Safety Inventories Service Levels
Safety InventoriesService Levelsutdallas.edu/ metin1
Inventory Cycles and Replenishment PoliciesinventoryAn inventorycycleQROPtimeLead TimesShortage When to reorder? How much to reorder? These decisions are related.Continuous Review: Order fixed quantity when inventory drops below Reorder Point– ROP (Reorder Point) meets the demand (DLT) during the lead time LT.– One has to figure out the ROP to meet (most of the) demand (most of the time).LTDi demand in period i 1.7.ROPDLT LT 7 Di 1utdallas.edu/ metin1 27i2
Safety Inventory and Service Measures Hold safety inventory» Desire for quick product availability Ease of search for another supplier. “I want it now” culture.» Demand uncertainty Short product life cycles: 18 months for Nokia cell phonesMeasures of demand uncertainty– Variance of demand– Ranges for demand Measures of product availability– Stockout, what happens?» Backorder (patient customer, unique product or big cost advantage) or Lost sales.– I. Cycle service level (CSL), % of cycles with no stockout– II. Product fill rate (FR), % of products sold from the shelf– Order fill rate, % of orders» Equivalent to product fill rate if orders contain one productutdallas.edu/ metin3
Service measures: CSL and FR are differentinventoryCSL is 0%, fill rate is almost 100%timeinventoryCSL is 0%, fill rate is almost 0%timeutdallas.edu/ metin4
Recall from StatisticsDLT: Demand During Lead timef i , Fi probability density and cumulative density functions for DiRi E ( Di ) Di f i ( Di )dDiVar ( Di ) i2 E{(Di Ri ) 2 } ( Di Ri ) 2 f i (Di )dDicov( Di , D j ) i2, j E{(Di Ri )( D j R j )} ( Di Ri )( D j R j ) f i , j ( Di , D j )dDi dD j i2, j /( i j ) correlation coefficien tCoefficien t of variation of D : cv Var ( D) / E ( D)LLi 1i 1E ( Di ) Ri .LLLLLLVar ( Di ) cov( Di , D j ) cov(Di , D j )i 1i 1 j 1i 12ii 1 j 1j iMostly Di Normal(mean Ri , variance i2 ).utdallas.edu/ metin5
Recall from StatisticsNormal Density yDensity function (pdf) at x :Prob MeanExcel statistical functions :x 95.44%normdist ( x, mean, st dev ,0)Cumulative function (cdf) at x :normdist ( x, mean, st dev ,1)99.74%1probInverse function of cdf at " prob":norminv ( prob, mean, st dev )normdist(x,mean,st dev,1)0utdallas.edu/ metinxnorminv(prob,mean,st dev)6
I. Cycle Service Level (CSL) and Safety Stock (ss)For example consider 10 cycles :3 cycles have stockouts1 1 0 1 1 1 0 1 0 1CSL Write 0 if a cycle has stockout,1 otherwise10CSL 0.7 Probabilit y that a single cycle has sufficient inventory4 cycles have stockouts1 1 0 1 1 0 0 1 0 1CSL 10CSL 0.6 Probabilit y that a single cycle has sufficient inventory[Sufficent Inventory] [Demand during LT ROP)CSL Cycle Service Level P( DLT ROP)ss ROP E( DLT )Safety stock (ss) is a general concept. It exists without lead time.It is the stock held minus the expected demand.utdallas.edu/ metin7
CSL and ss: Statistical ManipulationsSuppose that demands are identically and independently distributed.LLi 1i 1E ( Di ) LR and Var ( Di ) L 2LIf Di N ( R, ) then Di N ( LR, L 2 )2 P N ( R L, L i 1CSL P N ( R L, L R ) ROP22R ) R L ROP R L Taking out the mean 2N(R L,L ) R LROP R LR Dividing by theStDev P L RL R N ( 0 ,1) ROP R L Obtaining standard normal distribution P N (0,1) L R normdist ( ROP, R L, L R ,1) normdist ( ROP R L /( L R ),0,1,1)utdallas.edu/ metinss norminv(CSL,0,1) L 8
Finding CSL, ss and ROPExample [ROP CSL, ss]: R 2,500 /week; 500 L 4 weeks; ROP 16,000ss ROP – L R Stdev of demand during lead time L ROP R L ROP R L CSL P N (0,1) normdist ,0,1,1 normdist ROP, R L, L R ,1 L R L R Example [CSL ss, ROP]: R 2,500/week; 500; L 4 weeks; CSL 0.90.ss norminv(CSL 0.9,0,1) ( L 4)( 500)ROP R L ss 2500(4) norminv(CSL 0.9,0,1) 2 (500)Factors driving CSL and safety inventory– Replenishment lead time– Demand uncertaintyutdallas.edu/ metin9
II. Fill rate:Expected shortage per cycle ESC is the expected shortage per cycle ESC is not a percentage, it is the number of units, also see next page Demand ROP ifShortage 0if Demand ROPDemand ROPESC E (max{Demand during lead time - ROP,0}) ESC ( x ROP) f ( x)dxx ROP Fill rate : Proportion of customer demand satisfied from stockQ: Order quantityutdallas.edu/ metinESCFill rate fr 1 Q10
