《供应链系统设计与管理》课程授课教案（讲义）Chapter 02（Lecture 4）Multiple Order Opportunities

全英文课《DesigningandManagingSupplyChainSystem》授课教案Chapter2(Lecture4)MultipleOrderOpportunitiesOBJECTIVES(1) Understanding Review Policy(2) Calculating the order quantityTEACHING CONTENT2.2.5MultipleOrderOpportunitiesThe situations we considered above all focus on a single ordering or productionopportunity. This may be the case for fashion items where the selling season is shortand there is no second opportunity to reorder products based on realized customerdemand. In many practical situations, however, the decision maker may orderproducts repeatedly at any time during the year.To manage inventory effectively, the distributor needs to decide when and howmuch to order. We distinguish between two types of policies:1.Continuous review policy,in which inventory is reviewed continuously,andan order is placed when the inventory reaches a particular level, or reorder point. Thistypeofpolicyismostappropriatewheninventorycanbecontinuouslyreviewed-forexample,whencomputerizedinventorysystemsareused.2. Periodic review policy, in which the inventory level is reviewed at regularintervals and an appropriate quantity is ordered after each review. This type of policyis most appropriate for systems in which it is impossible or inconvenient to frequentlyreview inventory and place orders if necessary.2.2.6 Continuous Review PolicyWe first consider a system in which inventory is continuously reviewed. Such areview system typically provides a more responsive inventory management strategythan the one associated with a periodic review system (why?).Wemakethefollowing additional assumptions1.Daily demand is random and follows a normal distribution.In other words, weassume that the probabilistic forecast of daily demand follows the famous bell-shapedcurve. Note that we can completely describe normal demand by its average andstandard deviation.2. Every time the distributor places an order from the manufacturer, thedistributor pays a fixed cost, K, plus an amount proportional to the quantity ordered.3. Inventory holding cost is charged per item per unit time.4.Inventorylevel is continuouslyreviewed, and ifan orderisplaced,the orderarrives after the appropriate lead time.5. If a customer order arrives when there is no inventory on hand to fill the order(i.e., when the distributor is stocked out), the order is lost.6. The distributor specifies a required service level. The service level is the

全英文课《Designing and Managing Supply Chain System》 授课教案 Chapter 2(Lecture 4) Multiple Order Opportunities OBJECTIVES (1) Understanding Review Policy (2) Calculating the order quantity TEACHING CONTENT 2.2.5 Multiple Order Opportunities The situations we considered above all focus on a single ordering or production opportunity. This may be the case for fashion items where the selling season is short and there is no second opportunity to reorder products based on realized customer demand. In many practical situations, however, the decision maker may order products repeatedly at any time during the year. To manage inventory effectively, the distributor needs to decide when and how much to order. We distinguish between two types of policies: 1. Continuous review policy, in which inventory is reviewed continuously, and an order is placed when the inventory reaches a particular level, or reorder point. This type of policy is most appropriate when inventory can be continuously reviewed—for example, when computerized inventory systems are used. 2. Periodic review policy, in which the inventory level is reviewed at regular intervals and an appropriate quantity is ordered after each review. This type of policy is most appropriate for systems in which it is impossible or inconvenient to frequently review inventory and place orders if necessary. 2.2.6 Continuous Review Policy We first consider a system in which inventory is continuously reviewed. Such a review system typically provides a more responsive inventory management strategy than the one associated with a periodic review system (why?). We make the following additional assumptions. 1. Daily demand is random and follows a normal distribution. In other words, we assume that the probabilistic forecast of daily demand follows the famous bell-shaped curve. Note that we can completely describe normal demand by its average and standard deviation. 2. Every time the distributor places an order from the manufacturer, the distributor pays a fixed cost, K, plus an amount proportional to the quantity ordered. 3. Inventory holding cost is charged per item per unit time. 4. Inventory level is continuously reviewed, and if an order is placed, the order arrives after the appropriate lead time. 5. If a customer order arrives when there is no inventory on hand to fill the order (i.e., when the distributor is stocked out), the order is lost. 6. The distributor specifies a required service level. The service level is the

