The economic order quantity calculation often abbreviated to EOQ determines the unit quantity a business should order to minimize its inventory cost. In this context an order can refer to a purchase order where goods are purchased from a supplier or a production order where a business manufactures in different batch sizes.

## Inventory Costs

Inventory costs can be separated into three types.

**Product cost**– the cost of purchasing or manufacturing the product itself.**Ordering cost**– the cost involved in placing purchase orders or setting up production orders.**Holding cost**– the cost involved in holding inventory.

For the purposes of economic order quantity calculations it is assumed that the **unit** product cost remains constant irrespective of the order quantity and the **total** product cost depends only on demand for the product.

The ordering and holding costs however will vary depending on the size of the order. As the size of an order increases the frequency of ordering and therefore the total ordering costs decrease. In contrast the same increase in order size results in an increase in the average level of inventory held and a corresponding increase in the total holding costs.

It can be seen that as the size of the order changes the total ordering and holding costs move in opposite directions; the purpose of the economic order quantity calculation is to determine the order quantity which minimizes the total of these two costs.

## Ordering Cost

The total ordering cost depends on the cost of each order and the number of orders placed by the business during the period.

Ordering cost = Cost of each order x Number of orders

The cost of each order is denoted by the symbol K and is assumed to be fixed and represents all the costs associated with either placing a purchase order with a supplier such as administrative, transport, receiving and inspecting costs or, in the case of a manufacturing business, the production order set up costs necessary to manufacture the order quantity.

To calculate the number of orders we can simply divide the demand for the period by the quantity of units in each order. If the demand is denoted by D and the quantity is denoted by Q then the number of orders in the period is given by D/Q.

The inventory ordering cost formula can now be written as follows.

Ordering cost = K x D/Q

## Holding Cost

The total holding cost depends on the average number of units of inventory held by the business and the holding cost for each unit.

Holding cost = Cost of holding each unit x Number of units held

The cost of holding each unit is denoted by the symbol h and includes costs such as interest on finance used to fund the inventory working capital, warehousing, handling, administrative, insurance, and shrinkage, deterioration and obsolescence costs.

To calculate the number of units held it is assumed in the economic order quantity model that the inventory is delivered as soon as an order is placed. The inventory level of the product is therefore allowed to fall to zero before ordering the quantity Q. The average inventory level is therefore Q/2 units.

The inventory holding cost formula can now be written as follows.

Holding cost = h x Q/2

For simplicity, the holding cost h is often expressed as a percentage of the unit product cost. So for example if the unit product cost is 75 and the holding cost percentage is 40% then the inventory holding cost h is calculated as follows

h = Unit product cost x Holding cost % h = 75 x 40% = 30

## Total Cost

By combining the three costs the total inventory cost can be summarized as follows.

Inventory cost = Product cost + Ordering cost + Holding cost

The product cost is simply the unit cost of each product (c) multiplied by the unit demand for the product (D).

The inventory cost formula can now be stated as follows.

Inventory cost = c x D + K x D/Q + h x Q/2

The following should be noted.

- The order quantity Q had no effect on the product cost.
- The ordering cost
**decreases**as the quantity Q**increases**as fewer orders are placed - The holding cost
**increases**as the quantity Q**increases**as the average inventory held increases.

It can be seen intuitively that as the quantity Q changes the order costs and the holding costs move in the opposite direction and that at some value of Q the total costs will therefore be at a minimum, this value is our economic order quantity EOQ.

## Economic Order Quantity Formula

It can be shown mathematically by differentiating the cost formula above that the inventory costs are at a minimum when the ordering cost is equal to the holding cost.

Solving for Q when these costs are equal gives the economic order quantity formula as follows.

Ordering cost = Holding cost K x D/Q = h x Q/2 Q = Economic order quantity = (2 x D x K/h)^{1/2}

The economic order quantity is the value of Q which minimizes the total inventory cost as demonstrated in the diagram below.

## Economic Order Quantity Example

Suppose the following information is available for a component a business uses in its manufacturing process.

Annual demand D = 912

Unit cost c = 75

Order cost K = 95

Holding cost % = 40% of unit cost

Holding cost h = Unit cost x Holding cost %

Holding cost h = 75 x 40% = 30

The economic order quantity is calculated as follows.

Economic order quantity = (2 x D x K/h)^{1/2}Economic order quantity = (2 x 912 x 95/30)^{1/2}Economic order quantity = 76

The economic order quantity is 76 units. To minimize inventory costs the business should place purchase orders with the supplier or manufacture in production runs of 76 units.

At this quantity the total inventory cost is calculated as follows.

Inventory cost = c x D + K x D/Q + h x Q/2 Inventory cost = 75 x 912 + 95 x 912/76 + 30 x 76/2 Inventory cost = 68,400 + 1,140 + 1,140 = 70,680

It should be noted that at the economic order quantity the ordering cost is the same as the holding cost (1,140).

Our free Excel economic order quantity calculator is available to help carry out the calculations described in this post.

## Economic Order Quantity and Financial Projections

The management of inventory levels is a balancing act. Too little inventory can result in lost sales and delays for customers, whereas excessive inventory leads to an increase in the working capital requirements and can cause a drain on the cash resources of a business.

The economic order quantity equation is a management tool which can be used to calculate the optimal level of inventory to minimize total costs. By calculating the EOQ a business can arrive at an optimum inventory cost and use this as a guide when adjusting the inventory days assumption in the financial projections template.

## About the Author

Chartered accountant Michael Brown is the founder and CEO of Plan Projections. He has worked as an accountant and consultant for more than 25 years and has built financial models for all types of industries. He has been the CFO or controller of both small and medium sized companies and has run small businesses of his own. He has been a manager and an auditor with Deloitte, a big 4 accountancy firm, and holds a BSc from Loughborough University.