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Unformatted text preview: INVENTORY MANAGER'IENT w (t, Inventory management focuses on four basic questions. (1) How many units should
be ordered (or produced) at a given time? (2) At what point should inventory be
ordered (or produced)? (5) What inventory items warrant special attention? (4) Can
inventory cost changes be hedged? The remainder of the chapter is devoted to pro viding answers to these four questions. SelfTest Question
What four basic questions are addressed by the inventory manager? INVENTORY COSTS The goal of inventory management is to provide at the lowest total cost the inven
tories required to sustain operations. The ﬁrst step in inventory management is to '
identify all the costs involved in purchasing and maintaining inventories. Table 21—6
gives a listing of the typical costs that are associated with inventories. In the table,
we have broken down costs into three categories: those associated with carrying Table 216 Costs Associated with Inventories ' ' Approximate Annual
Cost as a Percentage
. of Inventory Value
I. Carrying Costs ,
Cost of capital tied up 12.0%
Storage and handling costs 0.5
Insurance 0.5
Property taxes 1.0
Depreciation and obSole'scence 12.0
Total 26.0%
II. Ordering, Shipping, and Receiving Costs 'i.
Cost of placing orders, including production and setup costs Varies
Shipping and handling costs 2.5%
III Costs of Running Short
Loss of sales Varies
Loss of customer goodwill Varies
Varies Disruption of production schedules Nore: These costs vary from ﬁrm to ﬁrm, from item to item, and also over time. The ﬁgures shown are U.S.
Department of Commerce estimates for an average manufacturing ﬁrm. Where costs vary so 'widely that no meaningful numbers can be assigned, we simply report "Varies." _________,_____.____.__..___———————————— 836. ~ Part VI ‘ ShortTerm Financial Management inventories, those associated with ordering and receiving inventories, and those as— sociated with running short of inventories,
Although they may well be the most important element, we shall at this point disregard the third category of costs—~the costs of running short. These costs are
dealt with by adding safety stocks, as we will discuss later. Similarly, we shall discuss
quantity discounts in a later section. The costs that remain for consideration at this
stage, then, are carrying costs and ordering, shipping, and receiving costs. k Carrying Costs
Carrying costs generally‘rise in direct proportion to the average amount of inventory carried. Inventories carried, in turn, depend on the frequency with which orders are placed. To illustrate, if a ﬁrm sells 8 units per year, and if it places equal—sized orders
N times per year, then S/N units will be purchased with each order. If the inventory
is used evenly over the year, and if no safety stocks are carried, then the average inventory, A, will be: _ Units per order S/N
= —_ 21
2 2 ( 1)
For example, if S = 120,000 units in a year, and N = 4, then the ﬁrm will order
30,000 units at a time, and its average inventory will be 15,000 units: ' A 12 0004 0000 ~
A =§ﬂ = —M = 5’— = 15,000 units. 2 2 2 Just after a shipment arrives, the inventory will be 30,000 units; just before the next
shipment arrives, it will be zero; and on average, 15,000 units will be carried.
Now assume the ﬁrm purchases its inventory, at a price P = $2 per unit. The
average inventory value is, thus, (PXA) = $2(15,000) = $30,000. If the ﬁrm has a
' cost of capital of 10 percent, it will incur $3,000 in ﬁnancing charges to carry the
inventory for one year. Further, assume that each year the ﬁrm incurs $2,000 of
storage costs (space, utilities, security, taxes, and so forth), that its inventory insur
ance costs are $500, and that it must mark down inventories by $1,000 because of
depreciation and obsolescence. The ﬁrm’s total costs of carrying the $30,000 average
inventory is thus $3,000 + $2,000 + $500 + $1,000 = $6,500, and the annual
percentage cost of carrying the inventory is $6,500/$30,000 = 0.217 = 21.7%.
