Chapter 13 PDF: Linking Product Availability to Profits

Here is the converted markdown format of the text in the images you sent: # Chapter 13 **Linking Product Availability to Profits** ## Learning Objectives After reading this chapter, you will be able to: * **13.1** Identify the factors affecting the optimal level of product availability. * *...

Here is the converted markdown format of the text in the images you sent: # Chapter 13 **Linking Product Availability to Profits** ## Learning Objectives After reading this chapter, you will be able to: * **13.1** Identify the factors affecting the optimal level of product availability. * **13.2** Evaluate the cycle service level that maximizes profits. * **13.3** Use basic managerial levers to improve supply chain profitability. * **13.4** Discuss how speed can improve profits in a supply chain. * **13.5** Understand conditions under which postponement is valuable in a supply chain. * **13.6** Allocate limited supply capacity among multiple products to maximize expected profits. In this chapter, we explore how the level of product availability offered to customers affects expected profits. The chapter examines factors that influence the optimal cycle service level. We discuss and demonstrate how different managerial levers can be used to match supply and demand to improve supply chain profitability. ## 13.1 Identify the factors affecting the optimal level of product availability. **Factors Affecting the Desired Level of Product Availability** The level of product availability is measured using the cycle service level or the fill rate, which are metrics for the amount of customer demand satisfied from available inventory. The level of product availability, also referred to as the customer service level, is one of the primary measures of a supply chain's responsiveness. A supply chain can use a high level of product availability to improve its responsiveness and attract customers, thus increasing revenue for the supply chain. However, a high level of product availability requires large inventories, which raise supply chain costs. Therefore, a supply chain must achieve a balance between the level of availability and the cost of inventory. The optimal level of product availability is one that maximizes expected supply chain profits. In the fourth quarter of 2008, U.S. inventories shot up by $6.2 billion because of the rapid decline in demand that hit retailers and manufacturers. For some manufacturers, the situation was exaggerated because of the excess inventory of raw materials, such as steel and plastics, that they had built up while anticipating price increases. Retailers were also hit hard, with some, such as Saks Fifth Avenue, slashing prices by 70 percent during the holiday season to spur demand. The excess inventories and the drop in demand led to several retailers, such as Steve & Barry's and Circuit City, declaring bankruptcy during this period. In contrast, Nintendo missed out on an estimated $1.3 billion in sales during the 2007 holiday season because of a failure to meet soaring global demand for its Wii video game console. These examples make clear that having too high or too low a level of product availability has a significant impact on supply chain profits. Whether the optimal level of availability is high or low depends on how a particular company believes it can maximize profits. Nordstrom has focused on providing a high level of product availability and has used its reputation for responsiveness to become a successful department store chain. However, prices at Nordstrom are higher than those at a discount store, where the level of product availability is lower. Power plants ensure that they (almost) never run out of fuel because a shutdown is extremely expensive, resulting in several days of lost production. Some power plants try to maintain several months' fuel supply to avoid any probability of running out. In contrast, most supermarkets carry only a few days' supply of product, and out-of-stock situations do occur with some frequency. The Internet allows a customer to easily shop at an alternative store if the first choice is out of stock. This competitive environment puts pressure on online retailers to increase their level of availability. Simultaneously, significant price competition has lowered