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254 Part III • Implementation and Controlling Results

APPENDIX 7-A

Economic Order Quantity

As noted in the chapter, in addition to having to pay for inventory when, or shortly after, it is acquired, there are other costs related to inventory. We must have physical space to store it, we may need to pay to insure it, and there are costs related to placing an order and having it shipped. A method called the economic order quantity (EOQ) considers all of these factors in calculating the inventory level at which additional inventory should be ordered.

The more inventory ordered at one time, the sooner we pay for inventory and the greater the costs for things such as inventory storage. These are called canying or holding costs. However, if we keep rela­ tively little inventory on hand, to keep carrying costs low, we will have to order inventory more often. That drives ordering costs up. EOQ balances these two fac­ tors to find the optimal amount to order.

There are two categories of carrying costs: capital costs and out-of-pocket costs. The capital cost is the cost related to having paid for inventory, as opposed to using those resources for alternative uses. At a minimum, this is the forgone interest that could have been earned on the money paid for inventory. Out-of-pocket costs are other costs related to holding inventory, including rent on space where inventory is kept, insurance and taxes on the value of inventory, the cost of annual inventory counts, the losses due to obsolescence and date-related expirations, and the costs of damage, loss, and theft.

Ordering costs include the cost of having an employee spend time placing orders, the shipping and handling charges for the orders, and the cost of correcting errors when orders are placed. The more orders, the more errors.

There is an offsetting dynamic in inventory man­ agement. The more orders per year, the less inventory that needs to be on hand at any given time, and there­ fore the lower the carrying cost. However, the more orders per year, the greater the amount the organiza­ tion spends on placing orders, shipping and handling costs, and error correction. The total costs of inventory are the sum of the amount paid for inventory, plus the carrying costs, plus the ordering costs:

Total Inventory Cost = Purchase Cost + Carrying Cost + Ordering Cost

The goal of inventory management is to minimize this total without reducing the quality of services the orga­ nization provides.

We will use TC to stand for the total inventory cost, P to stand for the purchase cost per unit, CC for the total carrying cost, and OC for the total order­ ing cost. N will stand for the total number of units of inventory ordered for the year. Therefore,

TC = (P X N) + CC + OC (7.A.1)

We will let C stand for the annual cost to carry one unit of inventory. The annual total carrying cost, CC, is then equal to the carrying cost per unit, C, multiplied by the average number of units on hand. Assume that Q is the number of units of inventory ordered each time an order is placed. On average at any given time we will have Q -;- 2 units on hand. If we start with Q units and use them until there are O units left, ori average we will have half of Q units on hand. Carrying costs are determined using the average number of units of inventory on hand. The carrying costs will therefore be as follows:

CC= C X Q 2

CQ 2

(7.A.2)

That is, the carrying costs per year (CC) will be equal to the carrying costs of one unit (C), multiplied by the aver­ age number of units on hand at any given time (Q/2).9

A formula can also be developed for ordering costs. We will let O stand for the cost of making one order. The total ordering cost, OC, is the cost of mak­ ing an order, 0, times the number of orders per year. Recall that the total number of units needed for the year is N and Q is the number of units in each order. Then N/Q is the number of orders placed per year. The ordering costs are as follows:

N OC = 0 X -

Q

ON Q

(7.A.3)

9This becomes somewhat more complex if a safety stock is kept on hand at all times. In such a case the CC is equal to C multiplied by the sum of Q + 2 plus the safety stock. However, this is not needed for the EOQ calculation. Safety stocks will not affect the economic order quantity, since they are projected to be constantly on hand regardless of the frequency or size of orders.

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Chapter 7 • Managing Short-Term Resources and Obligations 255

That is, the total cost of placing all orders for the year (0C) is the cost of making one order (0) multiplied by the number of orders per year (N/Q). For instance, suppose that Meals for the Homeless buys 2,000 sacks of rice each year (N = 2,000). If it orders 200 sacks at a time (Q = 200), it would have to make 10 orders per year (N/Q = 2,000/200 = 10).

We now can calculate the purchase cost of the inventory, the carrying costs, and the order­ ing cost. Suppose that Meals pays $2 per sack for rice. Each time it places an order, it takes a paid clerk about $8.075 worth of time to process the order. The delivery cost is $1 per order. This $9.075 is the only ordering cost. Meals could earn 8 percent in­ terest on its money. Therefore, the capital part of the carrying cost is $.16 per sack per year (8% X $2 price = $ .16). Other canying costs are determined to be $2.84 per sack per year. Therefore, the total car­ rying costs are $3 per sack per year. W hat is the total cost of inventory, assuming that there are 10 orders per year?

