Lean Thinking

Re-Order Point Problems Set 1: General

#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety Inventory1Notations

#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety InventoryROP Problemsa) If average demand per day is 20 units and lead time is 10 days. Assuming zero safety stock. Isafety=0, compute ROP.b) If average demand per day is 20 units and lead time is 10 days. Assuming 70 units of safety stock. Isafety=70, compute ROP?c) If average demand during the lead time is 200 and standard deviation of demand during lead time is 25. Compute ROP at 90% service level. Compute safety stock.d) If average demand per day is 20 units and standard deviation of daily demand is 5, and lead time is 16 days. Compute ROP at 90% service level. Compute safety stock.e) If demand per day is 20 units and lead time is 16 days and standard deviation of lead time is 4 days. Compute ROP at 90% service level. Compute safety stock.#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety InventoryROP; Fixed R, Fixed L, Zero Safety Stocka) If average demand per day is 20 units and lead time is 10 days. Assuming zero safety stock. Isafety = 0, L=10, R=20Compute ROP.

ROP = average demand during lead time + safety stockROP = LTD+ Isafety ROP = LTD + 0LTD = L RLTD = 20 10 = 200ROP = 200#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety InventoryROP; Fixed R, Fixed L, Isafety >0b) If average demand per day is 20 units and lead time is 10 days. Assuming 70 units of safety stock.Isafety = 70, L=10, R=20Compute ROP.

ROP = average demand during lead time + Safety StockROP = LTD + Isafety LTD = R L = 20 10 = 200Isafety = 70ROP = 200 + 70 = 270#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety InventoryROP; total demand during lead time is variablec) If average demand during the lead time is 200 and standard deviation of demand during lead time is 25. Compute ROP at 90% service level. Compute safety stock.

#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety InventorySafety Stock and ROPRisk of astockoutService levelProbability ofno stockoutSafety stock0zQuantityz-scaleEach Normal variable x is associated with a standard Normal Variable zx is Normal (Average x , Standard Deviation x) z is Normal (0,1)AveragedemandROPThere is a table for z which tells us Given any probability of not exceeding z. What is the value of z. Given any value for z. What is the probability of not exceeding z.#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety InventoryPopular z Values

RiskService level z value0.1 0.9 1.280.05 0.95 1.650.01 0.99 2.33

#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety Inventory8z = (x-Average x)/(Standard Deviation of x)x = Average x +z (Standard Deviation of x) = Average x = Standard Deviation of x x = +z ROP = LTD +z LTD ROP = LTD +Isafety

ROP and Isafety#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety InventoryROP and IsafetyLTD = Lead Time Demand (Demand during lead time)LTD = Average LTD = LR LTD = Standard Deviation of Lead Time DemandIsafety = zLTD

#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety InventoryROP; total demand during lead time is variablez = (X- )/X= +z ROP = LTD + IsafetyIsafety = z LTDROP=LTD + zLTDSL = 90%z = 1.28LTD = 200LTD = 25ROP= 200+ 1.28 25ROP = 200 + 32ROP = 232Isafety = 32

c) If average demand during the lead time is 200 and standard deviation of demand during lead time is 25. Compute ROP at 90% service level. Compute safety stock

#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety Inventoryd) If average demand per day is 20 units and standard deviation of demand is 5 per day, and lead time is 16 days. Compute ROP at 90% service level. Compute safety stock.

ROP; Variable R, Fixed L#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety InventoryPrevious Problem: If average demand during the lead time is 200 and standard deviation of demand during lead time is 25. Compute ROP at 90% service level. Compute safety stock.

This Problem: If average demand per day is 20 units and standard deviation of demand per day is 5, and lead time is 16 days. Compute ROP at 90% service level. Compute safety stock.