Deﬁning the annual percentage carrying cost as C, we can, in general, ﬁnd the
annual total carrying cost, TCC, as the percentage carrying COSt, C, times the price
per unit, P,_times the average number of units, A: *‘ TCC = Total carrying cost = (CXPXA). (212) In our example, TCC = (O.217)($2)(15,000) x $6,500. Chapter 21 Accounts Receivable and Inventory 857 Ordering Costs Although we assume that carrying costs are entirely variable and rise in direct propor
tion to the average size of inventories, ordering costs are. usually ﬁxed. For example,
the costs of placing and receiving an order—interoﬁﬁce memos, longdistance tele
phone calls, setting up a production run, and taking delivery— are essentially ﬁxed
regardless of the size of an order, so this part of inventory cost is simply the ﬁxed cost
of placing and receiving orders times the number of orders placed per year.9 We
deﬁne the ﬁxed costs associated with ordering inventories as F, and if we place N
orders per year, the total ordering cost is given by Equation 21—5: Total ordering cost = TOC = (FXN). ' ' (213) Here TOC = total ordering cost, F = ﬁxed costs per order, and N = number of orders placed per year. , ,
Equation 211 may be rewritten as N .=‘" S/2A, and then substituted into Equation 215: (214) Total ordering cost = T OC 2A To illustrate the use of Equation 214, if F = $100, 8 = 120,000 units, and A =
15,000 units, then TOC, the total annual ordering cost, is $400: 120,000
50,000 Toc = $100< ) = $100<4> = $400. Total Inventory Costs Total carrying cost, TCC, as deﬁned in Equation 21—2, and total ordering cost, TOC,
as deﬁned in Equation 214, may be combined to ﬁnd total inventory costs, TIC, as follows: Total inventory costs: TIC = . TCC + TOC 5 (215)
= (CXPXA) + 9Note that, in reality, both carrying and ordering costs can have variable and ﬁxed cost elements, at least
over certain ranges of average inventory. For example, security and utilities charges are probably ﬁxed
in the short run over a wide range of inventory levels. Similarly, labor costs in receiving inventory could
be tied to the quantity received, hence could be variable. To simplify matters, we treat all carrying costs
as variable and all ordering costs as ﬁxed. However, if these assumptions do not ﬁt the situation at hand,
the cost deﬁnitions can be changed. For example, one could add another term for shipping costs if there
are economies of scale in shipping, such that the cost of shipping a unit is smaller if shipments are larger.
However, in most situations, shipping costs are not sensitive to order size, so total shipping costs are
simply the shipping cost per unit times the units ordered (and sold) during the year. Under this condi—
tion, shipping costs are not influenced by inventory policy, hence they may be disregarded for purposes
of determining the optimal inventory level and the optimal order size. 838 Part VI ShortTerm Financial Management Recognizing that the average inventory carried is A = (2/2, or onehalf the size of
each order quantity, Q, we may rewrite Equation 215as follows: TIC = TCC + TOC _ 2 2 (216
— (exp) (2) + <F><Q>. Here we see that total carrying cost equals average inventory'in units, Q/Z, multie
plied by unit price, P, times the percentage annual carrying cost, C. Total ordering
cost equals the number of orders placed per year, S/Q, multiplied by the ﬁxed cost”
of placing and receiving an order, F. We will use this equation in the neXt section to develop the optimal inventory ordering quantity. SelfTest Questions What are the three categories of inventory costs? What are some speciﬁc inventory carrying costs? As deﬁned here, are these costs;
ﬁxed Or variable? ' . What are some inventory ordering costs? As deﬁned here, are these costs ﬁxed or"
