Supply Chain Management: Inventory Management
Supply Chain Management: InventoryManagementDonglei DuFaculty of Business Administration, University of New Brunswick, NB Canada FrederictonE3B 9Y2 (ddu@umbc.edu)Du (UNB)SCM1 / 83
Table of contents I1Introduction2Inventory Management3Inventory models4Economic Order Quantity (EOQ)EOQ modelWhen-to-order?5Economic Production Quantity (EPQ): model descriptionEPQ model6The Newsboy Problem-Unknown demand (probabilistic model)The newsvendor model7Multiple-period stochastic model: model description8Managing inventory in the supply chainDu (UNB)SCM2 / 83
Section 1IntroductionDu (UNB)SCM3 / 83
Outline I1Introduce some basic concepts in inventory managementInventory level (IL)Reorder point (ROP)Lead timeSafety stockContinuous review and periodic review systemsService level2Introduce some basic inventory models, both deterministic andprobabilistic.deterministic models12EOQ modelEPQ modelprobabilistic models12Du (UNB)Single-period model: Newsboy modelMultiple-period modelSCM4 / 83
Section 2Inventory ManagementDu (UNB)SCM5 / 83
What is Inventory? IInventory is a stock or store of goods or services, kept for use orsale in the future. There are four types of inventoryRaw materials & purchased partsPartially completed goods called work in progress (WIP)Finished-goods inventoriesGoods-in-transit to warehouses or customers (GIT)The motive for inventory: there are three motives for holdinginventory, similar to cash.Transaction motive: Economies of scale is achieved when thenumber of set-ups are reduced or the number of transactionsare minimized.Precautionary motive: hedge against uncertainty, includingdemand uncertainty, supply uncertaintySpeculative motive: hedge against price increases in materials orlaborDu (UNB)SCM6 / 83
What is inventory management?The objective of inventory is to achieve satisfactory levels ofcustomer service while keeping inventory costs within reasonablebounds.Level of customer service: (1) in-stock (fill) rate (2) number ofback orders (3) inventory turnover rate: the ratio of average costof goods sold to average inventory investmentInventory cost: cost of ordering and carrying trade-offDu (UNB)SCM7 / 83
An Example ILet us look at a typical supply chain consists of suppliers andmanufacturers, who convert raw materials into finishedmaterials, and distribution centers and warehouses, form whichfinished products are distributed to customers. This implies thatinventory appears in the supply chain in several forms:Finished productGoods-in-transit(GIP)Raw tion/purchasecostsDu (UNB)Transportationcostdistributors TransportationcostwarehousesInventory tomersInventory &warehousingcosts8 / 83
Section 3Inventory modelsDu (UNB)SCM9 / 83
Inventory Models IInventory models come in all shapes. We will focus on modelsfor only a single product at a single location. Essentially eachinventory model is determined by three key variables:demand:deterministic, that is, known exactly and in advance.random. Forecasting technique may be used in the latter casewhen historical data are available to estimate the average andstandard deviation of the demand (will introduce forecastingtechniques later).costs:order cost: the cost of product and the transportation costinventory holding cost: taxes and insurances, maintenance, etc.physical aspects of the system: the physical structure of theinventory system, such as single or multiple-warehouse.Usually, there are two basic decisions in all inventory models:Du (UNB)SCM10 / 83
Inventory Models II12How much to order?When to order?Du (UNB)SCM11 / 83
Section 4Economic Order Quantity (EOQ)Du (UNB)SCM12 / 83
Subsection 1EOQ modelDu (UNB)SCM13 / 83