If we can transform this problem into the previous problem, then we are done, because we already know how to solve the previous problem. ROP; Variable R, Fixed L#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety Inventoryd) If average demand per day is 20 units and standard deviation of demand is 5 per day, and lead time is 16 days. Compute ROP at 90% service level. Compute ss.What is the average demand during the lead timeWhat is standard deviation of demand during lead time

ROP; Variable R, Fixed R#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety Inventory

If Demand is variable and Lead time is fixed

L: Lead TimeR: Demand per period (per day, week, month)R: Average Demand per period (day, week, month)R: Standard deviation of demand (per period)LTD: Average Demand During Lead TimeLTD = L RLTD: Standard deviation of demand during lead time and of demand per period and fixed L#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety Inventory15

If demand is variable and Lead time is fixed

L: Lead Time = 16 daysR: Demand per day R: Average daily demand =20R: Standard deviation of daily demand =5LTD: Average Demand During Lead TimeLTD = L R = 16 20 = 320LTD: Standard deviation of demand during lead time and of demand per period and fixed L#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety Inventory16The Problem originally was: If average demand per day is 20 units and standard deviation of demand is 5 per day, and lead time is 16 days. Compute ROP at 90% service level. Compute safety stock.

We transformed it to: The average demand during the lead time is 320 and the standard deviation of demand during the lead time is 20. Compute ROP at 90% service level. Compute safety stock.

Which is the same as the previous problem: If average demand during the lead time is 200 and standard deviation of demand during lead time is 25. Compute ROP at 90% service level. Compute safety stock. Now It is Transformed#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety InventorySL = 90% z = 1.28x = + z ROP= LTD +z LTDLTD = 320LTD = 20ROP= 320+ 1.28 20ROP= 320 + 25.6ROP= 320 + 26ROP = 346Isafety = 26ROP; Variable R, Fixed L#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety Inventorye) If demand per day is 20 units and lead time is 16 days and standard deviation of lead time is 4 days. Compute ROP at 90% service level. Compute Isafety.What is the average demand during the lead timeWhat is standard deviation of demand during lead time

ROP; Variable R, Fixed L#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety Inventory

If Lead time is variable and Demand is fixed

L: Lead TimeL: Average Lead TimeL: Standard deviation of Lead timeR: Demand per periodLTD: Average Demand during lead timeLTD = L R LTD: Standard deviation of demand during lead time and of L and Fixed R#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety Inventory20

If Lead time is variable and Demand is fixed

L: Lead TimeL: Average Lead Time = 16 daysL: Standard deviation of Lead time = 4 daysR: Demand per period = 20 per dayLTD: Average Demand During Lead TimeLTD = 16 20 = 320 LTD: Standard deviation of demand during lead time and of L and Fixed R#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety Inventory21

LDT = 320 LTD = 80SL = 90%z =1.28ROP = LTD +zLTDROP = 320 +1.28(80)ROP = 320 + 102.4Isafety = 102.4 and of L and Fixed R#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety Inventory22

Average demand per day is 10 units and standard deviation of demand per day is 3 units. The lead time is 5 daysDemand per day is 10 units. The average lead time is 3 days and standard deviation of lead time is 1 days. Comparing the two problems

#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety Inventory23

Average lead time demand is 20,000 units. Standard deviation of lead time demand is estimated to be 5,000 units. The warehouse currently orders a 14-day supply, 28,000 units, each time the inventory level drops to 24,000 units. Suppose holding cost is $2 per unit per year. LTD = 20,000, LTD = 5,000, ROP = 24,000, H=$2, R = 2,000 / day, Q or EOQ = 28,000. a) Compute the service level. ROP = LTD + Isafety Isafety = 24,000 20,000 = 4,000ROP = LTD +z LTD Isafety = z LTD =4000z(5,000) = 4,000 z = 4,000/5,000 = 0.8z =0.8 P(z Z) = 0.7881 = 78.81%In 78.81 % of the order cycles, the warehouse will not have a stockout. Risk = 21.19%.Given Safety Inventory Compute Service Level#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety Inventory24

b) Compute the cycle inventory, and average inventory.At reorder point we order 28,000 units. Icycle = Q/2 = 28,000/2 = 14,000Isafety = 4,000 Average Inventory = I = Icycle + Isafety =14,000 + 4,000 = 18000.c) Compute the total holding costs per year.H(Average Inventory) = H(I) = 2 18,000 = 36,000/yeard) Compute the average flow time.R = 2,000 / day, I = 18,000RT = I 2000T = 18,000T = 99 daysGiven Safety Inventory Compute Service Level#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety Inventory25

