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Analysis for joint importance of components in a coherent system
European Journal of Operational Research 182 (2007) 282–299 www.elsevier.com/locate/ejor
Stochastics and Statistics
Analysis for joint importance of components in a coherent system Xueli Gao, Lirong Cui *, Jinlin Li School of Management and Economics, Beijing Institute of Technology, Beijing 100081, China Received 19 December 2005; accepted 19 July 2006 Available online 1 November 2006
Abstract JRI (Joint Reliability Importance) of two components is a measure of interaction of two components in a system for their contribution to the system reliability. It is defined as the rate at which the system reliability improves as the reliabilities of the two components improve. But, sometimes we may improve system reliability through improving reliabilities of three or more components. This article extends the concepts of JRI & JFI (Joint Failure Importance) of two components to multi-components, and establishes some relationships between JRI & JRI, JFI & JFI, and JFI & JRI. The paper also investigates the concept of Conditional Reliability Importance while the working states of certain components are known. Finally, the JRI of multi-components and Conditional Reliability Importance are analyzed in detail for a k-out-of-n:G system. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Reliability; Joint reliability importance; Joint failure importance; Conditional reliability importance
1. Introduction One purpose of system reliability analysis is to identify the weakness and/or the critical components in a system and to quantify the impact of component failures. The so-called ‘‘reliability importance’’ can be used for this purpose. So far various importance definitions have been introduced for system reliability analysis. These importance measures provide a numerical rank to determine which components are more important to system reliability improvement or more critical to system failure. Importance was first introduced by Birnbaum [1]. This index characterizes the rate at which the system reliability changes with respect to changes in the reliability of a given component; it is also defined as MRI (Marginal Reliability Importance). In order to evaluate the importance of different aspects for a system, a set of different importance measures was introduced, such as Structure Importance, Reliability Criticality Importance [2], Monte-Carlo Variance Importance [3]. Other measures of components and minimal cut sets importance in coherent systems were
*
Corresponding author. Tel.: +86 10 68948098; fax: +86 10 68912483. E-mail address: [email protected] (L. Cui).
0377-2217/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.07.022
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Notations c1, c2, . . . , cn components of system S considered system S * ci system with component ci working S ci system with component ci failed pi reliability of component ci qi 1 pi unreliability of component ci 1i, 0i event: pi = [1, 0], i.e., component ci is always [working, failed] p (p1, p2, . . . , pn), vector q (q1, q2, . . . , qn), vector Xi indicator for component ci : Xi = 1 if ci is working, Xi = 0 otherwise Xi vector: (X1, X2, . . . , Xi) h(p) system reliability as a function of p F(q) system unreliability as a function of q = (1 p1,1 p2, . . . , 1 pn) Min-path a minimal set of components whose working is sufficient to make the system work Min-cut a minimal set of components whose failure is sufficient to make the system fail Relevant component a component which appears in at least one min-path and/or min-cut S-Coherent system a system in which every component is relevant, and for which h(p) is non-decreasing in each pi (i.e., there is no component whose functioning causes the system to fail)
developed by Barlow and Prochan [4,5], Fussell and Vesely [6,7]. Xie and Bergman [8] obtained an importance measure by ordering the reduction of the expected yield due to the variation of the life-time distribution of the components, Hwang [9] reviewed structural component importance and introduced some new measures. Importance measures in multi-state systems were also introduced [10,11]. Importance measures can also be estimated by Monte Carlo simulation [12,13]. The discussion of importance is still a hot topic in reliability field recently, see for example, Levitin [14] and Meng [15]. These importance definitions have been well defined and widely used in reliability engineering practice. A component’s MRI is the rate at which the system reliability changes with respect to changes in the reliability of that component [5]. Improvements in reliability of components with the highest MRI cause the greatest increase in system reliability. This information can be used to determine which components should be improved first in order to make the largest improvement in system reliability. For example, one might wish to increase the reliability of the subsystem (such as computer, engine, valves, circuits etc) in the newest version of a system (such as ship, aircraft, vehicle, electronics etc.), MRI could provide an initial ranking of the components to be considered for improvement for a given system structure. In certain cases, the system structure indicates which components have the highest MRI: some examples from [5] include pure parallel and pure series systems, some more complicated examples include a consecutive k-out-of-n:G or a k-out-of-r-of-n:G linear system in [16]. However, MRI does not provide all information on how components affect the system reliability. In particular, MRI gives very little information about how the component reliabilities affect each other. For example, design engineers working within a fixed resource want to improve the system reliability through increasing some components’ reliabilities. For them to use MRI would require recalculation for all components with every tradeoff made or considered. The JRI can be used in this kind of situation, since it indicates roughly how components interact in determining system reliability. The JRI of pairs of components was introduced by Hong and Lie in [17], and the definition was extended to deal with the statistical dependence between components by Armstrong [18] and for a multi-state system by Wu [19]. It is also discussed in dual failure-mode [20] and k-out-of-n:G system [21]. In practice, we may need to investigate the associated effect of more than two components on the system reliability, failure or on some states of the system for system maintenance. Hence we use this investigation to improve the system reliability in certain cases and indicate which component is more critical in some situations. Inspired by the idea of JRI
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introduced by Hong and Lie, in this paper we extend the concept of two-component JRI to multi-components joint reliability importance. Also in [22], Enrico Zio defined DIMII which describes the interaction between two components. The need to extend DIMII to multi-components can be satisfied by multi-components JRI. We also define the concepts of MRI and JRI under the condition that the working states of some components are known: we call it conditional reliability importance. The organization of the paper is as follows. Section 2 gives the definition of JRI of multi-components and formulae for calculating it, and some relationships among JRI and MRI are studied. In Section 3, a similar treatment is given for JFI, and a detailed discussion for a k-out-of-n:G system is presented to illustrate the results obtained. Section 4 discusses conditional reliability importance and establishes some properties in some k-out-of-n:G systems. Finally, the conclusion is given in Section 5. All proofs can be found in the appendix. In this paper, we investigate coherent systems, and assume, (1) there are 2 states of each component; (2) failures of components are mutually statistically independent (hereinafter, for short, independent). 2. JRI of multi-components First, we recall that MRI is defined as the rate at which the system reliability improves as the componentreliability improves, i.e., MRI of component ci is, MRIðci Þ ¼
ohðpÞ : opi
Assuming independence between components, the above equation can be written as MRIðci Þ ¼
ohðpÞ oðpi hð1i ; pÞ þ qi hð0i ; pÞÞ ¼ ¼ hð1i ; pÞ hð0i ; pÞ; opi opi
where (1i, p) = (p1, . . . , pi1, 1i, pi+1, . . . , pn), (0i, p) = (p1, . . . , pi1, 0i,pi+1, . . . , pn). For a coherent system, the definition of JRI for two components introduced by Hong and Lie [13] is, JRIðc1 ; c2 Þ ¼
o2 hðpÞ : op1 op2
Hence, from this formula, JRI indicates how components interact in determining system reliability. For simplicity, without loss of generality, we denote the components of interest for the two-component case as c1,c2. For independent components, it is easy to get, ohðpÞ o2 hðpÞ o op1 oðhð11 ; pÞ hð01 ; pÞÞ JRIðc1 ; c2 Þ ¼ ¼ ¼ op1 op2 op2 op2 oðp2 hð11 ; 12 ; pÞ þ q2 hð11 ; 02 ; pÞ p2 hð01 ; 12 ; pÞ q2 hð01 ; 02 ; pÞÞ ¼ op2 ¼ hð11 ; 12 ; pÞ hð11 ; 02 ; pÞ hð01 ; 12 ; pÞ þ hð01 ; 02 pÞ; where h(11, 12, p) denotes h(p) under the fact that the components c1 and c2 are both in working states, the meanings for h(11, 02, p), h(01, 12, p) and h (01, 02, p) are similar. Some relationships are also established in the literature [13] between JRI & MRI, JRI & JFI, and JFI & MFI, which uncover the real practical implications of JMI and JFI for two components. This also is a base of our extension of JRI and JFI to multi-components. In order to understand our extensions in the following, we quote these earlier results here. Lemma 1 [13] ðaÞ JRIðc1 ; c2 Þ ¼ MRISc2 ðc1 Þ MRISc2 ðc1 Þ; ðbÞ JRIðc1 ; c2 Þ ¼ MRISc1 ðc2 Þ MRISc1 ðc2 Þ:
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Lemma 1, which is just the above derived expression in different notation, shows that JRI can be represented by MRI of each component in subsystems. We can also obtain the following corollary from Lemma 1, which indicates the practical implications of JRI and JFI. Corollary 1. JRI(c1, c2) < 0 is equivalent to MRISc2 ðc1 Þ < MRISc2 ðc1 Þ, which indicates that component c1 has higher MRI when component c2 is failed than when it is working; JRI(c1, c2) > 0 is equivalent to MRISc2 ðc1 Þ > MRISc2 ðc1 Þ, which indicates that component c1 has higher MRI when component c2 is working than when it is failed; JRI(c1, c2) = 0 is equivalent to MRISc2 ðc1 Þ ¼ MRISc2 ðc1 Þ, which indicates that component c1 has the same MRI when component c2 is working and when it is failed. Lemma 2 [13] ðaÞ JRIðc1 ; c2 Þ ¼ ðMRIðc1 Þ MRISc2 ðc1 ÞÞ=p2 ; ðbÞ JRIðc1 ; c2 Þ ¼ ðMRIðc2 Þ MRISc1 ðc2 ÞÞ=p1 : Through the lemmas listed above, we can measure the effect of the interaction of two components in determining system reliability. But how about the effect of interaction of more components on system reliability? To try to answer this question, a multi-component JRI is introduced, extending the definition of two-component JRI. Without loss of generality, we just consider components c1, c2, . . . , ck for our extensions below. Definition 1. The JRI of three components c1, c2 and c3 is, JRIðc1 ; c2 ; c3 Þ ¼
o3 hðpÞ : op1 op2 op3
Theorem 1. If components c1, c2 and c3 are independent, then JRIðc1 ; c2 ; c3 Þ ¼ hð11 ; 12 ; 13 ; pÞ hð11 ; 12 ; 03 ; pÞ hð11 ; 02 ; 13 ; pÞ hð01 ; 12 ; 13 ; pÞhð11 ; 02 ; 03 ; pÞ þ hð01 ; 12 ; 03 ; pÞ þ hð01 ; 02 ; 13 ; pÞ hð01 ; 02 ; 03 ; pÞ; where h(11, 12, 13, p) denotes h(p) under the fact that the components c1, c2 and c3 are all in working states. The other terms are defined similarly. The above theorem presents another way to calculate JRI, and also shows the relationship between some system reliabilities and JRI. The following definition and theorem extend the case of 3-component JRI to the case of multi-component JRI. Definition 2. The k component JRI is defined as, ok hðpÞ JRIðc1 ; . . . ; ck Þ ¼ Qk : i¼1 op i Theorem 2. For independent components c1, c2, . . . , ck, JRIðc1 ; . . . ; ck Þ ¼ hð11 ; 12 ; . . . ; 1k ; pÞ hð11 ; 12 ; . . . ; 0k ; pÞ hð01 ; 02 ; . . . ; 1k ; pÞ hð01 ; 02 ; . . . ; 0k ; pÞ: The sign before each function h follows the principles below: (1) While number k is odd, the sign before the function h is positive if the number of corresponding 1’s is odd, otherwise it is negative. (2) While number k is even, the sign before the function h is positive if the number of corresponding 1’s is even, otherwise it is negative. (3) The sign of last term h(01, . . . , 0k, p) is negative if k is odd, otherwise, it is positive.
