利用MINITAB做蒙特卡洛模擬

1、DoingMonteCarloSimulationinMinitabStatisticalSoftwareDoingMonteCarlosimulationsinMinitabStatisticalSoftwareisveryeasy.ThisarticleillustrateshowtouseMinitabforMonteCarlosimulationsusingbothaknownengineeringformulaandaDOEequation.byPaulSheehyandEstonMartzMonteCarlosimulationusesrepeatedrandomsamplingt

2、osimulatedataforagivenmathematicalmodelandevaluatetheoutcome.Thismethodwasinitiallyappliedbackinthe1940s,whenscientistsworkingontheatomicbombusedittocalculatetheprobabilitiesofonefissioninguraniumatomcausingafissionreactioninanother.Withuraniuminshortsupply,therewaslittleroomforexperimentaltrialande

3、rror.Thescientistsdiscoveredthataslongastheycreatedenoughsimulateddata,theycouldcomputereliableprobabilitiesandreducetheamountofuraniumneededfortesting.Today,simulateddataisroutinelyusedinsituationswhereresourcesarelimitedorgatheringrealdatawouldbetooexpensiveorimpractical.ByusingMinitabsabilitytoea

4、silycreaterandomdata,youcanuseMonteCarlosimulationto:Simulatetherangeofpossibleoutcomestoaidindecision-makingForecastfinancialresultsorestimateprojecttimelinesUnderstandthevariabilityinaprocessorsystemFindproblemswithinaprocessorsystemManageriskbyunderstandingcost/benefitrelationshipsStepsintheMonte

5、CarloApproachDependingonthenumberoffactorsinvolved,simulationscanbeverycomplex.Butatabasiclevel,allMonteCarlosimulationshavefoursimplesteps:1.IdentifytheTransferEquationTodoaMonteCarlosimulation,youneedaquantitativemodelofthebusinessactivity,plan,orprocessyouwishtoexplore.Themathematicalexpressionof

6、yourprocessiscalledthe“transferequation.”Thismaybeaknownengineeringorbusinessformula,oritmaybebasedonamodelcreatedfromadesignedexperiment(DOE)orregressionanalysis.2.DefinetheInputParametersForeachfactorinyourtransferequation,determinehowitsdataaredistributed.Someinputsmayfollowthenormaldistribution,

7、whileothersfollowatriangularoruniformdistribution.Youthenneedtodeterminedistributionparametersforeachinput.Forinstance,youwouldneedtospecifythemeanandstandarddeviationforinputsthatfollowanormaldistribution.3.CreateRandomDataTodovalidsimulation,youmustcreateaverylarge,randomdatasetforeachinputsomethi

8、ngontheorder100,000instances.Theserandomdatapointssimulatethevaluesthatwouldbeseenoveralongperiodforeachinput.Minitabcaneasilycreaterandomdatathatfollowalmostanydistributionyouarelikelytoencounter.4.SimulateandAnalyzeProcessOutputWiththesimulateddatainplace,youcanuseyourtransferequationtocalculatesi

9、mulatedoutcomes.Runningalargeenoughquantityofsimulatedinputdatathroughyourmodelwillgiveyouareliableindicationofwhattheprocesswilloutputovertime,giventheanticipatedvariationintheinputs.ThosearethestepsanyMonteCarlosimulationneedstofollow.HereshowtoapplytheminMinitab.MonteCarloUsingaKnownEngineeringFo

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

11、swillhaveastandarddeviationnogreaterthanml.Step1:IdentifytheTransferEquationThefirststepindoingaMonteCarlosimulationistodeterminethetransferequation.Inthiscase,youcansimplyuseanestablishedengineeringformulathatmeasurespumpflow:Flow(inml)=(D/2)2?L?RPMStep2:DefinetheInputParametersNowyoumustdefinethed

12、istributionandparametersofeachinputusedinthetransferequation.Thepumpspistondiameterandstrokelengthareknown,butyoumustcalculatethestrokes-per-minute(RPM)neededtoattainthedesired12ml/minuteflowrate.Volumepumpedperstrokeisgivenbythisequation:(D/2)2*LGivenD=andL=,eachstrokedisplacesml.Sotoachieveaflowof

13、12ml/minutetheRPMis.Basedontheperformanceofotherpumpsyourfacilityhasmanufactured,youcansaythatpistondiameterisnormallydistributedwithameanofcmandastandarddeviationofcm.Strokelengthisnormallydistributedwithameanofcmandastandarddeviationofcm.Finally,strokesperminuteisnormallydistributedwithameanofRPMa

14、ndastandarddeviationofRPM.Step3:CreateRandomDataNowyourereadytosetupthesimulationinMinitab.WithMinitabyoucaninstantaneouslycreate100,000rowsofsimulateddata.Startingwiththesimulatedpistondiameterdata,chooseCalcRandomDataNormal.Inthedialogbox,enter100,000inNumberofrowsofdatatogenerate,andenter“D”asthe

15、columninwhichtostorethedata.Enterthemeanandstandarddeviationforpistondiameterintheappropriatefields.PressOKtopopulatetheworksheetwith100,000datapointsrandomlysampledfromthespecifiednormaldistribution.ThensimplyrepeatthisprocessforStrokeLength(L)andStrokesperMinute(RPM).Step4:SimulateandAnalyzeProces

