《供应链系统设计与管理》课程授课教案（讲义）Chapter 02（Lecture 3）Single Stage Inventory Control

全英文课《DesigningandManagingSupplyChainSystem》授课教案Chapter2(Lecture3)Single StageInventoryControlOBJECTIVES(I)GraspingEconomicLot SizeModel(2) Learning the impact of demand uncertainty(3) Understanding the safe stockTEACHING CONTENT2.2 Single Stage Inventory Control2.2.1Economic Lot Size ModelThe model assumes the following:1.Demand is constant at a rate of D items per day2. Order quantities are fixed at Q items per order, that is, each time thewarehouseplacesanorder,itisforQitems.3. A fixed cost (setup cost), K, is incurred every time the warehouse places anorder.4.An inventory carrying cost, h, alsoreferred to as a holding cost, is accrued perunit held in inventory per day that the unit is held.5.Theleadtime,thetimethatelapsesbetweentheplacementofanorderand itsreceipt, iszero.6.Initial inventoryiszero7. The planning horizon is long (infinite).To find the optimal ordering policy in the economic lot size model, we considerthe inventory level as a function of time; see Figure 2-3. This is the so-calledsaw-toothed inventory pattern. We refer to the time between two successivereplenishments as a cycle time.Thus, total inventory cost in a cycle of length TisK+hTosince thefixed cost is charged once per order and holding cost can be viewed as theproduct of the per-unit, per-time-period holding cost, h, the average inventory level,Q/2,andthelengthofthecycle,TFIGURE 2-3 Inventory level as a function of timeOrdeSince the inventory level changes from Q to O during a cycle of length T, and

全英文课《Designing and Managing Supply Chain System》 授课教案 Chapter 2(Lecture 3) Single Stage Inventory Control OBJECTIVES (1) Grasping Economic Lot Size Model. (2) Learning the impact of demand uncertainty. (3) Understanding the safe stock. TEACHING CONTENT 2.2 Single Stage Inventory Control 2.2.1 Economic Lot Size Model The model assumes the following: 1. Demand is constant at a rate of D items per day. 2. Order quantities are fixed at Q items per order; that is, each time the warehouse places an order, it is for Q items. 3. A fixed cost (setup cost), K, is incurred every time the warehouse places an order.4. An inventory carrying cost, h, also referred to as a holding cost, is accrued per unit held in inventory per day that the unit is held. 5. The lead time, the time that elapses between the placement of an order and its receipt, is zero. 6. Initial inventory is zero. 7. The planning horizon is long (infinite). To find the optimal ordering policy in the economic lot size model, we consider the inventory level as a function of time; see Figure 2-3. This is the so-called saw-toothed inventory pattern. We refer to the time between two successive replenishments as a cycle time. Thus, total inventory cost in a cycle of length T is since the fixed cost is charged once per order and holding cost can be viewed as the product of the per-unit, per-time-period holding cost, h; the average inventory level, Q/2; and the length of the cycle, T. FIGURE 2-3 Inventory level as a function of time. Since the inventory level changes from Q to 0 during a cycle of length T, and