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Theorem 2 enables us to calculate the JRI based on the reliability function without any derivatives. Considering the relationship between JRI and MRI introduced in [13], and quoted in Lemmas 1 and 2 above, we establish similar relationships between multi-component JRIs below. These indicate the practical implications of the multi-component JRI. Corollary 2 JRIðc1 ; c2 ; c3 Þ ¼ JRISc3 ðc1 ; c2 Þ JRISc3 ðc1 ; c2 Þ: Similarly, we can obtain JRIðc1 ; . . . ; cn Þ ¼ JRIScn ðc1 ; . . . ; cn1 Þ JRIScn ðc1 ; . . . ; cn1 Þ. Based on the corollary above, the following conclusion is obtained. Corollary 3. JRI(c1, . . . , cn) < 0 is equivalent to JRIScn ðc1 ; . . . ; cn1 Þ < JRIScn ðc1 ; . . . ; cn1 Þ, which indicates that components c1, . . . , cn1 have higher JRI when component cn is failed than when it is working; JRI(c1, . . . ,cn) > 0 is equivalent to JRIScn ðc1 ; . . . ; cn1 Þ > JRIScn ðc1 ; . . . ; cn1 Þ, which indicates that components c1, . . . , cn1 have higher JRI when component cn is working than when it is failed; JRI(c1, . . . , cn) = 0 is equivalent to JRIScn ðc1 ; . . . ; cn1 Þ ¼ JRIScn ðc1 ; . . . ; cn1 Þ, which indicates that components c1, . . . , cn1 have the same JRI when component cn is working and when it is failed. Corollary 4 JRIðc1 ; c2 ; c3 Þ ¼ ðJRIðc1 ; c2 Þ JRISc3 ðc1 ; c2 ÞÞ=p3 : Similarly, we can obtain JRIðc1 ; . . . ; cn Þ ¼ ðJRIðc1 ; . . . ; cn1 Þ JRIScn ðc1 ; . . . ; cn1 ÞÞ=pn . According to these definitions and corollaries obtained above, we can measure the effect of multi-component interaction in determining system reliability. In order to illustrate that our extension has some real practical implications, a simple example, a series–parallel system with 4 components, is used (Fig. 1). From the definition of JRI for three components, in the system we can easily obtain JRIðc1 ; c2 ; c3 Þ ¼ 1 þ p4 < 0; JRISc3 ðc1 ; c2 Þ ¼ p4 < 0; JRISc3 ðc1 ; c2 Þ ¼ 1 < 0: From Corollary 2, we have JRIðc1 ; c2 ; c3 Þ ¼ JRISc3 ðc1 ; c2 Þ JRISc3 ðc1 ; c2 Þ: On the other hand, JRISc3 ðc1 ; c2 Þ ¼ MRISc3 c2 ðc1 Þ MRISc3 c2 ðc1 Þ < 0 so that component c1 is more important when c2 is failed and c3 is working than when both c2 and c3 are working, which is coincident with our intuition. From JRISc3 ðc1 ; c2 Þ ¼ p4 < 0, JRISc3 ðc1 ; c2 Þ ¼ MRISc3 c2 ðc1 Þ MRISc3 c2 ðc1 Þ < 0, we see that component c1 is more important when c2 and c3 are both failed than when c2 is working and c3 is failed, which is also coincident with our intuition. Combining the above results, we conclude that component c1 is more important when c2 is failed than when c2 is working regardless of whether c3 is working or failed. Similarly, we can also study recursively some practical implications of the JRI for multi-components in some complex systems. 1
3
2
4
Fig. 1. A series–parallel system I.
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2 4 1
6
7
5 3 Fig. 2. A series–parallel system II.
Corollary 5 (a) (b) (c) (d)
If If If If
components components components components
c3 c3 c3 c3
and and and and
(c1, c2) (c1, c2) (c1, c2) (c1, c2)
are are are are
in in in in
parallel, and (c1, c2) are in parallel, then JRI(c1, c2, c3) P 0. parallel, and (c1, c2) are in series, then JRI(c1, c2, c3) 6 0. series, and (c1, c2) are in parallel, then JRI(c1, c2, c3) 6 0. series, and (c1, c2) are in series, then JRI(c1, c2, c3) P 0.
Example 1. As an application, we use Corollary 5 in the following system. This is a series–parallel system consisting of 7 components (Fig. 2). All components are independent identically distributed, each of them has reliability p. From this example, we can conclude some useful results. From the definition of three component JRI we conclude, for general independent components, JRIðc1 ; c2 ; c4 Þ ¼ p3 p5 p6 p7 þ p3 p6 p7 þ p5 p6 p7 p6 p7 ; JRIðc1 ; c2 ; c3 Þ ¼ p4 p5 p6 p7 þ p4 p6 p7 þ p5 p6 p7 ; JRIðc1 ; c2 ; c6 Þ ¼ p3 p4 p5 p6 p3 p4 p7 þ p3 p5 p7 þ p4 p5 p7 p4 p7 p5 p7 ; JRIðc1 ; c2 ; c7 Þ ¼ p3 p4 p5 p6 þ p4 p5 p6 þ p3 p4 p6 þ p3 p5 p6 p5 p6 p4 p6 : Now, assuming identical reliabilities for all components, we get, JRIðc1 ; c2 ; c4 Þ ¼ JRIðc1 ; c2 ; c5 Þ ¼ p4 þ 2p3 p2 ¼ p2 ð2p 1 p2 Þ < 0; JRIðc1 ; c2 ; c3 Þ ¼ p4 þ 2p3 ¼ p3 ð2 pÞ > 0; JRIðc1 ; c2 ; c6 Þ ¼ p4 þ p3 2p2 ¼ p2 ðp p2 2Þ < 0; JRIðc1 ; c2 ; c7 Þ ¼ p4 þ 3p3 2p2 ¼ p2 ð3p p2 2Þ < 0; which illustrate the results of Corollary 5 in the special case of pi = p. 3. JFI of multi-components In this section, the idea of joint failure importance, JFI, is discussed. This is similar to JRI but the focus is on failure rather than working, which is used in Section 2. The JFI is the contribution of component failure to system failure. The definition of marginal failure importance, MFI, for component ci is [13] MFIðci Þ ¼
oF ðSÞ ; oqi
and the definition of JFI for two components c1 and c2 is [13] JFIðc1 ; c2 Þ ¼
o2 F ðSÞ : oq1 oq2
Definition 3. The definition of a multi-component JFI is extended as follows: on F ðSÞ JFIðc1 ; . . . ; cn Þ ¼ Qn : i¼1 oqi
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Lemma 3 [13]. For independent components, MFIðci Þ ¼
oF ðSÞ oðpi F ð1i ; pÞ þ qi F ð0i ; pÞÞ ¼ ¼ F ð0i ; pÞ F ð1i ; pÞ: oqi oqi
Lemma 4 [13]. The following alternative relationships hold between JFI & MFI: ðaÞ JFIðc1 ; c2 Þ ¼ MFISc1 ðc2 Þ MFISc1 ðc2 Þ; ðbÞ JFIðc1 ; c2 Þ ¼ MFISc2 ðc1 Þ MFISc2 ðc1 Þ: The above lemma can be used to give some implications for JFI, which are more useful in practical applications. Corollary 6. Similar to Lemma 4, the relationships between 3 components JFI and 2 components JFI, and n components JFI and n 1 components JFI, are obtained as follows: JFIðc1 ; c2 ; c3 Þ ¼ JFIScn ðc1 ; c2 Þ JFIScn ðc1 ; c2 Þ: Generally, for n P 3, JFIðc1 ; . . . ; cn Þ ¼ JFIScn ðc1 ; . . . ; cn1 Þ JFIScn ðc1 ; . . . ; cn1 Þ: Lemma 5 [13]. The following alternative relationships also hold between JFI & MFI: ðaÞ JFIðc1 ; c2 Þ ¼ ðMFIðc1 Þ MFISc2 ðc1 ÞÞ=p2 ; ðbÞ JFIðc1 ; c2 Þ ¼ ðMFIðc2 Þ MFISc1 ðc2 ÞÞ=p1 : Corollary 7. The relationship of 3 components JFI and 2 components JFI is as follows: JFIðc1 ; c2 ; c3 Þ ¼ ðJFIðc1 ; c2 Þ JFISc3 ðc1 ; c2 ÞÞ=q3 : The equation JFIðc1 ; . . . ; cn Þ ¼ ðJFIðc1 ; . . . ; cn1 Þ JFIScn ðc1 ; . . . ; cn1 ÞÞ=qn can also be easily obtained. We have found the relationships of MRI & MFI, JRI & JFI as follows: MFIðci Þ ¼ MRIðci Þ; JFIðc1 ; c2 Þ ¼ JRIðc1 ; c2 Þ: JFIðc1 ; c2 ; . . . ; cn Þ ¼ JRIðc1 ; c2 ; . . . ; cn Þ: The proofs, which can be done easily by using simple mathematical manipulations and induction, are omitted here. In order to illustrate some results obtained in Sections 2 and 3, another example is presented for 3 components in a k-out-of-n:G system. It is well-known that in a k-out-of-n:G system, which consists of n components, the system works if and only if at least k components are working. The following conclusion gives the simple closed-form for the JRI of a k-out-of-n:G system with independent identical components. Thus, hereinafter, we assume that pi = p, i = 1, . . . , n. Conclusion 1. The JRI of k-out-of-n:G system has the following simple form: JRIðc1 ; c2 ; c3 Þ ¼ pk3 qnk2
n3 n3 2 n3 2 pq þ p : q 2 k2 k1 k3
Note: The above formula assume k P 3. Let us consider a 3-out-of-5 system. From the formula given above, we have, JRIðc1 ; c2 ; c3 Þ ¼ 6p2 6p þ 1;
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and JRISc3 ðc1 ; c2 Þ ¼ 2p 3p2 : So we have JRISc3 ðc1 ; c2 Þ ¼ JRIðc1 ; c2 ; c3 Þ þ JRISc3 ðc1 ; c2 Þ ¼ 1 4p þ 3p2 . Fig. 3 illustrates JRISc3 ðc1 ; c2 Þ with various values of p. It can be easily seen, when p 2 ½0; 13Þ, JRISc3 ðc1 ; c2 Þ > 0. From Corollary 2, we have, MRISc3 c2 ðc1 Þ > MRISc3 c2 ðc1 Þ. This means that, when p 2 ½0; 13Þ, component c1 is more important when both c2 and c3 are working than when c2 is down and c3 is working; When p 2 ð13 ; 1Þ, JRISc3 ðc1 ; c2 Þ < 0. Also from corollary 2, we have, MRISc3 c2 ðc1 Þ < MRISc3 c2 ðc1 Þ. This means that, when p 2 ð13 ; 1Þ, component c1 is more important when c2 is down and c3 is working than when both c2 and c3 are working; When p ¼ 13, JRISc3 ðc1 ; c2 Þ ¼ 0. This means that for the two cases of c2 is down and c3 is working and both c2 and c3 are working, component c1 has the same importance. Fig. 4 illustrates JRISc3 ðc1 ; c2 Þ with various values of p. It can be seen, when p 2 ½0; 23Þ, JRISc3 ðc1 ; c2 Þ > 0. From corollary 2, we have, MRISc3 c2 ðc1 Þ > MRISc3 c2 ðc1 Þ. This means that, when p 2 ½0; 23Þ, component c1 is more important when c2 is working and c3 is down than when both c2 and c3 are down; When p 2 ð23 ; 1Þ, JRISc3 ðc1 ; c2 Þ < 0. Then, MRISc3 c2 ðc1 Þ < MRISc3 c2 ðc1 Þ. This means that, when p 2 ð23 ; 1Þ, component c1 is more important when both c2 and c3 are down than when c2 is working and c3 is down; When p ¼ 23, JRISc3 ðc1 ; c2 Þ ¼ 0. This means that component c1 has the same importance whether c2 is working and c3 is down or both c2 and c3 are down. In the following, we list some functions of JRI(c1, c2, c3) with respect to different values of n for fixed k = 3. When k = 3, n = 5, JRI(c1, c2, c3) = q2 4pq + p2, when k = 3, n = 6, JRI(c1, c2, c3) = q(q2 6pq + 3p2), when k = 3, n = 7, JRI(c1, c2, c3) = q2(q2 8pq + 6p2), when k = 3, n = 8, JRI(c1, c2, c3) = q3(q2 10pq + 10p2). From Fig. 5, we see that JRIn(c1, c2, c3) > JRIn+1 (c1, c2, c3) for fixed k = 3 when the component reliability p is near to 0 or 1. The following conclusion supports our intuition.
Fig. 3. JRISc3 ðc1 ; c2 Þ of a 3-out-of-5 system.
Fig. 4. JRISc3 ðc1 ; c2 Þ of a 3-out-of-5 system.
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Fig. 5. The curves of JRI in 3-out-of-n:G systems.
