# Economic Order Quantity Models

In a continuous, or fixed-order-quantity, system when inventory reaches a specific level, referred to as the reorder point, a fixed amount is ordered. The most widely used and traditional means for determining how much to order in a continuous system is the economic order quantity (EOQ) model, also referred to as the economic lotsize model. The earliest published derivation of the basic EOQ model formula in 1915 is credited to Ford Harris, an employee at Westinghouse.

The function of the EOQ model is to determine the optimal order size that minimizes total inventory costs. There are several variations of the EOQ model, depending on the assumptions made about the inventory system. We will describe two model versions, including the basic EOQ model and the EOQ model with noninstantaneous receipt.

## The Basic EOQ Model

The basic EOQ model is a formula for determining the optimal order size that minimizes the sum of carrying costs and ordering costs. The model formula is derived under a set of simplifying and restrictive assumptions, as follows:

• Demand is known with certainty and is constant over time.
• No shortages are allowed.
• Lead time for the receipt of orders is constant.
• The order quantity is received all at once.

These basic model assumptions are reflected in Figure 12.1, which describes the continuous-inventory order cycle system inherent in the EOQ model. An order quantity, Q, is received and is used up over time at a constant rate. When the inventory level decreases to the recorder point, R, a new order is placed; a period of time, referred to as the lead time, is required for delivery. The order is received all at once just at the moment when demand depletes the entire stock of inventory--the inventory level reaches 0--so there will be no shortages. This cycle is repeated continuously for the same order quantity, reorder point, and lead time.

As we mentioned, the economic order quantity is the order size that minimizes the sum of carrying costs and ordering costs. These two costs react inversely to each other. As the order size increases, fewer orders are required, causing the ordering cost to decline, whereas the average amount of inventory on hand will increase, resulting in an increase in carrying costs. Thus, in effect, the optimal order quantity represents a compromise between these two inversely related costs.

The total annual ordering cost is computed by multiplying the cost per order, designated as Co, times the number of orders per year. Since annual demand, D, is assumed to be known and to be constant, the number of orders will be D/Q, where Q is the order size and

The only variable in this equation is Q; both Co and D are constant parameters. Thus, the relative magnitude of the ordering cost is dependent upon the order size.

Total annual carrying cost is computed by multiplying the annual per-unit carrying cost, designated as Cc, times the average inventory level, determined by dividing the order size, Q, by 2: Q/2;

The total annual inventory cost is the sum of the ordering and carrying costs:

The graph in Figure 12.2 shows the inverse relationship between ordering cost and carrying cost, resulting in a convex total cost curve.

The optimal order quantity occurs at the point in Figure 12.2 where the total cost curve is at a minimum, which coincides exactly with the point where the carrying cost curve intersects the ordering cost curve. This enables us to determine the optimal value of Q by equating the two cost functions and solving for Q:

Alternatively, the optimal value of Q can be determined by differentiating the total cost curve with respect to Q, setting the resulting function equal to zero (the slope at the minimum point on the total cost curve), and solving for Q:

The total minimum cost is determined by substituting the value for the optimal order size, Qopt, into the total cost equation:

 EXAMPLE12.2 The Economic Order Quantity

The I-75 Carpet Discount Store in North Georgia stocks carpet in its warehouse and sells it through an adjoining showroom. The store keeps several brands and styles of carpet in stock; however, its biggest seller is Super Shag carpet. The store wants to determine the optimal order size and total inventory cost for this brand of carpet given an estimated annual demand of 10,000 yards of carpet, an annual carrying cost of \$0.75 per yard, and an ordering cost of \$150. The store would also like to know the number of orders that will be made annually and the time between orders (i.e., the order cycle) given that the store is open every day except Sunday, Thanksgiving Day, and Christmas Day (which is not on a Sunday).

SOLUTION:

Cc = \$0.75 per yard

Co = \$150

D = 10,000 yards

The optimal order size is

The total annual inventory cost is determined by substituting Qopt into the total cost formula:

The number of orders per year is computed as follows:

Given that the store is open 311 days annually (365 days minus 52 Sundays, Thanksgiving, and Christmas), the order cycle is

The optimal order quantity, determined in this example, and in general, is an approximate value, since it is based on estimates of carrying and ordering costs as well as uncertain demand (although all of these parameters are treated as known, certain values in the EOQ model). In practice it is acceptable to round the Q values off to the nearest whole number. The precision of a decimal place is generally not necessary. In addition, because the optimal order quantity is computed from a square root, errors or variations in the cost parameters and demand tend to be dampened. For instance, in Example 12.2, if the order cost had actually been 30 percent higher, or \$200, the resulting optimal order size would have varied only by a little under 10 percent (i.e., 2,190 yards instead of 2,000 yards). Variations in both inventory costs will tend to offset each other, since they have an inverse relationship. As a result, the EOQ model is relatively resilient to errors in the cost estimates and demand, or is robust, which has tended to enhance its popularity.