Economic Order Quantity (EOQ): Modeldescription IThe EOQ model is a simple deterministic model that illustratesthe trade-offs between ordering and inventory costs.Consider a single warehouse facing constant demand for a singleitem. The warehouse orders from the supplier, who is assumedto have an unlimited quantity of the product.The EOQ model assume the following scenario:Annual demand D is deterministic and occurs at a constantrate—constant demand rate: i.e., the same number of units istaken from inventory each period of time, such as 5 units perday, 25 units per week, 100 units per month and so on.Order quantity are fixed at Q units per order.Du (UNB)SCM14 / 83
Economic Order Quantity (EOQ): Modeldescription IIA fixed ordering cost co per order, not depending on thequantity ordered.A holding cost ch per unit per year, depending on the size ofthe inventory. Sometimes holding cost can usually be calculatedas follows:ch annual holding cost rate annual cost of holding one unit in inDu (UNB)SCM15 / 83
Shortages (such as stock-outs and backorder) are not permitted.There is no lead time or the lead time is constant.The planning horizon is longThe objective is to decide the order quantity Q per order so asto minimize the total annual cost, including holding andsetting-up costs.Du (UNB)SCM16 / 83
Economic Order Quantity (EOQ): AnalysisThe following graph illustrates the inventory level as a functionof time—a saw-toothed inventory pattern.Quality on imeleadtimeDu (UNB)SCM17 / 83
Inventory level (IL) is the quantity on hand, which is differentfrom inventory position (IP), which is equal to inventoryon-hand plus quantity on order minus backorder (if any).The maximum IL is Q, the minimum is 0, therefore the averageIL is Q2 .SinceAnnual holding cost Average inventory Annual holding costQ ch2Annual ordering cost Number of orders per year cost per ordDco QSoTotal annual cost annual holding cost annual setup costQD ch co2QDu (UNB)SCM18 / 83
The trade-off of holding and ordering costs is illustrated in thefollowing figure.Annual CostThe Total-Cost Curve is U-ShapedTC QDH S2QOrdering CostsQO (optimal order quantity)Du (UNB)SCMOrder Quantity(Q)19 / 83
Therefore, the optimal order quantity is achieved at the pointwhere the two costs meet.1The economic order quantity isrQ 22DcochThe cycle time (the time between two consecutive orders) isCycle Time 3Q DNumber of orders per yearNumber of orders per year Du (UNB)SCM1D cycle timeQ20 / 83
An Example IA local distributor for a national tire company expects to sellapproximately 9600 steel-belted radical tires of a certain size andtread design next year. Annual carrying cost is 16 per tire, andordering cost is 75 per order. The distributor operates 288 daysa year.1What is the EOQ?ppQ 2Dco /ch 2(9600)75/16 3002How many times per year would the store reorder if the ECQ isordered?D/Q 9600/300 32Du (UNB)SCM21 / 83
An Example II3What is the length of an order cycle if the EOQ is ordered?Q /D 300/9600 1/32 yearor1/32 288 9 working days4What is the total cost.T C (Q /2)ch (D/Q )co (300/2)16 (9600/300)75 4800Du (UNB)SCM22 / 83
Sensitivity analysisEstimating ordering and inventory costs is not an easy job, letalone accurate. Now let us see whether the EOQ model isrobust to the costs, i.e., whether the order quantity is stable ornot when the costs vary.Our claim is EOQ model is insensitive to small variations orerrors in the cost estimates.Du (UNB)SCM23 / 83
Subsection 2When-to-order?Du (UNB)SCM24 / 83
When-to-order: some conceptsThe reorder point (ROP) is defined to be the inventory level atwhich a new order should be placed.The when-to-order decision is usually expressed in terms of areorder point.We will consider deterministic and stochastic models separately.Du (UNB)SCM25 / 83
When-to-order: deterministic demand anddeterministic lead timeHere we consider a simple deterministic demand anddeterministic lead time model.Assume constant demand per period is d.Lead times is for a new order in terms of periods.ThenROP dNote that in our previous EOQ model, we assume 0 so thereorder point is ROP 0.Du (UNB)SCM26 / 83
Inventory levelROPlddl lead timeDu (UNB)dTimel lead timeSCM27 / 83