Conclusion 2. In a 3-out-of-n:G system, JRIn ðc1 ; c2 ; c3 Þ > JRInþ1 ðc1 ; c2 ; c3 Þ;
when p <
2 : n1
4. Conditional reliability importance Under the condition that some components are working or some components are down, or some parts of components are working and some parts of components are down, what is the reliability importance of some other components? Based on the definitions of MRI & JRI, we consider the reliability importance when the states of certain components are known. It is assumed that the system also works under these conditions. In fact, the decomposition of JRI or JFI in previous sections has provided some useful information on conditional importance. This is one reason that the conditional importance will be introduced in this section, which really uncovers the practical implications of JRI or JFI. Another reason for introducing the conditional importance is to meet the practical requirements such as maintenance and operating state monitoring etc. For example, if a system is in an operating state but some components are failed, one practical problem is to decide to which components we should pay more care in terms of maintenance. To answer this question, the conditional importance is more useful. In fact, the JRI or JFI can give similar useful information on this question. 4.1. Conditional MRI Given one component’s state, either working or failed, another component’s conditional MRI is given in the following two definitions. Definition 4. The conditional MRI of component cj, when component ci is working or down, is, MRIj ðpi ¼ zÞ ¼
ohðp1 ; p2 ; . . . ; pi1 ; zi ; piþ1 . . . ; pn Þ opj
ðj 6¼ iÞ:
For independent components, MRIj ðpi ¼ zÞ ¼ ¼
ohðp1 ; p2 ; . . . ; zi ; . . . ; pn Þ opj oðpj hðp1 ; . . . ; zi ; . . . ; pj1 ; 1j ; pjþ1 ; . . . ; pn Þ þ qj hðp1 ; . . . ; zi ; . . . ; pj1 ; 0j ; pjþ1 ; . . . ; pn ÞÞ opj
¼ hðp1 ; . . . ; zi ; . . . ; pj1 ; 1j ; pjþ1 ; . . . ; pn Þ hðp1 ; . . . ; zi ; . . . ; pj1 ; 0j ; pjþ1 ; . . . ; pn Þ; where zi = 1 (0) when the component ci is working (failed). Note that it does not matter here which out of i and j is larger.
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Based on the definition of JRI, we obtain the relationship between conditional MRI and JRI as follows: JRIðci ; cj Þ ¼ MRIj ðpi ¼ 1Þ MRIj ðpi ¼ 0Þ: Similarly, the other conditional MRI is given in the following. Definition 5. In a system with independent components, the conditional MRI of component ck when the states of components i and j are known, is given as follows: MRIk ðpi ¼ zi ; pj ¼ zj Þ ¼
ohðzi ; zj ; pk ; pÞ ; opk
for i 6¼ j 6¼ k:
It is easily shown that MRIk ðpi ¼ zi ; pj ¼ zj Þ ¼ hðzi ; zj ; 1k ; pÞ hðzi ; zj ; 0k ; pÞ;
for i 6¼ j 6¼ k;
where zi, zj denote the states of components ci, cj respectively. Other conditional MRIs can be similarly defined. Based on the definition of JRI, we also obtain the relationship between conditional MRI and JRI as follows: JRIðci ; cj ; ck Þ ¼ MRIk ðpi ¼ 1; pj ¼ 1Þ þ MRIk ðpi ¼ 0; pj ¼ 0Þ MRIk ðpi ¼ 0; pj ¼ 1Þ MRIk ðpi ¼ 1; pj ¼ 0Þ: The relationship we have given above can give us some useful information. The value of JRI(ci, cj, ck) is classified into three cases JRI(ci, cj, ck) > 0, JRI(ci, cj, ck) < 0 and JRI(ci, cj, ck) = 0. We only discuss one case of these three. Others can be discussed similarly. When JRI(ci, cj, ck) > 0, from the formula above we have, MRIk ðpi ¼ 1; pj ¼ 1Þ MRIk ðpi ¼ 0; pj ¼ 1Þ > MRIk ðpi ¼ 0; pj ¼ 0Þ MRIk ðpi ¼ 1; pj ¼ 0Þ: We note that if the right hand side of the above inequality is greater than zero, it implies component ck is more important when both ci and cj are down than when ci is working and cj is down, and ck is more important when both ci and cj are working than when ci is down and cj is working. Similarly we can analyze other situations, such as when the left hand side of the above inequality is less than zero. 4.2. Conditional JRI Similarly, the two component’s conditional JRI can be defined in the following. Definition 6. The conditional JRI of components ci and cj, when the state of component ck is given, is defined as, JRIðci ; cj Þðpk ¼ zÞ ¼
o2 hðp1 ; . . . ; zk ; . . . ; pn Þ opi opj
ðk 6¼ i; jÞ:
For independent components, it is easily shown that JRIðci ; cj Þðpk ¼ zÞ ¼
o2 hðp1 ; . . . ; zk ; . . . ; pn Þ opi opj
¼ hðzk ; 1i ; 1j ; pÞ hðzk ; 1i ; 0j ; pÞ hðzk ; 0i ; 1j ; pÞ þ hðzk ; 0i ; 0j ; pÞ; where zk = 1 (0) when component ck is working (failed). Also the following relationship can be found: JRIðci ; cj ; ck Þ ¼ JRIðci ; cj Þðpk ¼ 1Þ JRIðci ; cj Þðpk ¼ 0Þ:
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Definition 7. In a system with independent components, the conditional JRI of components ci and cj, when the states of components ct and cs are given, is, JRIðci ; cj Þðps ¼ zs ; pt ¼ zt Þ ¼
o2 hðzs ; zt ; pÞ ; opi opj
i 6¼ j 6¼ s 6¼ t
it is easily shown that JRIðci ; cj Þðps ¼ zs ; pt ¼ zt Þ ¼ hð1i ; 1j ; zs ; zt ; pÞ hð1i ; 0j ; zs ; zt ; pÞ hð0i ; 1j ; zs ; zt ; pÞ þ hð0i ; 0j ; zs ; zt ; pÞ; where zs and zt denote the states of components s and t. Based on the relationships derived above, we obtain the following result. Corollary 8. For a component independent system, the value of JRI of k components is between 2k1 and 2k1. 4.3. Conditional MRI and JRI in k-out-of-n:G systems with independent identical components In this subsection, we discuss some special cases of k-out-of-n:G systems in order to calculate their conditional MRI and conditional JRI. The results obtained can be used directly in practical applications of reliability analysis. We also illustrate these results with some figures to show their properties. Some examples are used to illustrate the results for some k-out-of-n:G systems. 4.3.1. Conditional MRI in k-out-of-n:G systems Example 2. In a k-out-of-n:G system, when the state of component ci is known, the conditional MRI of component cj is, MRIi ðpj ¼ 0Þ ¼ hðp1 ; . . . ; 0i ; . . . ; pj1 ; 1j ; pjþ1 ; . . . ; pn Þ hðp1 ; . . . ; 0i ; . . . ; pj1 ; 0j ; pjþ1 ; . . . ; pn Þ ! ! ! n2 n2 X X n2 r n2 r n 2 k1 nk1 n2r n2r ¼ p ð1 pÞ p ð1 pÞ ¼ p q : r r k1 r¼k r¼k1 In a 2-out-of-n:G system, the conditional MRI of component cj has the following properties: MRInj ðpi ¼ 0Þ 6 MRInþ1 j ðpi ¼ 0Þ; MRInj ðpi ¼ 0Þ > MRInþ1 j ðp i ¼ 0Þ;
1 ; n1 1 : when p > n1 when p 6
Because MRInj ðpi ¼ 0Þ MRIjnþ1 ðpi ¼ 0Þ ¼ ðn 2Þpqn1 ðn 1Þpqn ¼ pqn1 ððn 1Þp 1Þ; therefore, the above results are easily proved. From them we can obtain the information that, if one compo1 1 nent is down when p < n1 ðwhen p > n1 Þ, any another one component plays a more important role in a 1 2-out-of-n + 1(2-out-of-n) system than in a 2-out-of-n (2-out-of-n + 1) system; when p ¼ n1 , any other one component has the same importance in the above two systems. The following result can also be obtained: n 2 k2 nkþ2 MRIj ðpi ¼ 1Þ ¼ p q : k2 In a 2-out-of-n:G system, the conditional MRI of component cj has the following property, which can easily be proved: MRInj ðpi ¼ 1Þ > MRInþ1 j ðp i ¼ 1Þ:
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Example 3. In a k-out-of-n:G system, when the states of components cj and ck are known, the conditional MRI of component ci (i 5 j, k) is as follows: MRIi ðpj ¼ 0; pk ¼ 0Þ ¼ hð0j ; 0k ; 1i ; pÞ hð0j ; 0k ; 0i ; pÞ ! n3 n3 n3 X X ¼ pr ð1 pÞn3r r r¼k r¼k1
n3
! r
p ð1 pÞ
n3r
¼
n3
! pk1 qnk2 :
k1
r
Thus in 3-out-of-n:G system the conditional MRI of component ci has the following properties: 2 ; n2 2 : when p > n2
MRIni ðpj ¼ 0; pk ¼ 0Þ 6 MRInþ1 ðpj ¼ 0; pk ¼ 0Þ; i
when p 6
MRIni ðpj ¼ 0; pk ¼ 0Þ > MRIinþ1 ðpj ¼ 0; pk ¼ 0Þ; Because MRIni ðpj ¼ 0; pk ¼ 0Þ MRIinþ1 ðpj ¼ 0; pk ¼ 0Þ ¼ 2 n5
¼p q
ðn 3Þðn 4Þ ðn 2Þðn 3Þq 2
n3
! p2 qn5
n2
2 2 n5
¼p q
! p2 qn4
2 ðn 3Þððn 2Þp 2Þ ; 2
therefore, the above conclusions are true. Similarly, we have, MRIj ðp1 ¼ 1; p2 ¼ 1Þ ¼ hð11 ; 12 ; 1j ; pÞ hð11 ; 12 ; 0j ; pÞ ! n3 n3 n3 X X n3r ¼ pr ð1 pÞ r r¼k3 r¼k2
n3
! r
p ð1 pÞ r
n3r
n3 ¼
! pk3 qnk ;
k3
in 3-out-of-n:G system, MRIni ðpj ¼ 1; pk ¼ 1Þ > MRInþ1 ðpj ¼ 1; pk ¼ 1Þ: i And, MRIi ðpj ¼ 0; pk ¼ 1Þ ¼
n 3 k2 nk1 p q ; k2
in 3-out-of-n:G system, MRIni ðpj ¼ 0; pk ¼ 1Þ 6 MRInþ1 ðpj ¼ 0; pk ¼ 1Þ; i
when p 6
1 ; n2
MRIni ðpj ¼ 0; pk ¼ 1Þ > MRIinþ1 ðpj ¼ 0; pk ¼ 1Þ;
when p >
1 : n2
Fig. 6 shows the comparisons of the MRIi(pj = 1, pk = 1), MRIi(pj = 0, pk = 0) and MRIi(pj = 0, pk = 1) in a 3-out-of-5:G system. 4.3.2. Conditional JRI Example 4. In a k-out-of-n:G system, when the state of component ck is known, the conditional JRI of components ci and cj (i, j 5 k) is,
JRIðci ; cj Þðpk ¼ 0Þ ¼ pk2 qnk2
n3 k2
q
n3 k1
p :
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Fig. 6. Comparison curves of conditional MRI in a 3-out-of-5:G system.
It can also be obtained easily, JRIn ðci ; cj Þðpk ¼ 0Þ < JRInþ1 ðci ; cj Þðpk ¼ 0Þ;
when p <
1 : n2
Because in 3-out-of-n:G system, JRIn ðci ; cj Þðpk ¼ 0Þ JRInþ1 ðci ; cj Þðpk ¼ 0Þ n3 n3 n2 n2 n5 n4 ¼ pq q p pq q p 1 2 1 2 ðn 2Þðn 3Þ ðn 3Þðn 4Þ p ¼ pqn5 ððn 2Þp 1Þ; < pqn5 q þ 2 therefore, the conclusion is proved. Fig. 7 describes the various of conditional JRI(ci, cj)(pk = 0) of components ci and cj with k = 3, n = 5, 6, 7, 8. Similarly, we can get some results in a 3-out-of-n:G system, n3 n3 k3 nk1 JRIðci ; cj Þðpk ¼ 1Þ ¼ p q q p ; k3 k2 1 : JRIn ðci ; cj Þðpk ¼ 1Þ > JRInþ1 ðci ; cj Þðpk ¼ 1Þ; when p < n2
Fig. 7. Various curves of conditional JRI(ci, cj)(pk = 0).
