IEEE TRANSAaIONS ON RELIABILITY. VOL. 45, NO. 1, 1996 MARCH 59

Comparing the Importance of System Components by Some Structural Characteristics

Fan C. Meng Academia Sinica, Taipei

Key words - Structural importance measure, Reliability im- portance measure, Component criticality

Summary & Conclusions - Various results concerning impor- tance measures & rankings of system components are presented. The vectors (of component states) which appear useful in studying various component importance measures are decomposed. In par- ticular, a new method to compute Birnbaum importance measures of components is introduced. For some systems & components the Vesely-Fussell importance measure might not be suitable; thus a new importance measure is proposed. By using the notion of com- ponent criticality introduced by Boland, Proschan, Tong, several results are obtained which can be used to compare the relative im- portance of components when their reliabilities are unknown and/or when the calculation involved in assessing their importance measures become prohibitively extensive.

1. INTRODUCTION

Acronyms & Abbreviations

MCS minimal cut set@) MPS minimal path set(s) s.t. “such that” w.r.t. “with respect to”.

In the reliability-theory literature, measures of importance of system components have been investigated extensively. The various importance measures and rankings of system components can be classified as:

reliability importance structural importance

of components. Although the reliability importance measures are generally superior to the structural ones and are of primary concern, the probabilistic information required for their calcula- tion might not be available in practice. Furthermore, computa- tions involved in quantifying those importance measures can become prohibitively extensive for large complex systems. When such situations are encountered, structural measures (rankings) of component importance must be used, which can provide a fair basis to compare the relative importance among system components. For example, Butler [7] formulated an im- portance ranking among components based on MCS of the system, and showed that the so-called cut-importance ranking is consistent with the component ranking induced by Birnbaum reliability importance measure when the component reliabilities are equal and are close to 1. Later, Boland, Proschan, Tong [6] introduced the notion of structural criticality of components, and developed a procedure for optimally allocating components in s-coherent systems. More recently, Meng [9] characterized

the criticality ordering due to Boland et al [6], in terms of the MCSIMPS, and derived a relationship between the criticality ordering and Birnbaum reliability importance measure of components,

This paper presents more results in this direction. Section 2 introduces some basic concepts. Section 3 decomposes the vectors of component states, which is useful in studying various component importance measures. A new method to compute Birnbaum importance measures of system components is also introduced. Section 4 shows how the notion of component criticality [6] can be used to compare the various importance measures of system components. All proofs are in the appendix.

Assumptions

or failed. 1. Each component and the system are either functioning

2. Component failures are mutually s-independent. +

Notation

0,l [failed, functioning] state n number of components i

PI q1 C X

P ( . l , x) Z$ ( i ) , Zk( i; p ) Birnbaum [structural, reliability] importance

I$ , ( i ; p ) , ZVF ( i ; p ) Vesely-Fussell [structural, reliability]

index for component i ; i = 1,. . . ,n unless otherwise stated Pr {XI = 1) : reliability of component i Pr{X, = 0) : unreliability of component i; pI + qz = 1 { 1,2,. . . ,n} : index set of the components (xlr ..., xn): vector of states of components 1, ..., n (p , , . , . , p,) : reliability vector of the n components (x, ,... J-1, * J L + I , . . . , xn )

of component i [4]

importance of component i [8, 111 j U , j , ..., j ) , J = O , l h_((p) Pr{q5(X)=1): system reliability h ( p ) Pr{$(X)=O): system unreliability C, MCS k pr MPS r C ( i ) , P ( i ) C,(x) C, (x) IAl cardinality of set A E , ( i ) , E 2 ( i ) event: ‘there is at least 1 [MCS, MPS] which

contains component i’ and ‘all components other than i in the [MCS, MPS] are [failed, functioning]’ vector with elements x,, i E A set A with element i removed (0, 1)” - {0, 1): non-decreasing structure function of a binary s-coherent system of n components 1 - q5(1--x), for all x E (0,l)“: dual structure of q5

{(.J): XE {o,I}~-’}: set of 2”-’ vectors subset of W ( i ) in which component i is critical

[MCS, MPS] which contain component i {i: x, = O ) : cut set corresponding to cut vector x { i : x, = 1): path set corresponding to path vector x

