利用MINITAB做蒙特卡洛模拟

DoingMonteCarloSimulationinMinitabStatisticalSoftwareDoingMonteCarlosimulationsinMinitabStatisticalSoftwareisveryeasy.ThisarticleillustrateshowtouseMinitabforMonteCarlosimulationsusingbothaknownengineeringformulaandaDOEequation.byPaulSheehyandEstonMartzMonteCarlosimulationusesrepeatedrandomsamplingtosimulatedataforagivenmathematicalmodelandevaluatetheoutcome.Thismethodwasinitiallyappliedbackinthe1940s,whenscientistsworkingontheatomicbombusedittocalculatetheprobabilitiesofonefissioninguraniumatomcausingafissionreactioninanotherWithuraniuminshortsupply,therewaslittleroomforexperimentaltrialanderror.Thescientistsdiscoveredthataslongastheycreatedenoughsimulateddata,theycouldcomputereliableprobabilities-andreducetheamountofuraniumneededfortesting.Today,simulateddataisroutinelyusedinsituationswhereresourcesarelimitedorgatheringrealdatawouldbetooexpensiveorimpractical.ByusingMinitab’sabilitytoeasilycreaterandomdata,youcanuseMonteCarlosimulationto:Simulatetherangeofpossibleoutcomestoaidindecision-makingForecastfinancialresultsorestimateprojecttimelinesUnderstandthevariabilityinaprocessorsystemFindproblemswithinaprocessorsystemManageriskbyunderstandingcost/benefitrelationshipsStepsintheMonteCarloApproachDependingonthenumberoffactorsinvolved,simulationscanbeverycomplex.Butatabasiclevel,allMonteCarlosimulationshavefoursimplesteps:IdentifytheTransferEquationTodoaMonteCarlosimulation,youneedaquantitativemodelofthebusinessactivity,plan,orprocessyouwishtoexplore.Themathematicalexpressionofyourprocessiscalledthe“transferequation.”Thismaybeaknownengineeringorbusinessformula,oritmaybebasedonamodelcreatedfromadesignedexperiment(DOE)orregressionanalysis.DefinetheInputParametersForeachfactorinyourtransferequation,determinehowitsdataaredistributed.Someinputsmayfollowthenormaldistribution,whileothersfollowatriangularoruniformdistribution.Youthenneedtodeterminedistributionparametersforeachinput.Forinstance,youwouldneedtospecifythemeanandstandarddeviationforinputsthatfollowanormaldistribution.CreateRandomDataTodovalidsimulation,youmustcreateaverylarge,randomdatasetforeachinput—somethingontheorder100,000instances.Theserandomdatapointssimulatethevaluesthatwouldbeseenoveralongperiodforeachinput.Minitabcaneasilycreaterandomdatathatfollowalmostanydistributionyouarelikelytoencounter.SimulateandAnalyzeProcessOutputWiththesimulateddatainplace,youcanuseyourtransferequationtocalculatesimulatedoutcomes.Runningalargeenoughquantityofsimulatedinputdatathroughyourmodelwillgiveyouareliableindicationofwhattheprocesswilloutputovertime,giventheanticipatedvariationintheinputs.ThosearethestepsanyMonteCarlosimulationneedstofollow.Here’showtoapplytheminMinitab.MonteCarloUsingaKnownEngineeringFormulaAmanufacturingcompanyneedstoevaluatethedesignofaproposedproduct:asmallpistonpumpthatmustpump12mloffluidperminute.Youwanttoestimatetheprobableperformanceoverthousandsofpumps,givennaturalvariationinpistondiameter(D),strokelength(L),andstrokesperminute(RPM).Ideally,thepumpflowacrossthousandsofpumpswillhaveastandarddeviationnogreaterthan0.2ml.Step1:IdentifytheTransferEquationThefirststepindoingaMonteCarlosimulationistodeterminethetransferequation.Inthiscase,youcansimplyuseanestablishedengineeringformulathatmeasurespumpflow:Flow(inml)=n(D/2)2*L*RPMStep2:DefinetheInputParametersNowyoumustdefinethedistributionandparametersofeachinputusedinthetransferequation.Thepumpspistondiameterandstrokelengthareknown,butyoumustcalculatethestrokes-per-minute(RPM)neededtoattainthedesired12ml/minuteflowrate.Volumepumpedperstrokeisgivenbythisequation:n(D/2)2*LGivenD=0.8andL=2.5,eachstrokedisplaces1.256ml.Sotoachieveaflowof12ml/minutetheRPMis9.549.Basedontheperformanceofotherpumpsyourfacilityhasmanufactured,youcansaythatpistondiameterisnormallydistributedwithameanof0.8cmandastandarddeviationof0.003cm.Strokelengthisnormallydistributedwithameanof2.5cmandastandarddeviationof0.15cm.Finally,strokesperminuteisnormallydistributedwithameanof9.549RPMandastandarddeviationof0.17RPM.Step3:CreateRandomDataNowyou’rereadytosetupthesimulationinMinitab.WithMinitabyoucaninstantaneouslycreate100,000rowsofsimulateddata.Startingwiththesimulatedpistondiameterdata,chooseCalc>RandomData>Normal.Inthedialogbox,enter100,000inNumberofrowsofdatatogenerate,andenter“D”asthecolumninwhichtostorethedata.Enterthemeanandstandarddeviationforpistondiameterintheappropriatefields.PressOKtopopulatetheworksheetwith100,000datapointsrandomlysampledfromthespecifiednormaldistribution.ThensimplyrepeatthisprocessforStrokeLength(L)andStrokesperMinute(RPM).Step4:SimulateandAnalyzeProcessOutputNowcreateafourthcolumnintheworksheet,Flow,toholdtheresultsofyourprocessoutputcalculations.Withtherandomlygeneratedinputdatainplace,youcansetupMinitab'scalculatortocalculatetheoutputandstoreitintheFlowcolumn.GotoCalc>Calculator,andsetuptheflowequationlikethis:Minitabwillquicklycalculatetheoutputforeachrowofsimulateddata.