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And, k1 ; nk2 k2 ; JRIn ðci ; cj Þðps ¼ 1; pt ¼ 0Þ < JRInþ1 ðci ; cj Þðps ¼ 1; pt ¼ 0Þ; when p > n k1 n 4 k2 nk2 n 4 k1 nk3 JRIðci ; cj Þðps ¼ 0; pt ¼ 0Þ ¼ ; p q p q k2 k1 n 4 k4 nk n 4 k3 nk1 JRIðci ; cj Þðps ¼ 1; pt ¼ 1Þ ¼ ; p q p q k4 k3 n 4 k3 nk1 n 4 k2 nk2 JRIðci ; cj Þðps ¼ 1; pt ¼ 0Þ ¼ : p q p q k3 k2
JRIn ðci ; cj Þðps ¼ 0; pt ¼ 0Þ < JRInþ1 ðci ; cj Þðps ¼ 0; pt ¼ 0Þ;
when p >
In the above numerical examples, the results and figures present lots of properties of importance for k-outof-n:G systems, which can be a great help in understanding component importance, also they can guide reliability design and maintenance activities. 5. Conclusion This article extends the definition of the two-component JRI (JFI) to multi-component JRI (JFI). The multi-component JRI is the rate at which system reliability improves as multi-component reliability improves simultaneously. JRI of multi-components provides useful information about multi-component interactions in a reliability system. It also provides conditional importance for components, which expresses the practical implications for JRI or JFI. We applied the results obtained in this paper to some k-out-of-n:G systems, and through some numerical examples illustrated the definitions and properties of JRI, JFI and conditional importance. Also some relationships among them were obtained. The paper also investigated the concept of conditional reliability importance while certain components’ states are known. Finally, the conditional reliability importance is analyzed for some k-out-of-n:G systems. The results obtained can provide more information on components, which can guide reliability designs and maintenance activities. No doubt, the new component importance definitions, together with original concepts, supply further information for understanding the system in many situations. Acknowledgements This research is supported by the BIT grant and Program for New Century Excellent Talents in University, China (NCET-05-0180). The Authors thank the anonymous referees for their helpful suggestions which improved the presentation of this paper. Also the authors give thanks to professor Alan G. Hawkes for his English corrections. Appendix Proof of Theorem 1. By using the pivotal decomposition theorem, it is easily got, 2 o hðpÞ o op o3 hðpÞ oðhð11 ; 12 ; pÞ hð11 ; 02 ; pÞ hð01 ; 12 ; pÞ þ hð01 ; 02 ; pÞÞ 1 op2 JRIðc1 ; c2 ; c3 Þ ¼ ¼ ¼ op1 op2 op3 op3 op3 oðp3 hð11 ; 12 ; 13 ; pÞ þ q3 hð11 ; 12 ; 03 ; pÞ p3 hð11 ; 02 ; 13 ; pÞ q3 hð11 ; 02 ; 03 ; pÞÞ ¼ op3 oðp3 hð01 ; 12 ; 13 ; pÞ q3 hð01 ; 12 ; 03 ; pÞ þ p3 hð01 ; 02 ; 13 ; pÞ þ q3 hð01 ; 02 ; 13 ; pÞÞ þ op3 ¼ hð11 ; 12 ; 13 ; pÞ hð11 ; 12 ; 03 ; pÞ hð11 ; 02 ; 13 ; pÞ hð01 ; 12 ; 13 ; pÞ þ hð11 ; 02 ; 03 ; pÞ þ hð01 ; 12 ; 03 ; pÞ þ hð01 ; 02 ; 13 ; pÞ hð01 ; 02 ; 03 ; pÞ:
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Proof of Theorem 2. By using induction, we have, (1) k = 1, we have MRI(c1) = h(11, p) h(01, p). (2) k = 2, we have JRI(c1,c2) = h(11, 12, p) h(11, 02, p) h(01, 12, p) + h(01, 02, p). (3) We assume that the conclusion is true for k = i, and then, we will prove the conclusion is also correct for k = i + 1. From Definition 2, we have, JRIðc1 ; . . . ; ci Þ ¼ hð11 ; 12 ; . . . ; 1i ; pÞ hðXi ; pÞ hð01 ; 02 ; . . . ; 0i ; pÞ: We only prove one case that i is odd and the sum of 1s in the vector (Xi, p) is odd. Without loss of generality, we assume that the sign of the term h(Xi, p) is positive. We can get the JRI of components c1, c2 ,. . . , ci+1 as follows based on the above formula: JRIðc1 ; . . . ; ciþ1 Þ ¼ hð11 ; 12 ; . . . ; 1iþ1 ; pÞ þ þ hðXi ; 1iþ1 ; pÞ hðXi ; 0iþ1 ; pÞ þ þ hð01 ; 02 ; . . . ; 0iþ1 ; pÞ: If i is odd (even), then i + 1 is even (odd), the sum of 1s in the vector (Xi, 1i+1, p) is odd (even) if the sum of 1s in the vector (Xi, p) is even (odd), which tells us that the sign of h(Xi, 1i+1, p) is positive, the sign of h(Xi, 0i+1, p) is negative. The sign of h(01 ,. . . , 0i+1, p) is easily obtained. Thus the conclusion is correct for k = i + 1. h Proof of Corollary 2 JRIðc1 ; c2 ; c3 Þ ¼
2 o3 hðpÞ o hðpÞ ¼o op3 ¼ oJRIðc1 ; c2 Þ=op3 op1 op2 op3 op1 op2
¼ JRISc3 ðc1 ; c2 Þ JRISc3 ðc1 ; c2 Þ:
Proof of Corollary 4 JRIðc1 ; c2 Þ ¼
o2 h o2 ðp3 hðS c3 Þ þ q3 hðS c3 ÞÞ ¼ ¼ p3 JRISc3 ðc1 ; c2 Þ þ q3 JRISc3 ðc1 ; c2 Þ op1 op2 op1 op2
¼ p3 ðJRISc3 ðc1 ; c2 Þ JRISc3 ðc1 ; c2 ÞÞ þ JRISc3 ðc1 ; c2 Þ ¼ p3 JRIðc1 ; c2 ; c3 Þ þ JRISc3 ðc1 ; c2 Þ:
Proof of Corollary 5. (a) When components c3 and (c1, c2) are in parallel, the subsystem (c1, c2) is irrelevant in S * c3, which means JRISc3 ðc1 ; c2 Þ ¼ 0. We also have known that in system S c3, (c1, c2) are in parallel. So, JRISc3 ðc1 ; c2 Þ 6 0. From formula JRIðc1 ; c2 ; c3 Þ ¼ JRISc3 ðc1 ; c2 Þ JRISc3 ðc1 ; c2 Þ. We can conclude JRI(c1, c2, c3) P 0. Similarly, we can get results b, c and d. h Proof of Corollary 6. Based on the definition of the multi-component JFI, we have, on F ðSÞ on1 oF ðSÞ on1 ðF ð0n ; pÞ F ð1n ; pÞÞ on1 F ð0n ; pÞ on1 F ð1n ; pÞ JFIðc1 ; . . . ; cn Þ ¼ Qn ¼ Qn1 ¼ ¼ Qn1 Qn1 Qn1 oqi oqi oqi ð i oqi Þoqn i oqi i i i ¼ JFIScn ðc1 ; . . . ; cn1 Þ JFIScn ðc1 ; . . . ; cn1 Þ:
Proof of Corollary 7 JFIðc1 ; c2 Þ ¼
o2 F o2 ðp3 F ðS c3 Þ þ q3 F ðS c3 ÞÞ ¼ ¼ p3 JFISc3 ðc1 ; c2 Þ þ q3 JFISc3 ðc1 ; c2 Þ oq1 oq2 oq1 oq2
¼ q3 ðJFISc3 ðc1 ; c2 Þ JFISc3 ðc1 ; c2 ÞÞ þ JFISc3 ðc1 ; c2 Þ ¼ q3 JFIðc1 ; c2 ; c3 Þ þ JFISc3 ðc1 ; c2 Þ:
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Proof of Conclusion 1. For a k-out-of-n:G system with independent identical components, the system reliability is, n X n r nr hðp; k; nÞ ¼ p ð1 pÞ : r r¼k From the definition of the JRI for 3 components, we have, JRIðc1 ; c2 ; c3 Þ ¼ hð11 ; 12 ; 13 ; pÞ hð11 ; 12 ; 03 ; pÞ hð11 ; 02 ; 13 ; pÞ hð01 ; 12 ; 13 ; pÞhð11 ; 02 ; 03 ; pÞ þ hð01 ; 12 ; 03 ; pÞ þ hð01 ; 02 ; 13 ; pÞ hð01 ; 02 ; 03 ; pÞ: The system reliability of a k-out-of-n:G system given that the 3 components are functioning, is the same as that of a (k 3)-out-of-(n 3):G system. That is, n3 X n 3 r n3r hðp; k; njp1 ¼ p2 ¼ p3 ¼ 1Þ ¼ : pq r r¼k3 Similarly, hðp; k; njp1 ¼ p2 ¼ 1; p3 ¼ 0Þ ¼ hðp; k; njp2 ¼ p3 ¼ 1; p1 ¼ 0Þ ¼ hðp; k; njp1 ¼ 1; p2 ¼ 0; p3 ¼ 1Þ n3 X n 3 r n3r ¼ ; pq r r¼k2 hðp; k; njp1 ¼ 1; p2 ¼ 0; p3 ¼ 0Þ ¼ hðp; k; njp2 ¼ 1; p1 ¼ p3 ¼ 0Þ ¼ hðp; k; njp1 ¼ p2 ¼ 0; p3 ¼ 1Þ n3 X n 3 r n3r ¼ ; pq r r¼k1 n3 X n 3 r n3r : hðp; k; njp1 ¼ 0; p2 ¼ 0; p3 ¼ 0Þ ¼ pq r r¼k Therefore, the JRI of 3 components is as follows: ! ! ! n3 n3 n3 X X X n 3 r n3r n 3 r n3r n 3 r n3r JRIðc1 ; c2 ; c3 Þ ¼ pq pq pq r r r r¼k3 r¼k2 r¼k2 ! ! ! n3 n3 n3 X X X n 3 r n3r n 3 r n3r n 3 r n3r þ þ pq pq pq r r r r¼k1 r¼k2 r¼k1 ! ! n3 n3 X n 3 r n3r X n 3 r n3r þ pq pq r r r¼k r¼k1 ! ! ! ! n 3 k3 nk n 3 k2 nk1 n 3 k2 nk1 n 3 k1 nk2 ¼ þ p q p q p q p q k3 k2 k2 k1 ! ! ! n 3 k3 nk n 3 k2 nk1 n 3 k1 nk2 ¼ p q 2 p q þ p q k3 k2 k1 ! ! ! ! n3 n3 2 n3 2 k3 nk2 ¼p q pq þ p : q 2 k2 k1 k3 Proof of Conclusion 2. By using Conclusion 1, we get, JRIðc1 ; c2 ; c3 Þ ¼ pk3 qnk2
n3 n3 2 pq þ p ; q2 2 k1 k2 k3
n3
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in a 3-out-of-n:G system, JRIn ðc1 ; c2 ; c3 Þ ¼ q q 2ðn 3Þpq þ p2 ; 2 n2 2 p : JRInþ1 ðc1 ; c2 ; c3 Þ ¼ qn4 q2 2ðn 2Þpq þ 2 n5
2
n3
The above equations imply that JRInþ1 ðc1 ; c2 ; c3 Þ JRIn ðc1 ; c2 ; c3 Þ n2 2 n3 2 n4 2 n5 2 ¼q q 2ðn 2Þpq þ q 2ðn 3Þpq þ p q p 2 2 n2 2 n3 2 < qn4 q2 2ðn 2Þpq þ p q2 2ðn 3Þpq þ p 2 2 ¼ qn4 2pq þ ðn 3Þp2 ¼ qn4 pððn 1Þp 2Þ: 2 Therefore, JRIn > JRIn+1, when p < n1 . h
Proof of Corollary 8. The values of component ci’s Conditional reliability importance MRIi(pj = 1) and MRIi(pj = 0) are between [1, 1] obviously, and for independent components ci and cj, JRIðci ; cj Þ ¼ MRIi ðpj ¼ 1Þ MRIi ðpj ¼ 0Þ: Therefore, the value of JRI(ci, cj) is between [2, 2]. Similarly, in terms of the following relationship: JRIðci ; cj ; ck Þ ¼ JRIðci ; cj Þðpk ¼ 1Þ JRIðci ; cj Þðpk ¼ 0Þ ... JRIðc1 ; c2 ; . . . ; ck Þ ¼ JRIðc1 ; c2 ; . . . ; ck1 Þðpk ¼ 1Þ JRIðc1 ; c2 ; . . . ; ck1 Þðpk ¼ 0Þ: We can conclude that the value of JRI of k components is between 2k1 and 2k1.