prices online. Web retailers with excess inventory find it difficult to be profitable. Providing the optimal level of product availability is thus a key to success online. To understand the factors that influence the optimal level of product availability, consider L. L. Bean, a large mail-order company that sells apparel. One of the products L. L. Bean sells is ski jackets, for which the selling season is from November to February. The buyer at L. L. Bean currently purchases the entire season's supply of ski jackets from the manufacturer before the start of the selling season. Providing a high level of product availability requires the purchase of a large number of jackets. Although a high level of product availability is likely to satisfy all demand that arises, it is also likely to result in a large number of unsold jackets at the end of the season. Given that leftover jackets will have to be sold at a significant discount, L. L. Bean is likely to lose money on discounted jackets. In contrast, a low level of product availability is likely to result in few unsold jackets. However, it is quite likely that L. L. Bean will have to turn away customers who are willing to buy jackets, because they are sold out. In this scenario, L. L. Bean loses potential profit by losing customers. The buyer at L. L. Bean must balance the loss from having too many unsold jackets (if the number of jackets ordered is more than demand) and the lost profit from turning away customers (if the number of jackets ordered is less than demand) when deciding the level of product availability. L. L. Bean can perfectly match supply and demand if demand is predictable. Thus, deciding on an optimal level of product availability makes sense only in the context of demand uncertainty. Traditionally, many firms have forecast a consensus estimate of demand without any measure of uncertainty. In this setting, firms do not make a decision regarding the level of availability; they simply order the consensus forecast. Since the beginning of the twenty-first century, firms have developed a better appreciation for uncertainty and have started developing forecasts that include a measure of uncertainty. Incorporating uncertainty and deciding on the optimal level of product availability can increase profits relative to using a consensus forecast. L. L. Bean has a buying committee that decides on the quantity of each product to be ordered. Based on demand over the past few years, the buyers have estimated the demand distribution for a women's red ski parka to be as shown in *Table 13-1*. This is a deviation from its traditional practice of using the average historical demand as the consensus forecast. To simplify the discussion, we assume that all demand is in hundreds of parkas. The manufacturer also requires that L. L. Bean place orders in multiples of 100. In *Table 13-1*, $p_i$ is the probability that demand equals $D_i$, and $P_i$ is the probability that demand is less than or equal to $D_i$. From *Table 13-1*, we evaluate the expected demand for parkas as **Table 13-1 Demand Distribution for Parkas at L. L. Bean** $Expected\ demand = \sum{D_i p_i} = 1,026$ Under the old policy of ordering the expected value, the buyers would have ordered 1,000 parkas. However, demand is uncertain, and Table 13-1 shows that there is a 51 percent probability that demand will be 1,000 or less. Thus, a policy of ordering a thousand parkas results in a cycle service level of 51 percent at L. L. Bean. The buying committee must decide on an order size and cycle service level that maximizes the expected profits from the sale of parkas at L. L. Bean. The loss that L. L. Bean incurs from an unsold parka and the profit that it makes on each parka it sells influences the buying decision. Each parka costs L. L. Bean c = $45 and is priced in the catalog at p = $100. Any unsold parkas at the end of the season are sold at the outlet store for $50. Holding the parka in inventory and transporting it to the outlet store costs L. L. Bean $10. Thus, L. L. Bean recovers a salvage value of s = $40 for each parka that is unsold at the end of the season. L. L. Bean makes a profit of p - c = $55 on each parka it sells during the season and incurs a loss of c - s = $5 on each unsold parka that is sold at the outlet store. We ignore any impact that