The order quantity that minimizes annual holding and ordering costs for an item.

to plan, the firm will run out of units just as the next order arrives. Finally, because the inventory

*Q *level in this model goes from *Q* to 0 over and over again, the average inventory level is 2 .

**The Economic Order Quantity ( EOQ)**

How do managers of a continuous review system choose the order quantity (*Q*)? Is there a “best” order quantity, and if so, how do holding costs (*H*) and ordering costs (*S*) affect it? To understand the role of holding and ordering costs in a continuous review system, let’s see what happens if the order quantity is sliced in half, to *Q* as shown in Figure 11.8. The result: With quantity *Q* the manager ends up ordering twice as often, which doubles the company’s ordering costs. On the other hand, cutting the order quantity in half also halves the average inventory level, which low-ers holding costs.

The relationship between holding costs and ordering costs can be seen in the following equation:

Total holding and ordering cost for the year = total yearly holding cost | ||||||

+ total yearly ordering cost | ||||||

Q | D | |||||

= a | bH + | a | bS | (11.4) | ||

2 | Q |

Yearly holding cost is calculated by taking the average inventory level (*Q*/2) and multiply-ing it by the per-unit holding cost. Yearly ordering cost is calculated by calculating the number of times we order per year (*D*/*Q*) and multiplying this by the fixed ordering cost.

As Equation (11.4) suggests, there is a trade-off between yearly holding costs and ordering costs. Reducing the order quantity, *Q,* will decrease holding costs, but force the organization to order more often. Conversely, increasing *Q* will reduce the number of times an order must be placed, but result in higher average inventory levels.

Figure 11.9 shows graphically how yearly holding and ordering costs react as the order quantity, *Q*, varies. In addition to showing the cost curves for yearly holding costs and yearly ordering costs, Figure 11.9 includes a total cost curve that combines these two. If you look closely, you can see that the lowest point on the total cost curve also happens to be where yearly holding costs equal yearly ordering costs.

Figure 11.9 illustrates the **economic order quantity ( EOQ)** , the particular order quantity (

Q | D | |||||||

a | bH = | a | bS | |||||

2 | Q | |||||||

Q2 = | 2DS | |||||||

H | ||||||||

Q = | 2DS | = EOQ | (11.5) | |||||

H |

where:

*Q *=* *order quantity

*H *=* *annual holding cost per unit* D *=* *annual demand

*S *=* *ordering cost

**Figure 11.9**

**The Relationships among Yearly Holding Costs, Yearly Ordering Costs, and the Order Quantity, Q**

**EXAMPLE 11.2**

**Calculating the EOQ at**

**Boyer’s Department**

**Store**

**CHAPTER 11** • Managing Inventory throughout the Supply Chain **337**

Total | |||

Cost | (Q2 | (H | |

(QD | (S | ||

Order quantity (*Q*)

As Figure 11.9 shows, order quantities that are higher than the *EOQ* will result in annual holding costs that are higher than ordering costs. Conversely, order quantities that are lower than the *EOQ* will result in annual ordering costs that are higher than holding costs.

You are in charge of ordering items for Boyer’s Department Store, located in Seattle. For one of the products Boyer’s carries, the Hudson Valley Model Y ceiling fan, you have the following information:

Annual demand (*D*) = 4,000 fans a year

Annual holding cost (*H*) = +15 per fan

Ordering cost (*S*) = +50 per order

Your predecessor ordered fans four times a year, in quantities (*Q*) of 1,000. The result-ing annual holding and ordering costs were:

Holding costs for the year + ordering costs for the year

· (1,000 2)+15 + (4,000 1,000)+50

· +7,500 + +200 = +7,700

Because holding costs are much higher than ordering costs, we know that the *EOQ* must be much lower than 1,000 fans. In fact:

EOQ = | 2*4, 000*+50 | , which rounds to 163 fans per order |

+15 |

The number 163 seems strange, so let’s check to see if it results in lower annual costs:

Holding costs + ordering costs

· (163 2)+15 + (4,000 163)+50

· +1,222.50 + +1,226.99 = +2,449.49

Notice that holding costs and ordering costs are essentially equal, as we would expect. More important, *simply by ordering the right quantity*, you could reduce annual holding and ordering costs for this item by

+7,700 – +2,449 = +5,251

Now suppose Boyer’s carries 250 other products with cost and demand structures sim-ilar to that of the Hudson Valley Model Y ceiling fan. In that case, you might be able to save 250*+5,251 = +1,312,750 per year just by ordering the right quantities!

Of course, the *EOQ* has some limitations. Holding costs (*H*) and ordering costs (*S*) cannot always be estimated precisely, so managers may not always be able to calculate the true *EOQ*. However, as Figure 11.9 suggests, total holding and ordering costs are relatively flat over a wide range around the *EOQ*. So order quantities can be off a little and still yield total costs that are close to the minimum.

A more valid criticism of the *EOQ* is that it does not take into account volume discounts, which can be particularly important if suppliers offer steep discounts to encourage customers to order in large quantities. Later in the chapter, we examine how volume discounts affect the order quantity decision.

** ** **PART IV** • Planning and Controlling Operations and Supply Chains

Other factors that limit the application of the *EOQ* model include ordering costs that are not always fixed and demand rates that vary throughout the year. However, the *EOQ* is a good starting point for understanding the impact of order quantities on inventory-related costs.

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