全英文课《DesigningandManagingSupplyChainSystem》授课教案demand is constant at a rate of D units per unit time, it must be that Q= TD. Thus, wecan divide the cost above by T, or, equivalently, Q/D, to get the average total cost perunitoftime:碧+驾QUsing simple calculus, it is easy to show that the order quantity Q* thatminimizes the cost function above isQ-VKDThe simple model provides two important insights:1. An optimal policy balances inventory holding cost per unit time with setupcost per unit time.Indeed, setup cost per unit time=KD/Q,whileholding cost perunit time=hQ/2 (see Figure 2-4).Thus, as one increases the order quantity Qinventory holding costs per unit of time increase while setup costs per unit of timedecrease. The optimal order quantity is achieved at the point at which inventory setupcost per unit of time (KD/Q) equals inventory holding cost per unit of time (hQ/2)That is:KDhoQ2orQ=V平FIGURE2-4Economic lot sizemodel:total cost per unit time$160$140$120Totalcost$100Holdingcost8O$60Ordercost$40I$20S01,0001,5005000Orderquantity(numberofunits)2. Total inventory cost is insensitive to order quantities; that is, changes in orderquantitieshavea relatively small impact on annual setup costs and inventoryholdingcosts. To illustrate this issue, consider a decision maker that places an order quantityQthat is a multiple b of theoptimal orderquantityQ*.In other words,for a given b,thequantityorderedisQ=bQ*Thus,b=1impliesthatthedecisionmakerordersthe economic orderquantity.Ifb=1.2 (b=0.8),the decision makerorders 20percentmore(less)thantheoptimalorderquantity.Table2-1presentstheimpactof changesin b on total system cost. For example, if the decision maker orders 20 percent morethan the optimal order quantity (b = 1.2), then the increase in total inventory costrelativeto theoptimal total cost is nomorethan 1.6percent

全英文课《Designing and Managing Supply Chain System》 授课教案 demand is constant at a rate of D units per unit time, it must be that Q = TD. Thus, we can divide the cost above by T, or, equivalently, Q/D, to get the average total cost per unit of time: Using simple calculus, it is easy to show that the order quantity Q* that minimizes the cost function above is The simple model provides two important insights: 1. An optimal policy balances inventory holding cost per unit time with setup cost per unit time. Indeed, setup cost per unit time = KD/Q, while holding cost per unit time = hQ/2 (see Figure 2-4). Thus, as one increases the order quantity Q, inventory holding costs per unit of time increase while setup costs per unit of time decrease. The optimal order quantity is achieved at the point at which inventory setup cost per unit of time (KD/Q) equals inventory holding cost per unit of time (hQ/2). That is: or FIGURE 2-4 Economic lot size model: total cost per unit time 2. Total inventory cost is insensitive to order quantities; that is, changes in order quantities have a relatively small impact on annual setup costs and inventory holding costs. To illustrate this issue, consider a decision maker that places an order quantity Q that is a multiple b of the optimal order quantity Q*. In other words, for a given b, the quantity ordered is Q = bQ*. Thus, b = 1 implies that the decision maker orders the economic order quantity. If b = 1.2 (b = 0.8), the decision maker orders 20 percent more (less) than the optimal order quantity. Table 2-1 presents the impact of changes in b on total system cost. For example, if the decision maker orders 20 percent more than the optimal order quantity (b = 1.2), then the increase in total inventory cost relative to the optimal total cost is no more than 1.6 percent

全英文课《DesigningandManagingSupplyChainSystem》授课教案TABLE2-1SENSITIVITYANALYSIS100.50.80.91.11.21.52b2.5%0.5%0.4%1.6%8.9%25%25%Increase incost2.2.2.theEffect of Demand Uncertainty on inventoryTheprevious model illustrates thetrade-offs between setupcosts and inventoryholding costs. It ignores, however, issues such as demand uncertainty and forecastingIndeed, many companies treat the world as if it were predictable, making productionand inventory decisions based on forecasts ofthe demand made far in advance of theselling season.Althoughthesecompanies areawareofdemand uncertaintywhentheycreate a forecast, they design their planning processes as if the initial forecast was anaccuraterepresentation of reality.In this case, one needs to remember the followingprinciples ofall forecasts:1.Theforecast is alwayswrong2.Thelonger theforecast horizon,theworse theforecast.3.AggregateforecastsaremoreaccurateThus, the first principle implies that it is difficult to match supply and demand,and the second one implies that it is even more difficult if one needs to predictcustomer demandfora longperiodoftime,forexample,thenext12to18months.The third principle suggests, for instance, that while it is difficult to predict customerdemandforindividual SKUs,itismucheasiertopredictdemandacrossall SKUswithin one productfamily.This principle is an exampleof therisk pooling concept.2.2.3SinglePeriod ModelsTo better understand the impact of demand uncertainty, we consider a series ofincreasingly detailed and complex situations.To start, we consider a product that has ashort lifecycle and hence the firm has only one ordering opportunity. Thus, beforedemandoccurs,thefirmmustdecidehowmuchtostockinordertomeetdemand.Ifthefirmstocks toomuch, itwill be stuckwith excess inventory it has to disposeof.Ifthe firm stocks too little, it will forgo some sales, and thus some profits.Using historical data, the firm can typically identify a variety of demandscenarios and determine alikelihood or probability that each of these scenarios willoccur. Observe that given a specific inventory policy,the firm can determine the profitassociated with a particular scenario.Thus.given a specific order quantity.thefirmcanweight each scenario'sprofitbythelikelihoodthat it will occur and hencedetermine the average, or expected, profit for a particular ordering quantity. It is thusnatural forthefirm to order the quantity that maximizes theaverage profit.EXAMPLEConsider a company that designs, produces, and sells summer fashion items suchas swimsuits.About sixmonthsbefore summer,thecompanymustcommititselftospecific production quantities for all its products. Since there is no clear indication ofhow the market will respond to the new designs, the company needs to use various