16、sOutputNowcreateafourthcolumnintheworksheet,Flow,toholdtheresultsofyourprocessoutputcalculations.Withtherandomlygeneratedinputdatainplace,youcansetupMinitabscalculatortocalculatetheoutputandstoreitintheFlowcolumn.GotoCalcCalculator,andsetuptheflowequationlikethis:Minitabwillquicklycalculatetheoutput

17、foreachrowofsimulateddata.Nowyourereadytolookattheresults.SelectStatBasicStatisticsGraphicalSummaryandselecttheFlowcolumn.Minitabwillgenerateagraphicalsummarythatincludesfourgraphs:ahistogramofdatawithanoverlaidnormalcurve,boxplot,andconfidenceintervalsforthemeanandthemedian.Thegraphicalsummaryalsod

18、isplaysAnderson-DarlingNormalityTestresults,descriptivestatistics,andconfidenceintervalsforthemean,median,andstandarddeviation.ThegraphicalsummaryofyourMonteCarlosimulationoutputwilllooklikethis:Fortherandomdatageneratedtowritethisarticle,themeanflowrateisbasedon100,000samples.Onaverage,weareontarge

19、t,butthesmallestvaluewasresultsinastandarddeviationofandthelargestwas.Thatsquitearange.Thetransmittedvariation(ofallcomponents)ml,farexceedingthemltarget.Also,weseethatthemltargetfallsoutsideoftheconfidenceintervalforthestandarddeviation.Itlookslikethispumpdesignexhibitstoomuchvariationandneedstobef

20、urtherrefinedbeforeitgoesintoproduction;MonteCarlosimulationwithMinitabletusfindthatoutwithoutincurringtheexpenseofmanufacturingandtestingthousandsofprototypes.Lestyouwonderwhetherthesesimulatedresultsholdup,tryityourself!Creatingdifferentsetsofsimulatedrandomdatawillresultinminorvariations,buttheen

21、dresultanunacceptableamountofvariationintheflowratewillbeconsistenteverytime.ThatsthepoweroftheMonteCarlomethod.MonteCarloUsingaDOEResponseEquationWhatifyoudontknowwhatequationtouse,oryouaretryingtosimulatetheoutcomeofauniqueprocessAnelectronicsmanufacturerhasassignedyoutoimproveitselectrocleaningop

22、eration,whichpreparesmetalpartsforelectroplating.Electroplatingletsmanufacturerscoatrawmaterialswithalayerofadifferentmetaltoachievedesiredcharacteristics.Platingwillnotadheretoadirtysurface,sothecompanyhasacontinuous-flowelectrocleaningsystemthatconnectstoanautomaticelectroplatingmachine.Aconveyerd

23、ipseachpartintoabathwhichsendsvoltagethroughthepart,cleaningit.InadequatecleaningresultsinahighRootMeanSquareAverageRoughnessvalue,orRMS,andpoorsurfacefinish.ProperlycleanedpartshaveasmoothsurfaceandalowRMS.Tooptimizetheprocess,youcanadjusttwocriticalinputs:voltage(Vdc)andcurrentdensity(ASF).Foryour

24、electrocleaningmethod,thetypicalengineeringlimitsforVdcare3to12volts.Limitsforcurrentdensityare10to150ampspersquarefoot(ASF).Step1:IdentifytheTransferEquationYoucannotuseanestablishedtextbookformulaforthisprocess,butyoucansetupaResponseSurfaceDOEinMinitabtodeterminethetransferequation.Responsesurfac

25、eDOEsareoftenusedtooptimizetheresponsebyfindingthebestsettingsforavitalfewcontrollablefactors.Inthiscase,theresponsewillbethesurfacequalityofpartsaftertheyhavebeencleaned.TocreatearesponsesurfaceexperimentinMinitab,chooseStatDOEResponseSurfaceCreateResponseSurfaceDesign.Becausewehavetwofactorsdesign

26、,whichhas13runs.voltage(Vdc)andcurrentdensity(ASF)wellselectatwo-factorcentralcompositeAfterMinitabcreatesyourdesignedexperiment,youneedtoperformyour13experimentalruns,collectthedata,andrecordthesurfaceroughnessofthe13finishedparts.MinitabmakesiteasytoanalyzetheDOEresults,reducethemodel,andcheckassu

27、mptionsusingresidualplots.UsingthefinalmodelandMinitabyourvariables.Inthiscase,yousetvoltstosresponseoptimizer,youcanfindtheoptimumsettingsforandASFtotoobtainaroughnessvalueof.TheresponsesurfaceDOEyieldsthefollowingtransferequationfortheMonteCarlosimulation:22Roughness=?(Vdc)?(ASF)+(Vdc)+(ASF)Step2:

28、DefinetheInputParametersNowyoucansettheparametricdefinitionsforyourMonteCarlosimulationinputs.(Thestandarddeviationsmustbeknownorestimatedbasedonexistingprocessknowledge.)VoltsarenormallydistributedwithameanofVdcandastandarddeviationofVdc.AmpsperSquareFoot(ASF)arenormallydistributedwithameanofASFandastandarddeviationof3ASF.Step3:CreateRandomDataWiththeparametersdefined,itRandomDataNormalssimpletocreate100,000rowsofsimulateddataforourtwoinputsusingMinitabdialog.sCalcStep4:SimulateandAnalyzeProcessOutputNowwecanusetheCalculatortoenterourformula,followedbyStatB