X A

A\{i) 4

q5D(x) (C,q5) the system ~ ( i ) B ( i ) V,(i) E[(i) \B(i) , for 1=1,2

0018-9529/96/$5.00 01996 IEEE

60 lEEE TRANSACXFIOWS ON RELIABILITY, VOL. 45, WO. 1, 1996 MARCH

mi) y > x i 5 j

‘y, 2 x, for all i’ and ‘y l > x, for some i’ ‘+( l l ,Ol,x) 2 +(O,,lj,x), for all x’ and ‘strict ine- quality holds for some x’ - I { (o,,x): #‘(o,,x) =o; 3 c k E c(i) s.t. c k E co(o,,x)} I -

i f j +( I,,o~, XI = +(o,,I~,x), for all x 3 “there exists a”. ( 5 )

I {x: 4 (x) = 0 ) I

(6) Other, standard notation is given in “Information for Readers & Authors” at the rear of each issue.

& (i) = ( i ; x,. . . , x).

3. DECOMPOSITION OF STATE VECTORS Nomenclature [MCS, MPS]: If x is a minimal [cut, path] vector, then [Co(x), Cl(x)] is the [MCS, MPS] for (C,+). Component relevance: Component i is relevant to (C, +) iff ‘+( l , ,x )=l ’ and ‘+(O,,x)=O’ for some (-,,x) Critical vector: (.,,x) is a critical vector for component i iff

Series, parallel: The terms, series &parallel are used in their logic-diagram sense, irrespective of the schematic-diagram or physical-layout. More critical: Component i is more critical than component j for + iff i sj. [6] Permutation equivalent: Components i & j are permutation equivalent iff i z j .

+(l, ,x) = 1 - +(O,,x) = 1.

Decomposition of state vectors for a component is useful in analyzing the Birnbaum and the Vesely-Fussell importance measures. A method to compute the Birnbaum importance is introduced. For some systems & components the Vesely-Fussell importance measure is not suitable. Thus a new importance measure is proposed.

@(i) measures the proportion of critical vectors for com- ponent i; 1; (i) is the probability that component i is critical. Theorem 1 is useful in computing the Birnbaum importance measures (see examples 1 & 2).

Zheorem 1. If + ( 1 , ~ ) > + (O,,x), ie, ( .,J) is a critical vec- tor for component i, then there exists a MPS P, and a MCS Ck such that:

2. PRELIMINARIES a. P,nC, = {i),

b. x,=l for all s~P,.\{i), and x,=O for all s ~ C k \ ( i ) . +(1) = 1-4(0) = 1,

4 ie, the system is [functioning, failed] if all of its components are [functioning, failed]. Example 1.

Birnbaum defined 2 measures of component importance 141.

I&) 5 2 - ( n - 1 ) . I { ( * 1 J ) : + ( 1 , J ) > 4 (0 ,J)) I ,

Let +(x) = 1 - (1-xl~xz~x3)~(l-x1~x2~xq)~(1 -

The MPS & MCS of the system are: . x2.xg).

(1)

measures the importance of a component to the system. If @ ( i ) = O then component i is irrelevant or a

PI = (1, 2, 31,

dummy-component . p2 = (1, 2, 41,

m ; P ) = h ( L P ) - h(O,,p) = Pr{+(LX) > +(0,”(2) p , = ( 2 , 5);

is the probability that failure of component i coincides with system failure. c1 = (21,

c2 = (1, 51, @(i) = z;(i;(?h, ..., $5)) 121.

I

If+(x)=[O,l] thenxis a [cut,path] vector. Aminimal [cut,path] vector is a [cut, path] vector x for which d, (y) = [ 1 for all y > x, 0 for all y < XI.

A component can contribute to system failure without be- ing critical; so Vesely [ I l l and Fussell [8] proposed:

c3 = i 3 3 4, 5 ) .

Consider component 4. The MPS & MCS which contain com- Ponent 4 are p2 I% c3 respectively. Thus there is exactly 1 critical vector for component 4:

MENG: COMPARING THE IMPORTANCE OF SYSTEM COMPONENTS BY SOME STRUCTURAL CHARACTERISTICS

~

61

A common method could be used to obtain h (p ) ; then Zk (4;p) = ah(p)/aP4. 4

Example 2

Consider the bridge structure in figure 1.

[The logic of the structure is defined by its MPS & MCS]

Figure 1. Bridge Structure

The system MPS & MCS are:

PI = (1, 41,

p2 = ( 2 , 51,

p3 = (1, 3, 5 ) ,

p 4 = ( 2 , 3, 4);

c4 = (2, 3 , 4).

