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MANAGING INVENTORIES:
WHAT IS THE APPROPRIATE ORDER QUANTITY?

The management of inventories is an important task in nearly every type of organization,

from manufacturing firms to hospitals and restaurants. In many manufacturing businesses, in
fact, inventory is the single largest asset on the balance sheet. Inventory accounts for nearly 40
percent of the current assets of the typical manufacturing company and for 50 to 60 percent of
the current assets in wholesaling and retailing industries. Redesigning the elements of a firm’s
inventory management system is often the key aspect in improving a firm’s working-capital
position and its return on assets.

Fundamental questions come up in every inventory system—what is the appropriate

order quantity for replenishment of the items recently consumed? Should we order more than we
need in the near term in order to get a volume discount from this new supplier? Should we issue
a factory production order for several months’ supply of an item in order to spread large
machinery changeover and set-up costs across a large lot size? Should we be making more or
less than our historical standard batch size because the firm’s working capital costs have recently
fallen sharply? How does a change in the item’s manufacturing costs alter our order quantity
rules in our computer-based ordering system? Shouldn’t we be using an “economic order
quantity” lot size formulation for our ordering decisions? What are the relevant costs we should
be considering?

Ways to Categorize Inventory and Evaluate Inventory Planning Decisions

Before addressing the questions above, consider the categories used to distinguish
inventories in the firm’s financial records. The major accounting categories of inventory are:

1. Raw material: Components, subassemblies, or materials that have been purchased and
are waiting to be placed into production.
2. Work-in-process: Parts or products in various stages of completion throughout the
manufacturing operation, including raw material that has been released for initial
processing and partially processed units that have had labor and overhead added and are
being stored for use at a later time.

This note was prepared by Professors James R. Freeland and Robert D. Landel. Copyright © 2003 by the University
of Virginia Darden School Foundation, Charlottesville, VA. All rights reserved. To order copies, send an e-mail to
sales@dardenpublishing.com. No part of this publication may be reproduced, stored in a retrieval system, used in a
spreadsheet, or transmitted in any form or by any means—electronic, mechanical, photocopying, recording, or
otherwise—without the permission of the Darden School Foundation. Rev. 1/05.

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3. Finished goods: Completed goods that are being held for sale to customers.
4. Maintenance parts and supplies: Components used to maintain or repair processing
equipment.

The same physical good may be in a different category for different companies. For
example, steel sheeting may be a finished good for a steel manufacturer but is a semi-finished
good to a company that fabricates it into cabinets for consumer microwave ovens. In vertically
integrated companies, what may be a finished good for one division may be a raw material to
another division. This aspect explains why in the annual reports of many companies the balance
sheet does not separate out inventories into raw material, work-in-process, and finished goods.

Categorizing inventories according to the function they provide for the firm’s operations
helps a manager to evaluate inventory investment planning decisions using a costing approach.
Five major categories of function are:

1. Cycle-stock inventories: to allow for batching of purchased goods or manufactured

goods. Cycle stock represents the average inventory carried between reordering cycles.
For example, the firm may believe it is prudent to manufacture a particular item in a
batch size of 3000 units even when forecast usage is 300 units per week. Any number of
considerations could have driven this lot size decision of 3000 units. Another firm
manufacturing the same type of item with a similar weekly demand of 300 units could
choose to produce in a batch size of 600 units. The inventory that is carried between
order cycles is very different. How does one assess the better lot size decision made by
these two firms?
2. Pipeline inventories: to provide for transit of inventory. Pipeline stock is the inventory
moving from one point to another. Some firms seek to minimize the transit time and ship
with an air-express overnight mode. Another firm may be willing to experience a longer
transit time and ship by ground transportation. Which firm has made a better choice?
3. Safety-stock inventories: to protect against uncertainties in supply or demand. Because
of unanticipated fluctuations in demand, one firm may hold two extra pallets of finished
goods of an item, while a competitor with the same expected demand conditions could
chose to protect itself against demand uncertainties by holding four pallets of the item.
Which policy is better?
4. Seasonal inventories: to smooth a mismatch between demand and supply. Two firms
facing a similar back-to-school business high-volume demand season could choose very
different planning rules for meeting demand. Firm A might choose to produce at a
constant rate during the year, building seasonal inventory in anticipation of the large
demands in August. Another firm might be willing to alter its production work force
during the year to match the seasonal low to high swings in demand. Which planning
method is better?
5. Speculation inventories: to deal with special buying circumstances. Why do some firms
speculate and buy ahead while others do not venture into this area?

