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www.downloadslide.net 12 Inventory Management with Steady Demand LEARNING OBJECTIVES LO12-1 Evaluate optimal order quantities and performance measures with the EOQ model LO12-2 Recognize the presence of economies of scale in inventory management and understand the impact of product variety on inventory costs LO12-3 Evaluate optimal order quantities when there are quantity constraints or quantity discounts are offered CHAPTER OUTLINE Introduction 12.1 The Economic Order Quantity 12.2 Economies of Scale and Product Variety 12.3 Quantity Constraints and Discounts 12.1.1 The Economic Order Quantity Model 12.3.1 Quantity Constraints 12.1.2 EOQ Cost Function 12.3.2 Quantity Discounts 12.1.3 Optimal Order Quantity Conclusion 12.1.4 EOQ Cost and Cost per Unit Introduction Imagine (i) you are standing in the aisle of a grocery store, (ii) you are staring at the packages of ramen noodles, and (iii) you actually like to eat ramen noodles—they are tasty, convenient, and relatively cheap (What’s not to like?). You probably would grab a few packages, maybe even more than a handful, without much thought. But let’s think about what the “optimal” quantity to buy could be. You eat these noodles at a consistent pace. So you know that you will go through the packages pretty quickly. It is a hassle to go to the grocery store and to worry about remembering to buy noodles, so maybe you should stock up. That way you don’t have to deal with buying them for quite some time. On the other hand, you are a bit tight on cash at the moment and you dread explain© Ingram Publishing/SuperStock/RF ing to your roommate why you are filling up one entire cabinet in your kitchen with just noodles. Hence, if you buy only a few packages, you are likely to find yourself buying noodles frequently, while if you buy a cartful, you will find yourself storing them in odd places in your apartment. Given all of this, what is the “optimal” number of packages to buy? And to make your decision even 362 www.downloadslide.net Chapter Twelve Inventory Management with Steady Demand more complicated, if the ramen noodles happen to be on sale—buy four, get one free—how many would you buy? This chapter focuses on two reasons why inventory exists: batching due to fixed costs and price discounts. We begin with an inventory management model to rigorously analyze questions like “How many packages of ramen noodles should I buy?” (section 12.1). While it might be silly for you to take this data-driven and hyperanalytical approach with your grocery store purchases, firms do not have the luxury of being so casual with their inventory decisions. This is especially true given that firms incur substantial costs for holding inventory and generally must manage thousands of different products in each of their locations. In particular, we’ll analyze Walmart’s inventory purchasing decision for one analgesic, Extra Strength Tylenol caplets (24 count), manufactured by Johnson & Johnson. Beyond making specific decisions, our inventory model provides some managerial guidance with respect to the set of products offered to customers—what are the inventory consequences of adding or subtracting products from this set? As is so common in operations, we’ll see that there are economies of scale in inventory management (section 12.2). Finally, we deal with the issue of quantity constraints and quantity discounts. Quantity constraints restrict order quantities to some multiple of a standard size, such as a case. Quantity discounts give us the opportunity to order a large quantity to get a discount off the purchase price. Our inventory model allows us to precisely evaluate whether the quantity discount is a good deal or not (section 12.3). 12.1 The Economic Order Quantity © Jill Braaten RF/Jill Braaten/RF 363 www.downloadslide.net 364 Chapter Twelve Inventory Management with Steady Demand Walmart is the world’s largest retailer. One reason it became so big is because it is very good at managing its inventory. To help us get a handle on its capability, let’s focus on one of the many, many inventory decisions it must make. This decision takes place in one of its retail stores, a supercenter, in the United States. If you haven’t been to a supercenter, then you should know that these are big stores—a typical supercenter encompasses about 180,000 square feet (not including the parking lot), which is bigger than the combined area of three American football fields, or four acres of land. Given its size, a supercenter stocks tens of thousands of different