Marvin Rausand, March 19, 2004 System Reliability Theory (2nd ed.), Wiley, 2003 – p. 1/19
Chapter 5Component Importance
Marvin RausandDepartment of Production and Quality Engineering
Norwegian University of Science and Technology
Introduction
● Measures Covered
● Importance Depends On
Importance Measures
Marvin Rausand, March 19, 2004 System Reliability Theory (2nd ed.), Wiley, 2003 – p. 2/19
Measures Covered
The following component importance measures are definedand discussed in this chapter:
■ Birnbaum’s measure (and some variants)■ The improvement potential measure (and some variants)■ Risk achievement worth■ Risk reduction worth■ The criticality importance measure■ Fussell-Vesely’s measure
Introduction
● Measures Covered
● Importance Depends On
Importance Measures
Marvin Rausand, March 19, 2004 System Reliability Theory (2nd ed.), Wiley, 2003 – p. 3/19
Importance Depends On
The various measures are based on slightly differentinterpretations of the concept component importance.Intuitively, the importance of a component should depend ontwo 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 thecomponent reliability.
Introduction
Importance Measures
● Birnbaum’s Measure (1)
● Birnbaum’s Measure (2)
● Birnbaum’s Measure (3)
● Birnbaum’s Measure (4)
● Birnbaum’s Measure (5)
● Birnbaum’s Measure (6)
● Birnbaum’s Measure (7)
● Improvement Potential (1)
● Improvement Potential (2)
● Risk Achievement Worth
● Risk Reduction Worth
● Criticality Importance (1)
● Criticality Importance (2)
● Criticality Importance (3)
● Fussell-Vesely’s Measure
Marvin Rausand, March 19, 2004 System Reliability Theory (2nd ed.), Wiley, 2003 – p. 4/19
Importance Measures
Introduction
Importance Measures
● Birnbaum’s Measure (1)
● Birnbaum’s Measure (2)
● Birnbaum’s Measure (3)
● Birnbaum’s Measure (4)
● Birnbaum’s Measure (5)
● Birnbaum’s Measure (6)
● Birnbaum’s Measure (7)
● Improvement Potential (1)
● Improvement Potential (2)
● Risk Achievement Worth
● Risk Reduction Worth
● Criticality Importance (1)
● Criticality Importance (2)
● Criticality Importance (3)
● Fussell-Vesely’s Measure
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Birnbaum’s Measure (1)
Birnbaum (1969) proposed the following measure of thereliability importance of a component:
Birnbaum’s measure of importance of component i at time t is
IB(i | t) =∂h(p(t))
∂pi(t)
Birnbaum’s measure is thus obtained by partial differentiationof the system reliability with respect to pi(t). This approach iswell known from classical sensitivity analysis. If IB(i | t) islarge, a small change in the reliability of component i will resultin a comparatively large change in the system reliability at timet
Introduction
Importance Measures
● Birnbaum’s Measure (1)
● Birnbaum’s Measure (2)
● Birnbaum’s Measure (3)
● Birnbaum’s Measure (4)
● Birnbaum’s Measure (5)
● Birnbaum’s Measure (6)
● Birnbaum’s Measure (7)
● Improvement Potential (1)
● Improvement Potential (2)
● Risk Achievement Worth
● Risk Reduction Worth
● Criticality Importance (1)
● Criticality Importance (2)
● Criticality Importance (3)
● Fussell-Vesely’s Measure
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Birnbaum’s Measure (2)
By fault tree notation, Birnbaum’s measure may be written as
IB(i | t) =∂Q0(t)
∂qi(t)
where
qi(t) = 1 − pi(t)
Q0(t) = 1 − pS(t) = 1 − h(p(t))
Birnbaum’s measure is named after the Hungarian-Americanprofessor Zygmund William Birnbaum (1903-2000)
Introduction
Importance Measures
● Birnbaum’s Measure (1)
● Birnbaum’s Measure (2)
● Birnbaum’s Measure (3)
● Birnbaum’s Measure (4)
● Birnbaum’s Measure (5)
● Birnbaum’s Measure (6)
● Birnbaum’s Measure (7)
● Improvement Potential (1)
● Improvement Potential (2)
● Risk Achievement Worth
● Risk Reduction Worth
● Criticality Importance (1)
● Criticality Importance (2)
● Criticality Importance (3)
● Fussell-Vesely’s Measure
Marvin Rausand, March 19, 2004 System Reliability Theory (2nd ed.), Wiley, 2003 – p. 7/19
Birnbaum’s Measure (3)
By pivotal decomposition we have
h(p(t)) = pi(t) · h(1i,p(t)) + (1 − pi(t)) · h(0i,p(t))
= pi(t) · [h(1i,p(t)) − h(0i,p(t))] − h(0i,p(t))
Birnbaum’s measure can therefore we written as
IB(i | t) =∂h(p(t))
∂pi(t)= h(1i,p(t)) − h(0i,p(t))
