Page 230 - How To Solve Word Problems In Calculus

holding costs will be small, but the total ordering fees will be
large. We will use calculus to determine the lot size x of each
shipment that minimizes the total cost.
The cost incurred is C(x) = H(x) + O(x), where H(x)
represents the total holding cost and O(x) is the total ordering
cost.
Assuming that the demand for the product is uniform
throughout the year, the average yearly holding cost to store
a lot of size x is the same as if x/2 items were stored for the
entire year. Therefore

x

H(x) = (annual storage cost per item)
2
If q items are ordered in lots of size x, then the total number
of shipments necessary will be q/x. Consequently,
O(x) = (cost per order)(number of orders)

q
= (cost per order)
x
The total cost is then

C(x) = H(x) + O(x)

x
= (annual storage cost per item)
2

q
+ (cost per order)
x
Once the function C(x) is constructed we apply maximum-
minimum theory to determine the value of x that minimizes
total cost.

EXAMPLE 12
A home improvement company expects to sell 180 kitchen
sinks during the year. There is a $60 ordering fee for each ship-
ment of sinks and it costs $6 to hold a sink for a year. How
many sinks should the company order in each shipment to
minimize total cost?

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