利用MINITAB做蒙特卡洛模拟

1、DoingMonteCarloSimulationinMinitabStatisticalSoftwareDoingMonteCarlosimulationsinMinitabStatisticalSoftwareisveryeasy.ThisarticleillustrateshowtouseMinitabforMonteCarlosimulationsusingbothaknownengineeringformulaandaDOEequation.byPaulSheehyandEstonMartzMonteCarlosimulationusesrepeatedrandomsamplingt

2、osimulatedataforagivenmathematicalmodelandevaluatetheoutcome.Thismethodwasinitiallyappliedbackinthe1940s,whenscientistsworkingontheatomicbombusedittocalculatetheprobabilitiesofonefissioninguraniumatomcausingafissionreactioninanother.Withuraniuminshortsupply,therewaslittleroomforexperimentaltrialande

3、rror.Thescientistsdiscoveredthataslongastheycreatedenoughsimulateddata,theycouldcomputereliableprobabilitiesandreducetheamountofuraniumneededfortesting.Today,simulateddataisroutinelyusedinsituationswhereresourcesarelimitedorgatheringrealdatawouldbetooexpensiveorimpractical.ByusingMinitab'sabilit

4、ytoeasilycreaterandomdata,youcanuseMonteCarlosimulationto:SSimulatetherangeofpossibleoutcomestoaidindecision-makingfForecastfinancialresultsorestimateprojecttimelinesuUnderstandthevariabilityinaprocessorsystemFFindproblemswithinaprocessorsystemMManageriskbyunderstandingcost/benefitrelationshipsSteps

5、intheMonteCarloApproachDependingonthenumberoffactorsinvolved,simulationscanbeverycomplex.Butatabasiclevel,allMonteCarlosimulationshavefoursimplesteps:1. IdentifytheTransferEquationTodoaMonteCarlosimulation,youneedaquantitativemodelofthebusinessactivity,plan,orprocessyouwishtoexplore.Themathematicale

6、xpressionofyourprocessiscalledthetransferequation."Thismaybeaknownengineeringorbusinessformula,oritmaybebasedonamodelcreatedfromadesignedexperiment(DOE)orregressionanalysis.2. DefinetheInputParametersForeachfactorinyourtransferequation,determinehowitsdataaredistributed.Someinputsmayfollowthenor

7、maldistribution,whileothersfollowatriangularoruniformdistribution.Youthenneedtodeterminedistributionparametersforeachinput.Forinstance,youwouldneedtospecifythemeanandstandarddeviationforinputsthatfollowanormaldistribution.3. CreateRandomDataTodovalidsimulation,youmustcreateaverylarge,randomdatasetfo

8、reachinputsomethingontheorder100,000instances.Theserandomdatapointssimulatethevaluesthatwouldbeseenoveralongperiodforeachinput.Minitabcaneasilycreaterandomdatathatfollowalmostanydistributionyouarelikelytoencounter.4. SimulateandAnalyzeProcessOutputWiththesimulateddatainplace,youcanuseyourtransferequ

9、ationtocalculatesimulatedoutcomes.Runningalargeenoughquantityofsimulatedinputdatathroughyourmodelwillgiveyouareliableindicationofwhattheprocesswilloutputovertime,giventheanticipatedvariationintheinputs.ThosearethestepsanyMonteCarlosimulationneedstofollow.Here'showtoapplytheminMinitab.MonteCarloU

10、singaKnownEngineeringFormulaAmanufacturingcompanyneedstoevaluatethedesignofaproposedproduct:asmallpistonpumpthatmustpump12mloffluidperminute.Youwanttoestimatetheprobableperformanceoverthousandsofpumps,givennaturalvariationinpistondiameter(D),strokelength(L),andstrokesperminute(RPM).Ideally,thepumpfl

11、owacrossthousandsofpumpswillhaveastandarddeviationnogreaterthan0.2ml.Step1:IdentifytheTransferEquationThefirststepindoingaMonteCarlosimulationistodeterminethetransferequation.Inthiscase,youcansimplyuseanestablishedengineeringformulathatmeasurespumpflow:Flow(inml)=MD/2)2?L?RPMStep2:DefinetheInputPara