全英文课《Designing and Managing Supply Chain System》 授课教案 TABLE 2-1 SENSITIVITY ANALYSIS 2.2.2. the Effect of Demand Uncertainty on inventory The previous model illustrates the trade-offs between setup costs and inventory holding costs. It ignores, however, issues such as demand uncertainty and forecasting. Indeed, many companies treat the world as if it were predictable, making production and inventory decisions based on forecasts of the demand made far in advance of the selling season. Although these companies are aware of demand uncertainty when they create a forecast, they design their planning processes as if the initial forecast was an accurate representation of reality. In this case, one needs to remember the following principles of all forecasts: 1. The forecast is always wrong. 2. The longer the forecast horizon, the worse the forecast. 3. Aggregate forecasts are more accurate. Thus, the first principle implies that it is difficult to match supply and demand, and the second one implies that it is even more difficult if one needs to predict customer demand for a long period of time, for example, the next 12 to 18 months. The third principle suggests, for instance, that while it is difficult to predict customer demand for individual SKUs, it is much easier to predict demand across all SKUs within one product family. This principle is an example of the risk pooling concept. 2.2.3 Single Period Models To better understand the impact of demand uncertainty, we consider a series of increasingly detailed and complex situations. To start, we consider a product that has a short lifecycle and hence the firm has only one ordering opportunity. Thus, before demand occurs, the firm must decide how much to stock in order to meet demand. If the firm stocks too much, it will be stuck with excess inventory it has to dispose of. If the firm stocks too little, it will forgo some sales, and thus some profits. Using historical data, the firm can typically identify a variety of demand scenarios and determine alikelihood or probability that each of these scenarios will occur. Observe that given a specific inventory policy, the firm can determine the profit associated with a particular scenario. Thus, given a specific order quantity, the firm can weight each scenario's profit by the likelihood that it will occur and hence determine the average, or expected, profit for a particular ordering quantity. It is thus natural for the firm to order the quantity that maximizes the average profit. EXAMPLE Consider a company that designs, produces, and sells summer fashion items such as swimsuits. About six months before summer, the company must commit itself to specific production quantities for all its products. Since there is no clear indication of how the market will respond to the new designs, the company needs to use various