products. But let’s pick just one product—Extra Strength Tylenol (Tylenol) caplets (24 count), an analgesic made by Johnson & Johnson. (Actually, Tylenol is made by the McNeil Consumer Healthcare division of McNeil Inc., but McNeil is owned by Johnson & Johnson.) Walmart buys each bottle of Tylenol for $3.00. Walmart sells 624 bottles through this store every year and demand is relatively constant throughout the year. Walmart can place orders for more Tylenol whenever it wants. However, each order costs Walmart a fixed $6 no matter the quantity ordered. This $6 represents the costs related to the processing of the order and stocking the product on the shelf (which do not depend on the amount actually stocked). The other relevant cost is the holding cost of inventory. Walmart believes that it incurs a 25 percent inventory holding cost. This means that it incurs $0.25 in cost to hold a dollar of inventory for one year. For example, if a bottle of Tylenol spends one year in the supercenter (which it is unlikely to actually do, but let’s say it does), then Walmart would incur a cost of $3 × 0.25 = $0.75.1 12.1.1 The Economic Order Quantity Model Walmart can use the economic order quantity (EOQ) model for the management of its ­Tylenol inventory. The model is built on the following assumptions. • • • Economic order quantity (EOQ) model A model used to select an order quantity that minimizes the sum of ordering and inventory holding costs per unit of time. • • Order cost, K The fixed cost incurred per order, which is ­independent of the amount ordered. Holding cost per unit, h The cost to hold one item in inventory for one unit of time. 1 Demand occurs at a constant rate R. For Tylenol, R = 624 bottles per year, or 12 bottles per week (624/52). See Connections: Consumption for a discussion of this assumption. There is a fixed order cost K per order. The order cost per order is independent of the amount ordered. For Tylenol, it is K = $6 per order. There is a holding cost per unit per unit of time, h. This holding cost represents all of the costs associated with keeping a unit of inventory for a period of time. For example, it includes the opportunity cost of capital—if Walmart has $10B invested in inventory, then it cannot use that $10B to earn a rate of return with other investments (such as building new stores, investing in other companies, or expanding its e-commerce site, etc.). It also includes the cost to rent and maintain the building space that stores the inventory, among other costs. For a bottle of Tylenol, the holding cost is 25 percent of its value per year, which is h = 25% × $3.00 = $0.75 per year. Note that the holding cost per unit, $0.75, is the product of the holding cost percentage (e.g., 25 percent) and the purchase cost per unit, not the selling price per unit. This is because the opportunity cost of capital is based on the cost to buy the unit (i.e., the amount of cash “locked up” in inventory for a period of time) rather than the amount eventually earned on the unit (which is received after the unit is no longer in inventory). All demand is satisfied from inventory. We do not want to run out of inventory because doing so risks losing some sales, which is viewed as too costly. Inventory never spoils, degrades, or is lost. This assumption means that everything purchased is eventually sold. In other words, no matter how long an item remains in inventory, it is considered to be in “as good as new” condition. While it is true that there is an expiration date on medicines, it is safe to assume that Walmart sells all of its inventory well before any of it approaches its expiration date. These data are only representative to protect confidential information. www.downloadslide.net Chapter Twelve • • Inventory Management with Steady Demand 365 Orders are delivered with a reliable lead time. The lead time for an order is the time between when an order is placed with the supplier and when the order is received by the customer. The EOQ model assumes the supplier reliably delivers orders, which means that there is a constant delay between when the order is placed and when it is received. As a result, we can time orders to arrive at our facility so that we do not run out of inventory—that is, orders arrive just before inventory is about to run out. Walmart is its own supplier for its supercenters—it receives deliveries from one of Walmart’s distribution centers. Therefore, it is safe to assume that deliveries to the supercenter are indeed reliable. There is a purchase price per unit that is independent