Note that Birnbaum’s measure IB(i | t) of component i onlydepends on the structure of the system and the reliabilities ofthe other components. IB(i | t) is independent of the actualreliability pi(t) of component i. This may be regarded as aweakness of Birnbaum’s measure
Introduction
Importance Measures
● Birnbaum’s Measure (1)
● Birnbaum’s Measure (2)
● Birnbaum’s Measure (3)
● Birnbaum’s Measure (4)
● Birnbaum’s Measure (5)
● Birnbaum’s Measure (6)
● Birnbaum’s Measure (7)
● Improvement Potential (1)
● Improvement Potential (2)
● Risk Achievement Worth
● Risk Reduction Worth
● Criticality Importance (1)
● Criticality Importance (2)
● Criticality Importance (3)
● Fussell-Vesely’s Measure
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Birnbaum’s Measure (4)
Since h(·i,p(t)) = E[φ(·i,X(t)] we can write
IB(i | t) = E[φ(1i,X(t)] − E[φ(0i,X(t)]
= E[φ(1i,X(t) − φ(0i,X(t)]
When the structure is coherent [φ(1i,X(t)) − φ(0i,X(t))] canonly take on the values 0 and 1. Therefore
IB(i | t) = Pr(φ(1i,X(t)) − φ(0i,X(t)) = 1)
This is to say that IB(i | t) is equal to the probability that(1i,X(t)) is a critical path vector for component i at time t
Birnbaum’s measure is therefore the probability that thesystem is such a state at time t that component i is critical forthe system
Introduction
Importance Measures
● Birnbaum’s Measure (1)
● Birnbaum’s Measure (2)
● Birnbaum’s Measure (3)
● Birnbaum’s Measure (4)
● Birnbaum’s Measure (5)
● Birnbaum’s Measure (6)
● Birnbaum’s Measure (7)
● Improvement Potential (1)
● Improvement Potential (2)
● Risk Achievement Worth
● Risk Reduction Worth
● Criticality Importance (1)
● Criticality Importance (2)
● Criticality Importance (3)
● Fussell-Vesely’s Measure
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Birnbaum’s Measure (5)
Assume that component i has failure rate λi. In somesituations we may be interested in measuring how much thesystem reliability will change by making a small change to λi.The sensitivity of the system reliability with respect to changesin λi can obviously be measured by
∂h(p(t))
∂λi
=∂h(p(t))
∂pi(t)·∂pi(t)
∂λi
= IB(i | t) ·∂pi(t)
∂λi
A similar measure can be used for all parameters related to thecomponent reliability pi(t), for i = 1, 2, . . . , n. In some cases,several components in a system will have the same failure rateλ. To find the sensitivity of the system reliability with respect tochanges in λ, we can still use ∂h(p(t))/∂λ
Introduction
Importance Measures
● Birnbaum’s Measure (1)
● Birnbaum’s Measure (2)
● Birnbaum’s Measure (3)
● Birnbaum’s Measure (4)
● Birnbaum’s Measure (5)
● Birnbaum’s Measure (6)
● Birnbaum’s Measure (7)
● Improvement Potential (1)
● Improvement Potential (2)
● Risk Achievement Worth
● Risk Reduction Worth
● Criticality Importance (1)
● Criticality Importance (2)
● Criticality Importance (3)
● Fussell-Vesely’s Measure
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Birnbaum’s Measure (6)
Consider a system where component i has reliability pi(t) thatis a function of a parameter θi. The parameter θi may be thefailure rate, the repair rate, or the test frequency, of componenti. To improve the system reliability, we may want to change theparameter θi (by buying a higher quality component orchanging the maintenance strategy). Assume that we are ableto determine the cost of the improvement as a function of θi,that is, ci = c(θi), and that this function is strictly increasing ordecreasing such that we can find its inverse function. Theeffect of an extra investment related to component i may nowbe measured by
∂h(p(t))
∂ci
=∂h(p(t))
∂θi
·∂θi
∂ci
= IB(i | t) ·∂pi(t)
∂θi
·∂θi
∂ci
Introduction
Importance Measures
● Birnbaum’s Measure (1)
● Birnbaum’s Measure (2)
● Birnbaum’s Measure (3)
● Birnbaum’s Measure (4)
● Birnbaum’s Measure (5)
● Birnbaum’s Measure (6)
● Birnbaum’s Measure (7)
● Improvement Potential (1)
● Improvement Potential (2)
● Risk Achievement Worth
● Risk Reduction Worth
● Criticality Importance (1)
● Criticality Importance (2)
● Criticality Importance (3)
● Fussell-Vesely’s Measure
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Birnbaum’s Measure (7)
