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430 CHAPTER 10 INVENTORY MODELS

possible losses, c o ¼ E10 and c u ¼ E20, by the probability of obtaining the loss, we
can compute the expected value of the loss, or simply the expected loss (EL),
associated with the order quantity alternatives. So:
ELðQ ¼ 501Þ¼ c o Pðdemand 500Þ¼ E10ð0:50Þ¼ E5

ELðQ ¼ 500Þ¼ c u Pðdemand > 500Þ¼ E20ð0:50Þ¼ E10
Based on these expected losses, do you prefer an order quantity of 501 or 500 pairs
of shoes? Because the expected loss is greater for Q ¼ 500, and because we want to
avoid this higher cost or loss, we should make Q ¼ 501 the preferred decision. We
could now consider incrementing the order quantity one additional unit to Q ¼ 502
and repeating the expected loss calculations.
Although we could continue this unit-by-unit analysis, it would be time-consuming
and cumbersome. We would have to evaluate Q ¼ 502, Q ¼ 503, Q ¼ 504 and so on,
until we found the value of Q where the expected loss of ordering one incremental
unit is equal to the expected loss of not ordering one incremental unit; that is, the
optimal order quantity Q* occurs when the incremental analysis shows that:

ELðQ þ 1Þ¼ ELðQ Þ (10:29)
When this relationship holds, increasing the order quantity by one additional unit has
no economic advantage. Using the logic with which we computed the expected losses
for the order quantities of 501 and 500, the general expressions for EL(Q* + 1) and
EL(Q*) can be written:

ELðQ þ 1Þ¼ c o Pðdemand Q Þ (10:30)

ELðQ Þ¼ c u Pðdemand > Q Þ (10:31)
Because we know from basic probability that:

Pðdemand Q Þþ Pðdemand > Q Þ¼ 1 (10:32)
we can write:

Pðdemand > Q Þ¼ 1 Pðdemand Q Þ (10:33)
Using this expression, Equation (10.31) can be rewritten as:

ELðQ Þ¼ c u ½1 Pðdemand Q ފ (10:34)
Equations (10.30) and (10.34) can be used to show that EL(Q*+1) ¼ EL(Q*)
whenever:

c o Pðdemand Q Þ ¼ c u ½1 Pðdemand Q ފ (10:35)

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