variable? THE ECONOMIC, ORDERIIVG"QUANTI'IY (EOQ) MODEL Inventories are obviously necessary, but it is equally obvious that a ﬁrm’s proﬁtabil
ity will suffer if it has too much or too little inventory. How can we determine the
optimal inventory level? One commonly used approach is based on the economic ordering quantity (EOQ) model, which is described next. Derivation of the EOQ Model Figure 211 illustrates the basic premise on which the EOQ model is built, namely,
that some costs rise with larger inventories while other costs decline, and there is
an optimal order size (and associated average inventory) which minimizes the total
costs of inventories. First, as noted earlier, the average investment in inventories
. depends on how frequently orders are placed and the size of each order—if we
order every day, average inventories will be much smaller than if we order once a
year. Further, as Figure 211 shows, the ﬁrm’s carrying costs rise with larger orders: 1
Larger orders mean larger average inventories, so warehousing costs, interest on i
funds tied up in inventory, insurance, and obsolescence costs will all increase. How— .g
ever, ordering costs decline with larger orders and inventories: The cost of placing ‘ Orders, suppliers’ production setup costs, and order handling costs will all decline ;
if we order infrequently and consequently hold larger quantities. I
If the carrying and ordering cost curves in Figure 211 are added, the sum ,
represents total inventory costs, TIC. The point where the TIC is minimized repre
sents the economic: ordering quam‘z'ry (EOQ), and this, in turn, determines the opti l
' l
l mal average inventory level. Figure 211 Determination of the Optimal Order Quantity Costs of Ordering and
Carrying Inventories ($) ﬁg M Total Carrying Cost (TCC) if” Total Ordering Cost (TOO) We
" we» . a were 0 ' EOQ Order Size (Units) In The EOQ is found by differentiating Equation 21—6 with respect to ordering
quantity, Q, and setting the derivative equal to zero: . ——————'0. Now, solving for Q, we obtain: (exp) = (Ms)
2 Q2
‘ 2 = 20%)
Q (exp) _. 2(F)(’S) _
EOQ ‘ \/ (00)“ (21.7) EOQ i economic ordering quantity, or the optimum quantity to be ordered each time an
order is placed. Here C F = ﬁxed costs of placing and receiving an order.
S = annual sales in units.
‘ C = annual carrying costs expressed as a percentage of average inventory value. P = purchase price the ﬁrm must pay per unit of inventory. 840 Part VI ShortTerm Financial Management Equation 217 is the EOQ model.10 The assumptions of the model, which will be
relaxed shortly, include the following: (1) sales can be forecasted perfectly, (2) sales
are evenly distributed throughout the year, and (5) orders are received when ex—
pected. ~ EOQ Model Illustration To illustrate the EOQ model, consider the following data, supplied by Cotton‘ Tops,
Inc., a distributor of custom—designed Tshirts which sells to concessionaires at Daisy World: 8 = annual sales = 26,000 shirts: per year.
C = percentage carrying cost = 25 percent of inventory value. P = purchase price per shirt = $4.92 .per shirt. (The sales price is 359, but this is irrelevant
for our purposes here.)  F = ﬁxed cost per order = $1,000. Cotton Tops designs and distributes the shirts, but the
actual production is done by another company. The bulk of this $1,000 cost is the labor
cost for setting up the equipment for the production run, which the manufacturer bills
separately from the $4.92 cost per shirt. ‘ * Substitutingthese data into Equation 217, we obtain an EOQ of 6,500 units: _ 2(FXS) = (2)($1,000)(26,000) 
EOQ ‘ V (on?) (0.25‘—“‘xi4,92)
. = V42,276,425 == 6,500 units. With an EOQ of 6,500 shirts and annual usage of 26,000 shirts, Cotton Tops will