In a practical reliability study of a complex system, one of themost time-consuming tasks is to find adequate estimates forthe input parameters (failure rates, repair rates, etc.). In somecases, we may start with rather rough estimates, calculateBirnbaum’s measure of importance for the variouscomponents, or the parameter sensitivities, and then spend themost time finding high-quality data for the most importantcomponents. Components with a very low value of Birnbaum’smeasure will have a negligible effect on the system reliability,and extra efforts finding high-quality data for such componentsmay be considered a waste of time
Introduction
Importance Measures
● Birnbaum’s Measure (1)
● Birnbaum’s Measure (2)
● Birnbaum’s Measure (3)
● Birnbaum’s Measure (4)
● Birnbaum’s Measure (5)
● Birnbaum’s Measure (6)
● Birnbaum’s Measure (7)
● Improvement Potential (1)
● Improvement Potential (2)
● Risk Achievement Worth
● Risk Reduction Worth
● Criticality Importance (1)
● Criticality Importance (2)
● Criticality Importance (3)
● Fussell-Vesely’s Measure
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Improvement Potential (1)
The improvement potential of component i at time t is
IIP(i | t) = h(1i,p(t)) − h(p(t))
The improvement potential may be expressed as
IIP(i | t) = IB(i | t) · (1 − pi(t))
or, by using the fault tree notation
IIP(i | t) = IB(i | t) · qi(t)
Introduction
Importance Measures
● Birnbaum’s Measure (1)
● Birnbaum’s Measure (2)
● Birnbaum’s Measure (3)
● Birnbaum’s Measure (4)
● Birnbaum’s Measure (5)
● Birnbaum’s Measure (6)
● Birnbaum’s Measure (7)
● Improvement Potential (1)
● Improvement Potential (2)
● Risk Achievement Worth
● Risk Reduction Worth
● Criticality Importance (1)
● Criticality Importance (2)
● Criticality Importance (3)
● Fussell-Vesely’s Measure
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Improvement Potential (2)
IIP(i | t) is the difference between the system reliability with aperfect component i, and the system reliability with the actualcomponent i. In practice, it is not possible to improve thereliability pi(t) of component i to 100% reliability.
Let us assume that it is possible to improve pi(t) to new value
p(n)
i (t) representing, for example, the state of the art for thistype of components. We may then calculate the realistic orcredible improvement potential (CIP) of component i at time t,defined by
ICIP(i | t) = h(p(n)
i (t),p(t)) − h(p(t))
where h(p(n)
i (t),p(t)) denotes the system reliability whencomponent i is replaced by a new component with reliabilityp
(n)
i (t).
Introduction
Importance Measures
● Birnbaum’s Measure (1)
● Birnbaum’s Measure (2)
● Birnbaum’s Measure (3)
● Birnbaum’s Measure (4)
● Birnbaum’s Measure (5)
● Birnbaum’s Measure (6)
● Birnbaum’s Measure (7)
● Improvement Potential (1)
● Improvement Potential (2)
● Risk Achievement Worth
● Risk Reduction Worth
● Criticality Importance (1)
● Criticality Importance (2)
● Criticality Importance (3)
● Fussell-Vesely’s Measure
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Risk Achievement Worth
The importance measure risk achievement worth (RAW) ofcomponent i at time t is
IRAW(i | t) =1 − h(0i,p(t))
1 − h(p(t))
The RAW is the ratio of the (conditional) system unreliability ifcomponent i is not present (or if component i is always failed)with the actual system unreliability.
The RAW presents a measure of the worth of component i inachieving the present level of system reliability and indicatesthe importance of maintaining the current level of reliability forthe component.
Introduction
Importance Measures
● Birnbaum’s Measure (1)
● Birnbaum’s Measure (2)
● Birnbaum’s Measure (3)
● Birnbaum’s Measure (4)
● Birnbaum’s Measure (5)
● Birnbaum’s Measure (6)
● Birnbaum’s Measure (7)
● Improvement Potential (1)
● Improvement Potential (2)
● Risk Achievement Worth
● Risk Reduction Worth
● Criticality Importance (1)
● Criticality Importance (2)
● Criticality Importance (3)
● Fussell-Vesely’s Measure
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Risk Reduction Worth
The importance measure risk reduction worth (RRW) ofcomponent i at time t is
IRRW(i | t) =1 − h(p(t))
1 − h(1i,p(t))
The RRW is the ratio of the actual system unreliability with the(conditional) system unreliability if component i is replaced bya perfect component with pi(t) ≡ 1.