of the quantity purchased. In short, there are no quantity discounts. For example, Walmart pays J&J $3.00 per bottle whether Walmart purchases 1 bottle or 1000 bottles. In the last section of this chapter, we explore how to include quantity discounts in this decision. Table 12.1 summarizes the relevant parameters of the EOQ model applied to the Tylenol order quantity decision. It is critical that the various parameters use the same units. For example, the model gives the same answer whether demand and the holding cost per unit are expressed in terms of one year, one week, or one hour. However, it is critical that they are both defined for the same time period—the model gives an incorrect answer if the demand rate is expressed in units per year while the holding cost per unit is in dollars per week. Similarly, don’t measure the order cost in euros while holding cost per unit is measured in dollars. And if demand is expressed in bottles per year, then the holding cost per unit should be for one bottle held for one year, not one case of bottles or one pill. The goal of the EOQ model is to minimize the sum of ordering and holding costs per unit of time, which is referred to as the EOQ cost per unit of time: • • • Ordering cost per unit of time. This is the sum of all order costs in a period of time. For example, if Walmart places 12 orders per year, then its ordering cost per year is $72 ($6 per order × 12 orders per year). Holding cost per unit of time. This is the total cost to hold inventory for a period of time. For example, if Walmart holds 100 bottles, on average, in inventory throughout the year, then its holding cost for the year is $75 (100 bottles × $0.75 per bottle per year). EOQ cost per unit of time. This is the sum of ordering and holding costs per unit of time. Be careful to note the difference between the order cost per order (e.g., K = $6) and the ordering cost per unit of time (e.g., $72 per year if 12 orders are made each year). The first one is an input to the model and is not influenced by the decision—Walmart incurs an order cost of $6 per order whether it orders frequently or not. The second one is a result of the ­decision—the ordering cost per year depends on the order quantity decision. A similar distinction applies between the holding cost per unit (e.g., h = $0.75) and the holding cost per unit of time (e.g., $75 if 100 bottles are held in inventory, on average, throughout the year). The holding cost per unit does not depend on the order quantity decision, whereas the holding cost per year does depend on that decision (and therefore is not constant). TABLE 12.1 EOQ Model Parameters and Values for the Tylenol Product Sold through One of Walmart’s Distribution Centers EOQ Model Parameter Variable Walmart/Tylenol Value Demand rate (units per year) R 624 Order cost ($ per order) K $6 Holding cost per unit ($ per unit per year) h $0.75 Order quantity (units) Q To be determined Lead time The time between when an order is placed and when it is received. Process lead time is ­frequently used as an alternative term for flow time. Ordering cost per unit of time The sum of all fixed order costs in a period of time. Holding cost per unit of time The total cost to hold inventory for a period of time. EOQ cost per unit of time The sum of ordering and holding costs per unit of time. www.downloadslide.net 366 Chapter Twelve Inventory Management with Steady Demand Besides ordering and holding costs, it is also useful to define the purchasing cost per unit of time: • Purchasing cost The cost to ­purchase inventory in a period of time. Purchasing cost. This is the cost to purchase inventory in a period of time, such as $1872 per year ($3 per bottle × 624 bottles per year). Given that Walmart spends a sizable amount to buy Tylenol for this one supercenter each year ($1872), you might be surprised that the EOQ model does not include the purchasing cost in its objective. The reason is simple: Walmart’s annual purchasing cost does not depend on the quantity it purchases with each order. If Walmart orders 12 bottles per order, then Walmart makes 52 orders per year (624 bottles per year/12 bottles per order). If Walmart orders 48 ­bottles per order, then Walmart makes 13 orders per year (624 bottles per year/48 bottles per order). So the number of orders made in a year depends on the order quantity, but not the total quantity actually ordered. The total quantity ordered per year should match the demand per year because Walmart does not run out of stock and everything purchased is eventually sold. Hence, the