全英文课《DesigningandManagingSupplyChainSystem》授课教案probability of not stocking out during lead time. For example, the distributor mightwant to ensure that the proportion of lead times in which demand is met out of stockis 95 percent. Thus, the required service level is 95 percent in this case.To characterize the inventory policy that the distributor should use, we need thefollowinginformation:AVG =Average daily demand faced by the distributorSTD=Standard deviation of daily demand faced by the distributorL = Replenishment lead time from the supplier to the distributor in daysh = Cost of holding one unit of the product for one day at the distributora = service level. This implies that the probability of stocking out is 1 - aIn addition, we need to define the concept of inventory position. The inventoryposition at any point in time is the actual inventory at the warehouse plus itemsordered by the distributor that have not yet arrived minus items that are backordered.To describe the policy that the distributor should use, we recall the intuitiondeveloped when we considered a single period inventory model with initial inventory.Inthatmodel.when inventory wasbelowa certainlevel,we ordered enoughtoraisethe inventory up to another, higher level.For the continuous review model, weemploy a similar approach, known as a (Q, R) policy—whenever inventory level fallsto a reorder level R, place an order for Q units.The reorder level, R,consists of two components.Thefirst is the averageinventory during lead time, which is the product of average daily demand and the leadtime. This ensures that when the distributor places an order, the system has enoughinventory to cover expected demand during lead time. The average demand duringleadtimeisexactlyL ×AVGThe second component represents the safety stock, which is the amount ofinventory that the distributor needs to keep at the warehouse and in the pipeline toprotect against deviations from average demand during lead time. This quantity iscalculated as follows:ZXSTDXVLwherez is a constant, referred to as the safety factor.This constant is associated withthe service level. Thus, the reorder level is equal toTABLE2-2SERVICELEVELANDTHESERVICEFACTOR,ZService level93%94%95%97%98%99%99.9%90%91%92%96%1.291.652.05z1.341.411.481.561.751.882.333.08The safety factor z is chosen from statistical tables to ensure that the probabilityof stockouts during lead time is exactly 1 -a,This implies that the reorder levelmust satisfy:Pr(Demandduring lead time≥LXAVG+z×STD×VL)=1 -aTable 2-2 provides a list of z values for different values of the service level dWhat about the order quantity, Q? Although calculating the optimal Q for thismodel is not easy, the EOQ order quantity we developed previously is very effective

全英文课《Designing and Managing Supply Chain System》 授课教案 probability of not stocking out during lead time. For example, the distributor might want to ensure that the proportion of lead times in which demand is met out of stock is 95 percent. Thus, the required service level is 95 percent in this case. To characterize the inventory policy that the distributor should use, we need the following information: AVG = Average daily demand faced by the distributor STD = Standard deviation of daily demand faced by the distributor L = Replenishment lead time from the supplier to the distributor in days h = Cost of holding one unit of the product for one day at the distributor α = service level. This implies that the probability of stocking out is 1 - α In addition, we need to define the concept of inventory position. The inventory position at any point in time is the actual inventory at the warehouse plus items ordered by the distributor that have not yet arrived minus items that are backordered. To describe the policy that the distributor should use, we recall the intuition developed when we considered a single period inventory model with initial inventory. In that model, when inventory was below a certain level, we ordered enough to raise the inventory up to another, higher level. For the continuous review model, we employ a similar approach, known as a (Q, R) policy—whenever inventory level falls to a reorder level R, place an order for Q units. The reorder level, R, consists of two components. The first is the average inventory during lead time, which is the product of average daily demand and the lead time. This ensures that when the distributor places an order, the system has enough inventory to cover expected demand during lead time. The average demand during lead time is exactly The second component represents the safety stock, which is the amount of inventory that the distributor needs to keep at the warehouse and in the pipeline to protect against deviations from average demand during lead time. This quantity is calculated as follows: where z is a constant, referred to as the safety factor. This constant is associated with the service level. Thus, the reorder level is equal to TABLE 2-2 SERVICE LEVEL AND THE SERVICE FACTOR, Z The safety factor z is chosen from statistical tables to ensure that the probability of stockouts during lead time is exactly 1 - α. This implies that the reorder level must satisfy: Table 2-2 provides a list of z values for different values of the service level α. What about the order quantity, Q? Although calculating the optimal Q for this model is not easy, the EOQ order quantity we developed previously is very effective