An Example IMr. X takes two special tablets per day, which are delivered tohis home seven days after an order is called in. At what pointshould Mr. X reorder?Solution: In this case, the daily demand and lead time areboth constants (special random variable), andd 2 7ThereforeROP d 2 7 14i.e., Mr. X should reorder when there are 14 tabletsleft.Du (UNB)SCM28 / 83
When-to-order: stochastic demand and stochasticlead timeThe reorder point for stochastic demand and stochastic leadtime is the expected demand during lead time and an extrasafety stock, which serves to reduce the probability ofexperiencing a stockout during lead time. FormallyROP Expected demand during lead time Safety stockDu (UNB)SCM29 / 83
QuantitySafety stock (SS): stock that is held in excess of expecteddemand due to variable demand rate and/or variable lead time.The following figure illustrates how safety stock can reduce therisk of stockout during lead time.Maximum probable demandduring lead timeExpected demandduring lead timeROPSafety stockLTDu (UNB)SCMTime30 / 83
Service level (SL): the probability that demand will not exceedsupply during lead time (i.e., the probability of no stockout). Sothe risk of stockout is 1 SL.Service levelRisk ofa stockoutProbability ofno stockoutExpecteddemand0Du (UNB)SCMROPQuantitySafetystockzz-scale31 / 83
AssumptionWe assume1Daily demands are i.i.d. random variables;2Daily demands are independent of the lead time.The amount of safety stock depends on123The desired service level 1 α.Daily demand random (rate) variable Di for day i, and hencethe unit of Di is items/per day—suppose expectation andstandard deviation of Di are (µD , σD ).Lead time random variable L, and hence the unit of L isnumber of days—suppose expectation and standard deviationare (µL , σL ).So the lead time demand—demand during lead time, which is anew random variable with unit being number of items is equal toDL LXDi D1 · · · DLi 1Du (UNB)SCM32 / 83
Then, we use the compound random variable property to findthe expectation and variance of DL (note that here both Di ’sand L are random variable and we cannot apply the rule ofexpectation and rule of variance directly): L PE[DL ] EDi µL µD i 1 LP2V[DL ] VDi µL σD σL2 µ2Di 1Assume DL followsp a normal distribution, that isDL N (E[DL ], V[DL ]).Then ROP should be chosen such that the probability of nostockout is at least 1 α, that isP(DL ROP) 1 αA standard given-p-value-find-x-value calculation gives us thedesired ROP:qp2ROP E[DL ] zα V[DL ] µD µL zα µL σD µ2D σL2Du (UNB)SCM33 / 83
An Example IA restaurant used an average of 50 jars of a special sauce eachweek. Weekly usage of sauce has a standard deviation of 3 jars.The manger is willing to accept no more than a 10 percent riskof stockout during lead time, which is two weeks. Assume thedistribution of usage is Normal. What is the ROP?Solution: In this case, the weekly demand is random and thelead time is constant, and α 0.10, µD 50,µL 2, σD 3, σL 0. Thereforeq2ROP µD µL zα µL σD µ2D σL2p 2 50 1.28 2(3)2 105.43i.e., the ROP is approximately 106 jars.Du (UNB)SCM34 / 83
Section 5Economic Production Quantity (EPQ): modeldescriptionDu (UNB)SCM35 / 83
Subsection 1EPQ modelDu (UNB)SCM36 / 83
Economic Production Quantity (EPQ): modeldescription IEconomic Production Quantity (EPQ), also called EconomicManufacturing Quantity (EMQ), is similar to EOQ model withone difference: instead of orders received in a single delivery,units are received incrementally during production, that is,constant production rate.The assumptions of the basic EPQ model are the following:Annual demand D is deterministic at a fixed rate of d units/perday.Annual production P is deterministic at a fixed rate of punits/per day, where p d.The setup cost co now has a different meaning—it is usually thecost to prepare for a production rum, which is independent ofthe production lot size.Du (UNB)SCM37 / 83