order quantity decision does not actually influence Walmart’s annual purchasing cost for Tylenol. It is reasonable to ignore purchasing cost when deciding on an order quantity when there are no quantity discounts. However, if there are quantity discounts, like “5 percent off for a full truck order,” then purchasing cost does become relevant for the order quantity decision. That case is handled in section 12.3.2 on quantity discounts. Check Your Understanding 12.1 Question: A retailer purchases a snowblower for $400 and sells it for $600. The retailer incurs a 25 percent annual inventory holding cost. What is the holding cost per unit per year, h? Answer: 25% × $400 = $100 CONNECTIONS: Consumption One of the assumptions of the EOQ model is that the demand rate does not depend on the amount actually ordered. But there are several reasons to believe that this might not always be the case. Say you open your refrigerator and you see a pile of yogurt containers. Are you as likely to eat a yogurt in this case as compared to the time when you open the refrigerator and you see only a couple of yogurts? It turns out that most people don’t treat these two cases the same: When they see a stockpile of inventory, they are more likely to consume relative to when inventory is limited. And as you might expect, marketers have devised a strategy to take advantage of this behavior—temporary price promotions! In particular, one of the motivations of a price promotion strategy is to encourage consumers to “stock up” with the hope that doing so gets them to consume more. Inventory might influence your demand even if it isn’t in your refrigerator. Say you want to purchase a midsized sedan. You visit a local dealer, call it “Full,” and you see that it has 20 ­versions of its midsized sedan. Next, you visit another dealer, call it “Lean,” from a different car company and it has only one of its mid-size sedans on its lot. Do these inventory numbers influence your preference? They could. You might conclude that Full has so many cars because the car is popular, and it would only be popular if it were a well-designed and reliable vehicle. Or you might prefer the car from Lean because you infer that the car must be so popular that it cannot keep the car on its lot. In the context of cars, evidence suggests that Lean would be more likely to get your sale, which is called the “scarcity effect”: Limited inventory can increase demand because it is scarce. However, there are settings in which the opposite occurs. When www.downloadslide.net Chapter Twelve 367 Inventory Management with Steady Demand you go to a grocery store, do you prefer the last package of hamburger? Chances are “no”—if it is the last package, then it must be old. However, if the grocery store fills up an entire shelf with the same flavor of coffee, you are more likely to grab that version, probably because the large inventory “catches your eye,” thereby drawing your attention. Sources: Ailawadi, Kusum L., and Scott A. Neslin. “The Effect of Promotion on Consumption: Buying More and C ­ onsuming It Faster.” Journal of Marketing Research 35, no. 3 (August 1998), pp. 390–98. Cachon, Gerard; Santiago Gallino; and Marcelo Olivares. Does Adding Inventory Increase Sales? Evidence of a ­Scarcity Effect in U.S. Automobile Dealerships (June 28, 2013). Available at SSRN: http://ssrn.com/abstract=2286800 or http://dx.doi.org/10.2139/ ssrn.2286800 12.1.2 EOQ Cost Function Let’s begin to understand the trade-offs associated with the order quantity decision. ­Figure 12.1 displays how inventory at the DC changes over time for two different order quantities. In the top panel, the order quantity is 50 bottles per order, while in the bottom panel it is 150 bottles per order. We can see from Figure 12.1 that inventory follows a “saw-toothed” pattern: When an order arrives, inventory immediately spikes to a higher level, and then it begins to fall at a constant rate. The inventory jumps equal the order quantity Q, and inventory falls at the rate of demand R; that is, R equals the slope of each triangle in the sawtooth. The inventory pattern in Figure 12.1 indicates that new orders arrive just as inventory falls to zero. This ensures that customers can always find inventory but also keeps inventory as low Figure 12.1 150 Pattern of inventory (in bottles) if Walmart purchases 50 bottles with each order (top panel) or 150 bottles with each order (bottom panel) Inventory 100 50 Q R 0 Time 150 Inventory 100 Q 50 R 0 Time www.downloadslide.net 368 Chapter Twelve Inventory Management with Steady Demand