lack of parka availability has on future purchases by the customer. The expected profit from ordering 1,000 parkas is given as $Expected\ profit = \sum_{i=1}^{10} [D_i (p - c) – (1,000 – D_i) (c - s)]p_i + \sum_{i=11}^{17} 1,000(p - c)p_i$ $= [400 × 55 - 600 × 5] × 0.01 + [500 × 55 - 500 × 5] × 0.02$ $+[600 × 55 - 400 × 5] × 0.04 + [700 × 55 – 300 × 5] × 0.08$ $+[800 × 55 – 200 × 5] × 0.09 + [900 × 55 – 100 × 5] × 0.11$ $+[1000 × 55 - 0 × 5] × 0.16 + 1000 × 55 × 0.20 + 1000 × 55 × 0.11$ $+1000 × 55 × 0.10 + 1000 × 55 × 0.04 + 1000 × 55 × 0.02$ $+1000 × 55 × 0.01 + 1000 × 55 × 0.01$ $= $49,900 **Table 13-2 Expected Marginal Contribution of Each Additional 100 Parkas** To decide whether to order 1,100 parkas, the buying committee must determine the impact of buying the extra 100 units. If 1,100 parkas are ordered, the extra 100 are sold (for a profit of $5,500) if demand is 1,100 or higher. Otherwise, the extra 100 units are sent to the outlet store at a loss of $500. From Table 13-1, we see that there is a probability of 0.49 that demand is 1,100 or higher and a 0.51 probability that demand is 1,000 or less. Thus, we deduce the following: * Expected profit from the extra 100 parkas $= 5,500$ x $Prob(demand\ ≥ 1,100) - 500$ × $Prob(demand$ < $1,100)= $5,500 × 0.49 – $500 × 0.51 = $2,440 The total expected profit from ordering 1,100 parkas is thus $52,340, which is almost 5 percent higher than the expected profit from ordering 1,000 parkas. Using the same approach, we evaluate the marginal contribution of each additional 100 parkas as in **Table 13-2** (see worksheet Table 13-1, 2 in spreadsheet Chapter13-examples). Note that the expected marginal contribution is positive up to 1,300 parkas, but it is negative from that point on. Thus, the optimal order size is 1,300 parkas. From Table 13-2, we have * Expected profit from ordering 1,300 parkas = $49,900 + $2,440 + $1,240 + $580 = $54, 160 This is more than an 8 percent increase in profitability relative to the policy of ordering the expected value of 1,000 parkas. A plot of total expected profits versus the order quantity is shown in *Figure 13-1*. The optimal order quantity maximizes the expected profit. For L. L. Bean, the optimal order quantity is 1,300 parkas, which provides a CSL of 92 percent. Observe that with a CSL of 0.92, L. L. Bean has a fill rate that is much higher. If demand is 1,300 or less, L. L. Bean achieves a fill rate of 100 percent, because all demand is satisfied. If demand is more than 1,300 (say, D), part of the demand (D - 1,300) is not satisfied. In this case, a fill rate of 1,300/D is achieved. Overall, the fill rate achieved at L. L. Bean if 1,300 parkas are ordered is given by The image shows a simple line plot that portrays the expected profile at LL Bean as you approach "R", a certain Order Quantity. $fr = 1 × Prob (demand ≤ 1,300) + \sum_{D_i>1,300} (1,300/D_i)p_i$ $= 1 × 0.92 + (1300/1400) × 0.04 + (1300/1500) × 0.02$ $+(1300/1600) × 0.01 + (1300/1700) × 0.01$ $= 0.99$ Thus, with a policy of ordering 1,300 parkas, L. L. Bean satisfies, on average, 99 percent of its demand from parkas in inventory. Observe that there are two key parameters that influence the optimal cycle service level at L. L. Bean. The first parameter is the loss incurred by L. L. Bean for each unsold unit at the end of the selling season. We refer to this parameter as the *cost of overstocking* and denote it by $C_o$. L. L. Bean has a cost of overstocking of $C_o = c - s = $5. The second parameter is the margin of each jacket, which represents the opportunity lost by L. L. Bean when it misses a sale because it is out of stock of the jacket. We refer to this parameter as the *cost of understocking* and denote it by $C_u$. L. L. Bean has a cost of understocking of $C_u = p - s = $55. In general, the cost of understocking should include the margin lost from current sales, as well as future sales if the customer does not return. In summary, the two key factors that influence the optimal level of product availability are: * Cost of overstocking the product, $C_o$ * Cost of understocking the product, $C_u$ **Summary of Learning Objective 1** The optimal level of product availability is affected by the cost of overstocking and the cost of understocking. The cost of overstocking refers to the loss incurred when the supply