# Economic Order Quantity (EOQ)

Economic order quantity (EOQ) is the the order size which minimizes the sum of carrying costs and ordering costs of a company’s inventories.

The two most significant inventory management costs are ordering costs and carrying costs. Ordering costs are costs incurred on placing and receiving a new shipment of inventories. These include communication costs, transportation costs, transit insurance costs, inspection costs, accounting costs, etc. Carrying costs represent costs incurred on holding inventory in hand. These include opportunity cost of money held-up in inventories, storage costs such as warehouse rent, insurance, spoilage costs, etc.

Ordering costs and carrying costs are opposite in nature. To minimize its inventory carrying costs, a company must place small orders. But small order size means that the company must place more orders which increases its total ordering costs. Similarly, if a company wants to cut its ordering costs, it must reduce the number of orders placed which is possible only when order size is large. But increase in order size means that average inventory balance on hand will be high which increases total carrying costs for the period.

Instead of focusing on ordering costs or carrying costs individually, a company should attempt to reduce the sum of these costs. The economic order quantity (EOQ) model does just that.

## Formula

EOQ can be determined using the following equation:

$$\text{EOQ}\ =\sqrt{\frac{\text{2}\times \text{D}\times \text{O}}{\text{C}}}$$

Where D is the annual demand (in units), O is the cost per order and C is the annual carrying cost per unit.

### Understanding the Math

The EOQ formula can be derived as follows:

STEP 1: Total inventory costs are the sum of ordering costs and carrying costs:

$$\text{Total Inventory Costs}\ =\ \text{Ordering Costs}\ +\ \text{Carrying Costs}$$

STEP 2: The number of orders N in a period would equal annual demand D divided by the order size Q and the total ordering cost would be the product of cost per order O and number of orders N. This can be written as follows:

$$\text{Ordering Costs}\ =\text{O}\times \text{N}$$

$$\text{Ordering Costs}\ =\text{O}\times\frac{\text{D}}{\text{Q}}$$

STEP 3: The carrying costs depend on the average inventory on hand. We assume that when an order is received, total quantity is Q which runs down to 0 by the time the next order is received. Hence, average inventory balance equals quantity ordered divided by 2. Total carrying costs equal the product of carrying cost per unit C and average inventory balance:

$$\text{Carrying}\ \ \text{Costs}\ =\text{C}\times\frac{\text{Q}}{\text{2}}$$

STEP 4: Total inventory costs can be written as follows:

$$\text{TC}\ =\text{O}\times\frac{\text{D}}{\text{Q}}+\ \text{C}\times\frac{\text{Q}}{\text{2}}$$

Just a bit of mathematical manipulation:

$$\text{TC}\ =\text{O}\times \text{D}\times \text{Q}^{-\text{1}}+\ \frac{\text{C}}{\text{2}}\times \text{Q}$$

STEP 5: Total cost is minimized when the rate of change of total cost with respect to order size is 0. The rate of change of the above function equals its slope which in turn equals its first derivative.

Let’s obtain the first derivative of the total cost function with respect to Q:

$$\frac{\text{d}(\text{TC})}{\text{d}(\text{Q})}\ =\text{OD}\times-\text{1}\ \ ^{-\text{2}}+\frac{\text{C}}{\text{2}}\times\text{1}$$

STEP 4: Setting the rate of change equal to 0:

$$\frac{\text{d}(\text{TC})}{\text{d}(\text{Q})}\ =\text{0}$$

$$\text{0}\ =-\text{OD}\times \text{Q}^{-\text{2}}+\frac{\text{C}}{\text{2}}$$

We need to isolate the quantity Q expression on the left-hand side:

$$\text{OD}\times \text{Q}^{-\text{2}}=\frac{\text{C}}{\text{2}}$$

$$\text{Q}^{-\text{2}}=\frac{\text{C}}{\text{2}\times \text{O}\times \text{D}}$$

Taking inverse of both sides of the above equation, we get:

$$\text{Q}^\text{2}=\frac{\text{2}\times \text{O}\times \text{D}}{\text{C}}$$

Taking square root of both sides give us a value of Q at which the sum of carrying costs and ordering costs are minimized:

$$\text{Q}=\sqrt{\frac{\text{2}\times \text{O}\times \text{D}}{\text{C}}}$$

Bingo! We have the EOQ formula.

## Example

ABC Ltd. is engaged in sale of footballs. Its cost per order is $400 and its carrying cost per unit per annum is$10. The annual demand for the company’s product is 20,000 units.

Calculate the economic order quantity i.e. the optimal order size, total orders required during a year, total carrying cost and total ordering cost for the year.

### Solution

Based on the EOQ model, the company should set its order size at 1,265 units per order.

$$\text{EOQ}\ =\sqrt{\frac{\text{2}\times\text{20,000}\times\text{\400}}{\text{\10}}}=\text{1,265 units}$$

Annual demand is 20,000 units so the company will have to place 16 orders (= annual demand of 20,000 divided by order size of 1,265). Total ordering cost is hence $6,400 ($400 multiplied by 16).

Average inventory held is 632.5 (=(0+1,265)/2) which means total annual carrying costs would be $6,325 (i.e. 632.5 ×$10).

Annual ordering costs of $6,400 and annual carrying costs of$6,325 translates to total annual inventories management cost of $12,649. The following table shows that an order size 1,265 units is indeed optimal because if we change the orders size, total costs increase. Order Size Carrying Costs Ordering Costs Total Cost Calculation 1200 6,000.00 6,666.67 12,666.67 1,200/2 ×$10 + 20,000/1,200 × $400 1265 6,325.00 6,324.11 12,649.11 1,265/2 ×$10 + 20,000/1,265 × $400 1300 6,500.00 6,153.85 12,653.85 1,300/2 ×$10 + 20,000/1,300 × \$400