as possible. It is possible to time the arrival of orders “just so” because we assume the supplier delivers orders reliably on time. In addition to the saw-toothed pattern, we see from Figure 12.1 that there seems to be more inventory with the larger order quantity but more orders with the smaller order quantity. This identifies the key trade-off in the model: Large order quantities add to holding cost, but small quantities suffer from high ordering cost. So what exactly is the average amount of inventory for a given order quantity? Looking at the triangles in the upper panel of Figure 12.1, we see that inventory peaks at 50 bottles and then steadily falls to 0 over the course of a little over four weeks (50 bottles/12 bottles per week = 4.17 weeks). The average inventory over that time is just one-half of the peak inventory, or 25 bottles. In general, Order quantity __ Q ____________ ​Average inventory =    ​   ​ = ​   ​​ 2 2 Once we know the average inventory, we can evaluate the holding cost per unit of time: ​Holding cost per unit of time = __ ​1 ​ × h × Q​ 2 For example, if Walmart orders 50 bottles with each order, then Walmart has 25 bottles, on average, in its supercenter. It costs Walmart $0.75 per bottle per year to hold inventory. So Walmart’s holding cost per year for Tylenol in its supercenter would be 25 bottles × $0.75 per bottle per year = $18.75 per year. We also have to worry about ordering cost. For now, let’s assume Walmart orders 150 ­bottles at a time. We also know that Walmart sells 624 bottles per year. So Walmart needs to make 4.16 orders per year (624 bottles per year/150 bottles per order). If Walmart makes more orders per year (at 150 bottles each), then it purchases more each year than it can sell, leaving it with inventory that continues to grow in size. That doesn’t work. If Walmart makes fewer than 4.16 orders per year (again, at 150 bottles per order), then Walmart does not order enough each year to satisfy its demand, which is also not acceptable. So to balance supply with demand, Walmart needs to make precisely 4.16 orders per year. But how is that possible given that 4.16 isn’t a whole number? The answer is that it makes four orders in some years and five orders in other years because the end of the year doesn’t coincide exactly with the arrival of an order. On average, across years, Walmart makes 4.16 orders per year with an order quantity of 150 bottles. In general, if Walmart sells R bottles per year and orders Q bottles with each order, then Walmart needs to make R/Q orders per year (on average): ​Orders per unit of time = __ ​ R  ​​ Q Therefore, Walmart incurs the following ordering cost per unit of time: ​Ordering cost per unit of time = K × __ ​  R  ​​ Q The “unit of time” in the above equation is the same unit used to describe the demand rate R. For example, if R is bottles per year, then the equation above yields the ordering cost per year. We are now ready to write down an equation for the sum of ordering and holding costs per unit of time, which we also call the EOQ cost per unit of time: ​EOQ cost per unit of time = C(Q) = ​ K × __ ​  R  ​ ​+ ​(__ ​  1 ​ × h × Q)​​ ( Q) 2 Figure 12.2 plots the ordering, holding, and EOQ cost functions for various order quantities. Holding cost increases as a straight line as the order quantity gets larger. Ordering cost decreases as a curve as the order quantity increases. The sum of those costs, the EOQ cost, forms a U-shaped curve. If the order quantity is small, then ordering cost dominates and the EOQ cost is high. EOQ cost is also high if the order quantity is large, because then holding cost dominates. The trick is to find the “just right” order quantity. www.downloadslide.net Chapter Twelve 369 Inventory Management with Steady Demand 160 Figure 12.2 140 Ordering, holding, and EOQ costs per unit of time (per year) as a function of the order quantity Cost ($ per year) 120 100 EOQ cost, C(Q) 80 60 Holding cost = hQ/2 40 20 0 Ordering cost = KR/Q 0 50 100 150 Order quantity, Q (bottles) 200 250 Check Your Understanding 12.2 Question: A drill in a hardware store incurs a holding cost per unit of $10 per year. The store sells one of these drills per week all year long. Each time the store orders more drills, it orders a case that contains 12 drills. What is the average inventory of drills at this hardware store? Answer: The order quantity is Q = 12 and average inventory is Q/2 = 12/2 = 6. Question: What holding cost does the hardware store incur per year to stock this drill? Answer: Holding cost per unit of time = 0.5 × h × Q = 0.5 × $10 per year per drill × 12 drills = $60. Question: The hardware store incurs a $6 cost with each order. What ordering cost does the hardware store incur per year? Answer: The order cost per order is K = $6. The flow rate is R = 52 drills per year. The ­ordering cost per year = K × R/Q = $6 × 52 drills/12 drills = $26. 