place 26,000/6,500 = 4 orders per year. Notice that average inventory holdings de
pend directly on the EOQ: This relationship is illustrated graphically in Figure 21—2,
where we see that average inventory = EOQ/2. Immediately after an order is re
ceived, 6,500 shirts are in stock. The usage rate, or sales rate, is 500 shirts per week
(26,000/52 weeks), so inventories are drawn down by this amount each week. Thus,
the actual number of units held in inventory will vary from 6,500 shirts just after an
order is received to zero just before a new order arrives. With a 6,500 beginning
balance, a zero ending balance, and a uniform sales rate, inventories will average
onehalf the EOQ, or 5,250 shirts, during the year. At a cost of $4.92 per shirt, the
average investment in inventories will be (5,250)($4.92) z $516,000. If inventories
are ﬁnanced by bank loans, the loan will vary from a high of $32,000 to a low of 3290,
but the average amount outstanding over the course of a year will be $16,000. ' 10The EOQ model can also be written as ZCFXS) EOQ = C, , 841 Figure 212,“ Inventory Position without Safety Stock Units
(in Thousands) 8 Maximum Inventory Slope = Sales Rate
7 / = 6,500 = EOQ = 71.23 Shirts per Day
6.5 ‘5 3 Average
\ Inventory : 3,250
Order Point / = 1,000 1 — _ _ — _ __ I
I
I
I
I
I
I
ll EOQ
l
2 I
I
I
I o 2 4 l s 8 10.12.14 16 18 2o 22 24 2s 28
I I Weeks
\_,._1  Order Lead Time = 2 Weeks or 14 Days Notice that the EOQ, hence average inventory holdings, rises with the square
root of sales. Therefore, a given increase in sales will result in a lessthan—propor
tionate increase in inventories, so the inventory/sales ratio will tend to decline as a
ﬁrm grows. For example, Cotton Tops’ EOQ is 6,500 shirts at an annual sales level
of 26,000, and the average inventory is 5,250 shirts, or $16,000. However, if sales
were to increase by 100 percent, to 52,000 shirts per year, the EOQ would rise only
to 9,195 units, or by 41 percent, and the average inventory would rise by this same
percentage. This suggests that there are economies of scale in holding inventories.11 Finally, look at Cotton Tops’ total inventory costs for the year, assuming that the
EOQ is ordered each time. Using Equation 216, we ﬁnd total inventory costs are $8,000: '
TIC: ch ’ + TOC g E
‘ (CXP ) < 2 > + (P) (Q) « ’
6,500 26,000 $4,000 + $4,000 = $8,000 [I ll ll 11Note, however, that these scale economies relate to each particular item, not to the entire ﬁrm. Thus, a
large distributor with $500 million of sales might have a higher inventory/sales ratio than a much smaller
distributor if the small ﬁrm has only a few highsales—volume items while the large ﬁrm distributes a great many lowvolume items. Note these two points: (1) The $8,000 total inventory cost represents the total of
carrying costs and ordering Lcosts, but this amount does not include the
26,000($4.92) = $127,920 annual purchasing cost of the inventory itself. (2) As we
see both in Figure 211 and in the numbers just preceding, at the EOQ, total carrying
cost (TCC) equals total ordering cost (TOG). This property is not unique to our
Cotton Tops illustration; it always holds. ‘ Setting the Order Point If a twoweek lead time is required for production and shipping, what is Cotton
Tops’ order point level? If we use a 52week year, Cotton Tops sells 26,000/52 =
500 shirts per week. Thus, a two—week lag occurs between placing an order and
receiving goods, Cotton Tops,_;must place the order when there are 2600) '= 1,000
shirts on hand. During the two—week production and shipping period, the inventory
balance will continue to decline at the rate of 500 shirts per week, and the inventory balance will hit zero just as the order of new shirts arrives.