全英文课《DesigningandManagingSupplyChainSystem》授课教案toolstopredictdemand foreachdesign,and plan production and supplyaccordinglyInthissetting,thetrade-offsare clear:overestimatingcustomerdemand will result inunsold inventory, while underestimating customer demand will lead to inventorystockouts and loss of potential customers.To assist management in these decisions, the marketing department useshistorical data from thelastfive years, current economic conditions,and other factorstoconstructaprobabilisticforecast of thedemand forswimsuits.Theyhaveidentifiedseveralpossiblescenariosforsalesinthecomingseason,basedon suchfactorsaspossibleweather patterns and competitors'behavior,andassigned eachaprobability,or chance of occurring. For example, the marketing department believes that ascenario that leads to 8,o00 unit sales has an 11 percent chance of happening,otherscenarios leading todifferent sales levels have different probabilities ofoccurring.These scenarios are illustrated in Figure 2-5.This probabilistic forecast suggests thataverage demand is about 13,000 units, but there is a probability that demand will beeither larger than average or smaller than average.FIGURE2-5Probabilisticforecast.30%25%20%ied拉15%10%5Og8,00010,00012,00014,00016,00018,000Unit salesWehavethefollowingadditional information1. To start production, the manufacturer has to invest $100,000 independent ofthe amount produced. We refer to this cost as the fixed production cost.2.The variableproduction cost per unit equals $80.3. During the summer season, the selling price of a swimsuit is $125 per unit4.Any swimsuit not sold duringthe summer season is sold to a discount store fors20. We refer to this value as the salvage valueTo identify the best production quantity,the firm needs to understand therelationship between the production quantity,customer demand, and profit.Supposethemanufacturer produces 10.000units whiledemandends at12.000swimsuits.It iseasilyverifiedthatprofitequalsrevenuefromsummersalesminusthevariableproductioncostminusthefixed production cost.That is:Profit=125(10,000)-80(10,000)100,000=350,000On theother hand,if the companyproduces 10,oo0 swimsuits and demand isonly 8,00 units,profit equals revenue from summer sales plus salvagevalueminusthe variableproduction cost minus thefixed production cost.That is:Profit=125(8,000)+20(2,000)-80(10,000)-100,000=140,000Notice that based on the marketing department's forecast, the probability thatdemand is 8,000 units is 11 percent while the probability that demand is 12,000 units

全英文课《Designing and Managing Supply Chain System》 授课教案 tools to predict demand for each design, and plan production and supply accordingly. In this setting, the trade-offs are clear: overestimating customer demand will result in unsold inventory, while underestimating customer demand will lead to inventory stockouts and loss of potential customers。 To assist management in these decisions, the marketing department uses historical data from the last five years, current economic conditions, and other factors to construct a probabilistic forecast of the demand for swimsuits. They have identified several possible scenarios for sales in the coming season, based on such factors as possible weather patterns and competitors' behavior, and assigned each a probability, or chance of occurring. For example, the marketing department believes that a scenario that leads to 8,000 unit sales has an 11 percent chance of happening; other scenarios leading to different sales levels have different probabilities of occurring. These scenarios are illustrated in Figure 2-5. This probabilistic forecast suggests that average demand is about 13,000 units, but there is a probability that demand will be either larger than average or smaller than average. FIGURE 2-5 Probabilistic forecast. We have the following additional information: 1. To start production, the manufacturer has to invest $100,000 independent of the amount produced. We refer to this cost as the fixed production cost. 2. The variable production cost per unit equals $80. 3. During the summer season, the selling price of a swimsuit is $125 per unit. 4. Any swimsuit not sold during the summer season is sold to a discount store for $20. We refer to this value as the salvage value To identify the best production quantity, the firm needs to understand the relationship between the production quantity, customer demand, and profit. Suppose the manufacturer produces 10,000 units while demand ends at 12,000 swimsuits. It is easily verified that profit equals revenue from summer sales minus the variable production cost minus the fixed production cost. That is: On the other hand, if the company produces 10,000 swimsuits and demand is only 8,000 units, profit equals revenue from summer sales plus salvage value minus the variable production cost minus the fixed production cost. That is: Notice that based on the marketing department's forecast, the probability that demand is 8,000 units is 11 percent while the probability that demand is 12,000 units