全英文课《DesigningandManagingSupplyChainSystem》授课教案for this model. Recall from this model that the order quantity, Q, is calculated asfollows:Q-VKXAGIfthereisnovariabilityincustomerdemand,thedistributorwouldorderQitemswhentheinventoryisatlevelLXAvGsince ittakesLdaystoreceivetheorderHowever,there isvariability in demand,sothedistributorplaces anorderforQitemswhenever the inventory position is at the reorder level, R.Figure 2-9 illustrates the inventory level over time when this type of policy isimplemented. What is the average inventory level in this policy? Observe that,between two successive orders,the minimum level of inventory is achieved rightbefore receiving an order, while the maximum level of inventory is achievedimmediately after receiving the order. The expected level of inventory beforereceiving the order is the safety stock.ZXSTDXVLwhiletheexpectedlevelofinventoryimmediatelyafterreceivingtheorderisQ+XSTDXVLFIGURE 2-9 Inventory level as a function of time in a (Q, R) policyEXAMPLEConsider a distributor of TV sets that orders from a manufacturer and sells toretailers. Suppose the distributor of the TV sets is trying to set inventory policies atthe warehouse for one of the TV models. Assume that whenever the distributor placesan order for TV sets, there is a fixed ordering cost of $4,500, which is independent ofthe order size.The cost of a TV set to the distributor is $250 and annual inventoryholding cost is about 18 percent of the product cost.Replenishment time (ie..leadtime) is about two weeks.Table 2-3 provides data on the number of TV sets sold to retailers in each of thelast 12months.Given that the distributor would like to ensure 97 percent service level,what is the reorder level and the order quantity that the distributor should use?TABLE2-3HISTORICALDATAMonthSeptOctNov.Dec.Fob.Mar.Jan.Apr.MayJuneJulyAug198Sales20015210022128717615124630998156Table 2-3 implies that average monthly demand is 191.17 and the standarddeviationofmonthlydemandis66.53

全英文课《Designing and Managing Supply Chain System》 授课教案 for this model. Recall from this model that the order quantity, Q, is calculated as follows: If there is no variability in customer demand, the distributor would order Q items when the inventory is at level L×AVG since it takes L days to receive the order. However, there is variability in demand, so the distributor places an order for Q items whenever the inventory position is at the reorder level, R. Figure 2-9 illustrates the inventory level over time when this type of policy is implemented. What is the average inventory level in this policy? Observe that, between two successive orders, the minimum level of inventory is achieved right before receiving an order, while the maximum level of inventory is achieved immediately after receiving the order. The expected level of inventory before receiving the order is the safety stock. while the expected level of inventory immediately after receiving the order is FIGURE 2-9 Inventory level as a function of time in a (Q, R) policy EXAMPLE Consider a distributor of TV sets that orders from a manufacturer and sells to retailers. Suppose the distributor of the TV sets is trying to set inventory policies at the warehouse for one of the TV models. Assume that whenever the distributor places an order for TV sets, there is a fixed ordering cost of $4,500, which is independent of the order size. The cost of a TV set to the distributor is $250 and annual inventory holding cost is about 18 percent of the product cost. Replenishment time (i.e., lead time) is about two weeks. Table 2-3 provides data on the number of TV sets sold to retailers in each of the last 12 months. Given that the distributor would like to ensure 97 percent service level, what is the reorder level and the order quantity that the distributor should use? TABLE 2-3 HISTORICAL DATA Table 2-3 implies that average monthly demand is 191.17 and the standard deviation of monthly demand is 66.53