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Each of these five operational situations represents alternatives for dealing with the
planning and control of inventories. The choices made by a firm create functional inventories
that would be carried as inventory investments. Such investments may require out-of-pocket
costs to a firm. For example, typical components of inventory holding costs may include:

Capital costs: Interest on money tied up in the inventory;

Possession costs: Insurance, property taxes, obsolescence, spoilage, deterioration,

pilferage, warehouse labor, and information record keeping; and

Facility costs: Property taxes, insurance, rental fees, maintenance, equipment, and
labor.

While inventories do create inventory holding costs, there are circumstances where the
economic benefits are significantly greater than the carrying costs’ consequences. Volume
purchase discounts may be substantial. Large scrap losses and multiple hours of set-up expenses
associated with starting-up equipment can be averaged out over large batch sizes. Expensive
transportation may be cheaper than the costs of inventory tied up in the transport system.
Investments in better forecast information may reduce demand uncertainties and permit safety
stocks’ expenses.

Because inventories management decisions can often involve concrete tradeoffs among a
variety of cost factors, the “appropriate” decision is guided by a “cost analysis” of the various
consequences.

Exhibit 1 summarizes by function many of the tradeoffs in costs and the actions
management can take to improve the management of inventories in each of the five categories.
Breaking down the inventories in a business according to how many dollars of inventory are
being used for each function is often useful. Such a classification may, for example, assist one in
deciding what approaches might be most effective in determining how to meet the five functions
of inventories with the minimal dollar investment and lowest operational expenses.

Another useful way to categorize inventory is an ABC Classification. Each item in

inventory is classified as either an A part, a B part, or a C part, often by ranking the items based
on annual dollar sales. Thus, A items are considered the highest-valued items, while the C items
are the least valued. It is not unusual for 10 percent of the items that are classified as A to
account for 60 percent of the total-dollar value. The B items may account for 20 percent of the
items while being 30 percent of the total dollar value, and C items might account for 70 percent
of the items but only 10 percent of the dollar value. Thus, a business would be advised to pay
much more attention in terms of analysis and control to the A items than the others. This ABC
classification is sometimes referred to as a Pareto Analysis or the 80–20 Rule.

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Objectives in Managing Inventories

The purpose of this note is to address the planning that guides a firm’s decisions
regarding cycle-stock ordering activities as materials flow from suppliers through a firm’s
operations to customers. The objective is to develop and implement a management system that
can improve the cost effectiveness of the investments in inventory. This note addresses those
cost-element considerations that deal with the cycle-stock decisions. Other functional inventory
decisions are addressed in separate technical notes.

In managing inventories, there can be some competing objectives. Marketing would like
to have enough inventory to maximize customer service by always having products available for
customers. Finance looks to minimize the investment in inventory. Because the objectives can
be conflicting, operations managers must be cautious about pursuing coordination policies that
contribute to the overall success of the firm. The operations manager seeks actions that allow
multiple objectives to be approached. For example, in a manufacturing operation, if the time to
change over from production of one product to another can be reduced, it may be possible to
reduce the lot size, which reduces the investment in inventory. Smaller lot sizes could mean
shorter manufacturing lead-times, which would allow customer lead-times to be reduced and
that, in turn, should improve customer service. Finally, if change-over times are less, then more
time can be spent producing products rather than changing over the equipment, so operating
efficiency could be improved.

An Economic Approach to Determine Order Size

A large amount of information is reported in the literature of inventory management on

the development of rules for deciding how much to order (lot size) under various conditions.
Regardless of the particular situation, constructing a cost model of the problem that captures the
relevant characteristics of the situation shapes nearly all decision rules.