Economic Production Quantity (EPQ): modeldescription IIA holding cost ch per unit per year, depending on the size ofthe inventory.Shortages (such as stock-outs and backorder) are not permitted.There is no lead time or the lead time is constant.The planning horizon is longDu (UNB)SCM38 / 83
The following simple observation will be used shortly, whichbasically a unit transformation: year or day.ObservationDd PpThe EPQ model is designed for production situations in which,once an order is placed, production begins and a constantnumber of units is produced to inventory each day until theproduction run has been completed.We introduce the concept of lot size, which is the number ofunits in an order. In general, if we let Q denote the productionlot size, then the objective is to find the production lot size thatminimizes the total holding and ordering cots.Du (UNB)SCM39 / 83
Economic Production Quantity (EPQ): analysisDuring the production run, there are two activities (see thefollowing figure):Demand reduces the inventoryProduction adds to the inventoryinventory entoryImaxAverageInventorypp-dd0timeQpDu (UNB)SCM40 / 83
Now we establish our total cost model analogously as before.First, we consider the annual holding cost. Similarly as beforewe need to know the average inventory, which is equal toaverage inventory 1(maximum inventory minimum invento2 Q1(p d) 02p 1d1 Q2pSo the annual holding cost isAnnual holding cost (Average inventory)(Annual holding cost 1d 1 Qch2pDu (UNB)SCM41 / 83
Second, the annual setup cost has the same formula:Annual ordering cost (Number of orders per year)(cost per orD coQSo the total annual cost isTotal annual cost annual holding cost annual setuImaxDch co2Q 1dD1 Qch co2pQ 1DD1 Qch co2PQ {z}ObservationDu (UNB)SCM242 / 83
The trade-off between the holding and setup costs are similar asthe EOQ model. The optimal order quantity is achieved whenthese two costs meet:1The economic production quantity isrr2Dcop Q chp d2The run time (the production phase) isRun Time 3The cycle time (the time between two consecutive runs) isCycle Time 4Q pQ dThe maximum inventory isImaxDu (UNB) d 1 QpSCM43 / 83
An Example IA toy manufacturer uses 48000 rubber wheels per year. The firmmakes its own wheels at a rate of 800 per day. Carrying cost is 1 per wheel per year. Setup cost for a production run is 45.The firm operates 240 days per year. Determine the:1234The optimal run sizeMinimum total annual cost for carrying and setup.Cycle time for the optimal run size.Run time for the optimal run size.Solution: D 48000 wheels per year, and henced 48000/240 200 wheels per day. p 800 wheels per day,and hence P 800(240) 192000. co 45, ch 1 perwheel per year,1Q Du (UNB)q2(48000)451q800800 200SCM 2400 wheels.44 / 83
An Example II234Imax 2400800 (800 200) 1800ImaxT C 2 ch QD co 18002 1 480002400 45 1800Cycle Time 2400200 12 days.2400Run Time 800 3 days.Du (UNB)SCM45 / 83
Section 6The Newsboy Problem-Unknown demand(probabilistic model)Du (UNB)SCM46 / 83
Subsection 1The newsvendor modelDu (UNB)SCM47 / 83
The Newsboy Problem-Unknown demand(probabilistic model): introductionThe EOQ and EPQ models treat the world as if it werepredictable by using forecast techniques. However this is notaccurate in general.The newsboy model is a single-period inventory model, whereone order is placed for the product; at the end of the period, theproduct is either sold out, or a surplus of unsold items will besold for a salvage value. This happens in seasonal or perishableitems, such as newspapers, seasonal clothing etc. Because weonly order once for the period, the only inventory decision is howmuch to order. Of course, if the demand is deterministic, andknown, then the solution is trivial. So the demand is assumed tobe a random variable following some given probabilisticdistribution.Du (UNB)SCM48 / 83