chain has an unsold unit of the product. The cost of understocking refers to the current and future margin lost when the supply chain turns away one unit of demand because it is short of the product. ## **Test Your Understanding** **13.1.1** A high level of product availability requires \_\_\_\_\_\_\_\_, which raises supply chain costs. * **O** large inventories * **O** increased revenues * **O** reduced costs * **O** understocking the product **13.1.2** Whether the optimal level of product availability is high or low depends on where a particular company believes they can * **O** minimize cost. * **O** maximize revenue. * **O** maximize profits. * **O** maximize product availability. ## 13.2. Evaluate the cycle service level that maximizes profits. **Evaluating the Optimal Level of Product Availability** The optimal level of product availability maximizes expected profits and is evaluated based on the tradeoff between the cost of understocking and the cost of overstocking. In this section we develop formulae to calculate the optimal service level for both seasonal products (such as Christmas ornaments) as well as continuously stocked items (such as detergent). **Optimal Cycle Service Level for Seasonal Items with a Single Order in a Season** In this section, we focus attention on seasonal products such as ski jackets, for which all leftover items must be disposed of at the end of the season. The assumption is that the season starts with product from the order placed and any leftover inventory is disposed at a discounted price at the end of the season. Assume a retail price per unit of p, a cost of c, and a salvage value of s. We consider the following inputs: * $C_o$: Cost of overstocking by one unit, $C_o = c - s$ * $C_u$: Cost of understocking by one unit, $C_u = p - c$ * $CS\L^*$: Optimal cycle service level * $O^*$: Corresponding optimal order size $CSL^*$ is the probability that demand during the season will be at or below $O^*$. At the optimal cycle service level $CSL^*$, the marginal contribution of purchasing an additional unit is zero. If the order quantity is raised from $O^*$ to $O^* + 1$, the additional unit sells if demand is larger than $O^*$. This occurs with probability $1 – CSL^*$ and results in a contribution of $p - c$. We thus have * Expected benefit of purchasing extra unit = $(1 – CSL^*)(p - c)$ The additional unit remains unsold if demand is at or below $O^*$. This occurs with probability $CSL^*$ and results in a cost of c-s. We thus have * Expected cost of purchasing extra unit = $CSL^*(c - s)$ Thus, the expected marginal contribution of raising the order size from $O^*$ to $O^* + 1$ is given by * $(1 - CSL^*)(p - c) – CSL^*(c - s)$ Because the expected marginal contribution must be 0 at the optimal cycle service level, we have The optimal cycle service level as a function of the ratio of the cost of overstocking and the cost of understocking is shown in Figure 13-2. Observe that as this ratio gets smaller, the optimal level of product availability increases. This fact explains the difference in the level of product availability between a high-end store such as Nordstrom and a discount store. Nordstrom has higher margins and thus a higher cost of understocking. It should thus provide a higher level of product availability than a discount store with lower margins and, as a result, a lower cost of stocking out. A more rigorous derivation of Equation 13.1 is provided in **Appendix 13A**. The optimal $CSL^*$ is also referred to as the *critical fractile*. The resulting optimal order quantity maximizes the firm's expected profit. If demand during the season is normally distributed, with a mean of u and a standard deviation of o, the optimal order quantity is given by The image is of a function with C sub O over C sub U on the x axis and CSL* on the y axis. When demand is normally distributed, with a mean of u and a standard deviation of o, the expected profit from ordering O units is given by Expected profit $ (p- s)µFs ((O – μ)/σ))– (p – s)σfs ((O – μ)/σ) – O (c – s)F (O, μ, σ) + O (p – c) [1 – F(O, μ, σ)]$ The derivation of this formula is provided in **Appendix 13B and Appendix 13C**. Here, Fs is the standard normal cumulative distribution function and fs is the standard normal density function discussed in **Appendix 12A in Chapter 