Consider component 3. The MPS & MCS which contain com- ponent 3 are:

P3, P4, C,, C4. Hence there are two critical vectors for com- ponent 3 :

Thus, we can compute Z$( i;p) of the components without com- 4 puting h(p), which task is usually complicated.

In general,

&(i) = { ( . i ,x ) : 3 P r € P ( i ) s.t. P,CCl(li,X,}. (9)

Then,

9(Oi,x) = 0, if ( a i , x ) € E l ( i ) ;

9(l i ,x) = 1, if ( s i , x ) € E 2 ( i ) ;

B ( i ) = El ( i )nEz( i ) .

Following (4) & (8),

From the definitions,

V,(i)

= {(.,J): 9(o,,x)=i; 3 P , E P ( ~ ) s.t. prscl(i , ,x)) .

(13)

If a vector (.,,x) is not in one of the three disjoint sets B ( i ) , Vl (i), V2 (i) , then ( .,J) must be in:

Nl(i) = {(.&: 9(l,,x)=O; CkgCo(O,,x), for all C k € C ( i ) } ,

or (14)

NI (i) & N2 ( i ) are the non-essential vectors for component i, because:

1. If (.i,x)ENl(i), then 9(Oi,x) = +(li ,x) = 0, and w.r.t. (Oi,x) there is no MCS containing i such that all com- ponents in the MCS are failed. Thus, in this instance, if we were to restore some components to resume the system functioning, component i should be eliminated from the repair-list because its functioning cannot cause the system to function.

2 . If ( . i ,x) E N2 ( i ) , then similarly, if we were to turn off the system by cutting some components off, component i need not be considered.

Remark 1.

a. ‘ ( . i , x ) ~ ~ ( i ) , for 9’ iff ‘ ( - i , ~ - x ) ~ ~ ( i ) , for P’; b. ‘ ( . , , x ) € V l ( i ) , for 9’ iff ‘ ( - , , l - x ) € V 2 ( i ) , for c $ ~ ’ ;

c. ‘ ( - i , x ) € N l ( i ) , for 9’ iff ‘ ( - i , l - x ) € N 2 ( i ) , for 9D7. 4

IEEE TRANSACTIONS ON RELIABILITY. VOL. 45, NO. 1, 1996 MARCH 62

W( i) = B ( i ) U v, ( i) U v, ( i) U NI( i) U N2 ( i ) .

Eq (17) - (19) are based on (4), ( 5 ) , (12).

(16) Pr{4(X)=1} = Pr{V2(i)} + Pr{N2(i)} + p,.Pr{B(i)}.

(24)

As stated in fact 1, in a parallel system all components have the same Vesely-Fussell importance measure. In a parallel system - put P, = {i} in (13),

(17)

Pr{+(X)=O} = Pr{Vl(i)} + Pr{Nl(i)f + qi-Pr{B(i)) .

(19) Hence,

If component i is in [parallel, series] with the rest of the system then [Vl(i) =N,(i) =0, V2(i) = N2( i ) =0]. Hence, fact #1 follows from (18) & (19) [lo].

Fact 1. If component i is in parallel with the rest of the system then Zvp(i;p) = 1 irrespective of the values of pl, . . . ,pa. -4

Because I; (i;p) =ah (p)lap,, the Birnbaum importance measure of component i doesn't depend on p I itself. This is a drawback of the Birnbaum importance measure. What about the Vesely-Fussell importance measure? Theorem 2 follows from (18) & (19).

Pr{B(i)} + Pr{V2(i)} = 1; and from (23),

In a parallel system the component with the highest reliability is the most important one in the sense of I$F( .).

Example 3 illustrates how to use (18) to compute ZVF.

Example 3.

The system is 2-out-of-3:G, -

+(XI, x2, ~ 3 ) = 1 - (1 -XI .x2) * (1 -xl-x3). (1 -x2.x3)

The MCS are (1, 2}, (2, 31, (1, 3).

B(1) = {(e1,12,O3), (*1,0~,13)},

Vl ( 1) = ( - 1,02,03), from (7) & (12).

Hence,

Pr{B( 1)) + Pr{ V ( 1 ) } = 1 -p2 .p3.

