a demand for these 10 additional units exists, the firm earns a direct profit of 10($75
- $50) = $250, and it reduces its loss of goodwill by 10($4) = $40. Thus, the effective
profit = $250 + $40 = $290. If a demand for the 10 additional units does not exist, the firm
incurs a loss of 10($50 - $6) = $440. Let /"(sold) and P(not sold) = probability the 10 ad-
ditional units will be sold and will not be sold, respectively, and E(AP) = expected incre-
mental profit, $. Then E(AP) = 290 [P(sold)] - 440 [P(not sold)]. Set P(not sold) = 1 -
P(sold), giving E(AP) = 730 [P(sold)] - 440, Eq. a.
- Apply this equation to find the optimal inventory
From the preceding calculation procedure, E(P) is positive if X= 150; thus, the firm
should order at least 150 units. Assume X increases from 150 to 160. From Table 28,
P(sold) = 1 - 0.08 = 0.92. By Eq. a, E(AP) = 730(0.92) - 440 = $231.60 > O, and the in-
cremental investment is justified. Assume X increases from 160 to 170. Then P(sold) = 1
- (0.08 + 0.13) = 0.79. By Eq. a, E(AP) = 730(0.79) - 440 = $136.70 > O, and the incre-
mental investment is justified. Assume X increases from 170 to 180. Then P(sold) = 1 -
(0.08 + 0.13 + 0.20) = 0.59. By Eq. a, E(AP) - 730(0.59) - 440 = -$9.30 < O, and the in-
cremental investment is not justified. Thus, the firm should order 170 units.
- Devise a direct method of solution
Determine when E(AP) changes sign by setting E(AP) = 730 [P(sold)] - 440 = O, giving
P(sold) = 440/730 = 0.603. This is the lower limit of P(sold) if the incremental investment
is to be justified. Now, P(sold) first goes below this value when X= 180; thus, the expect-
ed profit is maximum when X = 170.
Related Calculations'. From the preceding calculation procedure, when X = 150,
E(P) = $3643.60; when X= 160, E(P) = $3875.20. Thus, when Xincreases from 150 to
160, E(AP) - $3875.20 - $3643.60 = $231.60, and this is the result obtained in step 2.
Similarly, from the preceding calculation procedure, when X increases from 160 to 170,
E(AP) = $4011.90 - $3875.20 = $136.70, and this is the result obtained above. The two
methods of solution yield consistent results.
The incremental-profit method is less time-consuming than the method followed in
the preceding calculation procedure, and it is particularly appropriate when the firm sets a
minimum acceptable rate of return. Thus, assume that the firm will undertake an invest-
ment only if the expected rate of return is 15 percent or more. When the firm orders 10
additional units, it undertakes an incremental investment of $440. This incremental in-
vestment is justified only if the expected incremental profit is at least $440(0.15), or $66.
SIMULATION OF COMMERCIAL ACTIVITY
BY THE MONTE CARLO TECHNIQUE
A firm sells and delivers a standard commodity. The terms of sale require that the firm
deliver the product within 1 day after an order is placed. In the past, the volume of orders
received averaged 3315 units per week, with the variation in volume shown in Table 30.
The firm currently employs a trucking company. But the firm contemplates purchas-
ing its own fleet of trucks to make deliveries. It is therefore necessary to decide how
many trucks are to be purchased. Several plans are under consideration. The shipping
facilities under plan A have an estimated average capacity of 3405 units per week.