12**. The expected profit from ordering O units is evaluated in Excel using Equations 12.22, 12.25, and 12.26, as follows: Example 13-1 illustrates the use of Equations 13.1 and 13.2 to obtain the optimal cycle service level and order quantity (see worksheet Example 13-1 in spreadsheet Chapter 13-examples). **Example 13-1 Evaluating the Optimal Service Level for Seasonal Items** The manager at Sportmart, a sporting goods store, has to decide on the number of skis to purchase for the winter season. Based on past demand data and weather forecasts for the year, management has forecast demand to be normally distributed, with a mean of µ = 350 and a standard deviation of o = 100. Each pair of skis costs c = $100 and retails for p = $250. Any unsold skis at the end of the season are disposed of for $85. Assume that it costs $5 to hold a pair of skis in inventory for the season. How many skis should the manager order to maximize expected profits? *Analysis:* In this case, we have * Salvage value $ = $85 - $5 =$80 * Cost of understocking $C_u = p - c = $250 - $100 = $150 * Cost of overstocking $C_o = c - s = $100 - $80 = $20 Using Equation 13.1, we deduce that the optimal CSL is $CSL^* = Prob (Demand ≤ O^*) = C_u / (C_u + C_o) = 150/(150 + 20) = 0.88$ Using Equation 13.2, the optimal order size is $O^* = NORMINV (CSL^*, μ, σ) = NORMINV (0.88, 350, 100) = 468$ Thus, it is optimal for the manager at Sportmart to order 468 pairs of skis, even though the expected number of sales is 350. In this case, because the cost of understocking is much higher than the cost of overstocking, management is better off ordering more than the expected value to cover for the uncertainty of demand. Using Equation 13.3, the expected profits from ordering $O^*$ units are $Expected \ profits = (p-s)µNORMDIST [(O^* – μ)/σ, 0, 1, 1] + (p - s)σNORMDIST [(O^* – μ)/σ, 0,1,0)]- O^* (c – s) NORMDIST (O^*, μ, σ, 1) + O^* (p - c) [1 - NORMDIST (O^*, μ, σ, 1)]$ $170\times350\times NORMDIST(1.18, 0, 1, 1)$ $170\times100\times NORMDIST(1.18, 0, 1, 0)$ $468\times20\times NORMDIST(468, 350, 100, 1)$ $+468\times150\times [1 \hspace{0.2cm} - \hspace{0.2cm}NORMDIST(468, 350, 100, 1)] = $49, 146 The expected profit from ordering 350 pairs of skis can be evaluated as $45,718. Thus, ordering 468 pairs results in an expected profit that is almost 8 percent higher than the profit obtained from ordering the expected value of 350 pairs. In the above example, if the manager ordered 420 (520) units instead of the optimal quantity of 468, expected profits would decrease by about 1 percent to $48,671 ($48,789). As a result, companies should avoid spending an inordinate amount of effort to get exact estimates of various costs used to evaluate optimal levels of product availability. A reasonable approximation of the costs will generally produce expected profits that are close to optimal. Firms' efforts to set levels of product availability often get bogged down in debate over the cost of stocking out. The sometimes controversial nature of this cost and its hard-to-quantify components (such as loss of customer goodwill) make it a difficult number for people from different functions to agree on. However, it is often not necessary to estimate a precise cost of stocking out. Using a range of the cost, a manager can identify appropriate levels of availability and the associated profits. Often, profits do not change significantly in the range, thus eliminating the need for a more precise estimation of the cost of stocking out. When O units are ordered, a firm is left with either too much or too little inventory, depending on demand. When demand is normally distributed, with expected value u and standard deviation 6, the expected quantity overstocked at the end of the season is given by $Expected \ overstock = (O-μ)Fs ((O – μ)/σ) - σfts(((O – ))μ)/σ$ The derivation of this formula is provided in **Appendix 13D**. The formula can be evaluated using Excel as follows: The expected quantity understocked at the end of the season is given by $Expected \ understock = (μ - O) [1 - Fs((O)/σ)] + σfts(((O – ))μ)/σ$ The image has: μ (mu): population variable σ (sigma): standard deviation of all the values in a population The derivation of this formula is provided in **Appendix 13E**. The formula can be evaluated using Excel as follows: Example 13-2 