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SINGLE-PERIOD INVENTORY MODEL WITH PROBABILISTIC DEMAND 431

Solving for P(demand Q*), we have:

Pðdemand Q Þ¼ c u (10:36)
c u þ c o

This expression provides the general condition for the optimal order quantity Q*in
the single-period inventory model.
In the Juliano Shoe Company problem c o ¼ E10 and c u ¼ E20. Thus, equation
(10.36) shows that the optimal order size for Juliano shoes must satisfy the following
condition:

Pðdemand Q Þ¼ c u ¼ 20 ¼ 20 ¼ 2
c u þ c o 20 þ 10 30 3
We can find the optimal order quantity Q* by referring to the probability
distribution shown in Figure 10.8 and finding the value of Q that will provide
2
P(demand Q*) = / 3 . To find this solution, we note that in the uniform distribu-
tion the probability is evenly distributed over the entire range of 350–650 pairs of
shoes. So, we can satisfy the expression for Q* by moving two-thirds of the way from
350 to 650. Because this range is 650 350 ¼ 300, we move 200 units from 350
toward 650. Doing so provides the optimal order quantity of 550 pairs of shoes.
In summary, the key to establishing an optimal order quantity for single-period
inventory models is to identify the probability distribution that describes the demand
for the item and the costs of overestimation and underestimation. Then, using the
information for the costs of overestimation and underestimation, Equation (10.36)
can be used to find the location of Q* in the probability distribution.

Arabian Car Rental
As another example of a single period inventory model with probabilistic demand,
let us consider the situation faced by the Arabian Car Rental company (ACR),
based in Saudi Arabia. ACR operates through the Middle East and must decide how
many cars to have available at each car rental location at specific points in time
through the year. We shall illustrate using its car rental depot in Dubai as an
example. Analysis has shown that one of its more lucrative markets consists of
tourists and expatriate workers who rent four-wheel drive cars for a long weekend.
Typically, customer demand for this type of vehicle follows a normal distribution
with a mean of 150 vehicles demanded and a standard deviation of 14 vehicles.
The ACR situation can benefit from use of a single-period inventory model. The
company must establish the number of four-wheel drive cars (4WD) to have available
prior to the weekend. Customer demand over the weekend will then result in either a
stockout or a surplus. Let us denote the number of 4WD available by Q.If Q is greater
than customer demand, ACR will have a surplus of cars. The cost of a surplus is the
cost of overestimating demand. This cost is set at E80 per car, which reflects, in part,
the opportunity cost of not having the car available for rent elsewhere.
If Q is less than customer demand, ACR will rent all available cars and experi-
ence a stock-out or shortage. A shortage results in an underestimation cost of E200
per car. This figure reflects the cost due to lost profit and the lost goodwill of not
having a car available for a customer. Given this information, how many 4WDs
should ACR make available for the weekend?
Using the cost of underestimation, c u ¼ E200, and the cost of overestimation,
c o ¼ E80, equation (10.36) indicates that the optimal order quantity must satisfy the
following condition:

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