h
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Stochastics and Statistics
Analysis for joint importance of components in a coherent system Xueli Gao, Lirong Cui *, Jinlin Li School of Management and Economics, Beijing Institute of Technology, Beijing 100081, China Received 19 December 2005; accepted 19 July 2006 Available online 1 November 2006
Abstract JRI (Joint Reliability Importance) of two components is a measure of interaction of two components in a system for their contribution to the system reliability. It is defined as the rate at which the system reliability improves as the reliabilities of the two components improve. But, sometimes we may improve system reliability through improving reliabilities of three or more components. This article extends the concepts of JRI & JFI (Joint Failure Importance) of two components to multi-components, and establishes some relationships between JRI & JRI, JFI & JFI, and JFI & JRI. The paper also investigates the concept of Conditional Reliability Importance while the working states of certain components are known. Finally, the JRI of multi-components and Conditional Reliability Importance are analyzed in detail for a k-out-of-n:G system. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Reliability; Joint reliability importance; Joint failure importance; Conditional reliability importance
1. Introduction One purpose of system reliability analysis is to identify the weakness and/or the critical components in a system and to quantify the impact of component failures. The so-called ‘‘reliability importance’’ can be used for this purpose. So far various importance definitions have been introduced for system reliability analysis. These importance measures provide a numerical rank to determine which components are more important to system reliability improvement or more critical to system failure. Importance was first introduced by Birnbaum [1]. This index characterizes the rate at which the system reliability changes with respect to changes in the reliability of a given component; it is also defined as MRI (Marginal Reliability Importance). In order to evaluate the importance of different aspects for a system, a set of different importance measures was introduced, such as Structure Importance, Reliability Criticality Importance [2], Monte-Carlo Variance Importance [3]. Other measures of components and minimal cut sets importance in coherent systems were
*
Corresponding author. Tel.: +86 10 68948098; fax: +86 10 68912483. E-mail address: [email protected] (L. Cui).
0377-2217/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.07.022
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Notations c1, c2, . . . , cn components of system S considered system S * ci system with component ci working S ci system with component ci failed pi reliability of component ci qi 1 pi unreliability of component ci 1i, 0i event: pi = [1, 0], i.e., component ci is always [working, failed] p (p1, p2, . . . , pn), vector q (q1, q2, . . . , qn), vector Xi indicator for component ci : Xi = 1 if ci is working, Xi = 0 otherwise Xi vector: (X1, X2, . . . , Xi) h(p) system reliability as a function of p F(q) system unreliability as a function of q = (1 p1,1 p2, . . . , 1 pn) Min-path a minimal set of components whose working is sufficient to make the system work Min-cut a minimal set of components whose failure is sufficient to make the system fail Relevant component a component which appears in at least one min-path and/or min-cut S-Coherent system a system in which every component is relevant, and for which h(p) is non-decreasing in each pi (i.e., there is no component whose functioning causes the system to fail)
developed by Barlow and Prochan [4,5], Fussell and Vesely [6,7]. Xie and Bergman [8] obtained an importance measure by ordering the reduction of the expected yield due to the variation of the life-time distribution of the components, Hwang [9] reviewed structural component importance and introduced some new measures. Importance measures in multi-state systems were also introduced [10,11]. Importance measures can also be estimated by Monte Carlo simulation [12,13]. The discussion of importance is still a hot topic in reliability field recently, see for example, Levitin [14] and Meng [15]. These importance definitions have been well defined and widely used in reliability engineering practice. A component’s MRI is the rate at which the system reliability changes with respect to changes in the reliability of that component [5]. Improvements in reliability of components with the highest MRI cause the greatest increase in system reliability. This information can be used to determine which components should be improved first in order to make the largest improvement in system reliability. For example, one might wish to increase the reliability of the subsystem (such as computer, engine, valves, circuits etc) in the newest version of a system (such as ship, aircraft, vehicle, electronics etc.), MRI could provide an initial ranking of the components to be considered for improvement for a given system structure. In certain cases, the system structure indicates which components have the highest MRI: some examples from [5] include pure parallel and pure series systems, some more complicated examples include a consecutive k-out-of-n:G or a k-out-of-r-of-n:G linear system in [16]. However, MRI does not provide all information on how components affect the system reliability. In particular, MRI gives very little information about how the component reliabilities affect each other. For example, design engineers working within a fixed resource want to improve the system reliability through increasing some components’ reliabilities. For them to use MRI would require recalculation for all components with every tradeoff made or considered. The JRI can be used in this kind of situation, since it indicates roughly how components interact in determining system reliability. The JRI of pairs of components was introduced by Hong and Lie in [17], and the definition was extended to deal with the statistical dependence between components by Armstrong [18] and for a multi-state system by Wu [19]. It is also discussed in dual failure-mode [20] and k-out-of-n:G system [21]. In practice, we may need to investigate the associated effect of more than two components on the system reliability, failure or on some states of the system for system maintenance. Hence we use this investigation to improve the system reliability in certain cases and indicate which component is more critical in some situations. Inspired by the idea of JRI
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introduced by Hong and Lie, in this paper we extend the concept of two-component JRI to multi-components joint reliability importance. Also in [22], Enrico Zio defined DIMII which describes the interaction between two components. The need to extend DIMII to multi-components can be satisfied by multi-components JRI. We also define the concepts of MRI and JRI under the condition that the working states of some components are known: we call it conditional reliability importance. The organization of the paper is as follows. Section 2 gives the definition of JRI of multi-components and formulae for calculating it, and some relationships among JRI and MRI are studied. In Section 3, a similar treatment is given for JFI, and a detailed discussion for a k-out-of-n:G system is presented to illustrate the results obtained. Section 4 discusses conditional reliability importance and establishes some properties in some k-out-of-n:G systems. Finally, the conclusion is given in Section 5. All proofs can be found in the appendix. In this paper, we investigate coherent systems, and assume, (1) there are 2 states of each component; (2) failures of components are mutually statistically independent (hereinafter, for short, independent). 2. JRI of multi-components First, we recall that MRI is defined as the rate at which the system reliability improves as the componentreliability improves, i.e., MRI of component ci is, MRIðci Þ ¼
ohðpÞ : opi
Assuming independence between components, the above equation can be written as MRIðci Þ ¼
ohðpÞ oðpi hð1i ; pÞ þ qi hð0i ; pÞÞ ¼ ¼ hð1i ; pÞ hð0i ; pÞ; opi opi
where (1i, p) = (p1, . . . , pi1, 1i, pi+1, . . . , pn), (0i, p) = (p1, . . . , pi1, 0i,pi+1, . . . , pn). For a coherent system, the definition of JRI for two components introduced by Hong and Lie [13] is, JRIðc1 ; c2 Þ ¼
o2 hðpÞ : op1 op2
Hence, from this formula, JRI indicates how components interact in determining system reliability. For simplicity, without loss of generality, we denote the components of interest for the two-component case as c1,c2. For independent components, it is easy to get, ohðpÞ o2 hðpÞ o op1 oðhð11 ; pÞ hð01 ; pÞÞ JRIðc1 ; c2 Þ ¼ ¼ ¼ op1 op2 op2 op2 oðp2 hð11 ; 12 ; pÞ þ q2 hð11 ; 02 ; pÞ p2 hð01 ; 12 ; pÞ q2 hð01 ; 02 ; pÞÞ ¼ op2 ¼ hð11 ; 12 ; pÞ hð11 ; 02 ; pÞ hð01 ; 12 ; pÞ þ hð01 ; 02 pÞ; where h(11, 12, p) denotes h(p) under the fact that the components c1 and c2 are both in working states, the meanings for h(11, 02, p), h(01, 12, p) and h (01, 02, p) are similar. Some relationships are also established in the literature [13] between JRI & MRI, JRI & JFI, and JFI & MFI, which uncover the real practical implications of JMI and JFI for two components. This also is a base of our extension of JRI and JFI to multi-components. In order to understand our extensions in the following, we quote these earlier results here. Lemma 1 [13] ðaÞ JRIðc1 ; c2 Þ ¼ MRISc2 ðc1 Þ MRISc2 ðc1 Þ; ðbÞ JRIðc1 ; c2 Þ ¼ MRISc1 ðc2 Þ MRISc1 ðc2 Þ:
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Lemma 1, which is just the above derived expression in different notation, shows that JRI can be represented by MRI of each component in subsystems. We can also obtain the following corollary from Lemma 1, which indicates the practical implications of JRI and JFI. Corollary 1. JRI(c1, c2) < 0 is equivalent to MRISc2 ðc1 Þ < MRISc2 ðc1 Þ, which indicates that component c1 has higher MRI when component c2 is failed than when it is working; JRI(c1, c2) > 0 is equivalent to MRISc2 ðc1 Þ > MRISc2 ðc1 Þ, which indicates that component c1 has higher MRI when component c2 is working than when it is failed; JRI(c1, c2) = 0 is equivalent to MRISc2 ðc1 Þ ¼ MRISc2 ðc1 Þ, which indicates that component c1 has the same MRI when component c2 is working and when it is failed. Lemma 2 [13] ðaÞ JRIðc1 ; c2 Þ ¼ ðMRIðc1 Þ MRISc2 ðc1 ÞÞ=p2 ; ðbÞ JRIðc1 ; c2 Þ ¼ ðMRIðc2 Þ MRISc1 ðc2 ÞÞ=p1 : Through the lemmas listed above, we can measure the effect of the interaction of two components in determining system reliability. But how about the effect of interaction of more components on system reliability? To try to answer this question, a multi-component JRI is introduced, extending the definition of two-component JRI. Without loss of generality, we just consider components c1, c2, . . . , ck for our extensions below. Definition 1. The JRI of three components c1, c2 and c3 is, JRIðc1 ; c2 ; c3 Þ ¼
o3 hðpÞ : op1 op2 op3
Theorem 1. If components c1, c2 and c3 are independent, then JRIðc1 ; c2 ; c3 Þ ¼ hð11 ; 12 ; 13 ; pÞ hð11 ; 12 ; 03 ; pÞ hð11 ; 02 ; 13 ; pÞ hð01 ; 12 ; 13 ; pÞhð11 ; 02 ; 03 ; pÞ þ hð01 ; 12 ; 03 ; pÞ þ hð01 ; 02 ; 13 ; pÞ hð01 ; 02 ; 03 ; pÞ; where h(11, 12, 13, p) denotes h(p) under the fact that the components c1, c2 and c3 are all in working states. The other terms are defined similarly. The above theorem presents another way to calculate JRI, and also shows the relationship between some system reliabilities and JRI. The following definition and theorem extend the case of 3-component JRI to the case of multi-component JRI. Definition 2. The k component JRI is defined as, ok hðpÞ JRIðc1 ; . . . ; ck Þ ¼ Qk : i¼1 op i Theorem 2. For independent components c1, c2, . . . , ck, JRIðc1 ; . . . ; ck Þ ¼ hð11 ; 12 ; . . . ; 1k ; pÞ hð11 ; 12 ; . . . ; 0k ; pÞ hð01 ; 02 ; . . . ; 1k ; pÞ hð01 ; 02 ; . . . ; 0k ; pÞ: The sign before each function h follows the principles below: (1) While number k is odd, the sign before the function h is positive if the number of corresponding 1’s is odd, otherwise it is negative. (2) While number k is even, the sign before the function h is positive if the number of corresponding 1’s is even, otherwise it is negative. (3) The sign of last term h(01, . . . , 0k, p) is negative if k is odd, otherwise, it is positive.