In some situations, mathematical procedures like differential calculus are applied to

obtain specific decision rules. The Economic Order Quantity rule (EOQ) presented in the
following pages is one such rule that is used in business situations. It is important to understand
the specific mathematical model used to develop the decision rules to avoid using the rules
incorrectly. Studying the mathematical model allows us to understand the assumptions that are
being used. We can then ask whether those assumptions are realistic in the specific situation of
interest. From a managerial perspective, the specific mathematical techniques used to determine
the decision rules are less important.

Ordering in batches or lots is often economically advantageous. This policy creates

cycle-stock inventory. Under the simplest assumption of level demand, the optimal rule is given
by the EOQ (Economic Order Quantity) formula, which is usually credited to Wilson because he
first developed the model around 1915.

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Choosing a lot size Q based on annual costs

The decision variable to be determined is the lot size quantity to be ordered (Q). The
important cost components are the costs associated with ordering and the costs associated with
carrying inventory. Typically, the costs of ordering are those costs that vary with the number of
orders placed and the costs associated with the quantity ordered. So let:

S = cost of ordering, independent of the size of the order size, and

C = cost per unit ordered.

If an item is being purchased, then the cost of ordering is calculated by adding up the
costs associated with preparing the order, processing the invoice, and receiving and inspecting
the item. If an item is being manufactured, then the cost of ordering is the cost incurred for
preparing the paperwork, setting up the machines, and any scrap created during the set-up
operation and testing activities.

Figure 1 shows the relationship between the ordering cost and the lot size Q. Having
large lot sizes can minimize ordering costs.

Each time an order of size Q is placed, the cost is represented by S + CQ. If R represents
the annual requirement for the item, then the Annual Costs for the ordering activity and the item
purchased (manufactured) is:

R R
Annual Costs = Number of Orders x (S + CQ) = (S + CQ) = S + CR
Q Q

Figure 1: Total Annual Costs versus Lot Size

Annual Cost Or Total Annual Cost (TAC)

(in dollars) de
r KC Q/2)
(R ing Cost (
S/ Co lding
Q) s t
v e nto ry Ho
In
Item Cost (CR)

Q*
Lot Size, Q

The costs of carrying (holding) inventory are typically modeled as the cost of carrying a
unit in inventory for a year times the average inventory level for the year. There are two
categories of inventory carrying cost; out-of-pocket expenses and opportunity costs. Out-of-

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pocket expenses include such items as the cost of storage space (rental or alternative-use value),
insurance costs, costs associated with obsolescence, spoilage, or theft, and taxes. Opportunity
costs are the costs of forgone opportunities for the money that is invested in inventory, i.e., the
money invested in inventory that could be used in other areas of the company to earn some
return.

Under the assumption of constant demand, the inventory level over time will appear as
the saw-toothed pattern in Figure 2. When an order of size Q arrives, the inventory is at the
maximum level (Q, the lot size) and then will drop to zero just before the next order is to arrive.
Thus, the average inventory level is Q/2. The average inventory is referred to as average cycle
stock, as it represents the inventory stock held between ordering cycles. The cost of carrying a
unit in inventory is typically modeled as per-unit cost of the item times some fraction that
represents the annual holding-cost charge.

Figure 2: Inventory Level versus Time

Inventory
Q Average Inventory Level
Level
2

Time

Let K = annual inventory carrying costs as a fraction of the unit cost. Thus, the annual
cost of holding inventory is:

Q
Annual Costs of Holding Inventory = KC
2

Recall the chart for the annual costs of holding inventory as a function of the lot size in
Figure 2. The inventory holding costs can be minimized by ordering in smaller Q lot sizes.

Thus, combining the several costs components—ordering, item annual cost, and
holding—the Total Annual Costs (TAC) as a function of the lot size Q can be written as:

R Q
TAC (Q) = S + CR + KC (1)
Q 2

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For any lot size (Q) quantity, the total annual cost can be calculated using equation (1).

We want a rule for Q that minimizes the total annual costs. It is apparent that the term
CR, the annual cost of the units, does not vary with the lot size Q in this case. (If there are price
discounts associated with purchasing in quantity, CR will vary.) Figure 1 shows the graph of
TAC versus Q.