12.1.3 Optimal Order Quantity Figure 12.2 suggests that the optimal order quantity is about 100 bottles. But we should be more precise. If you have taken a calculus class, then you might remember how to derive the formula for the minimum of a function (take the first derivative, set it equal to 0, and solve for Q). But if you didn’t take calculus or don’t remember the procedure, don’t worry; you can just use the following equation for the order quantity Q* that minimizes the sum of ordering and holding costs per unit of time (i.e., minimizes C(Q)): _ × R  ​​ ​​Q​ ​= ​ _________ ​ 2 × K ​ h This quantity Q* is called the economic order quantity (which gives the model its name). Looking at the equation we can confirm that Q* increases if the order cost, K, increases; in that case, it makes sense to order a larger quantity to reduce the number of orders placed per unit of time. Similarly, Q* decreases if the holding cost per unit, h, increases: If it costs more * √ © Ron Chapple Stock/FotoSearch/Glow Images/RF www.downloadslide.net 370 Chapter Twelve Inventory Management with Steady Demand Check Your Understanding 12.3 Question: Xootr procures handle caps from a Taiwanese supplier. The supplier charges $300 per order (no matter the order quantity) to cover customs fees and other expenses. Xootr requires 700 caps per week and the annual holding cost on each cap is $0.40. How many caps should Xootr order to minimize the sum of ordering and holding costs? Answer: Because the holding cost is given for one year, the demand rate should be for one year as well. The demand rate per year is 700 caps per week ​×​52 weeks per year = ​ ​ √ __________________ 2 × $300 × 36,400 36,400 caps per year. The EOQ is then ​​ __________________ ​        ​ ​​ = ​ ​ 7389. $0.40 to hold inventory, order a smaller quantity to reduce the average amount of inventory. And, finally, products with higher demand, R, have a higher order quantity, as you would expect. Let’s apply the EOQ formula to Walmart’s Tylenol order quantity: * √ ​Q​ ​= ​ _ _____________ 2 × $6 × 624 × R  ​ = ​    _________ ____________ ​  2 × K ​    ​   ​  ​ = 100​ h √ $0.75 Thus, to minimize the sum of ordering and holding costs, Walmart should order about 100 Tylenol bottles per order. This is consistent with the graph in Figure 12.2, but now we have an equation to find the optimal quantity. Looking further at Figure 12.2 you may notice that it appears that the holding cost curve and the ordering cost curve cross right about at the point that minimizes total cost, 100 ­bottles. This is actually not a coincidence. In the EOQ model, the quantity Q* that minimizes total cost is also the quantity at which the holding cost per unit of time equals the ordering cost per unit of time. 12.1.4 EOQ Cost and Cost per Unit We now know that the EOQ, Q*, minimizes the sum of the ordering and holding costs per unit of time; that is, the EOQ cost. So what is that minimum EOQ cost? We can calculate it by plugging the order quantity, Q*, into the EOQ cost equation, C(Q): 1 ​ × h × Q* ​ = ​ $6 × ____ ​C(Q * ) = ​ K × ___ ​  R  ​ ​ + ​(​ __ ​ 624 ​)​+ ​(__ ​  1 ​ × $0.75 × 100)​ = $74.94​ ) ( ( Q* ) 2 100 2 Hence, if each of Walmart’s orders for Tylenol contains 100 bottles, then Walmart incurs an EOQ cost of $74.94 per year. (We know that this cost is “per year” because we evaluated the cost function using the demand rate for one year and the holding cost per unit for one year.) Is $74.94 per year a lot? To answer that question, it is helpful to measure the $74.94 relative to something else. One approach is to measure it against the purchase cost per unit of time. Each year Walmart purchases $1872 of Tylenol ($3 per bottle × 624 bottles per year) for this supercenter. Hence, $74.94 is 4.0 percent ($74.94/$1872) of the purchase cost. Another approach is to allocate the cost to each unit sold: $74.94 per year/624 bottles per year = $0.12 per bottle. Hence, Walmart incurs about a total of 12 cents in ordering and holding costs per bottle. It shouldn’t come as a surprise that $0.12 per bottle is 4.0 percent ($0.12/$3) of the cost of the bottle: If the sum of the ordering and holding costs of 624 bottles is 4.0 percent of the purchase cost, then the sum of the ordering