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Theorem 2 enables us to calculate the JRI based on the reliability function without any derivatives. Considering the relationship between JRI and MRI introduced in [13], and quoted in Lemmas 1 and 2 above, we establish similar relationships between multi-component JRIs below. These indicate the practical implications of the multi-component JRI. Corollary 2 JRIðc1 ; c2 ; c3 Þ ¼ JRISc3 ðc1 ; c2 Þ JRISc3 ðc1 ; c2 Þ: Similarly, we can obtain JRIðc1 ; . . . ; cn Þ ¼ JRIScn ðc1 ; . . . ; cn1 Þ JRIScn ðc1 ; . . . ; cn1 Þ. Based on the corollary above, the following conclusion is obtained. Corollary 3. JRI(c1, . . . , cn) < 0 is equivalent to JRIScn ðc1 ; . . . ; cn1 Þ < JRIScn ðc1 ; . . . ; cn1 Þ, which indicates that components c1, . . . , cn1 have higher JRI when component cn is failed than when it is working; JRI(c1, . . . ,cn) > 0 is equivalent to JRIScn ðc1 ; . . . ; cn1 Þ > JRIScn ðc1 ; . . . ; cn1 Þ, which indicates that components c1, . . . , cn1 have higher JRI when component cn is working than when it is failed; JRI(c1, . . . , cn) = 0 is equivalent to JRIScn ðc1 ; . . . ; cn1 Þ ¼ JRIScn ðc1 ; . . . ; cn1 Þ, which indicates that components c1, . . . , cn1 have the same JRI when component cn is working and when it is failed. Corollary 4 JRIðc1 ; c2 ; c3 Þ ¼ ðJRIðc1 ; c2 Þ JRISc3 ðc1 ; c2 ÞÞ=p3 : Similarly, we can obtain JRIðc1 ; . . . ; cn Þ ¼ ðJRIðc1 ; . . . ; cn1 Þ JRIScn ðc1 ; . . . ; cn1 ÞÞ=pn . According to these definitions and corollaries obtained above, we can measure the effect of multi-component interaction in determining system reliability. In order to illustrate that our extension has some real practical implications, a simple example, a series–parallel system with 4 components, is used (Fig. 1). From the definition of JRI for three components, in the system we can easily obtain JRIðc1 ; c2 ; c3 Þ ¼ 1 þ p4 < 0; JRISc3 ðc1 ; c2 Þ ¼ p4 < 0; JRISc3 ðc1 ; c2 Þ ¼ 1 < 0: From Corollary 2, we have JRIðc1 ; c2 ; c3 Þ ¼ JRISc3 ðc1 ; c2 Þ JRISc3 ðc1 ; c2 Þ: On the other hand, JRISc3 ðc1 ; c2 Þ ¼ MRISc3 c2 ðc1 Þ MRISc3 c2 ðc1 Þ < 0 so that component c1 is more important when c2 is failed and c3 is working than when both c2 and c3 are working, which is coincident with our intuition. From JRISc3 ðc1 ; c2 Þ ¼ p4 < 0, JRISc3 ðc1 ; c2 Þ ¼ MRISc3 c2 ðc1 Þ MRISc3 c2 ðc1 Þ < 0, we see that component c1 is more important when c2 and c3 are both failed than when c2 is working and c3 is failed, which is also coincident with our intuition. Combining the above results, we conclude that component c1 is more important when c2 is failed than when c2 is working regardless of whether c3 is working or failed. Similarly, we can also study recursively some practical implications of the JRI for multi-components in some complex systems. 1
3
2
4
Fig. 1. A series–parallel system I.
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2 4 1
6
7
5 3 Fig. 2. A series–parallel system II.
Corollary 5 (a) (b) (c) (d)
If If If If
components components components components
c3 c3 c3 c3
and and and and
(c1, c2) (c1, c2) (c1, c2) (c1, c2)
are are are are
in in in in
parallel, and (c1, c2) are in parallel, then JRI(c1, c2, c3) P 0. parallel, and (c1, c2) are in series, then JRI(c1, c2, c3) 6 0. series, and (c1, c2) are in parallel, then JRI(c1, c2, c3) 6 0. series, and (c1, c2) are in series, then JRI(c1, c2, c3) P 0.
Example 1. As an application, we use Corollary 5 in the following system. This is a series–parallel system consisting of 7 components (Fig. 2). All components are independent identically distributed, each of them has reliability p. From this example, we can conclude some useful results. From the definition of three component JRI we conclude, for general independent components, JRIðc1 ; c2 ; c4 Þ ¼ p3 p5 p6 p7 þ p3 p6 p7 þ p5 p6 p7 p6 p7 ; JRIðc1 ; c2 ; c3 Þ ¼ p4 p5 p6 p7 þ p4 p6 p7 þ p5 p6 p7 ; JRIðc1 ; c2 ; c6 Þ ¼ p3 p4 p5 p6 p3 p4 p7 þ p3 p5 p7 þ p4 p5 p7 p4 p7 p5 p7 ; JRIðc1 ; c2 ; c7 Þ ¼ p3 p4 p5 p6 þ p4 p5 p6 þ p3 p4 p6 þ p3 p5 p6 p5 p6 p4 p6 : Now, assuming identical reliabilities for all components, we get, JRIðc1 ; c2 ; c4 Þ ¼ JRIðc1 ; c2 ; c5 Þ ¼ p4 þ 2p3 p2 ¼ p2 ð2p 1 p2 Þ < 0; JRIðc1 ; c2 ; c3 Þ ¼ p4 þ 2p3 ¼ p3 ð2 pÞ > 0; JRIðc1 ; c2 ; c6 Þ ¼ p4 þ p3 2p2 ¼ p2 ðp p2 2Þ < 0; JRIðc1 ; c2 ; c7 Þ ¼ p4 þ 3p3 2p2 ¼ p2 ð3p p2 2Þ < 0; which illustrate the results of Corollary 5 in the special case of pi = p. 3. JFI of multi-components In this section, the idea of joint failure importance, JFI, is discussed. This is similar to JRI but the focus is on failure rather than working, which is used in Section 2. The JFI is the contribution of component failure to system failure. The definition of marginal failure importance, MFI, for component ci is [13] MFIðci Þ ¼
oF ðSÞ ; oqi
and the definition of JFI for two components c1 and c2 is [13] JFIðc1 ; c2 Þ ¼
o2 F ðSÞ : oq1 oq2
Definition 3. The definition of a multi-component JFI is extended as follows: on F ðSÞ JFIðc1 ; . . . ; cn Þ ¼ Qn : i¼1 oqi
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Lemma 3 [13]. For independent components, MFIðci Þ ¼
oF ðSÞ oðpi F ð1i ; pÞ þ qi F ð0i ; pÞÞ ¼ ¼ F ð0i ; pÞ F ð1i ; pÞ: oqi oqi
Lemma 4 [13]. The following alternative relationships hold between JFI & MFI: ðaÞ JFIðc1 ; c2 Þ ¼ MFISc1 ðc2 Þ MFISc1 ðc2 Þ; ðbÞ JFIðc1 ; c2 Þ ¼ MFISc2 ðc1 Þ MFISc2 ðc1 Þ: The above lemma can be used to give some implications for JFI, which are more useful in practical applications. Corollary 6. Similar to Lemma 4, the relationships between 3 components JFI and 2 components JFI, and n components JFI and n 1 components JFI, are obtained as follows: JFIðc1 ; c2 ; c3 Þ ¼ JFIScn ðc1 ; c2 Þ JFIScn ðc1 ; c2 Þ: Generally, for n P 3, JFIðc1 ; . . . ; cn Þ ¼ JFIScn ðc1 ; . . . ; cn1 Þ JFIScn ðc1 ; . . . ; cn1 Þ: Lemma 5 [13]. The following alternative relationships also hold between JFI & MFI: ðaÞ JFIðc1 ; c2 Þ ¼ ðMFIðc1 Þ MFISc2 ðc1 ÞÞ=p2 ; ðbÞ JFIðc1 ; c2 Þ ¼ ðMFIðc2 Þ MFISc1 ðc2 ÞÞ=p1 : Corollary 7. The relationship of 3 components JFI and 2 components JFI is as follows: JFIðc1 ; c2 ; c3 Þ ¼ ðJFIðc1 ; c2 Þ JFISc3 ðc1 ; c2 ÞÞ=q3 : The equation JFIðc1 ; . . . ; cn Þ ¼ ðJFIðc1 ; . . . ; cn1 Þ JFIScn ðc1 ; . . . ; cn1 ÞÞ=qn can also be easily obtained. We have found the relationships of MRI & MFI, JRI & JFI as follows: MFIðci Þ ¼ MRIðci Þ; JFIðc1 ; c2 Þ ¼ JRIðc1 ; c2 Þ: JFIðc1 ; c2 ; . . . ; cn Þ ¼ JRIðc1 ; c2 ; . . . ; cn Þ: The proofs, which can be done easily by using simple mathematical manipulations and induction, are omitted here. In order to illustrate some results obtained in Sections 2 and 3, another example is presented for 3 components in a k-out-of-n:G system. It is well-known that in a k-out-of-n:G system, which consists of n components, the system works if and only if at least k components are working. The following conclusion gives the simple closed-form for the JRI of a k-out-of-n:G system with independent identical components. Thus, hereinafter, we assume that pi = p, i = 1, . . . , n. Conclusion 1. The JRI of k-out-of-n:G system has the following simple form: JRIðc1 ; c2 ; c3 Þ ¼ pk3 qnk2
n3 n3 2 n3 2 pq þ p : q 2 k2 k1 k3
Note: The above formula assume k P 3. Let us consider a 3-out-of-5 system. From the formula given above, we have, JRIðc1 ; c2 ; c3 Þ ¼ 6p2 6p þ 1;
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and JRISc3 ðc1 ; c2 Þ ¼ 2p 3p2 : So we have JRISc3 ðc1 ; c2 Þ ¼ JRIðc1 ; c2 ; c3 Þ þ JRISc3 ðc1 ; c2 Þ ¼ 1 4p þ 3p2 . Fig. 3 illustrates JRISc3 ðc1 ; c2 Þ with various values of p. It can be easily seen, when p 2 ½0; 13Þ, JRISc3 ðc1 ; c2 Þ > 0. From Corollary 2, we have, MRISc3 c2 ðc1 Þ > MRISc3 c2 ðc1 Þ. This means that, when p 2 ½0; 13Þ, component c1 is more important when both c2 and c3 are working than when c2 is down and c3 is working; When p 2 ð13 ; 1Þ, JRISc3 ðc1 ; c2 Þ < 0. Also from corollary 2, we have, MRISc3 c2 ðc1 Þ < MRISc3 c2 ðc1 Þ. This means that, when p 2 ð13 ; 1Þ, component c1 is more important when c2 is down and c3 is working than when both c2 and c3 are working; When p ¼ 13, JRISc3 ðc1 ; c2 Þ ¼ 0. This means that for the two cases of c2 is down and c3 is working and both c2 and c3 are working, component c1 has the same importance. Fig. 4 illustrates JRISc3 ðc1 ; c2 Þ with various values of p. It can be seen, when p 2 ½0; 23Þ, JRISc3 ðc1 ; c2 Þ > 0. From corollary 2, we have, MRISc3 c2 ðc1 Þ > MRISc3 c2 ðc1 Þ. This means that, when p 2 ½0; 23Þ, component c1 is more important when c2 is working and c3 is down than when both c2 and c3 are down; When p 2 ð23 ; 1Þ, JRISc3 ðc1 ; c2 Þ < 0. Then, MRISc3 c2 ðc1 Þ < MRISc3 c2 ðc1 Þ. This means that, when p 2 ð23 ; 1Þ, component c1 is more important when both c2 and c3 are down than when c2 is working and c3 is down; When p ¼ 23, JRISc3 ðc1 ; c2 Þ ¼ 0. This means that component c1 has the same importance whether c2 is working and c3 is down or both c2 and c3 are down. In the following, we list some functions of JRI(c1, c2, c3) with respect to different values of n for fixed k = 3. When k = 3, n = 5, JRI(c1, c2, c3) = q2 4pq + p2, when k = 3, n = 6, JRI(c1, c2, c3) = q(q2 6pq + 3p2), when k = 3, n = 7, JRI(c1, c2, c3) = q2(q2 8pq + 6p2), when k = 3, n = 8, JRI(c1, c2, c3) = q3(q2 10pq + 10p2). From Fig. 5, we see that JRIn(c1, c2, c3) > JRIn+1 (c1, c2, c3) for fixed k = 3 when the component reliability p is near to 0 or 1. The following conclusion supports our intuition.
Fig. 3. JRISc3 ðc1 ; c2 Þ of a 3-out-of-5 system.
Fig. 4. JRISc3 ðc1 ; c2 Þ of a 3-out-of-5 system.
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Fig. 5. The curves of JRI in 3-out-of-n:G systems.