12、metersNowyoumustdefinethedistributionandparametersofeachinputusedinthetransferequation.Thepump'spistondiameterandstrokelengthareknown,butyoumustcalculatethestrokes-per-minute(RPM)neededtoattainthedesired12ml/minuteflowrate.Volumepumpedperstrokeisgivenbythisequation:KD/2)2*LGivenD=0.8andL=2.5,eac

13、hstrokedisplaces1.256ml.Sotoachieveaflowof12ml/minutetheRPMis9.549.Basedontheperformanceofotherpumpsyourfacilityhasmanufactured,youcansaythatpistondiameterisnormallydistributedwithameanof0.8cmandastandarddeviationof0.003cm.Strokelengthisnormallydistributedwithameanof2.5cmandastandarddeviationof0.15c

14、m.Finally,strokesperminuteisnormallydistributedwithameanof9.549RPMandastandarddeviationof0.17RPM.Step3:CreateRandomDataNowyou'rereadytosetupthesimulationinMinitab.WithMinitabyoucaninstantaneouslycreate100,000rowsofsimulateddata.Startingwiththesimulatedpistondiameterdata,chooseCalc>RandomData&

15、gt;Normal.Inthedialogbox,enter100,000inNumberofrowsofdatatogenerate,andenterD”asthecolumninwhichtostorethedata.Enterthemeanandstandarddeviationforpistondiameterintheappropriatefields.PressOKtopopulatetheworksheetwith100,000datapointsrandomlysampledfromthespecifiednormaldistribution.1Thensimplyrepeat

16、thisprocessforStrokeLength(L)andStrokesperMinute(RPM).Step4:SimulateandAnalyzeProcessOutputNowcreateafourthcolumnintheworksheet,Flow,toholdtheresultsofyourprocessoutputcalculations.Withtherandomlygeneratedinput data in place, you can set up Minitabs calculator to calculate the output and store it in

17、 the Flow column. Go toCalc > Calculatorandsetuptheflowequationlikethis:and select the Flow column. Minitab willMinitabwillquicklycalculatetheoutputforeachrowofsimulateddata.Nowyou'rereadytolookattheresults.SelectStat>BasicStatistics>GraphicalSummarygenerateagraphicalsummarythatincludes

18、fourgraphs:ahistogramofdatawithanoverlaidnormalcurve,boxplot,andconfidenceintervalsforthemeanandthemedian.ThegraphicalsummaryalsodisplaysAnderson-DarlingNormalityTestresults,descriptivestatistics,andconfidenceintervalsforthemean,median,andstandarddeviation.5$5% Confkence Interval*NodtaHh 11g urn119W

19、12mnmn MQs the powerThegraphicalsummaryofyourMonteCarlosimulationoutputwilllooklikethis:SummaryforFlowAnderson-DttrtmghtormagTea:A-SQU4n«d2.MH«n1?Varwcfl0.573SAmmesOCMISMOKwtwfDm11339w10000aHlwmrn8,882I 就®ar琬e11A91Median1L995和1251115如95%GorrMenceIrwenrWlferM«n11,999U.D09第咻SiAdent

20、*MedianII 湖L2.OO295%ConM«nc«InurvalfarStiDwvJSJ0.760Fortherandomdatageneratedtowritethisarticle,themeanflowrateis12.004basedon100,000samples.Onaverage,weareontarget,butthesmallestvaluewas8.882andthelargestwas15.594.That'squitearange.Thetransmittedvariation(ofallcomponents)resultsinasta

21、ndarddeviationof0.757ml,farexceedingthe0.2mltarget.Also,weseethatthe0.2mltargetfallsoutsideoftheconfidenceintervalforthestandarddeviation.Itlookslikethispumpdesignexhibitstoomuchvariationandneedstobefurtherrefinedbeforeitgoesintoproduction;MonteCarlosimulationwithMinitabletusfindthatoutwithoutincurr