全英文课《DesigningandManagingSupplyChainSystem》授课教案Since lead time is two weeks, we transform the average and the standarddeviation to weekly values as follows:Average monthly demandAverageweeklydemand4.3whiledarddeviatiorMonthivystaStandard daviation of weekty demandV4.3These data are provided in Table 2-4. This allows us to calculate average demandduring lead time and safety stock using a safety factor z =1.9 (or, more precisely, 1.88)taken from Table 2-2 based on a 97 percent service level. The reorder point is simplythe sum of the average demand during lead time plus the safety stock. All these dataare presented in Table 2-4.TABLE2-4INVENTORYANALYSISStandargAverageAveragedeviationordemandauringsatotyReorderweekiystockpointdemandleadtimeweeklydemandValue44.5832.0889.1686.20176To determine the order quantity, Q, observe that the weekly inventory holdingcostper TV set is0.18 × 250 = 0.8752or 87 cents. This implies that the order quantity, Q, should be calculated asX445=679Q=V2X4That is, the distributor should place an order for 679 TV sets whenever theinventory level reaches 176 units. Finally, the average inventory level is equal to679/2+86.20=426which means that, on average, the distributor keeps at the warehouse about 1o (=426/44.58)weeksof supply2.2.8Periodic ReviewPolicyIn many real-life situations,the inventory level is reviewed periodically atregular intervals, and an appropriate quantity is ordered after each review. If theseintervals are relatively short (for example, daily), it may make sense to use amodifiedversionofthe(Q,R)policypresented above.Unfortunately,the(Q,R)policycan'tbedirectlyimplemented,sincetheinventorylevelmayfallbelowthereorderpointwhenthe warehouse places an order.To overcome this problem, define two inventory levelss and S, and during each inventory review, if the inventory position falls below s,order enough to raise the inventory position to S. We call this modified (Q, R) policyan (s, S)policy. Although it is difficult to determine the optimal values for s and S,one very effective approximation is to calculate the Q and R values as if this were acontinuous reviewmodel, sets equal toR,and S equal toR+QIf there isa larger time between successive reviews of inventory (weekly ormonthly, for example), it may make sense to always order after an inventory level

全英文课《Designing and Managing Supply Chain System》 授课教案 Since lead time is two weeks, we transform the average and the standard deviation to weekly values as follows: while These data are provided in Table 2-4. This allows us to calculate average demand during lead time and safety stock using a safety factor z = 1.9 (or, more precisely, 1.88) taken from Table 2-2 based on a 97 percent service level. The reorder point is simply the sum of the average demand during lead time plus the safety stock. All these data are presented in Table 2-4.TABLE 2-4 INVENTORY ANALYSIS To determine the order quantity, Q, observe that the weekly inventory holding cost per TV set is or 87 cents. This implies that the order quantity, Q, should be calculated as That is, the distributor should place an order for 679 TV sets whenever the inventory level reaches 176 units. Finally, the average inventory level is equal to which means that, on average, the distributor keeps at the warehouse about 10 (= 426/44.58) weeks of supply. 2.2.8 Periodic Review Policy In many real-life situations, the inventory level is reviewed periodically at regular intervals, and an appropriate quantity is ordered after each review. If these intervals are relatively short (for example, daily), it may make sense to use a modified version of the (Q, R) policy presented above. Unfortunately, the (Q, R) policy can't be directly implemented, since the inventory level may fall below the reorder point when the warehouse places an order. To overcome this problem, define two inventory levels s and S, and during each inventory review, if the inventory position falls below s, order enough to raise the inventory position to S. We call this modified (Q, R) policy an (s, S) policy. Although it is difficult to determine the optimal values for s and S, one very effective approximation is to calculate the Q and R values as if this were a continuous review model, set s equal to R, and S equal to R + Q If there is a larger time between successive reviews of inventory (weekly or monthly, for example), it may make sense to always order after an inventory level

