《船舶安全与可靠性理论》课程教学课件（英文讲义）Safety and Reliability Analysis Lecture 3/5

SafetyandReliabilityAnalysisLecture3Yiliu LiuDepartment of Production and Quality EngineeringNorwegian Universityof ScienceandTechnologyyiliu.liu@ntnu.noNTNU- TrondheimNorwegian University ofScience and Technologywww.ntnu.edu

1 Safety and Reliability Analysis Lecture 3 Yiliu Liu Department of Production and Quality Engineering Norwegian University of Science and Technology yiliu.liu@ntnu.no

COMPONENTIMPORTANCENTNU-TrondheimNorwegian University ofScienceandTechnologywww.ntnu.edu

2 COMPONENT IMPORTANCE

3ComponentimportanceThecomponentimportancemaybeusedtoProvideacoarserankingofthecomponentswithrespecttotheirinfluenceonthesystemreliability (ortopeventprobability inafaulttree)HelpfocusingonthetopcontributorstosystemunreliabilityHelprelaxforthelowestcontributorstosystemunreliabilityFocusonimprovementswiththegreatestreliabilityeffectIndicatesensitivityformodelparametersHelpfocusingreviewsandsensitivitystudiesGiveprioritiesforfault-findingincomplexsystemNTNU-TrondheimNorwegian UniversityofScienceand Technologywww.ntnu.edu

3 The component importance may be used to: • Provide a coarse ranking of the components with respect to their influence on the system reliability (or top event probability in a fault tree) • Help focusing on the top contributors to system unreliability • Help relax for the lowest contributors to system unreliability • Focus on improvements with the greatest reliability effect • Indicate sensitivity for model parameters • Help focusing reviews and sensitivity studies • Give priorities for fault‐finding in complex system Component importance

ImportancemeasuresThe following component importance measures are definedand discussed in this courseBirnbaum'smeasureTheimprovementpotentialmeasureRiskachievementworthRisk reductionworthThecriticalityimportancemeasureThe importance measures arenot always consistently defined in the literature.NTNU-Trondheim莎NorwegianLniversityofScience and Technologywww.ntnu.edu

4 The following component importance measures are defined and discussed in this course: – Birnbaum’s measure – The improvement potential measure – Risk achievement worth – Risk reduction worth – The criticality importance measure The importance measures are not always consistently defined in the literature. Importance measures

5ImportancemeasuresThevariousmeasuresarebased onslightlydifferent interpretationsoftheconceptcomponentimportance.Intuitively,theimportanceofacomponentshould dependontwofactors:ThelocationofthecomponentinthesystemThereliabilityofthecomponentinquestionand,perhaps,alsotheuncertaintyinourestimateofthecomponentreliabilityNTNU-TrondheimNorwegian University ofScience and Technologywww.ntnu.edu

5 The various measures are based on slightly different interpretations of the concept component importance. Intuitively, the importance of a component should depend on two factors: – The location of the component in the system – The reliability of the component in question and, perhaps, also the uncertainty in our estimate of the component reliability. Importance measures

FBirnbaummeasuresBirnbaum(1969)proposedthefollowingmeasureofthereliabilityimportanceofcomponentiattimet:ah(p(t))[B(ilt) =ap;(t)Birnbaum'smeasure is thereforeobtained as the partial derivativeof the system reliabilityh(p(t))withrespecttop;(t).ThisapproachiswellknownfromclassicalsensitivityanalysisIfB(ilt)islarge,a small changeinthereliabilityof component iresultsinacomparativelylargechangeinthesystemreliabilityattimetWhentakingthisderivative,thereliabilities of theothercomponents remain constant-onlytheeffectofvaryingp;(t)isstudied.NTNU-TrondheimNorwegian UniversityofScienceandTechnologywww.ntnu.edu

6 Birnbaum (1969) proposed the following measure of the reliability importance of component ݅ at time ݐ: ሻݐሺ௜݌߲ ሻሻݐሺ࢖ሺ ߲݄ൌ ݐ݅ ஻ܫ Birnbaum’s measure is therefore obtained as the partial derivative of the system reliability ݄ሺ࢖ሺݐሻሻ with respect to ݌௜ሺݐሻ. This approach is well known from classical sensitivity analysis. If ܫ ݅ ஻ݐ is large, a small change in the reliability of component ݅ results in a comparatively large change in the system reliability at time ݐ. When taking this derivative, the reliabilities of the other components remain constant – only the effect of varying ݌௜ሺݐሻ is studied. Birnbaum measures

BirnbaummeasuresIn the definition of Birnbaum's measure,the system reliability is denoted h(p(t))and thesystem reliability isthereforea function of the component reliabilities only,i.e.,ofp(t):(pi(t),p2(t),...Pn(t)).Thismeansthattheallthencomponentsmustbeindependent.This definition of Birnbaum's measure is therefore not useable when the components aredependent, e.g., when we have common-causefailures.NTNU-TrondheimNorwegianUniversityofScienceandTechnologywww.ntnu.edu

7 In the definition of Birnbaum’s measure, the system reliability is denoted h(p(t)) and the system reliability is therefore a function of the component reliabilities only, i.e., of ࢖ ݐ ൌ ሺ݌ଵ ݐ, ݌ଶ ݐ., ݌௡ሺݐሻሻ. This means that the all the ݊ components must be independent. This definition of Birnbaum’s measure is therefore not useable when the components are dependent, e.g., when we have common‐cause failures. Birnbaum measures