If Cotton Tops knew for certain that both the sales rate and the order lead time would never vary, it could operate exactly as shown in Figure 212. However, sales do change, and production and/or shipping delays are frequently encountered; to
guard against these events, the ﬁrm must carry additional inventories, or safety stocks, as discussed in the next section. SelfTest Questions What is the concept behind the EOQ model? What is the relationship between total carrying cost and total ordering cost at the
EOQ? What assumptions are inherent in the EOQ model as presented here? 509 MODEL ‘mNSIONS The basic EOQ model was derived under several restrictive assumptions. In this
section, we relax some of these assumptions and, in the process, extend the model to make it more useful. The Concept of Safety Stocks The concept of a safety stock is illustrated in Figure 213. First, note that the slope
of the sales line measures the expected rate of sales. The company expects to sell
, 500 shirts per week, but let us assume that the maximum likely sales rate is twice
this amount, or 1,000 units each week. Further, assume that Cotton Tops sets the
safety stock at 1,000 shirts, so it initially orders 7,500 shirts, the EOQ of 6,500 plus
the 1,000—unit safety stock. Subsequently, it reorders the EOQ whenever the inven Chapter 21 Accounts Receivable and Inventory 845 Figure 213 Inventory Position with Safety Stock Included Units
(Thousands) Maximum
Inventory r Average 24 26 28 301
Weeks 22 a 14 16 18 20 Lead Time tory level falls to 2,000 shirts, the safety stock of 1,000 shirts plus the 1,000 shirts
expected to be used while awaiting delivery of the order. ' . Notice that the company could, over the two—week delivery period, sell 1,000
units a week, or double its normal expected sales. This maximum rate of sales is
shown by the steeper dashed line in Figure 215. The condition that makes possible
this higher maximum sales rate is the safety stock of 1,000 shirts. The safety stock is also useful to guard against delays in receiving orders. The
expected delivery time is 2 weeks, but with a 1,000unit safety stock, the company
could maintain sales at the expected rate of 500 units per week for an additional 2
weeks if production or shipping delays held up an order. However, carrying a safety stock has a cost. The average inventory is now EOQ/Z
plus the safety stock, or 6,500/2 + 1,000 a 3,2so__ _+ 1,000 = 4,250 shirts, and. the
average inventory value is now (4,250X35492) = $20,910. This increase in average
inventory causes an increase in annual inventory carrying costs equal to (Safety stock)
(PXC) = 1,000(t4.92)(0.25) = $1,230. The optimal safety stock varies from situation to situation, but, in general, it
increases (1) with the uncertainty of demand forecasts, (2) with the costs (in terms
of lost sales and lost goodwill) that result from inventory shortages, and (5) with the
probability that delays will occur in receiving shipments. The optimum safety stock decreases as the cost of carrying this additional inventoryincreases.12 12For a more detailed discussion of safety stocks, see Arthur Snyder, “Principles of Inventory Manage
ment,” Financial Executive, April 1964, 13—21. 844 Part V1 ShortTerm financial Management Quantity Discounts Now suppose the Tshirt manufacturer offered Cotton Tops a quantity discount of 2.
percent on large orders. If the quantity discount applied to orders of 5,000 or more,
then Cotton Tops would continue to place the EOQ order of 6,500 shirts and take
the quantity discount. However, if the quantity discount required orders of 10,000
or more, then Cotton Tops’ inventory manager would have to compare the savings
in purchase price that would result if its ordering quantity were increased to 10,000
units with the increase in total inventory costs caused by the departure from the 6,500—unit EOQ. ,
First, consider the total costs associated with CottonTops’ EOQ of 6,500 units. We found earlier that total inventory costs are $8,000: TIC =* . ch + TOC
__ a ‘ ' _s_
 60
6,500 ' 26,000
— O.25($4.92)<T> + ($1,OOO)< 6,500)
x ' $4,000 +, $54,000 = $8,000 Now, what would the total inventory costs be if Cotton Tops ordered 10,000 units
instead of 6,500? The answer is $8,625: 26,000) 10000
= I ' 2 _’_
TIC 025($48 )< V 2 > + ($1,000)<10’OOO = $6,025 + $2,600 = $8,625. Notice that when the discount is taken, the price, P, is reduced by the amount of the
discount; the new price per unit would be O.98($4.92) = $4.82. Also note that when
the ordering quantity is increased, carrying costs increase because the ﬁrm is carry
ing a larger average inventory, but ordering costs decrease since the number of
orders per year decreases. Ifwe were to calculate total inventory costs at an ordering
quantity of 5,000, we would ﬁnd that carrying costs would be less than $4,000, and
ordering costs would be more than $4,000, but the total inventory costs would be
more than $8,000, since they are at a minimum when 6,500 units are ordered.15
Thus, inventory costs would increase by $8,625 — $8,000 = $625 if Cotton Tops
were to increase its order size to 10,000 shirts. However, this cost increase must be
compared wit/9 Cotton Tops’ savings 2f it ta/ees the discount. Taking the discount 15At an ordering quantity of 5,000 units, total inventory costs are $8,275:
5 000 26 000
= 0.2 4. 2 ’— + 1 OOO —’
TIC ( SM 9 )< 2 > (i, )(iOOO) = $3.075 + $35,200 = $8,275. would save 0.02($4.92) = $00984 per unit. Over the year, Cotton Tops orders
a 26,000 shirts, so the annual savings is $0.0984C26,000) z $2,558. Here is a summary: Reduction in purchase price = 0.02($4.92)(26,000) = $2,558
Increase in total inventory cost = 625
Net savings from taking discounts $1,955 Obviously, the company should order 10,000 units at a time and take advantage of
the quantity discount. Inﬂation Moderate inﬂation—say 3 percent per year—can largely be ignored for purposes
of inventory management, but higher rates of inﬂation must be explicitly considered.
