The Economic Order Quantity (EOQ) is the number of units that a company should add to inventory with each order to minimize the total costs of inventory—such as holding costs, order costs, and shortage costs. The EOQ is used as part of a continuous review inventory system in which the level of inventory is monitored at all times and a fixed quantity is ordered each time the inventory level reaches a specific reorder point. The EOQ provides a model for calculating the appropriate reorder point and the optimal reorder quantity to ensure the instantaneous replenishment of inventory with no shortages. It can be a valuable tool for small business owners who need to make decisions about how much inventory to keep on hand, how many items to order each time, and how often to reorder to incur the lowest possible costs.

The EOQ model assumes that demand is constant, and that inventory is depleted at a fixed rate until it reaches zero. At that point, a specific number of items arrive to return the inventory to its beginning level. Since the model assumes instantaneous replenishment, there are no inventory shortages or associated costs. Therefore, the cost of inventory under the EOQ model involves a tradeoff between inventory holding costs (the cost of storage, as well as the cost of tying up capital in inventory rather than investing it or using it for other purposes) and order costs (any fees associated with placing orders, such as delivery charges). Ordering a large amount at one time will increase a small business's holding costs, while making more frequent orders of fewer items will reduce holding costs but increase order costs. The EOQ model finds the quantity that minimizes the sum of these costs.

The basic EOQ relationship is shown below. Let us look at it assuming we have a painter using 3,500 gallons of paint per year, paying \$5 a gallon, a \$15 fixed charge every time he/she orders, and an inventory cost per gallon held averaging \$3 per gallon per year.

The relationship is TC = PD + HQ/2 + SD/Q '¦ where

• TC is the total annual inventory cost—to be calculated.
• P is the price per unit paid—assume \$5 per unit.
• D is the total number of units purchased in a year—assume 3,500 units.
• H is the holding cost per unit per year—assume \$3 per unit per annum.
• Q is the quantity ordered each time an order is placed—initially assume 350 gallons per order.
• S is the fixed cost of each order—assume \$15 per order.

Calculating TC with these values, we get a total inventory cost of \$18,175 for the year. Notice that the main variable in this equation is the quantity ordered, Q. The painter might decide to purchase a smaller quantity. If he or she does so, more orders will mean more fixed order expenses (represented by S) because more orders are handles—but lower holding charges (represented by H): less room will be required to hold the paint and less money tied up in the paint. Assuming the painter buys 200 gallons at a time instead of 350, the TC will drop to \$18,063 a year for a savings of \$112 a year. Encouraged by this, the painter lowers his/her purchases to 150 at a time. But now the results are unfavorable. Total costs are now \$18,075. Where is the optimal purchase quantity to be found?

The EOQ formula produces the answer. The ideal order quantity comes about when the two parts of the main relationship (shown above)—"HQ/2" and the "SD/Q"—are equal. We can calculate the order quantity as follows: Multiply total units by the fixed ordering costs (3,500 Ã— \$15) and get 52,500; multiply that number by 2 and get 105,000. Divide that number by the holding cost (\$3) and get 35,000. Take the square root of that and get 187. That number is then Q.

In the next step, HQ/2 translates to 281, and SD/Q also comes to 281. Using 187 for Q in the main relationship, we get a total annual inventory cost of \$18,061, the lowest cost possible with the unit and pricing factors shown in the example above.

Thus EOQ is defined by the formula: EOQ = square root of 2DS/H. The number we get, 187 in this case, divided into 3,500 units, suggests that the painter should purchase paint 19 times in the year, buying 187 gallons at a time.

The EOQ will sometimes change as a result of quantity discounts offered by some suppliers as an incentive to customers who place larger orders. For example, a certain supplier may charge \$20 per unit on orders of less than 100 units and only \$18 per unit on orders over 100 units. To determine whether it makes sense to take advantage of a quantity discount when reordering inventory, a small business owner must compute the EOQ using the formula (Q = the square root of 2DS/H), compute the total cost of inventory for the EOQ and for all price break points above it, and then select the order quantity that provides the minimum total cost.

For example, say that the painter can order 200 gallons or more for \$4.75 per gallon, with all other factors in the computation remaining the same. He must compare the total costs of taking this approach to the total costs under the EOQ. Using the total cost formula outlined above, the painter would find TC = PD + HQ/2 + SD/Q = (5 Ã— 3,500) + (3 Ã— 187)/2 + (15 Ã— 3,500)/187 = \$18,061 for the EOQ. Ordering the higher quantity and receiving the price discount would yield a total cost of (4.75 Ã— 3,500) + (3 Ã— 200)/2 + (15 Ã— 3,500)/200 = \$17,187. In other words, the painter can save \$875 per year by taking advantage of the price break and making 17.5 orders per year of 200 units each.

EOQ calculations are rarely as simple as this example shows. Here the intent is to explain the main principle of the formula. The small business with a large and frequently turning inventory may be well served by looking around for inventory software which applies the EOQ concept more complexly to real-world situations to help purchasing decisions more dynamically.

## BIBLIOGRAPHY

"Accounting Software." Financial Executive. October 2002.

Balakrishnan, Antaram, Michael S. Pangburn, and Euthemia Stavrulaki. "Stack Them High, Let 'Em Fly." Management Science. May 2004.

Khouja, Moutaz and Sungjune Park. "Optimal Lot Sizing Under Continuous Price Decrease." Omega. December 2003.

Piasecki, Dave. "Optimizing Economic Order Quantity." IIE Solutions. January 2001.

Wang, Kung-Jeng, Hui-Ming Wee, Shin-Feng Gao, and Shen-Lian Chung. "Production and Inventory Control with Chaotic Demands." Omega. April 2005.

Woolsey, Robert E.D. and Ruth Maurer. Inventory Control (For People Who Really Have to Do It). Lionheart Publishing, March 2001.