10.1.1 Economic_Order_Quantity_from_Brigham_Gapenski

10.1.1 Economic_Order_Quantity_from_Brigham_Gapenski -...

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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. Self-Test 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 21-6 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 set-up 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 ‘ Short-Term 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). (21-2) 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, long-distance 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). ' ' (21-3) Here TOC = total ordering cost, F = fixed costs per order, and N = number of orders placed per year. , , Equation 21-1 may be rewritten as N .=‘" S/2A, and then substituted into Equation 21-5: (21-4) Total ordering cost = T OC 2A To illustrate the use of Equation 21-4, 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 21-4, may be combined to find total inventory costs, TIC, as follows: Total inventory costs: TIC = . TCC + TOC 5 (21-5) = (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 Short-Term Financial Management Recognizing that the average inventory carried is A = (2/2, or one-half the size of each order quantity, Q, we may rewrite Equation 21-5-as follows: TIC = TCC + TOC _ 2 2 (21-6 — (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. Self-Test 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 21-1 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 21-1 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 21-1 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 21-1 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 Short-Term Financial Management Equation 21-7 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 T-shirts 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 21-7, 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 one-half 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 21-2,“ 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 less-than—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 21-6, 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 high-sales—volume items while the large firm distributes a great many low-volume 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 21-1 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 two-week lead time is required for production and shipping, what is Cotton Tops’ order point level? If we use a 52-week 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 21-2. 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. Self-Test 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 21-3. 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 21-3 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 21-5. 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 1,000-unit 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 Short-Term financial Management Quantity Discounts Now suppose the T-shirt 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 21-7, 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 Short-Term Financial Management Table 21-7 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 21-7 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. Self-Test 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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This note was uploaded on 12/23/2009 for the course BCOM FINC 202 taught by Professor Warwickanderson during the Spring '09 term at Canterbury.

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