Reorem 2. ZvF(i;p) is decreasing in pz. 4

We refer to Boland & El-Neweihi [ 5 ] , Natvig [lo], and Bergman [3] for other importance measures of system com- ponents. In general the most appropriate importance measure of components for a given situation depends highly on: a) the type of desired improvement for the system, and b) the system structure. For structures & components as described in fact #1, the Vesely-Fussell measure might not be suitable. Another short- coming of the Vesely-Fussell measure is that it doesn't con- sider the contribution of components to system success. These considerations motivate a new component importance measure,

Z$F ( i; ?h, . . . , %) reduces to the structural importance measure, zVF(1 ;p ) = q1 . ( 1 -p2 .p3) /,f&),

I+$( i ) = h ( p ) = Pl'P2 + P2'P3 + Pl'P3 - 2Pl*P2.P3. -4

1 { (lz,x): 4 (l,,x)= 1; 3 P , E P ( i ) s.t. P,c_ C1( lz,x)} 1 l{x: +(x)=lll 4. CRITICALITY OF SYSTEM COMPONENTS

The main result is: If two components can be ordered by their structural criticality (see Nomenclature), then their relative ZvF(.) or IbF( .) can be readily determined by imposing some

(21)

Thus,

minor conditions. Lemmas 1 & 2 concern structural ordering.

Lemma 1. Let i sj. Then, (22)

4

MENG: COMPARING THE IMPORTANCE OF SYSTEM COMPONENTS BY SOME STRUCTURAL CHARACTERISTICS

which can be directly concluded from theorem 3. 4

Corollaries 1 & 2 are special cases of theorem 3. Corollary 2 is in contrast to fact 1, and is analogous to [ 1 : theorem 3.81. Corollary 1. Let i s j for 4. Then,

Corollary 2. Let component i be in series with the rest of the system. Then,

Iv~( i ;p ) 2 ZvF(j;p), for all 0 < p < 1 satisfying pi 5 pi.

The strict inequality holds for some 0 < p < 1 satisfying p, 5 pi unless component j is also a 1-component-cut. 4

For many well-known systems, such as k-out-of-n, all com- ponents are permutation equivalent. Two permutation equivalent components have the same Birnbaum importance measures if they have the same reliability.

Different results pertain to the Vesely-Fussell importance measure:

~

63

If i 4 j then condition #a or #b holds [9: theorem 2.41:

a. C ( i ) = CO’);

b. If ‘ck E c(i)’ and ‘ck e CO’) then c k u G > \ ( i > E CO’)’. 4

Theorem 4. Let i A j , and 0 < p < 1. Then either,

a. zVF(i;p) = ZVF(j;p), for all p ,

b. IvF(~;P) > IvF(~;P), iffpi < pj. 4

Examples 1 & 3 illustrate theorem 4 as follows.

Example 1 (continued)

Components 3 & 4 satisfy theorem 4, condition #a. Then,

IvF(4;p) = IvF(3;p) directly follows from theorem 4, condi- tion #a. 4

Example 3 (continued)

Similarly to example 1 (continued),

hence,

IVF(1;p) > zVF(2;p) if p1 < p2 directly follows theorem 4, condition #b because in a k-out-of-n:G system condition #b holds for all i, j E { 1,2,. . . ,n} and hence the component with the lowest reliability possesses the largest Vesely-Fussell impor- tance measure. 4

Consider Z$F(.). Lemma 3, theorems 5 & 6 , and cor- ollary 3 arise from duality.

Lemma 3. Let i 5 j . Then,

a. (ej,lj,x) E V 2 ( j ) * (.j,lj,x) E V2(i)UB(i) .

b. (.,,Oi,x) E V2U) * ( . j ,Oj,x) E V2(i). 4

Theorem 5 . Let iJ E {1,2 ,..., n} and 0 < p < 1. Then,

i 5 j * ZGF (i;p) > ZGF ( j ;p) for all p satisfying p, 2 pj. 4

Corollary 3. Let component i be in parallel with the rest of the system, and 0 < p < 1. Then,

Z&(i;p) 2 Z$F(j;p) holds for all p satisfying p, 2 pJ. The strict inequality holds for some p satisfying pL 2 pJ unless com-

4

Theorem 6 . Let i

ponent j is also a 1-component-path.

j and 0 < p < 1. Then either

a. IGF(i;p) = Z$F(j;p) for all p , or

b. Z&(i;p) > Z&(j;p) iff pi > p p 4

64 lEEE TRANSAaIONS ON RELIABILITY. VOL. 45, NO. 1. I996 MARCH

APPENDIX Case 1. If +(l,,O,,x)=l, then (.,,O,,x) E B ( i ) . Case 2. If+ ( l,,O,,x) =0, we then show ( .,,O,,x) E Vl ( i )

to prove the assertion. If #i in [9: theorem 2.21 holds then i E C,, and hence (. ,,O,,x) E VI (i) . If #ii or #iii in [9: theorem 2.21 holds. Then +(l,,O,,Ock\b},l)=O. Because C, is a MCS, there must exist a

A 1 Proof of Theorem 1

Let $ ( l , , x ) = l and +(O,,x)=o. Let +(1,,IA,OB)=1 and +(O,,lA,OB) =O. Then,

+(O,,lA,lB2,OB1) = O for some MCS BIU {i},

where A , 5 A and B, 5 B.