MassPC Inc. produces a 4-week supply of its PC Pal model when stock on hand drops to 525 units. It takes 1 week to produce a batch. Orders average 400 units per week, and standard deviation of forecast errors is estimated at 125 units.Problem 7.1 Normal tableUp to the first digitafterdecimalSecond digitafter decimalProbabilityz

LTD = N(400, 125).ROP = LTD + IsafetyROP = LTD +z LTD525= 400+z(125)125 = 125zz = 125/125 = 1ROP =550Q =L =1 weekLTD =400/week4(400)#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety Inventory26Safety StockService Level

Increasing service level increases the required safety inventory more than proportionately.Problem 7.1b) Compute Isafety for 80%, 90%, 95%, 99% service levels.

#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety InventorySelecting service level and safety inventory level is an important strategic decision.A firm may choose high quality service in terms of product availability. Or The firm may choose to be a low cost supplier by holding down inventory costs. In either case, it is positioning itself along the quality vs. cost trade-off curve.This graph is a perfect example to show how a strategic position could be operationalized. Problem 7.1 CostQuality

#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety Inventory

Within 100 time intervals, stockouts occur in 20. Service Level = 80/100 = 80%.Probability of stock-out = 20%.Risk = # of stockout intervals/ Total # of intervals Service Level = 1- (# of stockout intervals/ Total # of intervals) Suppose that in each time interval in which a stockout occurred, the number of units by which we were short. Suppose that cumulative demand during the 100 time intervals was 15,000 units and the total number of units short in the 20 intervals with stockouts was 1,500 units. Fill rate = (15,000-1,500)/15,000 = 13,500/15,000 = 90%.Fill Rate = Expected Sales / Expected Demand Fill Rate = (1- Expected Stockout )/ Expected DemandService Level, Fill Rate#Ardavan Asef-Vaziri Sep-2012Flow Variability; Safety Inventory29Sheet1

z0.000.010.020.030.040.050.060.070.080.0900.50000.50400.50800.51200.51600.51990.52390.52790.53190.53590.10.53980.54380.54780.55170.55570.55960.56360.56750.57140.57530.20.57930.58320.58710.59100.59480.59870.60260.60640.61030.61410.30.61790.62170.62550.62930.63310.63680.64060.64430.64800.65170.40.65540.65910.66280.66640.67000.67360.67720.68080.68440.68790.50.69150.69500.69850.70190.70540.70880.71230.71570.71900.72240.60.72570.72910.73240.73570.73890.74220.74540.74860.75170.75490.70.75800.76110.76420.76730.77040.77340.77640.77940.78230.78520.80.78810.79100.79390.79670.79950.80230.80510.80780.81060.81330.90.81590.81860.82120.82380.82640.82890.83150.83400.83650.838910.84130.84380.84610.84850.85080.85310.85540.85770.85990.86211.10.86430.86650.86860.87080.87290.87490.87700.87900.88100.88301.20.88490.88690.88880.89070.89250.89440.89620.89800.89970.90151.30.90320.90490.90660.90820.90990.91150.91310.91470.91620.91771.40.91920.92070.92220.92360.92510.92650.92790.92920.93060.93191.50.93320.93450.93570.93700.93820.93940.94060.94180.94290.94411.60.94520.94630.94740.94840.94950.95050.95150.95250.95350.95451.70.95540.95640.95730.95820.95910.95990.96080.96160.96250.96331.80.96410.96490.96560.96640.96710.96780.96860.96930.96990.97061.90.97130.97190.97260.97320.97380.97440.97500.97560.97610.976720.97720.97780.97830.97880.97930.97980.98030.98080.98120.98172.10.98210.98260.98300.98340.98380.98420.98460.98500.98540.98572.20.98610.98640.98680.98710.98750.98780.98810.98840.98870.98902.30.98930.98960.98980.99010.99040.99060.99090.99110.99130.99162.40.99180.99200.99220.99250.99270.99290.99310.99320.99340.99362.50.99380.99400.99410.99430.99450.99460.99480.99490.99510.99522.60.99530.99550.99560.99570.99590.99600.99610.99620.99630.99642.70.99650.99660.99670.99680.99690.99700.99710.99720.99730.99742.80.99740.99750.99760.99770.99770.99780.99790.99790.99800.99812.90.99810.99820.99820.99830.99840.99840.99850.99850.99860.998630.99870.99870.99870.99880.99880.99890.99890.99890.99900.9990