全英文课《DesigningandManagingSupplyChainSystem》授课教案is27percent.Thus,producing10,000swimsuitsleadstoaprofitof$350,000withprobability of 27 percent and a profit of $140,000 with probability of 11 percent. Insimilar fashion, one can calculate the profit associated with each scenario given thatthemanufacturer produces 10,000 swimsuits.This allowsus to determine theexpected (or average)profitassociatedwithproducing10,o00 units.This expectedprofit is the total profit of all the scenarios weighted by the probability that eachscenario will occur.We, of course, would like to find the order quantity that maximizes averageprofit.Figure2-6 plots the average profit as afunction of the production quantity.Itshows that the optimal production quantity, or the quantity that maximizes averageprofit, is about 12,000.FIGURE2-6Average profit as a function of production quantity$400,00$100.0098,00012,00016,0020,000CoeInterestingly,the orderquantity that maximizes total expected profit is notnecessarily equal to the average demand. Indeed, in the previous example, the orderquantity that maximizes total expected profit is 12,000 units while average demand is13,000So what is therelationship between the optimal order,or production, quantityand average demand? Should the optimal order quantity always be less than averagedemand,asintheprevious example?Toanswerthesequestions,wecomparethemarginal profit and the marginal cost of ordering an additional unit. If an additionalunit is sold, the marginal profit is the difference between the selling price per unit andthe variable ordering (or production) cost per unit, and if an additional unit is not soldduringthesellingseason,themarginal costisthedifferencebetweenthevariableproduction cost and the salvage value per unit. If the cost of not selling an additionalunit is largerthan the profitfrom selling an additional unit,the optimal quantity ingeneral will be less than average demand, while if the reverse is true, the optimalorder quantityingeneral will begreaterthantheaveragedemandOf course, this is only true if minimizing average profit is in fact the goal of thefirm.Aswithothertypesofinvestments,investmentininventorycarriesdownsiderisks if sales do not meet expectations, and upsiderewards if demand exceedsexpectations.Interestingly,it is possibleto characterizetheupsidepotential anddownsiderisks inourmodel andthusassistmanagementininventoryinvestmentdecisions.Once again, consider the previous example.Figure2-6 plots the average profit asa function of the production quantity. As mentioned above, it shows that the optimalproduction quantity,that is,the quantitythat maximizes average profit, is about

全英文课《Designing and Managing Supply Chain System》 授课教案 is 27 percent. Thus, producing 10,000 swimsuits leads to a profit of $350,000 with probability of 27 percent and a profit of $140,000 with probability of 11 percent. In similar fashion, one can calculate the profit associated with each scenario given that the manufacturer produces 10,000 swimsuits. This allows us to determine the expected (or average) profit associated with producing 10,000 units. This expected profit is the total profit of all the scenarios weighted by the probability that each scenario will occur. We, of course, would like to find the order quantity that maximizes average profit. Figure 2-6 plots the average profit as a function of the production quantity. It shows that the optimal production quantity, or the quantity that maximizes average profit, is about 12,000. FIGURE 2-6 Average profit as a function of production quantity. Interestingly, the order quantity that maximizes total expected profit is not necessarily equal to the average demand. Indeed, in the previous example, the order quantity that maximizes total expected profit is 12,000 units while average demand is 13,000. So what is the relationship between the optimal order, or production, quantity and average demand? Should the optimal order quantity always be less than average demand, as in the previous example? To answer these questions, we compare the marginal profit and the marginal cost of ordering an additional unit. If an additional unit is sold, the marginal profit is the difference between the selling price per unit and the variable ordering (or production) cost per unit, and if an additional unit is not sold during the selling season, the marginal cost is the difference between the variable production cost and the salvage value per unit. If the cost of not selling an additional unit is larger than the profit from selling an additional unit, the optimal quantity in general will be less than average demand, while if the reverse is true, the optimal order quantity in general will be greater than the average demand. Of course, this is only true if minimizing average profit is in fact the goal of the firm. As with other types of investments, investment in inventory carries downside risks if sales do not meet expectations, and upside rewards if demand exceeds expectations. Interestingly, it is possible to characterize the upside potential and downside risks in our model and thus assist management in inventory investment decisions. Once again, consider the previous example. Figure 2-6 plots the average profit as a function of the production quantity. As mentioned above, it shows that the optimal production quantity, that is, the quantity that maximizes average profit, is about