22、ingtheexpenseofmanufacturingandtestingthousandsofprototypes.Lestyouwonderwhetherthesesimulatedresultsholdup,tryityourself!Creatingdifferentsetsofsimulatedrandomdatawillresultinminorvariations,buttheendresultanunacceptableamountofvariationintheflowratewillbeconsistenteverytime.ThatoftheMonteCarlometh

23、od.MonteCarloUsingaDOEResponseEquationWhatifyoudon'tknowwhatequationtouse,oryouaretryingtosimulatetheoutcomeofauniqueprocess?prepares metal parts for electroplating.Anelectronicsmanufacturerhasassignedyoutoimproveitselectrocleaningoperation,whichElectroplatingletsmanufacturerscoatrawmaterialswit

24、halayerofadifferentmetaltoachievedesiredcharacteristics.Platingwillnotadheretoadirtysurface,sothecompanyhasacontinuous-flowelectrocleaningsystemthatconnectstoanautomaticelectroplatingmachine.Aconveyerdipseachpartintoabathwhichsendsvoltagethroughthepart,cleaningit.InadequatecleaningresultsinahighRoot

25、MeanSquareAverageRoughnessvalue,orRMS,andpoorsurfacefinish.ProperlycleanedpartshaveasmoothsurfaceandalowRMS.Tooptimizetheprocess,youcanadjusttwocriticalinputs:voltage(Vdc)andcurrentdensity(ASF).Foryourelectrocleaningmethod,thetypicalengineeringlimitsforVdcare3to12volts.Limitsforcurrentdensityare10to

26、150ampspersquarefoot(ASF).Step1:IdentifytheTransferEquationYoucannotuseanestablishedtextbookformulaforthisprocess,butyoucansetupaResponseSurfaceDOEinMinitabtodeterminethetransferequation.ResponsesurfaceDOEsareoftenusedtooptimizetheresponsebyfindingthebestsettingsfora"vitalfew"controllablef

27、actors.Inthiscase,theresponsewillbethesurfacequalityofpartsaftertheyhavebeencleaned.TocreatearesponsesurfaceexperimentinMinitab,chooseStat>DOE>ResponseSurface>CreateResponseSurfaceDesignBecausewehavetwofactorsvoltage(Vdc)andcurrentdensity(ASF)we'llselectatwo-factorcentralcompositedesign

28、,whichhas13runs.AfterMinitabcreatesyourdesignedexperiment,youneedtoperformyour13experimentalruns,collectthedata,andrecordthesurfaceroughnessofthe13finishedparts.MinitabmakesiteasytoanalyzetheDOEresults,reducethemodel,andcheckassumptionsusingresidualplots.UsingthefinalmodelandMinitab'sresponseopt

29、imizer,youcanfindtheoptimumsettingsforyourvariables.Inthiscase,yousetvoltsto7.74andASFto77.8toobtainaroughnessvalueof39.4.TheresponsesurfaceDOEyieldsthefollowingtransferequationfortheMonteCarlosimulation:Roughness=957.8-189.4(Vdc)-4.81(ASF)+12.26(Vdc2)+0.0309(ASF2)Step2:DefinetheInputParametersNowyo

30、ucansettheparametricdefinitionsforyourMonteCarlosimulationinputs.(Thestandarddeviationsmustbeknownorestimatedbasedonexistingprocessknowledge.)Voltsarenormallydistributedwithameanof7.74Vdcandastandarddeviationof0.14Vdc.AmpsperSquareFoot(ASF)arenormallydistributedwithameanof77.8ASFandastandarddeviatio

31、nof3ASF.Step3:CreateRandomDataWiththeparametersdefined,it'ssimpletocreate100,000rowsofsimulateddataforourtwoinputsusingMinitab'sCalc>RandomData>Normaldialog.Step4:SimulateandAnalyzeProcessOutputSummaryforRMSroughness95% CorrtMenctt IntervalsA-Squared4627.S?<0.00?MW甄蝴0,271Skewness2.01510KyrtositN100000MW上我国wti旭39佃Med on里71。3nj QuvtileMaxiitmoi幅1兆An&oCwling Normality Test95%Ligv闻 for Mea