Chapter 5 Component Importance - NTNU

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

[email protected]

http://www.ntnu.no/

Introduction

● Measures Covered

● Importance Depends On

Importance Measures


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

http://www.ntnu.no/ross/srt

Introduction

● Measures Covered

● Importance Depends On

Importance Measures


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)







● Improvement Potential (1)


● Risk Achievement Worth

● Risk Reduction Worth

● Criticality Importance (1)



● Fussell-Vesely’s Measure


Importance Measures


Introduction

Importance Measures

















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


















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


















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


















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


















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


















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


















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

















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

















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

















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

















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

















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


















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


















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

















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


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Chapter 5 Component Importance - NTNU