全英文课《DesigningandManagingSupplyChainSystem》授课教案12,000.Thefigurealsoindicates thatproducing9,000unitsorproducing16,000unitswill leadtoabout thesameaverageprofit of $294,000.If,forsomereason,wehadtochoose between producing 9,000 units and 16,000 units, which one should wechoose?To answer this question, we need to better understand the risk associated withcertain decisions.For this purpose,weconstruct afrequencyhistogram (seeFigure2-7) that provides information about potential profit for the two given productionquantities,9,000units and 16,000units.Forinstance,consider profitwhen theproduction quantity is 16,00 units.The graph shows that the distribution of profit isnot symmetrical. Losses of s220,000 happen about 11 percent of the time whileprofits of at least $410,000 happen 50 percent of the time. On the other hand, afrequency histogram of the profit when the production quantity is 9,oo0units showsthatthedistributionhasonlytwopossibleoutcomes.Profitiseither$200o,o00withprobability of about 11 percent, or s305,000 with probability of about 89 percentThus,while producing16,000 units has thesame average profit as producing 9,000units, the possible risk on the onehand, and possiblereward on the other hand,increasesas weincreasetheproductionsizeFIGURE2-7Afrequencyhistogramofprofit100%80%Kaigeord60%VQ=9.00040%Q=1600020%7$300$100$100$300$500(ncetna)Observations:1. The optimal order quantity is not necessarily equal to forecast, or average,demand. Indeed, the optimal quantity depends on the relationship between marginalprofit achieved from selling an additional unit and marginal cost. More importantly,thefixed costhasno impactontheproductionquantity,onlyonthedecisionwhetherto produce or not. Thus, given a decision to produce, the production quantity is thesame independently of the fixed production cost.2.As the order quantity increases, average profit typically increases until theproduction quantity reaches a certain value, after which the average profit startsdecreasing.3. As we increase the production quantity, the riskthat is, the probability oflargelosses-alwaysincreases.Atthesametime,theprobabilityoflargegainsalsoincreases. This is the risk/reward trade offQUESTIONS1.How does the demand uncertainty impact inventory?2.How firms cope with huge variability in customer demand

全英文课《Designing and Managing Supply Chain System》 授课教案 12,000. The figure also indicates that producing 9,000 units or producing 16,000 units will lead to about the same average profit of $294,000. If, for some reason, we had to choose between producing 9,000 units and 16,000 units, which one should we choose? To answer this question, we need to better understand the risk associated with certain decisions. For this purpose, we construct a frequency histogram (see Figure 2-7) that provides information about potential profit for the two given production quantities, 9,000 units and 16,000 units. For instance, consider profit when the production quantity is 16,000 units. The graph shows that the distribution of profit is not symmetrical. Losses of $220,000 happen about 11 percent of the time while profits of at least $410,000 happen 50 percent of the time. On the other hand, a frequency histogram of the profit when the production quantity is 9,000 units shows that the distribution has only two possible outcomes. Profit is either $200,000 with probability of about 11 percent, or $305,000 with probability of about 89 percent. Thus, while producing 16,000 units has the same average profit as producing 9,000 units, the possible risk on the one hand, and possible reward on the other hand, increases as we increase the production size FIGURE 2-7 A frequency histogram of profit Observations: 1. The optimal order quantity is not necessarily equal to forecast, or average, demand. Indeed, the optimal quantity depends on the relationship between marginal profit achieved from selling an additional unit and marginal cost. More importantly, the fixed cost has no impact on the production quantity, only on the decision whether to produce or not. Thus, given a decision to produce, the production quantity is the same independently of the fixed production cost. 2. As the order quantity increases, average profit typically increases until the production quantity reaches a certain value, after which the average profit starts decreasing. 3. As we increase the production quantity, the risk—that is, the probability of large losses—always increases. At the same time, the probability of large gains also increases. This is the risk/reward trade off QUESTIONS 1.How does the demand uncertainty impact inventory? 2.How firms cope with huge variability in customer demand