The Newsboy Problem-Unknown demand(probabilistic model): model descriptionThe assumptions of the basic Newsboy model are the following:Demand X is a random variable following a probabilisticdistribution with cumulative distribution function FX anddensity function fX .Order quantity Q units per order.The purchasing wholesale price is w per unit.The selling price is p per unit.The salvage price is s.The planning horizon is one-period.Du (UNB)SCM49 / 83
The objective is to decide the order quantity Q such that theexpected total cost is minimized, including the long run totalshortage and excess costs (see below). There are two oppositedriving forces in this model.1shortage cost—unrealized profit per item, which is theopportunity loss for underestimating the demand (Q X):cs selling price per unit purchasing price per unit p w2excess cost—the difference between purchase and salvage cost,which is the loss of overestimating demand (Q X):ce purchasing price per unit salvage price per unit w s3The expected total cost model is given byEX (T C) ce EX (max{Q X, 0}) cs EX (max{X Q, 0})Z QZ ce(Q x)f (x)dx cs(x Q)f (x)dx0Du (UNB)QSCM50 / 83
4The optimal solution Q depends on the underlying demanddistribution.Du (UNB)SCM51 / 83
we define the concept of service level (SL), usually denoted as α,as the probability that demand will not exceed the order quantity:csSL α P (X Q ) FX (Q )cs cewhere F is the cumulative probability function. SoQ FX 1 (α)Du (UNB)SCM52 / 83
The following figures illustration the service level for uniformdemand distribution.cecsBalance pointSL 0.50Q*ceBalance pointcsSL 0.75Q*Du (UNB)SCM53 / 83
A little note is necessary. When the order quantity is discreterather than continuous (and hence X is a discrete randomvariable rather than a continuous one), the optimal Q satisfyingthe last equation usually does not coincide with a feasible orderquantities—since it may be a fractional number. The solution isto order round up to the next highest order quantity (see thefigures and example below for further illustration).Cumulative probability service levelSL SL cscs ceQ*Du (UNB)cscs ceXSCM54 / 83
An Example IThe owner of a newsstand wants to determine the number ofCanada Post that must be stocked at the start of each day. Itcosts 30 cents to buy a copy, and the owner sells it for 75 cents.The sale of the newspaper typically occurs between 7:00 and8:am. Newspapers left at the end of the day are recycled for anincome of 5 cents a copy. How many copies should the ownerstock every morning in order to minimize the total cost (ormaximize the profit), assuming that the demand for the day canbe approximated by12A normal distribution withdeviation 20 copiesA discrete pdf defined asD200f(D) 0.1F(D) 0.1Du (UNB)mean 300 copies and standard2200.20.3SCM3000.40.73200.20.93400.11.055 / 83
An Example IISolution: w 30, p 75, s 5. Socs p w 75 30 45, ce w s 30 5 25.Therefore the service levelSL Du (UNB)cs45 0.6428cs ce70SCM56 / 83
An Example I12The probability of 0.6428 gives the z-value 0.36. Thereforeoptimal order Q 300 0.36(20) 307.2 or approximately308 for normal distribution.Since the service level 0.3 SL 0.7. The optimal orderQ 300 copies for discrete distribution.Du (UNB)SCM57 / 83
Section 7Multiple-period stochastic model: modeldescriptionDu (UNB)SCM58 / 83
Multiple-period stochastic model: modeldescriptionPreviously we consider the Newsvendor model, a single-periodstochastic model. Now we consider a stochastic multiple periodmodel.The inventory system operates continuously with manyrepeating periods or cycles.The lead time for a new order is L days and L is a randomvariable in general, reflecting the variation o
Table of contentsI 1 Introduction 2 Inventory Management 3 Inventory models 4 Economic Order Quantity (EOQ) EOQ model When-to-order? 5 Economic Production Quantity (EPQ): model description EPQ model 6 The Newsboy Problem-Unknown demand (probabilistic model) The newsvendor model 7 Multiple-period stochastic model: model description 8 Managing inventory in the supply chain
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