and holding costs for one of those 624 bottles is also 4.0 percent of its purchase cost. Is 4.0 percent a lot? At first glance it doesn’t seem like much. But remember that a retailer’s final profit is often only about 1 percent to 3 percent of its sales revenue. So 4 percent is the same order of magnitude as the firm’s profit. Hence, it is important to do inventory management well. For instance, if inventory costs were to double, to 8 percent, then this could turn this product from a profitable one to an unprofitable one. www.downloadslide.net Chapter Twelve Inventory Management with Steady Demand 371 Check Your Understanding 12.4 Question: Xootr procures wheels from a Chinese supplier. The supplier charges $200 per order (no matter the order quantity) to cover customs fees and other expenses at $3 per wheel. Xootr requires 80,000 wheels per year and its annual holding cost percentage is 40 percent. Suppose Xootr orders 8000 wheels with each order. What is the sum of its ordering and inventory holding costs per year? Answer: The holding cost per unit per year is $3 × 0.4 = $1.20. The sum of the ordering and inventory holding costs per year is then 80,000 R 1 1 ​C(Q) = ( ​ K × __ ​   ​)​ + ( ​ __ ​   ​ × h × Q)​ = ( ​ $200 × ______ ​   ​ ​ + ​ __ ​   ​ × $1.20 × 8000)​ = $6800​ 2 8000 ) (2 Q Question: If Xootr orders 5000 wheels per order, then the sum of its ordering and holding costs per year is $6200. What would be the ordering and holding cost per wheel? Answer: The ordering and holding cost per wheel is $6200 per year/80,000 wheels per year = $0.078. 12.2 Economies of Scale and Product Variety The fixed ordering cost feels somewhat like the fixed setup time discussed in Chapter 7, Process Interruptions. Recall that a setup time is a fixed amount of time for a process to get ready for production. No production occurs during the setup time and the setup time is the same duration no matter how many units are produced in the batch after the setup. When the setup time is substantial, it is intuitive that it makes sense to produce in large batches. In that way, the setup time is in some sense divided among the many units in the batch and the capacity of the process increases as the batch size gets larger. As a result, processes with setup times are said to exhibit economies of scale: They become more efficient when larger quantities are produced. The EOQ model does not have a setup time, but it does have a fixed order cost. Walmart incurs a cost of $6 per order no matter if it orders one bottle or 1000 bottles. So it is best to order in sufficiently large quantities. However, the order quantity also cannot be completely unaligned with demand. Ordering 1000 bottles at a time is potentially okay if Walmart is selling thousands of bottles per year. But if it is only selling a couple hundred bottles per year, then an order quantity of 1000 bottles is likely to incur a crazy high inventory holding cost. To illustrate the economies of scale created by the fixed order cost, let’s apply the EOQ model to the Tylenol ordering decision for different demand rates. Table 12.2 evaluates the total of ordering and holding costs for five different scenarios. The middle one, scenario III, is the base case that we have already analyzed. It has an annual demand rate of 624 bottles, an EOQ of 100 bottles, and an annual EOQ cost that is 4 percent of the annual purchase cost. The other four scenarios either ramp up or ramp down demand by some multiple. In particular, annual demand in scenarios I and II is only one-fourth or one-half, respectively, of the base, while in scenarios IV and V demand is either twice or four times as large as the base scenario. Let’s first focus on how the demand rate influences the EOQ. As we would expect, as demand increases, the optimal order quantity increases. But it doesn’t increase at the same rate of demand. For example, if the demand rate increases from 156 bottles per year (scenario I) to 624 bottles per year (scenario III), demand has increased by a factor of four (624/156 = 4), but the EOQ only doubles, from 50 to 100 bottles. Similarly, scenario V has 16 times more demand than ­scenario I (2496 versus 156 bottles per year), yet the EOQ in scenario V is only four times greater. The annual EOQ cost also increases as demand increases. But like the EOQ, it doesn’t increase at the same rate as demand. As a result, the annual EOQ cost becomes a smaller and Economies of scale Describes a relationship between operational efficiency and demand in which greater demand leads to a more efficient process.