Shortage, Inventory and Demand during Lead TimeUpside ROPdown0Shortage DLT-ROPShortageLT0DemandDuring LT0ROP0LTutdallas.edu/ metinDLT: DemandDuring LT0ROPInventory ROP ROP-DLTUpsideInventory downDLT: DemandDuring LTDemandDuring LT011
Expected shortage per cycle First let us study shortage during the lead timeExpected shortage E(max( 0, DLT ROP)) Example: d1 9 with prob p1 1/4 ROP 10, D d 2 10 with prob p2 2/4 , Expected Shortage? d 11 with prob p 1/4 3 3 3Expected shortage max{0, (d i ROP)} pi i 111 (d ROP)}P( D d )d 10121 1 max{0, (9 - 10)} max{0, (10 - 10)} max{0, (11 - 10)} 444 4utdallas.edu/ metin12
Expected shortage per cycle Example:ROP 10, D Uniform(6,12), Expected Shortage?D 12 1 10 2 11 D21 12 2Expected shortage ( D 10) dD 10 D 10(12) 10(10) 66 26 2 6 2 D 10D 1012 ( 48 ( 50)) / 6 2/6 1/3If demand is normal: ss ss ESC ss 1 normdist ,0,11, L normdist ,0,1,0 L L utdallas.edu/ metinDoes ESC decrease or increase with ss, L?Does ESC decrease or increase with expected value of demand?13
Finding the Fill Rate and Safety InventoryExample [ss fr] : 500; L 2 weeks; ss 1000; Q 10,000; Fill Rate (fr) ? ss ESC ss 1 normdist ,0,1,1 L ss L normdist ,0,1,0 L ESC 1000(1 normdist (1000 / 707,0,1,1) 707nomdist (1000 / 707,0,1,0).ESC 2513.fr (Q - ESC)/Q (10,000 - 25.13)/10,000 0.9975.Example [fr ss]: If desired fill rate is 0.975, how much safety inventory to hold?Clearly ESC (1 - fr)Q 250Try some values of ss or use goalseek function of Excel to solve ss ss 250 ss 1 normdist ,0,1,1 707normdist ,0,1,0 707 707 utdallas.edu/ metin14
Evaluating Safety Inventory For Given Fill RateSafety inventory is very sensitive to fill rate. Is fr 100% possible?Fill Rate97.5%98.0%98.5%99.0%99.5%67183321499767Safety Stock Safety inventory: If Safety inventory is up, – Fill Rate is up– Cycle Service Level is up.Lot size: If Lot size Q is up, (i) Cycle Service Level does not change. Reorder point, demand during lead timespecify Cycle Service Level. (ii) Expected shortage per cycle does not change. Safety stock and the variability of thedemand during the lead time specify the Expected Shortage per Cycle. Fill rate is up. To Cut Down the Safety Inventory Supply Side: Reduce the Supplier Lead Time––– utdallas.edu/ metinFaster transportation, such as Air shipped semiconductors from TaiwanBetter coordination, information exchange, advance retailer demand informationto prepare the supplier, such as Textiles; Obermeyer caseSpace out orders equally as much as possibleDemand Side: Reduce uncertainty of the demand– Contracts– Better forecasting to reduce demand variability– Collaborative forecasting: Manufacturers with the retailers.15
Lead Time VariabilityUncertain Lead Time:LE ( Di ) LRi 1L Average lead time.s 2 Variance of lead timeLVar ( Di ) L 2 R 2 s 2 : L2i 1The formulae do not change: ss norminv (CSL;0,1) L 2 R 2 s 2 ss ss ESC ss 1 normdist ;0,1,1 L nomdist ;0,1,0 L L Example: R 2,500/day; 500, L 7 days; CSL 0.90.StDev of LTSafety Stock (ss)Jump in ssutdallas.edu/ metin012341695362566289760 12927 16109 19298-1930300331323167531826318916
Summary Service levels– Cycle service level– Fill rateutdallas.edu/ metin
ROP R L L L ROP R L L ROP R L CSL P N Example [CSL ss, ROP]: R 2,500/week; 500; L 4 weeks; CSL 0.90. Factors driving CSL and safety inventory –Replenishment lead time –Demand uncertainty 2500(4) norminv(CSL 0.9,0,1) 2 (500) norminv(CSL 0.9,0,1) ( 4)( 500) ROP R L ss ss L
technical analysis of GHG inventory data and review of the inventories. They require that the annual national GHG inventories be transparent, consistent, comparable, complete and accurate. Application of these principles allows for more
AASB 102 4 COMPARISON Comparison with IAS 2 AASB 102 Inventories incorporates IAS 2 Inventories issued by the International Accounting Standards Board (IASB). Australian-specific paragraphs (which are not included in IAS 2) are identified with the prefix “Aus” or
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