If the rate of inﬂation in the types of goods the ﬁrm stocks tends to be relatively
constant, it can be dealt with quite easily—simply deduct the expected annual rate
of inﬂation from the carrying cost percentage, C, in Equation 217, and use this
modiﬁed version of the EOQ model to establish the working stock. The reason for
making this deduction is that inﬂation causes the value of the inventory to rise, thus
offsetting somewhat the effects of depreciation and other carrying costs factors. Since
C will now be smaller, the calculated EOQ, and the average inventory, will increase.
However, the higher the rate of inﬂation, the higher are interest rates, and this factor
will cause C to increase, thus lowering the EOQ and average inventories. On balance, there is no evidence that inﬂation either raises or lowers the opti
mal inventories of ﬁrms in the aggregate. Inﬂation should still be explicitly consid
ered, however, for it will raise the individual ﬁrm’s optimal holdings if the rate of
inﬂation for its own inventories is abOVe average (and is greater than the effects of inﬂation on interest rates), and vice versa. Seasonal Demand For most ﬁrms, it is unrealistic to assume that the demand for an inventory item is
uniform throughout the year. What happens when there is seasonal demand, as
would hold true for an ice cream company? Here the standard annual EOQ model
is obviously not appropriate. However, it does provide a point of departure for
setting inventory parameters, which are then modiﬁed to ﬁt the particular seasonal
pattern. The procedure here is to divide the year into the seasons in which annu—
alized sales are relatively constant, say the summer, the spring and fall, and the
winter. Then, the EOQ model can be applied separately to each period. During the
transitions between seasons, inventories would be either run down or else built up with special seasonal orders. EOQ Range Thus far, we have interpreted the EOQ, and the resulting. inventory variables, as
single point estimates. It can be easily demonstrated that small deviations from the 846 Part VI ShortTerm Financial Management Table 217 EOQ Sensitivity Analysis \E Percentage Ordering Total Inventory Deviation
Quantity Costs from Optimal
m 3,000 , $10,512 +31.4% 4,000 8,960 , + 12.0 5,000 8,275 + 3.4 6,000 8,023 ‘ + 0.3 6,500 8,000 0.0 7,000 8,019 + 0.2 8,000 11:." 1‘, 8,170 + 2.1 9,000 ’ 8,425 + 5.5 10,000 8,750 + 9.4 EOQ do not appreciably affect total inventory costs, and, consequently, that the op
timal ordering quantity should be Viewed more as a range than as a single value.14 To illustrate this point, we can examine the sensitivity of total inventory costs to
ordering quantity for Cotton Tops, Inc. Table 217 contains the results of our sensi
tivity analysis. We conclude that the ordering quantity could range from 5,000 to
8,000 units without affecting total inventory costs by more than 3.4 percent. Thus,
we see that managers can adjust the orderingquantity within a fairly wide range
without fear of signiﬁcantly increasing total inventory costs. SelfTest Questions Why are inventory safety stocks required? Conceptually, how would you evaluate a'quantity discount offer from a supplier?
What impact does inﬂation haveon the EOQ? Can the EOQ model be used when a company faces seasonal demand ﬂuctuations?
What is the impact of minor deviations from the EOQ on total inventory costs? ...
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 Spring '09
 WarwickAnderson
 eoq model, Cotton Tops

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