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Fuzzyarithmeticbasedreliabilityallocationapproachduringearlydesignanddevelopment初期阶段的设计和发展中基于模糊算法的可靠度分配方法V.Sriramdas,S.K.Chaturvedi,H.Gargama1.Introduction2.Factorsbasedconventionalreliabilityallocationmethod3.Fuzzynumbersandarithmetic4.Themethodology5.Illustrativeexample6.Conclusions1.IntroductionReliabilityallocationisanimportantanditerativetaskduringthedesignanddevelopmentactivitiesofanyengineeringsystem.Itisalsodifficulttaskbecauseoftheobscuredandincompletedesigndetailsandanumberoffactorshavetoconsiderindesignprocess.Duringthedesignphaseofasystemwithaspecifiedtargetreliabilitylevel,thereliabilitylevelsofthesubsystemsaffecttheoverallsystemreliability.Therefore,aproperreliabilityallocationmethodneedstobeadoptedtoallocatethetargetsystemreliabilitytoitsconstituentsubsystemsproportionately.42.FactorsbasedconventionalreliabilityallocationmethodInthisallocationmethod,thetargetreliabilitybasedonthereliabilityfactorsisapportionedtosubsystemsforwhichnopredictedreliabilityvaluesareknown.TherelationshipbetweenapportionedreliabilityofithsubsystemRi（子系统可靠度）andtargetsystemreliabilityR*（总系统可靠度）isdefinedwithaweightagefactorwi.Ri=(R*)wi(1)whereweightagefactorwi（权重因子）canbeexpressedwithproportionalityfactorZi（比例因子）as：wi=Zi/∑Zi(2)52.1.Complexity（复杂度）Thecomplexityfactorvariesfromsubsystemtosubsystemwithinasystemandismeasuredintermsofnumberofactivecomponentsthatasubsystemiscomposedof.Thenumberofcomponentsinasubsystemhasadirectbearingonthereliabilityofthesubsystem.Thus,complexityhasastrongimpactonthereliabilityallocation.Thefailurerateofthesubsystemwithhighcomplexityisgenerallygoingtobehigh.So,thefailurerateisallocatedproportionaltothecomplexityofthesubsystem.Hence,Zi∝Ki,whereKiisthecomplexityfactorfortheitssubsystem.1.Multiplefunctionalrelationshipswiththeothergroups.2.Numberofcomponentscomprisingsubsystem.62.2.Cost（成本）Foralargesystem,thecostincrementforreliabilityimprovementisrelativelyhigh.Thedemonstrationofahighreliabilityvalueforacostlysystemmaybeextremelyuneconomical.Hence,Zi∝Coi,whereCoiisthecostfactorfortheitssubsystem.2.3.State-of-the-art（工艺状态）Whenthecomponenthasbeenavailableforalongtime,itisquitedifficulttofurtherimprovethereliabilityofacomponentevenifthereliabilityisconsiderablylowerthandesired.Hence,Zi∝1/Si,whereSiisthestate-of-the-artfactorfortheitssubsystem.72.4.Criticality（临界值）Criticalityisanotherveryimportantfactorinreliabilityallocation.Itislogical,higherreliabilitytargetshouldbeallocatedtothefunctionallycriticalsub-systemsandthusZiisproportionaltocriticality.Hence,Zi∝1/Cri,whereCriisthecriticalityfactorfortheitssubsystem.2.5.Timeofoperation（运行时间）Theremaybesomesubsystemswhicharerequiredtobeoperatedforaperiodlessthanthemissiontime.So,forthesubsystemswithoperatingtimelessthanthemissiontime,itisonlylogicaltoallocaterelativelylowerreliability.Hence,Zi∝1/Ti,whereTiisthetimeofoperationfactorfortheithsubsystem2.6.Maintenance（维护）Acomponentwhichisperiodicallymaintainedoronewhichisregularlymonitoredorcheckedandrepairedasnecessarywillhave,onanaveragehigheravailabilitythanonewhichisnotmaintainedHence,Zi∝Mi,whereMiisthemaintenancefactorfortheitssubsystem.Theprocessofallocationofrelativescalesiscarriedoutasateamexercise,comprisingofexperiencedmembersfromtheeachofthesubsystemidentified.Fromthepreviousdiscussionsinthissection,afterconsiderationofvariousfactors,formulaforproportionalityfactor(Zi)as:3.FuzzynumbersandarithmeticDefinition:Afuzzynumberisafuzzysubsetthatisbothconvex,andnormal.Themostcommonlyusedfuzzynumbersaretriangularandtrapezoidalfuzzynumbers,parameterizedby(a,b,c),and(a,b,c,d),respectively,themembershipfunctionsofthesenumbersaredefinedbelow:Thenthestandardoperationsontrapezoidalfuzzynumbersareexpressedas:Addition:Subtraction:Multiplication:Division:Defuzzification（解模糊化）istheunderlyingreasonthatonecannotcomparefuzzynumbersdirectly.Althoughmanyauthorsproposedtheirfavoritemethods,thereisnouniversalconsensus.Eachmethodincludescomputingacrispvalue,tobeusedforcomparison.Thisassignmentofarealvaluetoafuzzynumberiscalleddefuzzification.Itcantakemanyforms,butthemoststandarddefuzzificationisthroughcomputingthecentroid(计算模糊重心).3.1.FuzzydivisionbyusinglinearprogrammingLetbetwotrapezoidalfuzzynumbersparameterizedby(l1,c1,c11,r1),and(l2,c2,c22,r2),wherel1andl2,c1andc2,c11andc22,andr1andr2denotesleftendpoints,leftcenterpoints,rightcenterpoints,andrightendpoints,respectively.Theresultingfuzzynumberscanbewrittenasfollows:Theconstraintsandobjectivefunctionofthelinearprogrammingproblemforthetrapezoidalfuzzydivisionaredefinedasfollows:Thefirstconstraintisconstructedbasedontheleftspreads.TheleftspreadvalueovercentervalueforshouldbeequaltoorlessthandivisionofthesameratioscalculatedforInasimilarmanner,thesecondconstraintisconstructedasfollows:Thethirdandfourthconstraintscanbedefinedbasedonthedefinitionoffuzzynumbers.Theleftendpointofafuzzynumbershouldbelessthantheleftcentervalue.Themathematicalexpressionforthethirdconstraintisgivenbelow:Therightendpointofafuzzynumbershouldbegreaterthantherightcentervalue.Themathematicalexpressionforth