# How to Calculate the Economic Order Quantity (EOQ)

Inventory costs include not only the price of goods, but also costs associated with placing orders and physically holding stock on or off location. Your yearly ordering and carrying costs depend on the annual consumer demand for your merchandise (**D**), number of items per order (**Q**), the fixed ordering costs per shipment (**Fo**), and the stocking cost per item (**Fc**).

Since the only variable you can control is the number of items per order, **Q**, it makes sense to choose this number in such a way that the inventory ordering and carrying costs are minimized. The value of **Q** that minimizes costs is called the Economic Order Quantity, or the EOQ for short. You can use calculus to find the EOQ, or use the calculator on the left to compute the optimal value of **Q** when you input the fixed inventory costs.

### Minimizing the Ordering and Carrying Cost Equation

The total annual inventory ordering and holding cost equation isFo*D/Q + Fc*Q/2

If we consider D, Fo, and Fc to be fixed constants and Q to be the variable, then the inventory costs are simply a function of Q:

I(Q) = Fo*D/Q + Fc*Q/2.

To find the value of Q that minimizes this equation, we must take the derivative of I(Q).

I'(Q) = -Fo*D/Q² + Fc/2

If we set this equal to 0 and solve for Q we get

0 = -Fo*D/Q² + Fc/2

Fo*D/Q² = Fc/2

Fo*D = Q²*Fc/2

2*Fo*D/Fc = Q²

sqrt(2*Fo*D/Fc) = Q

Thus, the EOQ is sqrt(2*Fo*D/Fc).

### Example

The fixed costs of placing an order are $120 per order, the cost to hold an item in stock is $0.80, and the annual demand for the item is 10,000 units. Given that D = 10000, Fo = 120, and Fc = 0.8, the EOQ issqrt(2*120*10000/0.8) = sqrt(3000000) = 1732.051

EOQ must be an integer since we cannot order a fractional number of items, therefore the EOQ is either 1732 ot 1733. Plugging both of these values of Q into the I(Q) gives us

I(1732) = $1385.64

I(1733) = $1385.64

So either value yields the minimal cost.

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