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432 CHAPTER 10 INVENTORY MODELS

Figure 10.9 Probability Distribution of Demand for the ACR Problem Showing the
Location of Q*

P (demand Q*) = 0.7143

= 14

150
Q* = 158

Pðdemand Q Þ¼ c u ¼ 200 ¼ 0:7143
ðc u þ c o Þ 200 þ 80
We can use the normal probability distribution for demand as shown in Figure 10.9
to find the order quantity that satisfies the condition that P(demand Q*) ¼ 0.7143.
From Appendix A, we see that 0.7143 of the area in the left tail of the normal
probability distribution occurs at z ¼ 0.57 standard deviations above the mean.
With a mean demand of ¼ 150 automobiles and a standard deviation of s ¼ 14
automobiles, we have:
EXCEL file
Q ¼ þ 0:57
SINGLE-PERIOD
¼ 150 þ 0:57ð14Þ¼ 158
An example of a single- So, ACR should plan to have 158 4WDs available for the weekend. Note that in this
period inventory model case the cost of overestimation is less than the cost of underestimation. So ACR is
with probabilistic demand willing to risk a higher probability of overestimating demand and hence a higher
described by a normal
probability distribution is probability of a surplus. In fact, ACR’s optimal order quantity has a 0.7143 proba-
considered in Problem 15. bility of a surplus and a 1 0.7143 ¼ 0.2857 probability of a stock-out. As a result,
the probability is 0.2857 that all 158 4WDs will be rented during the weekend.

NOTES AND COMMENTS

1 In any probabilistic inventory model, the 2 In the single-period inventory model, the value
assumption about the probability distribution for of c u /(c u + c o ) plays a critical role in selecting
demand is critical and can affect the the order quantity [see Equation (10.36)].
recommended inventory decision. In the Whenever c u ¼ c o , c u /(c u + c o ) equals 0.50; in
problems presented in this section, we used the this case, we should select an order quantity
uniform and the normal probability distributions to corresponding to the median demand. With
describe demand. In some situations, other this choice, a stock-out is just as likely as a
probability distributions may be more surplus because the two costs are equal.
appropriate. In using probabilistic inventory However, whenever c u < c o , a smaller order
models, we must exercise care in selecting the quantity will be recommended. In this case, the
probability distribution that most realistically smaller order quantity will provide a higher
describes demand. probability of a stock-out; however, the more

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