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SigmaR&L (2)111420142201132074201050

SigmaR&L (3)111410132108310114101150

Q2+SR/day15R7.5L9SL95%z1.64LTD22.5Isafety38ROP173WeeksInv20IniInv300

0030038191122883827281383122693848261385242373869228387212073882817938915164138101614838111813038129121381316105381410953815296638168583817164238182121381920301382032983821172813822028138231426738242724038252221838262219638271118538282316213829171453830141313831141173832151023833109238341874383507438361262383718443838243203839203003840929138411727438421825638431124538447238384513225384614211384720191384812179384920159138501714238512112138522695385368938542564385513513856232838571414385814038591728338602281386115266386211255386317238386427211386523188386601883867261621386825137386921116387091073871218638722264387315493874123738751819387625038771728338783125238791823438802520938812118838821517313883181553884151403885151253886151103887159538881481388914673890254238911923389217306389323283389418265389516249389621228389714214389816198389916182381001117113810113158381022313538103171183810428903810520703810611593810711483810844438109133138110832338111233003811228272381135267381146261381151824338116122313811715216381181020638119520138120141873812113174381221815613812313143381241313038125131173812616101381271685381281174381291856381301343381311924381324320381332829238134162763813519257381361724038137142263813823203381391418938140131763814119157138142415338143131403814401403814581323814615117381479108381483105381492481381509723815114358381521634238153193233815483153815573083815618290381572288381582626238159152473816028219381612919038162161743816320154138164161383816528110381667103381672281381681962381691646381701333381715283817213153817320295381742227338175142593817616243381771323038178822238179112113818020191381812516613818210156381836150381842512538185131123818617953818717783818821573818924333819015183819163123819213299381931728238194028238195192633819610253381975248381981023838199312073820019188382011217638202121641382032513938204231163820541123820615973820711863820848238209181382102754382116483821221327382139318382141230638215182883821619269382172224738218624138219172243822017207382212218538222118438223161681382241615238225221303822614116382271210438228158938229266338230558382312137382321918382331130738234292783823521257382369248382379239382381122838239132153824011204382411319138242211701382436164382442813638245261103824612983824723753824811643824924403825017233825181538252143013825322993825415284382557277382560277382571276382581825838259232353826013222382611920338262181853826315170138264281423826518124382661710738267188938268187138269373438270628382711612382721703827322278382741126738275232443827612433827718225382788217382793118638280171691382811215738282121453828331114382841210238285237938286176238287174538288639382891425382901730838291132953829242913829330261382949252382952522738296242033829714189382983186382992416213830091533830112141383021612538303171083830422863830519673830685938307154438308251938309123073831010297383111428338312122713831318253383142422938315022938316212083831715193383181717638319161601383201514538321121333832223110383231694383241975383251065383262045383272520383282903832926274383302924538331272183833223195383332816713833421653833520145383363810738337510238338119138339167538340768383413335383421421383431530638344202863834552813834692723834720252383481523738349312063835052013835111190383526184383532116313835414149383551413538356201153835719963835813833835926573836075038361232738362101738363262913836415276383651426238

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Sheet1Prob 7.4SL% changeZLTDIs =zLTD% changeROP = LTD +Is0.81000.84125105100505Deamand (R)01234567891011121314150.91131.28125160152560Frequency0001325167685413520.951191.64125206195606Porbability0000.01923076920.05769230770.03846153850.09615384620.01923076920.11538461540.13461538460.11538461540.15384615380.09615384620.07692307690.01923076920.05769230770.991242.33125291276691Cumulative0000.01923076920.07692307690.11538461540.21153846150.23076923080.34615384620.48076923080.59615384620.750.84615384620.92307692310.94230769231

Sheet1

Is =zLTD

Sheet2Deamand (r)100110120130140150160170180190200Porbability0.020.050.080.090.110.160.200.150.080.050.01Prob(R r)1.000.980.930.850.760.650.490.290.140.060.0125.59.611.7106.4135.2946400.81.51.60.94.89.6106.4

Sheet3