InventoryManagementIVSafetyStock

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1、Inventory Management IV Safety StockInventory Management IV Safety StockLecture 7ESD.260 Fall 2003 CapliceAssumptions: Basic EOQ Model Demand? Constant vs Variable? Known vs Random? Continuous vs Discrete Lead time Instantaneous? Constant or Variable (deterministic/stochastic) Dependence of items ?

2、Independent Correlated? Indentured ? Review Time Continuous vs Periodic Number of Echelons? One vs Many Capacity / Resources? Unlimited vs Limited Discounts? None? All Units or Incremental Excess Demand? None? All orders are backordered? Lost orders? Substitution Perishability? None? Uniform with ti

3、me Planning Horizon? Single Period? Finite Period? Infinite Number of Items? One? Many MIT Center for Transportation & Logistics - ESD.260 2 ? Chris Caplice, MITFundamental Purpose of InventoryTo buffer uncertainties in: -supply, -demand, and/or -transportationthe firm carries safety stocks.To c

4、apture scale economies in: -purchasing, -production, and/or -transportationthe firm carries cycle stocks.MIT Center for Transportation & Logistics - ESD.260 3 ? Chris Caplice, MITCycle Stock and Safety StockMIT Center for Transportation & Logistics - ESD.260 4 ? Chris Caplice, MITCycleStockC

5、ycleStockCycleStockSafety StockOn HandWhat should my inventory policy be? (how much to order when)What should my safety stock be?What are my relevant costs?TimePreview: Safety Stock Logic MIT Center for Transportation & Logistics - ESD.260 5 ? Chris Caplice, MITPSOCbSLKNKEUSTCRRFRDetermining the

6、 Reorder PointMIT Center for Transportation & Logistics - ESD.260 6 ? Chris Caplice, MITR = 9d + kReorder PointEstimated demand over the lead time (Ft) Safety Stock k = SS factor= RMSENoteWe usually pick k for desired stock out probability 2. Safety Stock = R dDefine Some TermsMIT Center for Tra

7、nsportation & Logistics - ESD.260 7 ? Chris Caplice, MIT Safety Stock Factor (k) Amount of inventory required in terms of standard deviations worth of forecast error Stockout Probability = Pd> R The probability of running out of inventory during lead time Service Level = PdR = 1-PSO The proba

8、bility of NOT running out of inventory during lead timeService Level and Stockout ProbabilityMIT Center for Transportation & Logistics - ESD.260 8 ? Chris Caplice, MITWhere d iidN(d=10,=25)SS = kProbability of StockoutService LevelDemandForecasted Demand (d)Reorder PointProb (x)Cumulative Normal

9、 DistributionMIT Center for Transportation & Logistics - ESD.260 9 ? Chris Caplice, MITWhere d iidN(d=10,=25)Prob (x)DemandProbability of Stockout = 1-SLService LevelReorder PointFinding SL from a Given KMIT Center for Transportation & Logistics - ESD.260 10 ? Chris Caplice, MITUsing a Table

10、 of Cumulative Normal Probabilities . . .0.82640.82380.82120.81860.81590.90.73890.73570.73240.72910.72570.60.77040.76730.76420.76110.75800.70.67000.66640.66280.65910.65540.40.70540.70190.698850.69500.69150.50.51600.51200.50800.50400.50000.00.55570.55170.54780.54380.53980.10.59480.59100.58710.58320.5

11、7930.20.63310.62930.62550.62170.61790.31.00.8K0.85080.84850.84610.84380.84130.79950.79670.79390.79100.78810.040.030.020.010.00If I select a K = 0.42. . . then my Service Level is this value.Or, in Excel, use the function: SL=NORMDIST(k+d, d, , TRUE)K Factor versus Service LevelMIT Center for Transpo

12、rtation & Logistics - ESD.260 11 ? Chris Caplice, MITK Factor versus Service LevelService Level %K Factor Safety Stock and Service Level MIT Center for Transportation & Logistics - ESD.260 12 ? Chris Caplice, MITExample: if d iidNormal (d=100,=10) What should my SS & R be? Reorder Point

13、Safety StockkSLPSO123232.33.99.01117171.65.95.05113 131.28.90.10100 00.50.50Service Level What is exactly is Service Level?MIT Center for Transportation & Logistics - ESD.260 13 ? Chris Caplice, MITIS NOTISService Level . . . . Probability that a stockout does not occur during lead timeThe fract

14、ion of demand satisfied from stockProbability that stock is availableFill rate for or availability of an item Fraction of time that stock is available in the systemProbability that all demand will be satisfied from on hand stock w/in any given replenishment cycleFraction of order cycles during which

15、 no SOsoccurSo, how do I find Item Fill Rate?MIT Center for Transportation & Logistics - ESD.260 14 ? Chris Caplice, MITFill Rate Fraction of demand met with on-hand inventory Based on each replenishment cycleOrderQuantity E UnitsShortFillRateOrderQuantity=But, how do I find Expected Units Short

16、? More difficult Need to calculate a partial expectation:Expected Units ShortConsider both continuous and discrete casesMIT Center for Transportation & Logistics - ESD.260 15 ? Chris Caplice, MITLooking for expected units short per replenishment cycle. 1 2 3 4 5 6 7 8 XWhat is EUS if R=5?1/8 PxF

17、or normal distribution wehave a nice result:EUS = NkWhere Nk = Normal Unit Loss Function Found in tables or formulaThe Nk TableMIT Center for Transportation & Logistics - ESD.260 16 ? Chris Caplice, MITA Table of Unit Normal Loss Integrals .09328.09503.09680.09860.10040.9.1334 .1358.1381.1405.14

18、290.7.1580 .1606.1633.1659.16870.6.1120 .1140.1160.1181.12020.81.00.50.40.30.20.10.0K.3328 .3373.3418.3464.3509.2904 .2944.2986.3027.3069.2518 .2555.2592.2630.2668.2169.2203.2236.2270.2304.1857 .1887.1917.1947.1978.07716.07866.08019.08174.08332.3793 .3841.3890.3940.3989.04.03 .02.01.00Item Fill Rate

19、MIT Center for Transportation & Logistics - ESD.260 17 ? Chris Caplice, MITOrderQuantity E UnitsShortFillRateOrderQuantity=FRSo, now we can look for the k that achieves our desired fill rate.Item Fill Rate ExampleFind fill rate for R=250, 300, 350, and 500 where, Q = 500 units and demand N(d=200

20、, =100)MIT Center for Transportation & Logistics - ESD.260 18 ? Chris Caplice, MIT.98338.33.0833.16.841.0300.0003.0293.1978Nk.032.9319.78Nks.9999.001.9993.0500.9941.07.931.5350.9604.31.690.5250FRPSOSLkRNote: Fill rate is usually much higher than service level Fill rate depends on both R and Q Q determines the number of exposures for an item