Conclusion 2. In a 3-out-of-n:G system, JRIn ðc1 ; c2 ; c3 Þ > JRInþ1 ðc1 ; c2 ; c3 Þ;
when p <
2 : n1
4. Conditional reliability importance Under the condition that some components are working or some components are down, or some parts of components are working and some parts of components are down, what is the reliability importance of some other components? Based on the definitions of MRI & JRI, we consider the reliability importance when the states of certain components are known. It is assumed that the system also works under these conditions. In fact, the decomposition of JRI or JFI in previous sections has provided some useful information on conditional importance. This is one reason that the conditional importance will be introduced in this section, which really uncovers the practical implications of JRI or JFI. Another reason for introducing the conditional importance is to meet the practical requirements such as maintenance and operating state monitoring etc. For example, if a system is in an operating state but some components are failed, one practical problem is to decide to which components we should pay more care in terms of maintenance. To answer this question, the conditional importance is more useful. In fact, the JRI or JFI can give similar useful information on this question. 4.1. Conditional MRI Given one component’s state, either working or failed, another component’s conditional MRI is given in the following two definitions. Definition 4. The conditional MRI of component cj, when component ci is working or down, is, MRIj ðpi ¼ zÞ ¼
ohðp1 ; p2 ; . . . ; pi1 ; zi ; piþ1 . . . ; pn Þ opj
ðj 6¼ iÞ:
For independent components, MRIj ðpi ¼ zÞ ¼ ¼
ohðp1 ; p2 ; . . . ; zi ; . . . ; pn Þ opj oðpj hðp1 ; . . . ; zi ; . . . ; pj1 ; 1j ; pjþ1 ; . . . ; pn Þ þ qj hðp1 ; . . . ; zi ; . . . ; pj1 ; 0j ; pjþ1 ; . . . ; pn ÞÞ opj
¼ hðp1 ; . . . ; zi ; . . . ; pj1 ; 1j ; pjþ1 ; . . . ; pn Þ hðp1 ; . . . ; zi ; . . . ; pj1 ; 0j ; pjþ1 ; . . . ; pn Þ; where zi = 1 (0) when the component ci is working (failed). Note that it does not matter here which out of i and j is larger.
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Based on the definition of JRI, we obtain the relationship between conditional MRI and JRI as follows: JRIðci ; cj Þ ¼ MRIj ðpi ¼ 1Þ MRIj ðpi ¼ 0Þ: Similarly, the other conditional MRI is given in the following. Definition 5. In a system with independent components, the conditional MRI of component ck when the states of components i and j are known, is given as follows: MRIk ðpi ¼ zi ; pj ¼ zj Þ ¼
ohðzi ; zj ; pk ; pÞ ; opk
for i 6¼ j 6¼ k:
It is easily shown that MRIk ðpi ¼ zi ; pj ¼ zj Þ ¼ hðzi ; zj ; 1k ; pÞ hðzi ; zj ; 0k ; pÞ;
for i 6¼ j 6¼ k;
where zi, zj denote the states of components ci, cj respectively. Other conditional MRIs can be similarly defined. Based on the definition of JRI, we also obtain the relationship between conditional MRI and JRI as follows: JRIðci ; cj ; ck Þ ¼ MRIk ðpi ¼ 1; pj ¼ 1Þ þ MRIk ðpi ¼ 0; pj ¼ 0Þ MRIk ðpi ¼ 0; pj ¼ 1Þ MRIk ðpi ¼ 1; pj ¼ 0Þ: The relationship we have given above can give us some useful information. The value of JRI(ci, cj, ck) is classified into three cases JRI(ci, cj, ck) > 0, JRI(ci, cj, ck) < 0 and JRI(ci, cj, ck) = 0. We only discuss one case of these three. Others can be discussed similarly. When JRI(ci, cj, ck) > 0, from the formula above we have, MRIk ðpi ¼ 1; pj ¼ 1Þ MRIk ðpi ¼ 0; pj ¼ 1Þ > MRIk ðpi ¼ 0; pj ¼ 0Þ MRIk ðpi ¼ 1; pj ¼ 0Þ: We note that if the right hand side of the above inequality is greater than zero, it implies component ck is more important when both ci and cj are down than when ci is working and cj is down, and ck is more important when both ci and cj are working than when ci is down and cj is working. Similarly we can analyze other situations, such as when the left hand side of the above inequality is less than zero. 4.2. Conditional JRI Similarly, the two component’s conditional JRI can be defined in the following. Definition 6. The conditional JRI of components ci and cj, when the state of component ck is given, is defined as, JRIðci ; cj Þðpk ¼ zÞ ¼
o2 hðp1 ; . . . ; zk ; . . . ; pn Þ opi opj
ðk 6¼ i; jÞ:
For independent components, it is easily shown that JRIðci ; cj Þðpk ¼ zÞ ¼
o2 hðp1 ; . . . ; zk ; . . . ; pn Þ opi opj
¼ hðzk ; 1i ; 1j ; pÞ hðzk ; 1i ; 0j ; pÞ hðzk ; 0i ; 1j ; pÞ þ hðzk ; 0i ; 0j ; pÞ; where zk = 1 (0) when component ck is working (failed). Also the following relationship can be found: JRIðci ; cj ; ck Þ ¼ JRIðci ; cj Þðpk ¼ 1Þ JRIðci ; cj Þðpk ¼ 0Þ:
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Definition 7. In a system with independent components, the conditional JRI of components ci and cj, when the states of components ct and cs are given, is, JRIðci ; cj Þðps ¼ zs ; pt ¼ zt Þ ¼
o2 hðzs ; zt ; pÞ ; opi opj
i 6¼ j 6¼ s 6¼ t
it is easily shown that JRIðci ; cj Þðps ¼ zs ; pt ¼ zt Þ ¼ hð1i ; 1j ; zs ; zt ; pÞ hð1i ; 0j ; zs ; zt ; pÞ hð0i ; 1j ; zs ; zt ; pÞ þ hð0i ; 0j ; zs ; zt ; pÞ; where zs and zt denote the states of components s and t. Based on the relationships derived above, we obtain the following result. Corollary 8. For a component independent system, the value of JRI of k components is between 2k1 and 2k1. 4.3. Conditional MRI and JRI in k-out-of-n:G systems with independent identical components In this subsection, we discuss some special cases of k-out-of-n:G systems in order to calculate their conditional MRI and conditional JRI. The results obtained can be used directly in practical applications of reliability analysis. We also illustrate these results with some figures to show their properties. Some examples are used to illustrate the results for some k-out-of-n:G systems. 4.3.1. Conditional MRI in k-out-of-n:G systems Example 2. In a k-out-of-n:G system, when the state of component ci is known, the conditional MRI of component cj is, MRIi ðpj ¼ 0Þ ¼ hðp1 ; . . . ; 0i ; . . . ; pj1 ; 1j ; pjþ1 ; . . . ; pn Þ hðp1 ; . . . ; 0i ; . . . ; pj1 ; 0j ; pjþ1 ; . . . ; pn Þ ! ! ! n2 n2 X X n2 r n2 r n 2 k1 nk1 n2r n2r ¼ p ð1 pÞ p ð1 pÞ ¼ p q : r r k1 r¼k r¼k1 In a 2-out-of-n:G system, the conditional MRI of component cj has the following properties: MRInj ðpi ¼ 0Þ 6 MRInþ1 j ðpi ¼ 0Þ; MRInj ðpi ¼ 0Þ > MRInþ1 j ðp i ¼ 0Þ;
1 ; n1 1 : when p > n1 when p 6
Because MRInj ðpi ¼ 0Þ MRIjnþ1 ðpi ¼ 0Þ ¼ ðn 2Þpqn1 ðn 1Þpqn ¼ pqn1 ððn 1Þp 1Þ; therefore, the above results are easily proved. From them we can obtain the information that, if one compo1 1 nent is down when p < n1 ðwhen p > n1 Þ, any another one component plays a more important role in a 1 2-out-of-n + 1(2-out-of-n) system than in a 2-out-of-n (2-out-of-n + 1) system; when p ¼ n1 , any other one component has the same importance in the above two systems. The following result can also be obtained: n 2 k2 nkþ2 MRIj ðpi ¼ 1Þ ¼ p q : k2 In a 2-out-of-n:G system, the conditional MRI of component cj has the following property, which can easily be proved: MRInj ðpi ¼ 1Þ > MRInþ1 j ðp i ¼ 1Þ:
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Example 3. In a k-out-of-n:G system, when the states of components cj and ck are known, the conditional MRI of component ci (i 5 j, k) is as follows: MRIi ðpj ¼ 0; pk ¼ 0Þ ¼ hð0j ; 0k ; 1i ; pÞ hð0j ; 0k ; 0i ; pÞ ! n3 n3 n3 X X ¼ pr ð1 pÞn3r r r¼k r¼k1
n3
! r
p ð1 pÞ
n3r
¼
n3
! pk1 qnk2 :
k1
r
Thus in 3-out-of-n:G system the conditional MRI of component ci has the following properties: 2 ; n2 2 : when p > n2
MRIni ðpj ¼ 0; pk ¼ 0Þ 6 MRInþ1 ðpj ¼ 0; pk ¼ 0Þ; i
when p 6
MRIni ðpj ¼ 0; pk ¼ 0Þ > MRIinþ1 ðpj ¼ 0; pk ¼ 0Þ; Because MRIni ðpj ¼ 0; pk ¼ 0Þ MRIinþ1 ðpj ¼ 0; pk ¼ 0Þ ¼ 2 n5
¼p q
ðn 3Þðn 4Þ ðn 2Þðn 3Þq 2
n3
! p2 qn5
n2
2 2 n5
¼p q
! p2 qn4
2 ðn 3Þððn 2Þp 2Þ ; 2
therefore, the above conclusions are true. Similarly, we have, MRIj ðp1 ¼ 1; p2 ¼ 1Þ ¼ hð11 ; 12 ; 1j ; pÞ hð11 ; 12 ; 0j ; pÞ ! n3 n3 n3 X X n3r ¼ pr ð1 pÞ r r¼k3 r¼k2
n3
! r
p ð1 pÞ r
n3r
n3 ¼
! pk3 qnk ;
k3
in 3-out-of-n:G system, MRIni ðpj ¼ 1; pk ¼ 1Þ > MRInþ1 ðpj ¼ 1; pk ¼ 1Þ: i And, MRIi ðpj ¼ 0; pk ¼ 1Þ ¼
n 3 k2 nk1 p q ; k2
in 3-out-of-n:G system, MRIni ðpj ¼ 0; pk ¼ 1Þ 6 MRInþ1 ðpj ¼ 0; pk ¼ 1Þ; i
when p 6
1 ; n2
MRIni ðpj ¼ 0; pk ¼ 1Þ > MRIinþ1 ðpj ¼ 0; pk ¼ 1Þ;
when p >
1 : n2
Fig. 6 shows the comparisons of the MRIi(pj = 1, pk = 1), MRIi(pj = 0, pk = 0) and MRIi(pj = 0, pk = 1) in a 3-out-of-5:G system. 4.3.2. Conditional JRI Example 4. In a k-out-of-n:G system, when the state of component ck is known, the conditional JRI of components ci and cj (i, j 5 k) is,
JRIðci ; cj Þðpk ¼ 0Þ ¼ pk2 qnk2
n3 k2
q
n3 k1
p :
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Fig. 6. Comparison curves of conditional MRI in a 3-out-of-5:G system.
It can also be obtained easily, JRIn ðci ; cj Þðpk ¼ 0Þ < JRInþ1 ðci ; cj Þðpk ¼ 0Þ;
when p <
1 : n2
Because in 3-out-of-n:G system, JRIn ðci ; cj Þðpk ¼ 0Þ JRInþ1 ðci ; cj Þðpk ¼ 0Þ n3 n3 n2 n2 n5 n4 ¼ pq q p pq q p 1 2 1 2 ðn 2Þðn 3Þ ðn 3Þðn 4Þ p ¼ pqn5 ððn 2Þp 1Þ; < pqn5 q þ 2 therefore, the conclusion is proved. Fig. 7 describes the various of conditional JRI(ci, cj)(pk = 0) of components ci and cj with k = 3, n = 5, 6, 7, 8. Similarly, we can get some results in a 3-out-of-n:G system, n3 n3 k3 nk1 JRIðci ; cj Þðpk ¼ 1Þ ¼ p q q p ; k3 k2 1 : JRIn ðci ; cj Þðpk ¼ 1Þ > JRInþ1 ðci ; cj Þðpk ¼ 1Þ; when p < n2
Fig. 7. Various curves of conditional JRI(ci, cj)(pk = 0).
