# Economic Order Quantity (EOQ)

By Yuriy Smirnov Ph.D.

## Definition

Economic order quantity or EOQ model is the equation that helps compute order quantity of inventory accompanied by the minimum total holding and ordering costs.

## Formula

The economic order quantity is derived from the total cost equation,

 TC = p × D + D × K + H × Q Q 2

where p is a unit price, D is annual demand quantity, K is ordering cost, H is holding cost per unit, and Q is order quantity.

To find order quantity with the minimum total cost, we should calculate the derivative of the total cost function with respect to variable “Q” and set it equal to zero assuming that other variables are constant.

 0 = - D × K + H Q2 2
 Q2 = 2 × D × K H

Solving the equation with respect to variable “Q” will give the economic order quantity.

## Assumptions of EOQ model

If the economic order quantity model is applied, the following assumptions should be met:

1. The rate of demand is constant, and total demand is known in advance.
2. The ordering cost is constant.
3. The unit price of inventory is constant, i.e., no discount is applied depending on order quantity.
4. Delivery time is constant.
5. Replacement of defective units is instantaneous.
6. There is no safety stock level, i.e., the minimum stock level is zero.
7. Restocking is made by the whole batch.

## Holding cost vs. ordering cost

The holding or carrying cost is the total cost for keeping and maintaining inventories in storage. Common examples are a rent fee for the storage space, depreciation, labor cost to operate storage, materials, equipment and its maintenance cost, shrinkage of stock, security expense, insurance, cost of capital, and other direct costs.

Ordering cost refers to the cost related to shipping and handling a new order, e.g., communication and transportation cost, and insurance. Please note that ordering cost doesn’t include cost of goods!

Holding cost and ordering cost move in opposite directions. To reduce ordering cost per unit, we should increase order quantity, but such a scenario leads to an increase in holding cost; in turn, reducing holding costs requires a smaller order quantity, which leads to an increase in ordering cost. This relationship is shown in the graph below.

## Economic order quantity calculation example

A company manufacturing building materials has an annual demand in concrete of 150,000 tons. The price is \$425 per ton, the ordering cost is \$3,750, and the annual holding cost per ton is \$48.25.

Let’s put all the data available in the formula above.

Thus, the economic order quantity of 4,829 tons provides the minimum total holding and ordering cost. To prove this, we calculate the total cost for EOQ and order quantity of 4,500 tons and 5,500 tons using the total cost equation above.

 TC = \$425 × 150,000 + 150,000 × \$3,750 + \$48.25 × 4,829 = \$63,982,983 4,829 2
 TC = \$425 × 150,000 + 150,000 × \$3,750 + \$48.25 × 4,500 = \$63,983,563 4,500 2
 TC = \$425 × 150,000 + 150,000 × \$3,750 + \$48.25 × 5,500 = \$63,984,960 5,500 2

As we can see, the EOQ model is always the best solution.

## Graph

If basic assumptions of the model are met, the graph of inventory consumption and restocking looks as follows:

The maximum stock balance equals the economic order quantity, and it is consuming at a constant rate until reaching zero. At this time, restocking is made by the whole batch.