Nowyou’rereadytolookattheresults.SelectStat>BasicStatistics>GraphicalSummaryandselecttheFlowcolumn.Minitabwillgenerateagraphicalsummarythatincludesfourgraphs:ahistogramofdatawithanoverlaidnormalcurve,boxplot,andconfidenceintervalsforthemeanandthemedian.ThegraphicalsummaryalsodisplaysAnderson-DarlingNormalityTestresults,descriptivestatistics,andconfidenceintervalsforthemean,median,andstandarddeviation.ThegraphicalsummaryofyourMonteCarlosimulationoutputwilllooklikethis:Fortherandomdatageneratedtowritethisarticle,themeanflowrateis12.004basedon100,000samples.Onaverage,weareontarget,butthesmallestvaluewas8.882andthelargestwas15.594.Thatsquitearange.Thetransmittedvariation(ofallcomponents)resultsinastandarddeviationof0.757ml,farexceedingthe0.2mltarget.Also,weseethatthe0.2mltargetfallsoutsideoftheconfidenceintervalforthestandarddeviation.Itlookslikethispumpdesignexhibitstoomuchvariationandneedstobefurtherrefinedbeforeitgoesintoproduction;MonteCarlosimulationwithMinitabletusfindthatoutwithoutincurringtheexpenseofmanufacturingandtestingthousandsofprototypes.Lestyouwonderwhetherthesesimulatedresultsholdup,tryityourself!Creatingdifferentsetsofsimulatedrandomdatawillresultinminorvariations,buttheendresul—anunacceptableamountofvariationintheflowrate-willbeconsistenteverytime.ThatsthepoweroftheMonteCarlomethod.MonteCarloUsingaDOEResponseEquationWhatifyoudon’tknowwhatequationtouse,oryouaretryingtosimulatetheoutcomeofauniqueprocess?Anelectronicsmanufacturerhasassignedyoutoimproveitselectrocleaningoperation,whichpreparesmetalpartsforelectroplating.Electroplatingletsmanufacturerscoatrawmaterialswithalayerofadifferentmetaltoachievedesiredcharacteristics.Platingwillnotadheretoadirtysurface,sothecompanyhasacontinuous-flowelectrocleaningsystemthatconnectstoanautomaticelectroplatingmachine.Aconveyerdipseachpartintoabathwhichsendsvoltagethroughthepart,cleaningit.InadequatecleaningresultsinahighRootMeanSquareAverageRoughnessvalue,orRMS,andpoorsurfacefinish.ProperlycleanedpartshaveasmoothsurfaceandalowRMS.Tooptimizetheprocess,youcanadjusttwocriticalinputs:voltage(Vdc)andcurrentdensity(ASF).Foryourelectrocleaningmethod,thetypicalengineeringlimitsforVdcare3to12volts.Limitsforcurrentdensityare10to150ampspersquarefoot(ASF).Step1:IdentifytheTransferEquationYoucannotuseanestablishedtextbookformulaforthisprocess,butyoucansetupaResponseSurfaceDOEinMinitabtodeterminethetransferequation.ResponsesurfaceDOEsareoftenusedtooptimizetheresponsebyfindingthebestsettingsfora"vitalfew"controllablefactors.Inthiscase,theresponsewillbethesurfacequalityofpartsaftertheyhavebeencleaned.TocreatearesponsesurfaceexperimentinMinitab,chooseStat>DOE>ResponseSurface>CreateResponseSurfaceDesign.Becausewehavetwofactors—voltage(Vdc)andcurrentdensity(ASF)—we’llselectatwo-factorcentralcompositedesign,whichhas13runs.AfterMinitabcreatesyourdesignedexperiment,youneedtoperformyour13experimentalruns,collectthedata,andrecordthesurfaceroughnessofthe13finishedparts.MinitabmakesiteasytoanalyzetheDOEresults,reducethemodel,andcheckassumptionsusingresidualplots.UsingthefinalmodelandMinitab'sresponseoptimizer,youcanfindtheoptimumsettingsforyourvariables.Inthiscase,yousetvoltsto7.74andASFto77.8toobtainaroughnessvalueof39.4.TheresponsesurfaceDOEyieldsthefollowingtransferequationfortheMonteCarlosimulation:Roughness=957.8-189.4(Vdc)-4.81(ASF)+12.26(VdC2)+0.0309(ASF2)Step2:DefinetheInputParametersNowyoucansettheparametricdefinitionsforyourMonteCarlosimulationinputs.(Thestandarddeviationsmustbeknownorestimatedbasedonexistingprocessknowledge.)Voltsarenormallydistributedwithameanof7.74Vdcandastandarddeviationof0.14Vdc.AmpsperSquareFoot(ASF)arenormallydistributedwithameanof77.8ASFandastandarddeviationof3ASF.Step3:CreateRandomDataWiththeparametersdefined,it'ssimpletocreate100,000rowsofsimulateddataforourtwoinputsusingMinitab'sCalc>RandomData>Normaldialog.Step4:SimulateandAnalyzeProcessOutputNowwecanusetheCalculatortoenterourformula,followedbyStat>BasicStatistics>GraphicalSummary.AnciewHXarlingTestAnciewHXarlingTestA-Squared4fi27.57Mbsh神确StDevamifl2.01S-10-KUIW545NHKXKJOMtinirmjml^tQtisrtile弛彻Median39.710JrdQtisitile尬MSMaximium45.135W5腕Mw&dsoeJritervalfar祢9%&Ofi