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And, k1 ; nk2 k2 ; JRIn ðci ; cj Þðps ¼ 1; pt ¼ 0Þ < JRInþ1 ðci ; cj Þðps ¼ 1; pt ¼ 0Þ; when p > n k1 n 4 k2 nk2 n 4 k1 nk3 JRIðci ; cj Þðps ¼ 0; pt ¼ 0Þ ¼ ; p q p q k2 k1 n 4 k4 nk n 4 k3 nk1 JRIðci ; cj Þðps ¼ 1; pt ¼ 1Þ ¼ ; p q p q k4 k3 n 4 k3 nk1 n 4 k2 nk2 JRIðci ; cj Þðps ¼ 1; pt ¼ 0Þ ¼ : p q p q k3 k2
JRIn ðci ; cj Þðps ¼ 0; pt ¼ 0Þ < JRInþ1 ðci ; cj Þðps ¼ 0; pt ¼ 0Þ;
when p >
In the above numerical examples, the results and figures present lots of properties of importance for k-outof-n:G systems, which can be a great help in understanding component importance, also they can guide reliability design and maintenance activities. 5. Conclusion This article extends the definition of the two-component JRI (JFI) to multi-component JRI (JFI). The multi-component JRI is the rate at which system reliability improves as multi-component reliability improves simultaneously. JRI of multi-components provides useful information about multi-component interactions in a reliability system. It also provides conditional importance for components, which expresses the practical implications for JRI or JFI. We applied the results obtained in this paper to some k-out-of-n:G systems, and through some numerical examples illustrated the definitions and properties of JRI, JFI and conditional importance. Also some relationships among them were obtained. The paper also investigated the concept of conditional reliability importance while certain components’ states are known. Finally, the conditional reliability importance is analyzed for some k-out-of-n:G systems. The results obtained can provide more information on components, which can guide reliability designs and maintenance activities. No doubt, the new component importance definitions, together with original concepts, supply further information for understanding the system in many situations. Acknowledgements This research is supported by the BIT grant and Program for New Century Excellent Talents in University, China (NCET-05-0180). The Authors thank the anonymous referees for their helpful suggestions which improved the presentation of this paper. Also the authors give thanks to professor Alan G. Hawkes for his English corrections. Appendix Proof of Theorem 1. By using the pivotal decomposition theorem, it is easily got, 2 o hðpÞ o op o3 hðpÞ oðhð11 ; 12 ; pÞ hð11 ; 02 ; pÞ hð01 ; 12 ; pÞ þ hð01 ; 02 ; pÞÞ 1 op2 JRIðc1 ; c2 ; c3 Þ ¼ ¼ ¼ op1 op2 op3 op3 op3 oðp3 hð11 ; 12 ; 13 ; pÞ þ q3 hð11 ; 12 ; 03 ; pÞ p3 hð11 ; 02 ; 13 ; pÞ q3 hð11 ; 02 ; 03 ; pÞÞ ¼ op3 oðp3 hð01 ; 12 ; 13 ; pÞ q3 hð01 ; 12 ; 03 ; pÞ þ p3 hð01 ; 02 ; 13 ; pÞ þ q3 hð01 ; 02 ; 13 ; pÞÞ þ op3 ¼ hð11 ; 12 ; 13 ; pÞ hð11 ; 12 ; 03 ; pÞ hð11 ; 02 ; 13 ; pÞ hð01 ; 12 ; 13 ; pÞ þ hð11 ; 02 ; 03 ; pÞ þ hð01 ; 12 ; 03 ; pÞ þ hð01 ; 02 ; 13 ; pÞ hð01 ; 02 ; 03 ; pÞ:
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Proof of Theorem 2. By using induction, we have, (1) k = 1, we have MRI(c1) = h(11, p) h(01, p). (2) k = 2, we have JRI(c1,c2) = h(11, 12, p) h(11, 02, p) h(01, 12, p) + h(01, 02, p). (3) We assume that the conclusion is true for k = i, and then, we will prove the conclusion is also correct for k = i + 1. From Definition 2, we have, JRIðc1 ; . . . ; ci Þ ¼ hð11 ; 12 ; . . . ; 1i ; pÞ hðXi ; pÞ hð01 ; 02 ; . . . ; 0i ; pÞ: We only prove one case that i is odd and the sum of 1s in the vector (Xi, p) is odd. Without loss of generality, we assume that the sign of the term h(Xi, p) is positive. We can get the JRI of components c1, c2 ,. . . , ci+1 as follows based on the above formula: JRIðc1 ; . . . ; ciþ1 Þ ¼ hð11 ; 12 ; . . . ; 1iþ1 ; pÞ þ þ hðXi ; 1iþ1 ; pÞ hðXi ; 0iþ1 ; pÞ þ þ hð01 ; 02 ; . . . ; 0iþ1 ; pÞ: If i is odd (even), then i + 1 is even (odd), the sum of 1s in the vector (Xi, 1i+1, p) is odd (even) if the sum of 1s in the vector (Xi, p) is even (odd), which tells us that the sign of h(Xi, 1i+1, p) is positive, the sign of h(Xi, 0i+1, p) is negative. The sign of h(01 ,. . . , 0i+1, p) is easily obtained. Thus the conclusion is correct for k = i + 1. h Proof of Corollary 2 JRIðc1 ; c2 ; c3 Þ ¼
2 o3 hðpÞ o hðpÞ ¼o op3 ¼ oJRIðc1 ; c2 Þ=op3 op1 op2 op3 op1 op2
¼ JRISc3 ðc1 ; c2 Þ JRISc3 ðc1 ; c2 Þ:
Proof of Corollary 4 JRIðc1 ; c2 Þ ¼
o2 h o2 ðp3 hðS c3 Þ þ q3 hðS c3 ÞÞ ¼ ¼ p3 JRISc3 ðc1 ; c2 Þ þ q3 JRISc3 ðc1 ; c2 Þ op1 op2 op1 op2
¼ p3 ðJRISc3 ðc1 ; c2 Þ JRISc3 ðc1 ; c2 ÞÞ þ JRISc3 ðc1 ; c2 Þ ¼ p3 JRIðc1 ; c2 ; c3 Þ þ JRISc3 ðc1 ; c2 Þ:
Proof of Corollary 5. (a) When components c3 and (c1, c2) are in parallel, the subsystem (c1, c2) is irrelevant in S * c3, which means JRISc3 ðc1 ; c2 Þ ¼ 0. We also have known that in system S c3, (c1, c2) are in parallel. So, JRISc3 ðc1 ; c2 Þ 6 0. From formula JRIðc1 ; c2 ; c3 Þ ¼ JRISc3 ðc1 ; c2 Þ JRISc3 ðc1 ; c2 Þ. We can conclude JRI(c1, c2, c3) P 0. Similarly, we can get results b, c and d. h Proof of Corollary 6. Based on the definition of the multi-component JFI, we have, on F ðSÞ on1 oF ðSÞ on1 ðF ð0n ; pÞ F ð1n ; pÞÞ on1 F ð0n ; pÞ on1 F ð1n ; pÞ JFIðc1 ; . . . ; cn Þ ¼ Qn ¼ Qn1 ¼ ¼ Qn1 Qn1 Qn1 oqi oqi oqi ð i oqi Þoqn i oqi i i i ¼ JFIScn ðc1 ; . . . ; cn1 Þ JFIScn ðc1 ; . . . ; cn1 Þ:
Proof of Corollary 7 JFIðc1 ; c2 Þ ¼
o2 F o2 ðp3 F ðS c3 Þ þ q3 F ðS c3 ÞÞ ¼ ¼ p3 JFISc3 ðc1 ; c2 Þ þ q3 JFISc3 ðc1 ; c2 Þ oq1 oq2 oq1 oq2
¼ q3 ðJFISc3 ðc1 ; c2 Þ JFISc3 ðc1 ; c2 ÞÞ þ JFISc3 ðc1 ; c2 Þ ¼ q3 JFIðc1 ; c2 ; c3 Þ þ JFISc3 ðc1 ; c2 Þ:
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Proof of Conclusion 1. For a k-out-of-n:G system with independent identical components, the system reliability is, n X n r nr hðp; k; nÞ ¼ p ð1 pÞ : r r¼k From the definition of the JRI for 3 components, we have, JRIðc1 ; c2 ; c3 Þ ¼ hð11 ; 12 ; 13 ; pÞ hð11 ; 12 ; 03 ; pÞ hð11 ; 02 ; 13 ; pÞ hð01 ; 12 ; 13 ; pÞhð11 ; 02 ; 03 ; pÞ þ hð01 ; 12 ; 03 ; pÞ þ hð01 ; 02 ; 13 ; pÞ hð01 ; 02 ; 03 ; pÞ: The system reliability of a k-out-of-n:G system given that the 3 components are functioning, is the same as that of a (k 3)-out-of-(n 3):G system. That is, n3 X n 3 r n3r hðp; k; njp1 ¼ p2 ¼ p3 ¼ 1Þ ¼ : pq r r¼k3 Similarly, hðp; k; njp1 ¼ p2 ¼ 1; p3 ¼ 0Þ ¼ hðp; k; njp2 ¼ p3 ¼ 1; p1 ¼ 0Þ ¼ hðp; k; njp1 ¼ 1; p2 ¼ 0; p3 ¼ 1Þ n3 X n 3 r n3r ¼ ; pq r r¼k2 hðp; k; njp1 ¼ 1; p2 ¼ 0; p3 ¼ 0Þ ¼ hðp; k; njp2 ¼ 1; p1 ¼ p3 ¼ 0Þ ¼ hðp; k; njp1 ¼ p2 ¼ 0; p3 ¼ 1Þ n3 X n 3 r n3r ¼ ; pq r r¼k1 n3 X n 3 r n3r : hðp; k; njp1 ¼ 0; p2 ¼ 0; p3 ¼ 0Þ ¼ pq r r¼k Therefore, the JRI of 3 components is as follows: ! ! ! n3 n3 n3 X X X n 3 r n3r n 3 r n3r n 3 r n3r JRIðc1 ; c2 ; c3 Þ ¼ pq pq pq r r r r¼k3 r¼k2 r¼k2 ! ! ! n3 n3 n3 X X X n 3 r n3r n 3 r n3r n 3 r n3r þ þ pq pq pq r r r r¼k1 r¼k2 r¼k1 ! ! n3 n3 X n 3 r n3r X n 3 r n3r þ pq pq r r r¼k r¼k1 ! ! ! ! n 3 k3 nk n 3 k2 nk1 n 3 k2 nk1 n 3 k1 nk2 ¼ þ p q p q p q p q k3 k2 k2 k1 ! ! ! n 3 k3 nk n 3 k2 nk1 n 3 k1 nk2 ¼ p q 2 p q þ p q k3 k2 k1 ! ! ! ! n3 n3 2 n3 2 k3 nk2 ¼p q pq þ p : q 2 k2 k1 k3 Proof of Conclusion 2. By using Conclusion 1, we get, JRIðc1 ; c2 ; c3 Þ ¼ pk3 qnk2
n3 n3 2 pq þ p ; q2 2 k1 k2 k3
n3
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in a 3-out-of-n:G system, JRIn ðc1 ; c2 ; c3 Þ ¼ q q 2ðn 3Þpq þ p2 ; 2 n2 2 p : JRInþ1 ðc1 ; c2 ; c3 Þ ¼ qn4 q2 2ðn 2Þpq þ 2 n5
2
n3
The above equations imply that JRInþ1 ðc1 ; c2 ; c3 Þ JRIn ðc1 ; c2 ; c3 Þ n2 2 n3 2 n4 2 n5 2 ¼q q 2ðn 2Þpq þ q 2ðn 3Þpq þ p q p 2 2 n2 2 n3 2 < qn4 q2 2ðn 2Þpq þ p q2 2ðn 3Þpq þ p 2 2 ¼ qn4 2pq þ ðn 3Þp2 ¼ qn4 pððn 1Þp 2Þ: 2 Therefore, JRIn > JRIn+1, when p < n1 . h
Proof of Corollary 8. The values of component ci’s Conditional reliability importance MRIi(pj = 1) and MRIi(pj = 0) are between [1, 1] obviously, and for independent components ci and cj, JRIðci ; cj Þ ¼ MRIi ðpj ¼ 1Þ MRIi ðpj ¼ 0Þ: Therefore, the value of JRI(ci, cj) is between [2, 2]. Similarly, in terms of the following relationship: JRIðci ; cj ; ck Þ ¼ JRIðci ; cj Þðpk ¼ 1Þ JRIðci ; cj Þðpk ¼ 0Þ ... JRIðc1 ; c2 ; . . . ; ck Þ ¼ JRIðc1 ; c2 ; . . . ; ck1 Þðpk ¼ 1Þ JRIðc1 ; c2 ; . . . ; ck1 Þðpk ¼ 0Þ: We can conclude that the value of JRI of k components is between 2k1 and 2k1.
h
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