TC = (P X N) + CC + OC (7.A.1)

The first part of the equation to be calculated is the purchase cost of the inventory:

P X N = $2 X 2,000 = $4,000

Next, we need to find the carrying cost:

CC= C x Q _ CQ

2 - 2

CC = $3 X 200

2 = $300

Finally, we need the ordering cost:

OC = 0 X N ON

Q

$9.075 X 2,000 = $90.75oc =

200

So the total costs are as follows:

(7.A.2)

(7.A.3)

TC= (P X N) + CC + OC (7.A.1) TC = $4,000 + $300 + $90 .75 = $4,390.75

However, it was arbitrarily decided that there would be 10 orders of 200 sacks each. The EOQ model is designed to determine the optimal number to

order at one time. The formula to determine the opti­ mal number to order at one time is as follows:

Q* = �2 �N (7.A.4)

where Q* is the optimal amount to order each time.

Q* = 2 X $9.075 X 2,000

$3 = 110

This result differs from the 200 sacks per order that we used earlier. If we use this result, how will it af­ fect total costs? The purchase cost will still be $4,000 . However, the canying costs and ordering costs will change:

CC= C x Q _ CQ

2 - -2-

cc = $3 X 110

2 = $165

Finally, we need the ordering cost:

N ON OC =OX-=­

Q Q

OC = $9.075 X 2,000

= $l65 110

So, the total costs are as follows:

(7.A.2)

(7.A.3)

TC = (P X N) + CC + OC (7.A.1) TC = $4,000 + $165 + $165 = $4,330

The new total cost of $4,330 represents a cost decrease of $60 .75. Relative to the total cost, this may not seem to be a great savings. However, if you put aside the purchase cost of the invento1y, the carrying and ordering costs have fallen from $390.75 to $330. This is more than a 15 percent savings. Across all inventory items for an organization, this could amount to a substantial dollar amount of savings.

It is not coincidental that the ordering cost equals the canying cost. The total cost is minimized at the point where these two costs are exactly the same!

It is important that EOQ calculations only include relevant costs. Carrying and ordering costs that are relevant are those that vary as a result of our EOQ decision. That is, if ordering more or less frequently will affect a cost, it is relevant and should be included in the calculation. For example,

256 Part Ill • Implementation and Controlling Results

ordering less frequently will likely increase capital costs related to interest. It would also likely affect shipping and handling, so these are relevant costs that belong in the calculation. By contrast, the cost of the purchasing department manager will probably not change with the number of orders. Therefore, none of that manager's salary should be included in the ordering costs.

The basic EOQ model, as presented here, involves making a number of assumptions that are often not true. For example, it involves assuming that any number of units can be purchased. In some cases, an item might only be sold in certain quantities, such as hundreds or dozens. Another assumption is that the price per unit does not change if we order differing numbers of units with each order. It is possible that we might get a quantity discount for large orders. Such a discount could offset some of the higher carrying cost related to large orders.

Another assumption is that we will use up our last unit of an item just when the next shipment arrives. A delay in processing, however, could cause inventory to arrive late, and we might run out of cer­ tain items. To avoid negative consequences of such stockouts, we might want to keep a safety stock on

hand. How large should that safety stock be? That will depend on how long it takes to get more invento1y if we statt to run out. It also depends on how serious the consequences of running out are. Is it life or death, or merely an inconvenience?

One of the greatest difficulties in employing EOQ is determining the carrying and ordering costs. In most cases, however, at least the major components of such costs-for example, the amount of labor needed to place an order--can be calculated. The purpose of this discussion of EOQ is to familiarize the reader with the basic concept of inventoty management. Many more sophisticated issues, such as those noted here, are addressed in more advanced books on managerial accounting and on operations management. Some of these are included in the list of readings at the end of the chapter.

Inventory models are a part of any efficient man­ agement operation that invests dollars in inventory. Public, health, and not-for-profit organizations have often considered their inventories to be of nominal value. However, the costs of ordering and carrying invento1y are sometimes surprisingly high, and use of a tool such as EOQ should at least be examined for potential savings.

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Key Terms from This Appendix

carrying costs of inventory. Capital costs and out-of pocket costs related to holding inventory. Capital cost represents the lost interest because money is tied up in inventory. Out-of-pocket costs include such expenses as insurance on the value of inventory, annual inspections, and obsolescence of inventory.

economic order quantity (EOQ). Approach to determine the balance between ordering costs and

carrying costs; optimal number of units of invento1y to be ordered each time an order is placed.

holding costs. See carrying costs of inventory.

ordering costs. Includes those costs associated with an order of inventory such as clerk time for preparation of a purchase order.

stockout costs. Costs incurred when an inventory item is not available but is needed.