User:YuriTomura

In the operations of a company, sometimes managers have to determine the optimal balance between numbers of orders of inventory per year and the economic order quantity in order to minimize inventory costs. To do this, one must determine the cost of placing an order as well as the cost of one unit of inventory. For example, a snowboard company expects to sell 5,000 boards during the coming year. We must assume sales occur at a constant rate throughout the year. And yearly carrying costs, which are based on the average number of snowboards in stock during the year is $20 per snowboard. The cost to place an order with the manufacturer is $60.

So, to determine the economic order quantity, let x = the number of snowboards in each order and let r = the number of orders per year. Therefore, we know that r*x must equal 5,000.
From here, we can conclude that r = 5000/x.

The cost function, C, is equal to 20x/2 + 60r.
Now, we can substitute r = 5000/x into our cost function, which simplifies to 10x + 300000/x.

We must then take the derivative of the cost function, giving us 10 - 300000/x^2.

Make the derivative equal to zero to find the critical points and solve for x, which, in this case, is:
x = sqrt(300000/10) = 173.205

Now that we have a value for x, we must then prove that this is a minimum using a sign table.

Therefore, we know that this is the economic order quantity and that the snowboarding company will receive 173 snowboards per order throughout the year.

So, in this inventory control problem, we have used a type of calculus that is commonly used in business to determine the economic order quantity (EOQ), which is something that store managers have to, or should have to do, each year.