Differential calculus can now be used to find the formula. Taking the derivative of TAC
with respect to Q and setting it equal to zero yields the minimum-cost solution:

d (TAC ) R KC
=0= - 2 S +
dQ Q 2

The economic order quantity is given by Q*. * 2 RS

Q=
KC
As an example, consider a company that sells replacement lamp bulbs. Demand for the
bulbs is level throughout the year and has been forecast as 2,000 units. The cost of preparing an
order is $14, and the unit cost of the bulbs is $4. The inventory holding charge is 35 percent per
year of the unit cost. Thus, we have:

R= 2000 (annual requirements)

S= 14 (cost of ordering)
C= 4 (cost per unit ordered)
K= .35 (annual inventory carrying cost as a fraction of unit cost)

The optimal lot size is:

* 2(2000) (14)
Q = = 200
(.35) (4)

Thus if this order policy is used, the average-cycle inventory level will be Q*/2 or 100
units. We will place R/Q = 2000/200 = 10 orders per year assuming the actual demand is
constant over the year (2000 units).

Using equation (1) and Q*, the total annual cost will be $8,280.

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Analysis of assumptions

Many assumptions have been implicitly made in developing the EOQ formula. One key
assumption is that the demand is level throughout the year. No seasonality, trend, or lumpiness
in demand is allowed for. Although not many real situations can meet this assumption, we
should not reject the EOQ formula as unrealistic yet. We will soon see that, even if this
assumption is not exactly met, the EOQ is often a good lot size.

Another assumption we have made is that there are no quantity discounts. The unit cost
is C dollars per unit regardless of how many we order. This assumption may also be unrealistic
in some situations, but we can modify the analysis to accommodate discounts.

The formula developed for the EOQ assumes that the incremental cost of placing an
additional order is S. However, in many situations the total annual costs associated with
ordering are more like that shown in Figure 3. A slight incremental cost is associated with each
order for the paper, postage, phone calls, etc, but over a broad range, the costs associated with
the labor of preparing and receiving a single order remain fixed. Only when we must add or
drop a person does the cost of ordering jump up or down. If we use only the actual incremental
cost and ignore the labor costs, then the cost of ordering will be very low and using the EOQ
formula will dictate that many orders be placed, thus requiring much more labor with very high
labor costs.

Figure 3: Annual Order Costs vs. Number of Orders per Year

$95k

$80k
$70k
Total Costs of $60k
Ordering Per Year
$45k
$40k

$20k

1000 2000 3000 4000

50% Fewer Present Number 50% More

Orders of Orders Orders

Number of Orders Per Year

Actual Cost
Approximation to Actual Cost

There are two possible ways to solve this dilemma. First, if the people who prepare the
orders and receive the orders can productively do other work, then one is justified in using their

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wage rate times only as the amount of time used to prepare and receive an order in the order cost.
If there is no other work that can be performed by the people who prepare and receive orders,
then one can use a linear approximation between the total cost of labor for ordering for a 50
percent increase in the number of orders per year and a 50 percent decrease in the number of
orders per year. For example, in Figure 3 we see that at present 2,500 orders per year are being
prepared. The current annual costs associated with ordering are $70,000, of which $60,000 is for
labor costs and $10,000 for paper, phone calls, etc. If 200 additional orders are placed per year,
the incremental cost of an order is only $4. Using a $4 value for S in the EOQ formula,
however, would probably result in the total orders increasing by more than 500. Instead, as an
approximation, we would compute what the incremental cost per order is between a 50 percent
increase and a 50 percent decrease in the number of orders:

$95,000 - $45,000
= $20/order
3,750 - 1,250

The numerator is the difference between the annual costs of having a 50 percent increase
in the number of orders and a 50 percent decrease. Using a value for S of $20 in the EOQ
formula would yield results that are more realistic.

