日本オペレーションズ・リサーチ学会
2021年 秋季研究発表会

2-D-6

A Note on Component Criticality Importance Measuresof Reliability Models under Epistemic Uncertainty
05001387
05000255
01013754
01307065

1.

Hiroshima University
Ritsumeikan University
Hiroshima University
Hiroshima University

Introduction

Fault tree (FT) as one of the model-based
evaluation methods plays an important role inreliability and represents the relationship between component failures and system failure.
FT provides some quantitative measures for thesystem reliability from component reliabilities,
which brings convenience when maintenance thesystem. However, actual systems often consistof multiple components, which brings insights
into system maintenance, e.g., which component
should be maintened ﬁrst. However, actual systems often consist of multiple components, whichcauses high cost of maintenance. Importanceanalysis is the well-known analysis method for
estimating the importance of system componentsthat contributes to the whole system, aiming atﬁguring out the important or critical components
by the order of their importance. As far as concerned, criticality importance (CI) measure plays
as one of the center roles in estimating components importance. Since the computation of importance measure requires the model parameters
estimated from data samples or expert’s knowledge. Another problem should be taken into account is that since CI measure is the degree ofcomponents failure to system failure, the maintenance of components is from low importanceof CI measure. However, the limitation of numbers of data may bring the unpredictable impact
on the estimation accuracy of model parameters,
which is called epistemic uncertainty. Due tothe uncertainty in the model parameters, the estimated importance measure could be diﬀerent
from what it should be. In general, the uncertainty can be solved by applying the Bayes estimation, but the credibility of the results is hardto measure. So in this paper, we want to present
an importance analysis method through the stationary analysis of continues-time Markov chain

ZHANG Jiahao
ZHENG Junjun
OKAMURA Hiroyuki
DOHI Tadashi(CTMC) based on CI measure in FT, aiming
at ensuring whether the component importancewhen model parameters contain uncertainty is reliable or not.

2.

Fault TreeConsider a FT to represent the relationshipbetween system and component failures. The
FT can be represented by a structure function.
Let F (x1 , . . . , xn ) be a structure function where
x1 , . . . , xn are the states of basic events. ThenF (x1 , . . . , xn ) means the state of top event. In
the structure function, AND and OR gates are
expressed by x1 x2 and 1−(1−x1 )(1−x2 ), respectively. Note that this is slightly diﬀerent from a
simple boolean function in the Boolean algebra.
By using such expression, the system reliability
Rs can be obtained by Rs = F (R1 , . . . , Rn ) where
R1 , . . . , Rn are component reliabilities.

3.

Importance under Epistemic Uncertainty in FTConsider the CI measure in FT, for i-th component, the deﬁnition of CI measure can be obtained by;
IiCR =∂F (R1 (t), . . . , Rn (t))
Ri (t).
∂Ri (t)F (R1 (t), . . . , Rn (t))(1)

Since the uncertainty in model parameters canaﬀect the ﬁnal results, the uncertainty analysis
should be focused on. In the Bayes theory, let themodel parameters involve uncertainty as the random variable, assume that the prior probability
density function fΛ (λ) of the failure rate Λ, andgiven the independent and identically distributed(i.i.d.) samples D = (t1 , · · · , tn ) as observed failure times of a component in the system. Theeﬀects from data samples can be obtained fromthe posterior density function of the failure ratefΛ (λ|D). Assume the reliability of the components follows the formula R(λ; t), the mean of∏1 − i∈H (1 − Ri ) where H = {A, B, C, D}, andthe component reliabilities can be computed by∫ ∞the well-known formular R(λ; t) = e−λt . the failE[R(λ; t)] =R(λ; t)fΛ (λ|D)dλ.
(2)
0ure rate λ of component i ∈ H are set as λA =On the other hand, although Bayes estimation 1/2, 50, 000, λB = 1/480, 000, λC = 1/670, 000,
can help to estimate importance measure when
λD = 1/120, 000. Also, the uncertainty of model
model parameters involve uncertainty, the estiparameters is injected by set the number of sammated results should be veriﬁed. We know that ples as 5.
CI measure indicates how much components failTable 1 shows the CI measure for epistemic unure devote to the system failure, which means
certaintyof system reliability. It’s obvious thatthat the larger the CI measure, the more critical
the results when model parameters contain unthe corresponding component is. However, due tocertainty are diﬀerent from the ones without unthe uncertainty, the value of CI measure can bechanged, which means there exists a probability certainty, even the ranks of importance are diﬀerthat the CI measure of component i can be larger
ent. But whether the result is credible or not, wethan component i (without uncertainty compo- apply the stationary solution of CTMC to verify.
nent i is smaller than component j). The dynamic By using the Eq. (3) to (4), the transition rateof this relationship can be captured by CTMC,
matrix can be computed and the probability ofwhere the change probability can be regarded as aimportance rank can be obtained from Table 2.
transition rate. Consider a FT composed by only
In the table, ’rank1’ means the most critical comAND∏gates, which means F (R1 (t) , . . . , Rn (t)) =ponent that devotes to the system failure, the1 − ni=1 (1 − Ri (t)). Then the probability when
component i is critical than j can be computed
probability of ’rank1’ means the probability ofby;
components being the most critical component.
From the Table 2, we can see that component A
pi,j = P (IiCR > IjCR ) = P (Ri (t) > Rj (t)) = P (Λi > Λj )has the highest probability to become the most∫ ∞critical component, and by the comparison of the=F̄Λi (λ|D)dFΛj (λ|D),(3)0results between Table 1 and 2, the rank of components become the ’rank1’ or ’rank2’ has the samewhere FΛ (λ|D) is the posterior cumulative distribution function (c.d.f.) of Λ, and F̄Λ (λ|D) =rank of the CI measure rank with uncertainty.
1 − FΛ (λ|D). Obviously, pj,i = 1 − pi,j . Assume
Although there is no ’Rank4’, it can be obtainedthat there exist n components. Deﬁning the trancomponent D is the last critical component in thesition rate matrix (i.e., inﬁnitesimal generator) of system.
the underlying CTMC in terms of pi,j as follows,R(λ; t) is deﬁned as;

pi,i
 ..
Q= .
pn,i···..
.
···

pi,n.. . Table 1: CI measure with uncertainty.
(4)

pn,n

∑n

where pi,i = − k̸=i pi,k . Since the dynamics
of the transition process can be captured by theabove CTMC, the probability that a component
is most critical among the current component’s
set can be computed by solving the linear equation πQ = 0, π1 = 1.

4.

Numerical Illustration

In this section we consider CI measure whenmodel parameters contain uncertainty in FT. In
particular, the results are veriﬁed through thestationary analysis of CTMC. Assume a FT composed of four components with a AND gate, thesystem reliability can be computed as RS =Component
ABC
D

Without uncertainty
Importancerank
3.24e-121
6.22e-133
8.68e-132
1.55e-13
4

Sample size = 5
Importancerank
3.73e-1211.45e-122
1.00e-123
2.64e-13
4

Table 2: Probability of importance rank.
Component
ABCDRank19.02e-01
7.15e-02
2.56e-02
6.03e-04

Rank2

Rank3

9.03e-01
7.15e-02
2.56e-02

9.24e-01
7.56e-02

References[1] R.M. Fricks, and K.S. Trivedi, “Importanceanalysis with Markov chains,” In: Proc. Annual Reliability and Maintainability Symposium, IEEE, pp. 89-95, 2003.