8BirnbaummeasuresConsider a series structure of two independent components,1 and 2,with componentreliabilitiespi andp2,respectively.Assume that pi>p2,i.e.,component1isthemostreliable of the two.The reliabilityof the seriessystem is h(p(t))=PiP2ah(p)1.Birnbaum'smeasureofcomponent1isB(1)==P2apiah(p)=p12.Birnbaum's measure of component2 is IB(2)=ap2Thismeansthat[B(2)>[B(1)andwecanconcludethatwhenusingBirnbaum'smeasure,themost importantcomponent inaseriesstructureistheonewiththelowestreliability.To improve a series structure,we should therefore improve the"weakest"component, i.e.,the component withthelowest reliability.NTNU-TrondheimDNorwegian University ofScienceandTechnologywww.ntnu.edu

8 Consider a series structure of two independent components, 1 and 2, with component reliabilities ݌ଵ and ݌ଶ, respectively. Assume that ݌ଵ ൐ ݌ଶ, i.e., component 1 is the most reliable of the two. The reliability of the series system is ݄ ࢖ ݐ ൌ݌ଵ݌ଶ. 1. Birnbaum’s measure of component 1 is ܫ ஻1 ൌ డ௛ሺ࢖ሻ డ௣భ ൌ ݌ଶ 2. Birnbaum’s measure of component 2 is ܫ ஻2 ൌ డ௛ሺ࢖ሻ డ௣మ ൌ ݌ଵ This means that ܫ ஻2 ൐ܫ ஻1 and we can conclude that when using Birnbaum’s measure, the most important component in a series structure is the one with the lowest reliability. To improve a series structure, we should therefore improve the “weakest” component, i.e., the component with the lowest reliability. Birnbaum measures

0BirnbaummeasuresConsider a parallel structure of two independent components,1 and 2,with componentreliabilitiespi andp2,respectively.Assume thatpi>p2,i.e.,component1isthemostreliableof thetwo.The reliabilityof the seriessystem is h(p(t))=pi+p2-pip2.ah(p)=1-P21.Birnbaum'smeasureofcomponent1isB(1)apiah(p)= 1-p12.Birnbaum'smeasure of component2isB(2)1ap2This means that [B(1)>[B(2) and we can conclude that when using Birnbaum's measure,themost importantcomponent inaparallelstructure istheonewiththehighest reliability.Toimproveaparallel structure,we shouldthereforeimprovethe“strongest"component,i.e.,thecomponentwiththehighestreliabilityNTNU-TrondheimNorwegian University ofScienceandTechnologywww.ntnu.edu

9 Consider a parallel structure of two independent components, 1 and 2, with component reliabilities ݌ଵ and ݌ଶ, respectively. Assume that ݌ଵ ൐ ݌ଶ, i.e., component 1 is the most reliable of the two. The reliability of the series system is ݄ ࢖ ݐ ൌ݌ଵ ൅ ݌ଶ െ ݌ଵ݌ଶ. 1. Birnbaum’s measure of component 1 is ܫ ஻1 ൌ డ௛ሺ࢖ሻ డ௣భ ൌ1െ݌ଶ 2. Birnbaum’s measure of component 2 is ܫ ஻2 ൌ డ௛ሺ࢖ሻ డ௣మ ൌ1െ݌ଵ This means that ܫ ஻1 ൐ܫ ஻2 and we can conclude that when using Birnbaum’s measure, the most important component in a parallel structure is the one with the highest reliability. To improve a parallel structure, we should therefore improve the “strongest” component, i.e., the component with the highest reliability. Birnbaum measures

10BirnbaummeasuresBypivotaldecomposition,wehaveh(p(t)) =p;(t)-h(1li,p(t))+ (1 -p;(t)) -h(Oi,p(t))Birnbaum'smeasurecanthereforewewrittenasah(p(t))2 = h(1;,p(t) - h(0i,p(t)[B(ilt) =ap;(t)whereh(1.p(t))isthesystemreliabilitywhenweknowthatcomponentiisfunctioningandh(Op(t))isthesystemreliabilitywhenweknowthat component i isnotfunctioning.Thisleads toa very simple way of calculating IB(ilt)-as illustrated by the example on thenextslide.Most computer programs for fault tree analysis computes Birnbaum's measure by thisapproach.NTNU-TrondheimNorwegian University ofScience and Technologywww.ntnu.edu

10 By pivotal decomposition, we have ݐ ࢖ ,௜0·݄ ሻ ݐ ௜݌ െ ሺ1 ൅ ݐ ࢖ ,௜1 ·݄ ݐ ௜݌ൌ ݐ ࢖ ݄ Birnbaum’s measure can therefore we written as ݐ ࢖ ,௜0݄ െ ݐ ࢖ ,௜1 ݄ ൌ ሻݐሺ௜݌߲ ሻሻݐሺ࢖ሺ ߲݄ൌ ݐ݅ ஻ܫ where ݄ 1௜, ࢖ ݐ is the system reliability when we know that component ݅ is functioning and ݄ 0௜, ࢖ ݐ is the system reliability when we know that component ݅ is not functioning. This leads to a very simple way of calculating ܫ ݅ ஻ݐ – as illustrated by the example on the next slide. Most computer programs for fault tree analysis computes Birnbaum’s measure by this approach. Birnbaum measures