In some applications, failure of a “component” may be anoperator error or some external event. If such “components”can be removed from the system, for example, by canceling anoperator intervention, this may be regarded as replacementwith a perfect component.
Introduction
Importance Measures
● Birnbaum’s Measure (1)
● Birnbaum’s Measure (2)
● Birnbaum’s Measure (3)
● Birnbaum’s Measure (4)
● Birnbaum’s Measure (5)
● Birnbaum’s Measure (6)
● Birnbaum’s Measure (7)
● Improvement Potential (1)
● Improvement Potential (2)
● Risk Achievement Worth
● Risk Reduction Worth
● Criticality Importance (1)
● Criticality Importance (2)
● Criticality Importance (3)
● Fussell-Vesely’s Measure
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Criticality Importance (1)
The component importance measure criticality importanceICR(i | t) of component i at time t is the probability thatcomponent i is critical for the system and is failed at time t,when we know that the system is failed at time t.
ICR(i | t) =IB(i | t) · (1 − pi(t))
1 − h(p(t))
By using the fault tree notation, ICR(i | t) may be written as
ICR(i | t) =IB(i | t) · qi(t)
Q0(t)
Introduction
Importance Measures
● Birnbaum’s Measure (1)
● Birnbaum’s Measure (2)
● Birnbaum’s Measure (3)
● Birnbaum’s Measure (4)
● Birnbaum’s Measure (5)
● Birnbaum’s Measure (6)
● Birnbaum’s Measure (7)
● Improvement Potential (1)
● Improvement Potential (2)
● Risk Achievement Worth
● Risk Reduction Worth
● Criticality Importance (1)
● Criticality Importance (2)
● Criticality Importance (3)
● Fussell-Vesely’s Measure
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Criticality Importance (2)
Let C(1i,X(t)) denote the event that the system at time t is ina state where component i is critical. We know that
Pr(C(1i,X(t))) = IB(i | t)
The probability that component i is critical for the system andat the same time is failed at time t is
Pr(C(1i,X(t)) ∩ (Xi(t) = 0)) = IB(i | t) · (1 − pi(t))
When we know that the system is in a failed state at time t,then
Pr(C(1i,X(t)) ∩ (Xi(t) = 0) | φ(X(t)) = 0)
Introduction
Importance Measures
● Birnbaum’s Measure (1)
● Birnbaum’s Measure (2)
● Birnbaum’s Measure (3)
● Birnbaum’s Measure (4)
● Birnbaum’s Measure (5)
● Birnbaum’s Measure (6)
● Birnbaum’s Measure (7)
● Improvement Potential (1)
● Improvement Potential (2)
● Risk Achievement Worth
● Risk Reduction Worth
● Criticality Importance (1)
● Criticality Importance (2)
● Criticality Importance (3)
● Fussell-Vesely’s Measure
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Criticality Importance (3)
Since the event C(1i,X(t)) ∩ (X(t) = 0) implies thatφ(X(t)) = 0), we get
Pr(C(1i,X(t)) ∩ (Xi(t) = 0))
Pr(φ(X(t)) = 0)=
IB(i | t) · (1 − pi(t))
1 − h(p(t))
ICR(i | t) is therefore the probability that component i hascaused system failure, when we know that the system is failedat time t. For component i to cause system failure, componenti must be critical, and then fail.
When component i is repaired, the system will start functioningagain. This is why the criticality importance measure may beused to prioritize maintenance actions in complex systems
Introduction
Importance Measures
● Birnbaum’s Measure (1)
● Birnbaum’s Measure (2)
● Birnbaum’s Measure (3)
● Birnbaum’s Measure (4)
● Birnbaum’s Measure (5)
● Birnbaum’s Measure (6)
● Birnbaum’s Measure (7)
● Improvement Potential (1)
● Improvement Potential (2)
● Risk Achievement Worth
● Risk Reduction Worth
● Criticality Importance (1)
● Criticality Importance (2)
● Criticality Importance (3)
● Fussell-Vesely’s Measure
Marvin Rausand, March 19, 2004 System Reliability Theory (2nd ed.), Wiley, 2003 – p. 19/19
Fussell-Vesely’s Measure
Fussell-Vesely’s measure of importance, IFV(i | t) is theprobability that at least one minimal cut set that containscomponent i is failed at time t, given that the system is failed attime t.
Fussell-Vesely’s measure can be approximated by
IFV(i | t) ≈1 −
∏mi
j=1(1 − (Q̌ij(t))
Q0(t)≈
∑mi
j=1 Q̌ij(t)
Q0(t)
where Q̌ij(t)) denotes the probability that minimal cut set j
among those containing component i is failed at time t