illustrates the use of Equations 13.4 and 13.5 to evaluate the quantity expected to be overstocked and understocked as a result of an ordering policy (see worksheet Example 13-2). **Example 13-2 Evaluating Expected Overstock and Understock** Demand for skis at Sportmart is normally distributed with a mean of µ = 350 and a standard deviation of σ = 100. The manager has decided to order 450 pairs of skis for the upcoming season. Evaluate the expected overstock and understock as a result of this policy. *Analysis:* We have an order size O = 450. An overstock results if demand during the season is less than 450. The expected overstock can be obtained using Equation 13.4 as * $ Expected \ overstock = (Ο – μ)NORMDIST ((O – μ)/σ, 0, 1, 1) + σNORMDIST ((O – μ)/σ, 0, 1, 0) $ = (450 – 350)NORMDIST [(450 – 350)/100, 0, 1, 1] + 100 NORMDIST [(450 – 350)/100, 0, 1, 0] = 108 Thus, the policy of ordering 450 pairs of skis results in an expected overstock of 108 pairs. An understock occurs if demand during the season is higher than 450 pairs. The expected understock can be evaluated using Equation 13.5 as follows: $Expected\ understock= (μ – Ο) (1 – NORMDIST [(Ο – μ)/σ, 0, 1, 1]) + σNORMDIST [(O – μ)/σ, 0, 1, 0)]$ =(350-450)(1- NORMDIST [(450-350)/100, 0, 1, 1]) + 100 NORMDIST [(450-350)/100, 0, 1, 0]=8 Thus, the policy of ordering 450 pairs results in an expected understock of 8 pairs. Note that both the expected understock and overstock are positive. This result may seem counterintuitive, but it makes sense because the values used to calculate an expected understock or overstock are always greater than or equal to zero. For example, if demand is 500 and 450 jackets are in inventory, there is an understock of 50 and an overstock of 0 (not -50). This guarantees that the expected value of each will be greater than or equal to zero. **One-Time Orders in the Presence of Quantity Discounts** In this section, we consider a buyer who has to make a single order when the seller offers a price discount based on the quantity purchased. Such a situation may arise in the context of seasonal items such as apparel, for which the manufacturer offers a lower price per unit if order quantities exceed a given threshold. Such decisions also arise at the end of the life cycle for a product or spare parts. Future demand for the product or spare parts is uncertain, and the buyer has a single opportunity to order. The buyer must account for the discount when selecting the order size. Consider a retailer of spare parts that has one last chance to order parts before the manufacturer stops production. The part has a retail price per unit of p, a cost to the retailer (without discount) of c, and a salvage value of s. The manufacturer has offered a discounted price of $c_d$ if the retailer orders at least K units. The retailer can make its order size decision using the following steps: 1. Using $C_o = c − s$ and $C_¹ = p - c$, evaluate the optimal cycle service level $CSL^*$ and order size $O^*$ without a discount using Equations 13.1 and 13.2, respectively. Evaluate the expected profit from ordering $O^*$ using Equation 13.3. 2. Using $C_o = c_d - s$ and $C_¹ = p – c_d$, evaluate the optimal cycle service level $CSL_¹$ and order size $O_¹$ with a discount using Equations 13.1 and 13.2, respectively. If $O*_d > K$, evaluate the expected profit from ordering $O*_d$ units using Equation 13.3. If $O*_d < K$, evaluate the expected profit from ordering K units using Equation 13.3. * Order $O^*$ units if the profit in step 1 is higher. If the profit in step 2 is higher, order $O_d^*$ units if $O_d^* > K$ or K units if $O*_d < K$. We illustrate the procedure in Example 13-3 (see worksheet Example 13-3). **Example 13-3 Evaluating Service Level with Quantity Discounts** SparesRUs, an auto parts retailer, must decide on the order size for a 20-year-old model of brakes. The manufacturer plans to discontinue production of these brakes after this last production run. SparesRUs has forecast remaining demand for the brakes to be normally distributed, with a mean of 150 and a standard deviation of 40. The brakes have a retail price of $200. Any unsold brakes are useless and have no salvage value. The manufacturer plans to sell each brake for $50 if the order is for less than 200 brakes and $45 if the order is for at