## The EOQ Model with Noninstantaneous Receipt

A variation of the basic EOQ model is the noninstantaneous receipt model, also referred to as the gradual usage and production lot-size model. In this EOQ model the assumption that orders are received all at once is relaxed. The order quantity is received gradually over time, and the inventory level is depleted at the same time it is being replenished. This situation is most commonly found when the inventory user is also the producer, as in a manufacturing operation where a part is produced to use in a larger assembly. This situation also can occur when orders are delivered gradually over time or when the retailer is also the producer.

The noninstantaneous receipt model is shown graphically in Figure 12.3. The inventory level is gradually replenished as an order is received. In the basic EOQ model, average inventory was half the maximum inventory level, or Q/2, but in this model variation, the maximum inventory level is not simply Q; it is an amount somewhat lower than Q, adjusted for the fact the order quantity is depleted during the order receipt period.

In order to determine the average inventory level, we define the following parameters unique to this model:

p = daily rate at which the order is received over time, also known as the production rate

d = the daily rate at which inventory is demanded

The demand rate cannot exceed the production rate, since we are still assuming that no shortages are possible, and, if d = p, there is no order size, since items are used as fast as they are produced. For this model the production rate must exceed the demand rate, or p > d.

Observing Figure 12.3, the time required to receive an order is the order quantity divided by the rate at which the order is received, or Q/p. For example, if the order size is 100 units and the production rate, p, is 20 units per day, the order will be received in 5 days. The amount of inventory that will be depleted or used up during this time period is determined by multiplying by the demand rate: (Q/p)d. For example, if it takes 5 days to receive the order and during this time inventory is depleted at the rate of 2 units per day, then 10 units are used. As a result, the maximum amount of inventory on hand is the order size minus the amount depleted during the receipt period, computed as

Since this is the maximum inventory level, the average inventory level is determined by dividing this amount by 2:

The total carrying cost using this function for average inventory is

Thus the total annual inventory cost is determined according to the following formula:

Solving this function for the optimal value Q,

 EXAMPLE12.3 The EOQ Model with Noninstantaneous Receipt

Assume that the I-75 Outlet Store has its own manufacturing facility in which it produces Super Shag carpet. The ordering cost, Co, is the cost of setting up the production process to make Super Shag carpet. Recall Cc = \$0.75 per yard and D = 10,000 yards per year. The manufacturing facility operates the same days the store is open (i.e., 311 days) and produces 150 yards of the carpet per day. Determine the optimal order size, total inventory cost, the length of time to receive an order, the number of orders per year, and the maximum inventory level.

SOLUTION:

The optimal order size is determined as follows:

This value is substituted into the following formula to determine total minimum annual inventory cost:

The length of time to receive an order for this type of manufacturing operation is commonly called the length of the production run.

The number of orders per year is actually the number of production runs that will be made:

Finally, the maximum inventory level is

## Computer Solution of EOQ Models with Excel and Excel OM

EOQ analysis can be done with Excel and Excel OM (as well as with POM for Windows). The Excel screen for the noninstantaneous receipt model of Example 12.3 is shown in Figure 12.2. The formula for the optimal value of Q is contained in cell D10 as shown on the formula bar at the top of the screen.

Excel OM also includes a set of spreadsheet macros for inventory that includes EOQ analysis. After the Excel OM menu is accessed by clicking on "OM" on the menu bar at the top of the spreadsheet, click on "Inventory." Figure 12.3 shows the Excel OM spreadsheet for the noninstantaneous receipt model in Example 12.3.

12-8. Explain how the order quantity is determined using the basic EOQ model.

12-9. What are the assumptions of the basic EOQ model and to what extent do they limit the usefulness of the model?

12-10. How are the reorder point and lead time related in inventory analysis?

12-11. Describe how the noninstantaneous receipt model differs from the basic EOQ model.

12-12. In the noninstantaneous-receipt EOQ model, what would be the effect of the production rate becoming increasingly large as the demand rate became increasingly small, until the ratio d/p was negligible?