全英文课《DesigningandManagingSupplyChainSystem》授课教案review.Sinceanorderis placed after each inventoryreview,thefixed cost ofplacingan order is a sunk cost and hence can be ignored;presumably,thefixed cost was usedto determine the review interval. The quantity ordered arrives after the appropriateleadtimeWhat inventorypolicy shouldthewarehouseuseinthiscase?Sincefixed costdoesnotplayarole inthis environment,the inventory policy is characterizedbyasingle parameter, the base-stock level. That is, the warehouse determines a targetinventory level, the base-stock level, and each review period, the inventory position isreviewed and the warehouse orders enough to raise the inventory position to thebase-stock level.What is an effective base-stock level? For this purpose, let r be the length of thereview period-we assume that orders are placed every r periods of time. As before, Lis the lead time, AVG is the average daily demand faced by the warehouse, and STDisthestandarddeviationofthisdailydemand.Observe that at the timethe warehouseplaces an order, this order raises theinventorypositiontothebasestocklevel.Thislevel oftheinventoryposition shouldbe enough to protect the warehouse against shortages until the next order arrives.Since the next order arrives after a period of r + L days, the current order should beenough to cover demand during a period of r + L days.FIGURE2-1o Inventory level as a function of time in aperiodic reviewpolicyadThus, the base-stock level should include two components: average demandduring an interval of r + L days, which is equal to(+L)XAVGand the safety stock,which isthe amount of inventorythatthe warehouseneeds tokeep to protect against deviations from average demand during a period of r + L days.This quantity is calculated as follows:XSTDXV+Lwherez isthe safetyfactorFigure 2-10 illustrates the inventory level over time when this type of policy isimplemented.What is the average inventory level in this case? As before, themaximum inventory level is achieved immediately after receiving an order, while theminimum level of inventory is achieved just before receiving an order. It is easy to seethat the expected level of inventory after receiving an order is equal toTXAVG+:XSTDXVF+Lwhile the expected level of inventory before an order arrives is just the safety stock,2XSTDXVF+L

全英文课《Designing and Managing Supply Chain System》 授课教案 review. Since an order is placed after each inventory review, the fixed cost of placing an order is a sunk cost and hence can be ignored; presumably, the fixed cost was used to determine the review interval. The quantity ordered arrives after the appropriate lead time. What inventory policy should the warehouse use in this case? Since fixed cost does not play a role in this environment, the inventory policy is characterized by a single parameter, the base-stock level. That is, the warehouse determines a target inventory level, the base-stock level, and each review period, the inventory position is reviewed and the warehouse orders enough to raise the inventory position to the base-stock level. What is an effective base-stock level? For this purpose, let r be the length of the review period—we assume that orders are placed every r periods of time. As before, L is the lead time, AVG is the average daily demand faced by the warehouse, and STD is the standard deviation of this daily demand. Observe that at the time the warehouse places an order, this order raises the inventory position to the basestock level. This level of the inventory position should be enough to protect the warehouse against shortages until the next order arrives. Since the next order arrives after a period of r + L days, the current order should be enough to cover demand during a period of r + L days. FIGURE 2-10 Inventory level as a function of time in a periodic review policy Thus, the base-stock level should include two components: average demand during an interval of r + L days, which is equal to and the safety stock, which is the amount of inventory that the warehouse needs to keep to protect against deviations from average demand during a period of r + L days. This quantity is calculated as follows: where z is the safety factor. Figure 2-10 illustrates the inventory level over time when this type of policy is implemented. What is the average inventory level in this case? As before, the maximum inventory level is achieved immediately after receiving an order, while the minimum level of inventory is achieved just before receiving an order. It is easy to see that the expected level of inventory after receiving an order is equal to while the expected level of inventory before an order arrives is just the safety stock,