least 200 brakes. How many brakes should SparesRUs order? *Analysis:* In step 1, we calculate the optimal order quantity at the regular price c = $50: * Cost of understocking = $C_u = p - c = $200 - $50 = $150 * Cost of overstocking $C_o = c - s = $50 - $0 = $50 Using Equation 13.1, the optimal CSL without a discount is $CSL^* = Prob (Demand ≤ O^*) = C_u / (C_u + C_o) = 150/(150+50) = 0.75$ Using Equation 13.2, the optimal order size is $O^* = NORMINV (CSL^*, μ, σ) = NORMINV (0.75, 150, 40) = 177$ Using Equation 13.3, the expected profit if SparesRUs does not go after the discount is Expected profit from ordering 177 units = $19,958 In step 2, we consider the discount price $c_d = $45 and obtain * Cost of understocking Cu = $p-Cd$ = $200-$45 = $155 * Cost of overstocking $C_o = Cd – s = $45-$0 = $45 Using Equation 13.1, we deduce that the optimal CSL with the discount price is $CSLi^* = Prob (Demand ≤ O_d^*) = C_u / (C_u + C_o) = 155/(155+45) = 0.775$ Using Equation 13.2, the optimal order size at the discount price is: $O_d^* = NORMINV (CSLi^*, μ, σ) = NORMINV (0.775, 150, 40) = 180$ Given that 180 < 200, the retailer must order at least 200 brakes to benefit from the discount. Thus, we calculate the expected profit from ordering 200 units using Equation 13.3 as Expected profits from ordering 200 units at $45 each = $20,595 It is thus optimal for SparesRUs to order 200 brakes to take advantage of the quantity discount. The expected overstock can be calculated using Equation 13.4 to be 52. **Desired Cycle Service Level for Continuously Stocked Items** In this section, we focus on products such as detergent that are ordered repeatedly by a retail store such as Walmart. Walmart uses safety inventory to increase the level of availability and decrease the probability of stocking out between successive deliveries. If detergent is left over in a replenishment cycle, it can be sold in the next cycle. It does not have to be disposed of at a lower cost. However, a holding cost is incurred as the product is carried from one cycle to the next. The manager at Walmart is faced with the issue of deciding the CSL to aim for. Two extreme scenarios should be considered: 1. All demand that arises when the product is out of stock is backlogged and filled later, when inventories are replenished. 2. All demand arising when the product is out of stock is lost. The reality in most instances is somewhere in between, with some of the demand lost and other customers returning when the product is in stock. We consider both extreme cases. We assume that demand per unit time is normally distributed, along with the following inputs: DEMAND DURING STOCKOUT IS BACKLOGGED We first consider the case in which all demand arising when the product is out of stock is backlogged. Because no demand is lost, minimizing costs becomes equivalent to maximizing profits. As an example, consider a Walmart store selling detergent. The store manager offers a rain check at a discount of C₁ to each customer wanting to buy detergent when it is out of stock. Assume that the rain check ensures that all these customers return when inventory is replenished. Thus, Cu is the backlogging or understocking cost per unit. If the store manager increases the level of safety inventory, more orders are satisfied from stock, resulting in lower backlogs. This decreases the backlogging or understocking cost. However, the cost of holding inventory increases. We start by considering the costs and benefits of holding an additional unit of safety inventory in each replenishment cycle. If the safety inventory is increased from ss (which provides a cycle service level, CSL) to ss + 1, the supply chain incurs cost to hold the additional unit of inventory for a replenishment cycle (which has duration Q/D). The additional unit of safety inventory is beneficial (the benefit equals the cost of understocking C₁) if demand during the replenishment cycle is such that more than ss units of safety inventory are consumed \[**this happens with probability (1 –** $CSL$**)**]. We thus have the following: Increased cost per replenishment cycle of additional safety inventory of 1 unit = $(Q/D)H$ Benefit per replenishment cycle of additional safety inventory of 1 unit = $(1 – CSL)Cu$ In this case, the optimal cycle service level is obtained by equating the