14ReliabilityAnalysis(1)

ReliabilityAnalysisTotalQualityManagement-UniversityofMichigan1PatHammett1ReliabilityAnalysis2ReliabilitynReliabilityDefined:Reliabilityistheprobabilitythatasystemorproductwillperforminasatisfactorymannerforagivenperiodoftimewhenoperatedunderspecifiedconditions.nReliabilityAnalysis:Analysisofsystemfailuresandrepairsto:nincreasethedesignlifeofaproduct.nreducethelikelihoodoffailures.nreducemanufacturingprocessdowntime.neliminatesafetyconcerns.ReliabilityAnalysisTotalQualityManagement-UniversityofMichigan2PatHammett3ReliabilityandSixSigmaProjectsnConsiderthefollowingdefects:nHighwarrantycostsduetoexcessivereplacementofcertainproductcomponents.nLatedeliveryofproductsduetoexcessivemachinedowntime.nAneffectivemethodtoexaminetheeffectsofthesedefectsisReliabilityAnalysis.4ProductTypes/FailureConditions1.Non-repairableitems{replace}(e.g.,lightbulb)ninterestedinprobabilityoffirstandonlyfailure(hazardrate)nAnalyzeMTTF(mean-time-to-failure)2.Repairableitems(e.g.,copymachine)ninterestedinprobabilitythatafailurewilloccuroversometimeperiodnAnalyzeMTBF(mean-time-between-failures)ReliabilityAnalysisTotalQualityManagement-UniversityofMichigan3PatHammett5ReliabilityAssessmentnTodescribefailurepatternsinordertopredictfutureissues,weneedstatisticaldistributions.nCommondistributionsfornon-repairabledevices:nWeibullnExponential*isspecialcasewherefailurerateisconstant.nLog-normalnNormal*nCommondistributionsforrepairabledevices:nPoissonnNonhomogenousPoissonProcess(NHPP)6FailurePatternforProductLifeCycle-BathtubCurveBathtubCurveλ,(failurerateorhazardrate)ReliabilityAnalysisTotalQualityManagement-UniversityofMichigan4PatHammett7ConstantFailureRateModelsExample:nSupposeyousell1000televisionson1/1/00andyourfailurerateis10%everyyear.Calculatetheexpectednumberoffailureseachtimeperiod.#ofUnitsLeftTimePeriod,yrFailureRate#ofFailures100010.1010090020.109081030.10??40.10?8DistributionofFailuresnWhatdistributionfunctionbestmodelsthe#offailuresiffailurerateisassumedconstant?(example:0.1/year)0.020.040.060.080.0100.0120.002468101214161820timeperiod,t#ofFailuresReliabilityAnalysisTotalQualityManagement-UniversityofMichigan5PatHammett9ReliabilityPrediction-ExponentialnConstantfailureratet=time,λ=failurerateorfrequencyoffailuresMean-time-to-failure(MTTF)=1/λ;nExample:fuseis“asgoodasnew”overusefullifeMTTFt-e)(==-tetRl10R(t)-ReliabilityFunctionnProbabilityunitwillnotfailinfirst6months=.9512nProbabilityunitwillnotfailafter2years=.8WhatisR(10)?TimePeriodR(t)0.50.951210.90481.50.860720.81872.50.778830.740800.20.40.60.8105101520timeperiod,yearR(t)ReliabilityAnalysisTotalQualityManagement-UniversityofMichigan6PatHammett11TypeI*censoringtoestimateλnRunasampleofpartsforafixedtime.nTimetofailure=fixedtime.nIfpartdoesnotfail,stoptestatfixedtimeassignfixedtimevalue(censoreddata).nLetλ=#failures/totaltimenSupposeyoutest10unitsusinga600-hourreliabilitytestthatsimulatesusefullife.Fromthetest,unit1failsafter75hours,Unit4failsafter125hours,Unit7failsafter130hours,Unit9failsafter325hours,Unit10failsafter525hours,theother5donotfail.ourfailures/h001196.)600(5525325130125755=+++++=l*TypeIIcensoringiswhenyouterminatetestafterfixednumberoffailures.12ExponentialReliabilityExamplenR(t)=e-λtnWhatistheprobabilityaunitwillnotfailafter10hours?nWhatistheprobabilityunitwillfailafter10hours?ReliabilityAnalysisTotalQualityManagement-UniversityofMichigan7PatHammett13GeneralNon-repairableModelnUseWeibullModelforPredictionnReliability{Pr(NotFail)}--R(t)=e-(λt)bnWhere:nt-timeperiod(note:timemaybereplacedbyatimesurrogatesuchas#ofmiles,#copies)nλ-1/MTTFor1/MTBFnb-shapeparameterb1;decreasingfailureratewithusageb=1;constantfailurerateb1;increasingfailurerate3b4;approachessymmetrical-normal(b=3.5)14Reliability:NormalDistributionnForsomeproducts,failuresfollowanormaldistribution(“wearingeffect”).nExample:productssuchasdrillbitsmayhaveanormallydistributedfailuredependingon“wear-out”nTousenormaldistribution,weneedtwoparameterestimates:naveragetimetofailure(MTTF),nstandarddeviationoffailuretime.ReliabilityAnalysisTotalQualityManagement-UniversityofMichigan8PatHammett15CalculatingReliabilityUsingNormalDistributionnTodeterminethereliabilityattimet,wemustfirstcomputetheZvalueattimet.)(Prob1)(tZZtR-=sm-=tZtt=timeperiodμ=MTTFσ=standarddeviationoftimetofailuretProductFails(t)ProductOK16ReliabilityExamplenSupposeadrillbithasaMTTFof100withastandarddeviationof25.Whatisthereliabilityofthetoollasting50hoursiffailuresfollowanormaldistribution?t=50hours,MTTF(μ)=100hours,σ=25hoursZt=(50-100)/25=-2.0Usingexcel:=normdist(t,MTTF,sigma,true)Pr(fail)=normdist(50,100,25,true)=0.02275Pr(notfail)=R(50)=1-0.02275=0.97725ReliabilityAnalysisTotalQualityManagement-UniversityofMichigan9PatHammett17IdentifyingWarrantyPeriodsnGivenfailuredistribution,acommonapplicationofreliabilityassessmentistoidentifyawarrantyperiod.nEssentially,Identifyadesiredreliabilityandsolvefort.18DeterminingWarrantyGivenReliabilityGoal(Exponential)Supposeλ=.0001,whatwarrantyperiodisneededtoachieveareliabilitygoalof0.99.(findt)AssumingExpo