全英文课《DesigningandManagingSupplyChainSystem》授课教案Hence, the average inventory level is the average of these two values, which is equaltoEXAVG,xSTDVr+L2.2.9. Service Level OptimizationSo far we have assumed that the objective of this inventory optimization is todetermine the optimal inventory policy given a specific service level target. Thequestion, of course, is how the facility should decide on the appropriate level ofservice. Sometimes this is determined by the downstream customer. In other words,the retailer can require the facility, for example, the supplier, to maintain a specificlevel of service and the supplier will use that target to manage its own inventoryIn other cases, the facility has the flexibility to choose the appropriate level ofservice. The trades-offs, presented in Figure 2-1l, are clear: everything else beingequal, the higher the service level, the higher the inventory level. Similarly, for thesame inventory level,the longerthe lead timeto thefacility,thelowerthelevel ofservice provided by the facility.Finally,the marginal impact on service leveldecreases with inventory level. That is. the lower the inventory level. the higher theimpactofaunitof inventoryonservicelevelandhenceonexpectedprofit.Thus, one possible strategy, used in retailing, to determine service level for eachSKU is to focus on maximizing expected profit across all, or some, of their productsThat is, given a target service level across all products, we determine service level foreach SKU so as to maximize expected profit. Everything else being equal, servicelevel will be higher for products with1. High profit margin.2.Highvolume.3. Low variability.4. Short lead time.FIGURE2-11 Service level versus inventory level as a function of lead time100.0099.00%98.00%97.00%96.00%95.00%t94.00%X93.00%92.00%91.00%90390440490540InventoryLead time=2--Lead time=4FIGURE2-12ServiceleveloptimizationbySKU

全英文课《Designing and Managing Supply Chain System》 授课教案 Hence, the average inventory level is the average of these two values, which is equal to 2.2.9. Service Level Optimization So far we have assumed that the objective of this inventory optimization is to determine the optimal inventory policy given a specific service level target. The question, of course, is how the facility should decide on the appropriate level of service. Sometimes this is determined by the downstream customer. In other words, the retailer can require the facility, for example, the supplier, to maintain a specific level of service and the supplier will use that target to manage its own inventory. In other cases, the facility has the flexibility to choose the appropriate level of service. The trades-offs, presented in Figure 2-11, are clear: everything else being equal, the higher the service level, the higher the inventory level. Similarly, for the same inventory level, the longer the lead time to the facility, the lower the level of service provided by the facility. Finally, the marginal impact on service level decreases with inventory level. That is, the lower the inventory level, the higher the impact of a unit of inventory on service level and hence on expected profit. Thus, one possible strategy, used in retailing, to determine service level for each SKU is to focus on maximizing expected profit across all, or some, of their products. That is, given a target service level across all products, we determine service level for each SKU so as to maximize expected profit. Everything else being equal, service level will be higher for products with 1. High profit margin. 2. High volume. 3. Low variability. 4. Short lead time. FIGURE 2-11 Service level versus inventory level as a function of lead time. FIGURE 2-12 Service level optimization by SKU

全英文课《DesigningandManagingSupplyChainSystem》授课教案Sskusurahnnargin3,5003003.5000O20C ameart35.410OFR>9%FR95-9%OFR×955The impact of profit optimization on service level is nicely illustrated by Figure2-12.The vertical axis represents annual sales while the horizontal axis representsdemand variability. Each circle is associated with a product and the size of the circleis proportional to profit margin. The target inventory level is 95 percent across allproducts. As you can see, the service level is higher than 99 percent (brown-coloredcircles) for many products with high profit margin, high volume, and low variabilityOntheotherhand,servicelevelislowerthan95percentforproductswithlowprofitmargin,lowvolume,andhighvariabilityQUESTIONS1.Whattherelationshipis between serviceand inventorylevels?2.What impact lead time and lead time variability have on inventory levels?

全英文课《Designing and Managing Supply Chain System》 授课教案 The impact of profit optimization on service level is nicely illustrated by Figure 2-12. The vertical axis represents annual sales while the horizontal axis represents demand variability. Each circle is associated with a product and the size of the circle is proportional to profit margin. The target inventory level is 95 percent across all products. As you can see, the service level is higher than 99 percent (brown-colored circles) for many products with high profit margin, high volume, and low variability. On the other hand, service level is lower than 95 percent for products with low profit margin, low volume, and high variability. QUESTIONS 1.What the relationship is between service and inventory levels? 2.What impact lead time and lead time variability have on inventory levels?