Thus, (.j,Oj,~) E Vl(i).

A.5 Proof of Theorem 3

Q. E. D.

Q. E.D.

A.2 Proof of Theorem 2

A.3 Proof of Lemma 1 ( - , , l i ,oc~\~~}, l ) e BO’)UVIV,O’).

#a follows from, Clearly,

#b follows from, We thus have shown that,

MENG: COMPARING THE IMPORTANCE OF SYSTEM COMPONENTS BY SOME STRUCTURAL CHARACTERISTICS 65

A.1 Proof of Theorem 4 [l] R.E. Barlow, F. Proschan, “Importance of system components and fault tree events”, Stochastic Processes and Their Applications, vol3, 1975,

[2] R.E. Barlow, F. Proschan, Statistical Theory of Reliability and Life Testing, 1981; To Begin With (Silver Spring, MD).

[3] B. Bergman, “On reliability theory and its applications”, Scandinavian J. Statistics, vol 12, 1985, pp 1-41.

[4] Z.W. Birnbaum, “On the importance of different components in a multicomponent system”, Multivariate Analysis II (P.R. Krishnaiah, Ed), 1969, pp 581-592; Academic Press.

The then from (A-4) cfk (A-5). Q*E.D* [5] P.J. Boland, E. El-Neweihi, “Measures of component importance in

The proof of #a is trivial. pp 153-173. Suppose #b holds. (A-2) becomes,

Pr((.,,a,,x) E B(j )UVl( j )} =Pr((.i,uj,x) E B( i )UVl ( i )} .

(A-5)

A.8 Proof of Lemma 3

a. If i 5 j for 4 then i

This implies (by lemma 2),

( . i ,Oj, l-x}) E v ~ ( ~ ) u B ( ~ ) for 4D.

reliability theory”, Computers and Operations Research, vol 22, num

[6] P.J. Boland, F. Proschan, Y.L. Tong, “Optimal arrangement of com- ponents via pairwise rearrangements”, Naval Research Logistics, vol36,

[7] D.A. Butler, “A complete importance ranking for components of binary coherent systems with extensions to multi-state systems”, Naval Research Logistics Quarterly, vol 4, 1979, pp 565-578.

4, 1995, pp 455-463.

5 j for q5D. Hence, by remark 1, 1989, pp 807-815.

( - j , ~ i , ~ - x ) E v1(j) for +D.

Again by remark 1,

( e i , l j , x ) E V 2 ( i ) U B ( i ) for 4.

b. The proof is similar to #a.

[8] J.B. Fussell, “How to hand-calculate system reliability and safety characteristics”, IEEE Trans. Reliability, vol R-24, 1975 Aug, pp

[9] F.C. Meng, “Comparing criticality of nodes via minimal cut (path) sets for coherent systems”, Probability in the Engineering & Informational Sciences, vol 8, num 1, 1994, pp 79-87.

[lo] B. Natvig, “New light on measures of importance of system com- ponents”, Scandinavian Journal of Statistics, vol 12, 1985, pp 43-54.

[ l l ] W.E. Vesely, “A time dependent methodology for fault tree evaluation”,

169-174.

Nuclear Engineering Design, vol 13, 1970, pp 337-360. Q. E. D.

A.9 Proof of Theorem 5 AUTHOR

From (23) and lemmas l &L 2, the can be Obtained Dr. Fan C. Meng; Inst. of Statistical Science; Academia Sinica; Taipei 11529,

Internet (e-mail): fcmeng@stat.sinica.edu.tw Fan Chin Meng is an Associate Research Fellow of the Institute of

Statistical Science, Academia Sinica. He received his PhD (1989) from the Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago. His research interests include reliability theory and applied probability.

by using a similar argument to the proof of theorem 3. TAIWAN - R.O.C.

A.10 Proof of Theorem 6

The result is easily obtained by using a standard argument of duality.

Manuscript received 1995 April 10. REFERENCES

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