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FIU EIN 5346 - Safety Stock

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Assumptions: Basic EOQ ModelFundamental Purpose of InventoryCycle Stock and Safety StockPreview: Safety Stock LogicDetermining the Reorder PointDefine Some TermsService Level and Stockout ProbabilityCumulative Normal DistributionFinding SL from a Given KK Factor versus Service LevelSafety Stock and Service LevelService LevelSo, how do I find Item Fill Rate?Expected Units ShortThe N[k] TableItem Fill RateItem Fill Rate ExampleInventory Management IVSafety StockCapliceLecture 7ESD.260 Fall 2003© Chris Caplice, MIT2MIT Center for Transportation & Logistics – ESD.260Assumptions: Basic EOQ ModelDemand Constant vs Variable Known vs Random Continuous vs Discrete Lead time Instantaneous  Constant or Variable (deterministic/stochastic)Dependence of items Independent Correlated IndenturedReview Time Continuous vs PeriodicNumber of Echelons One vs ManyCapacity / Resources Unlimited vs LimitedDiscounts None All Units or IncrementalExcess Demand None All orders are backordered All orders are lost SubstitutionPerishability None Uniform with timePlanning Horizon Single Period Finite Period InfiniteNumber of Items One Many© Chris Caplice, MIT3MIT Center for Transportation & Logistics – ESD.260Fundamental Purpose of InventoryTo buffer uncertainties in:- supply,- demand, and/or- transportationthe firm carries safety stocks.To capture scale economies in:- purchasing,- production, and/or- transportationthe firm carries cycle stocks.© Chris Caplice, MIT4MIT Center for Transportation & Logistics – ESD.260Cycle Stock and Safety StockCycleStockCycleStockCycleStockSafety StockTimeOn HandWhat should my inventory policy be? (how much to order when)What should my safety stock be?What are my relevant costs?© Chris Caplice, MIT5MIT Center for Transportation & Logistics – ESD.260Preview: Safety Stock LogicP[SO] TC[R]E[US]N[K]FRRKSLCb© Chris Caplice, MIT6MIT Center for Transportation & Logistics – ESD.260Determining the Reorder PointReorder PointEstimated demand over the lead time (Ft)Safety Stock k = SS factorσ = RMSE R = d' + kσ()'Rdkσ−=Note1. We usually pick k for desired stock out probability 2. Safety Stock = R – d’© Chris Caplice, MIT7MIT Center for Transportation & Logistics – ESD.260Define Some TermsSafety Stock Factor (k) Amount of inventory required in terms of standard deviations worth of forecast errorStockout Probability = P[d > R]The probability of running out of inventory during lead timeService Level = P[d ≤ R] = 1- P[SO]The probability of NOT running out of inventory during lead time© Chris Caplice, MIT8MIT Center for Transportation & Logistics – ESD.260Service Level and Stockout ProbabilityNormal Distribution0.0%0.2%0.4%0.6%0.8%1.0%1.2%1.4%1.6%1.8%-100-88-76-64-52-40-28-16-4820324456688092DemandProb (x)Reorder PointProbability of StockoutService LevelWhere d ~ iid N(d’=10,σ=25)Forecasted Demand (d’)SS = kσ© Chris Caplice, MIT9MIT Center for Transportation & Logistics – ESD.260Cumulative Normal DistributionCumulative Normal Distribution0.0%20.0%40.0%60.0%80.0%100.0%120.0%-100-86-72-58-44-30-16-212264054688296DemandProb (x)Where d ~ iid N(d’=10,σ=25)Reorder PointProbability of Stockout = 1-SLService Level© Chris Caplice, MIT10MIT Center for Transportation & Logistics – ESD.260Finding SL from a Given KUsing a Table of Cumulative Normal Probabilities . . .K 0.00 0.01 0.02 0.03 0.04 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 . . . then my Service Level is this value.If I select a K = 0.42Or, in Excel, use the function:SL=NORMDIST(kσ+d’, d’, σ, TRUE)© Chris Caplice, MIT11MIT Center for Transportation & Logistics – ESD.260K Factor versus Service Level© Chris Caplice, MIT12MIT Center for Transportation & Logistics – ESD.260Safety Stock and Service LevelExample: if d ~ iid Normal (d’=100,σ=10)What should my SS & R be? P[SO] SL k Safety Stock Reorder Point .50 .50 0 0 100 .10 .90 1.28 13 113 .05 .95 1.65 17 117 .01 .99 2.33 23 123© Chris Caplice, MIT13MIT Center for Transportation & Logistics – ESD.260Service LevelWhat is exactly is Service Level?Service Level . . . . ISIS NOTProbability that a stockout does not occur during lead timeThe fraction of demand satisfied from stockProbability that stock is availableFraction of order cycles during which no SO’s occurProbability that all demand will be satisfied from on hand stock w/in any given replenishment cycleFraction of time that stock is available in the systemFill rate for or availability of an item© Chris Caplice, MIT14MIT Center for Transportation & Logistics – ESD.260So, how do I find Item Fill Rate?Fill Rate Fraction of demand met with on-hand inventory Based on each replenishment cycle[]OrderQuantity E UnitsShortFillRateOrderQuantity−=But, how do I find Expected Units Short? More difficult Need to calculate a partial expectation:© Chris Caplice, MIT15MIT Center for Transportation & Logistics – ESD.260Expected Units Short()[][]xREUS x R p x∞==−∑()[] ()oxooREUS x Rfxdx∞=−∫Consider both continuous and discrete casesLooking for expected units short per replenishment cycle. xP[x]12 3456 781/8For normal distribution wehave a nice result:E[US] = σN[k]Where N[k] = Normal Unit Loss Function Found in tables or formulaWhat is E[US] if R=5?© Chris Caplice, MIT16MIT Center for Transportation & Logistics – ESD.260The N[k] TableA Table of Unit Normal Loss IntegralsK .00 .01 .02 .03 .04 0.0 .3989 .3940 .3890 .3841 .3793 0.1 .3509 .3464 .3418 .3373 .3328 0.2 .3069 .3027 .2986 .2944 .2904 0.3 .2668 .2630 .2592 .2555 .2518 0.4 .2304 .2270 .2236 .2203 .2169 0.5 .1978 .1947 .1917 .1887 .1857 0.6 .1687 .1659 .1633 .1606 .1580 0.7 .1429 .1405 .1381 .1358 .1334 0.8 .1202 .1181 .1160 .1140 .1120 0.9 .1004 .09860 .09680 .09503 .09328 1.0 .08332 .08174 .08019 .07866 .07716© Chris Caplice, MIT17MIT Center for Transportation & Logistics – ESD.260Item Fill Rate()[][] []111[]OrderQuantity E UnitsShortFillRate FROrderQuantityEUS N kFRQQFR QNkσσ−===−


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FIU EIN 5346 - Safety Stock

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