A large office equipment store sells graphing calculators at a constant rate of $10$ per business day. The

store is open $300$ days per year, and so sells $3000$ calculators per year. Each calculator has a value of $$40.$

Currently, the manager is ordering $1000$ calculators at a time, $3$ times per year. Although her cost of placing

orders is low, she thinks she has too much money tied up in inventory and the costs of holding inventory

could be lower. She has hired you to mathematically analyze the problem and write a report giving your

recommendation of how many calculators to order at a time and how many times to order per year to order.

The manager has provided you with the following information concerning the costs of running the business.

The cost $($labor and materials$)$ of placing an order is $$24\mathrm{.}00$ per order, regardless of how much is ordered

at a time. Since the demand is $3000$ calculators per year and they order $1000$ at a time, ordering costs are

currently

$(3000/1000)$ $$24=$ $$72\mathrm{.}00$ per year.

Holding inventory also costs money; capital is tied up$,$ and the inventory must be stored safely and insured.

These costs annually amount to $25\%$ of the value of the inventory $($i$.$e$.,$ to hold $$100\mathrm{.}00$ worth of inventory

in stock for one year has an inventory holding cost of $$25\mathrm{.}00).$ Since the store sells calculators at a constant

rate throughout the year, the average inventory is one$-$half the size of the order; that is since currently, she

is ordering $1000$ calculators at a time, on average they have $500$ in inventory. Each one is valued at $$40\mathrm{.}00,$

so their current inventory holding costs are

$25\%$ $$40\mathrm{.}00500=$ $$5000.$

Currently, the combined cost of ordering and holding inventory is $$5072.$ The manager would like to mini$-$

mize these COMBINED costs of ordering and holding inventory.

Your report should include:

$$ The ordering quantity that results in the lowest total annual cost

$$ The total annual costs of ordering and holding inventory at this order quantity

$$ How much of a saving this is to the store

$$ Explanations as to how you arrived at your conclusions, including the computations needed