Finally, the model developed for the EOQ lot size assumed that the inventory carrying
charge is proportional to the unit cost of the item. However, there are certainly situations in
which inventory charges are not proportional to the dollar value but instead are proportional to
the maximum number of units in the inventory, such as storage costs. In this case, the annual
costs of holding inventory are:

Q
KC + VQ
2

where V is the dollar cost per unit in inventory. Thus, a new expression for total annual
cost could be formulated, and a slightly different formula for EOQ would result.

By carefully analyzing the model used to derive the EOQ formula, we can determine the
important assumptions made by the formula.

Sensitivity analysis

In practice, to specify precisely the cost and demand parameters for use in the EOQ
formula may be difficult. In this section, we will show how sensitive the EOQ formula is to
errors in specification.

Suppose instead of ordering the optimal lot size Q*, we incorrectly specify some of the
cost or demand inputs and arrive at nonoptimal order quantity Q′ where:

Q ′ = (1 + f ) Q* , and

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f is the fraction deviation of Q from Q*. Now, if we compute the percentage error we are
making by using a nonoptimal order quantity, we get:

Annual Ordering and Inventory Costs Using Q′ - Annual Ordering and Inventory Costs Using Q*
Error = x 100
Annual Ordering and Inventory Costs Using Q*

R Q' R Q*
S + KC − S − KC
Q' 2 Q* 2
= x 100
R Q*
S + KC
Q* 2

(2)
⎡ 1 ⎤
= 50 ⎢ f - 1 +
⎣ 1 + f ⎥⎦

Expression (2) results by substituting for Q' and Q* in the expression above and by
considerable manipulation and simplification. Expression (2) says that, for example, if Q is 50
percent greater than optimal (f = .5), then the percentage cost error is:

⎡ 1 ⎤
= 50 ⎢ .5 - 1 + ⎥⎦ = 8.33%
⎣ 1 + .5

Figure 4 shows the percentage cost error as a function of f. We see that, even if we err in
specifying the correct input parameters and thus use a nonoptimal order quantity, the error we
make in terms of cost is relatively small.
Figure 4: Percentage Error in Cost as a Function of Deviation for EOQ

26
24
22
20
18
16
Percentage
14
Cost Error
12
10
8
6
4
2
0
-0.50 -0.25 0.00 0.25 0.50 0.75 1.00

f (fraction deviation from Q*)

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Exhibit 1

MANAGING INVENTORIES:
WHAT IS THE APPROPRIATE ORDER QUANTITY?

Inventory Categories by Function

Class Function Tradeoff with the Management Actions

Cost
of Holding Inventory

Pipeline To provide for the movement Cost of moving, cost of Change transit or delay
of materials, work-in-process, operating, and cost of times, invest in materials-
and finished goods from one material-handling handling equipment,
place to another. To reflect the equipment. improve scheduling and
time delays in producing, loading practices.
handling, and distributing a
product.

Cycle To allow for batching of Costs associated with Reduce the costs associated
(Lot Size) purchased goods or ordering and receiving with ordering or setting up.
manufactured goods. Cycle purchased lots or
stock represents the stock held setting-up manufactured Reduce elements of inventory
between reordering cycles. process. Quantity holding costs. Eliminate
The order costs that do not discount opportunity. supplier minimum order size.
change with the order size can Reduce forecast errors.
be spread over more units, and
thus the average cost can be
reduced as the lot size is
increased. In some cases,
learning effects also result in a
decrease in average cost as the
lot size increases. In
purchasing, quantity discounts
can be realized.

Buffer To protect against uncertainties Customer service, downtime Change customer-service

(Safety) in supply or demand. To costs, overtime costs, and levels, reduce lead times,
handle unanticipated shortage costs. and reduce forecast errors.
fluctuations during the
replenishment lead-time which
cannot or are not handled
through corrective actions.

Anticipation To cover anticipated changes Cost of varying production Smooth out changes in
in demand or supply such as activity. demand, varying the
those caused by seasonal production capacity by
demands, promotions, plant overtime, undertime, hiring,
vacations, and planned laying off, subcontracting,
shutdowns. etc. Buy ahead or draw
down inventories.

Speculation To deal with anticipated Savings in price. Change suppliers.

price increases or decreases.

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