Stochastics and Statistics
The reliability importance of components and prime implicants in coherent and non-coherent systems including total-order interactions

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Abstract

In the management of complex systems, knowledge of how components contribute to system performance is essential to the correct allocation of resources. Recent works have renewed interest in the properties of the joint (J) and differential (D) reliability importance measures. However, a common background for these importance measures has not been developed yet. In this work, we build a unified framework for the utilization of J and D in both coherent and non-coherent systems. We show that the reliability function of any system is multilinear and its Taylor expansion is exact at an order T. We then introduce a total order importance measure (DT) that coincides with the exact portion of the change in system reliability associated with any (finite or infinitesimal) change in component reliabilities. We show that DT synthesizes the Birnbaum, joint and differential importance of all orders in one unique indicator. We propose an algorithm that enables the numerical estimation of DT by varying one probability at a time, making it suitable in the analysis of complex systems. Findings demonstrate that the simultaneous utilization of DT and J provides reliability analysts with a complete dissection of system performance.

Introduction

Reliability engineers face critical operational decisions such as the determination of optimal maintenance, inspection and replacement policies (Ozekici, 1988, Dogramaci and Fraiman, 2004, Kubzin and Strusevich, 2006, Chun, 2008, Castro, 2009)1, the assignment of components to graded quality assurance programs, the categorization of system structures and components (Cheok et al., 1998, Vesely, 1998, Borgonovo, 2008). These decision problems are most often characterized by trade-offs between safety levels, system performance and economic viability. As Dillon et al. (2003) state, managers of complex systems (projects) “face a challenge when deciding how to allocate scarce resources to minimize the risks of project (system) failure. As resource constraints become tighter, balancing these failure risks is more critical, less intuitive, and can benefit from the power of quantitative analysis (Dillon et al., 2003, p. 354).”

Quantitative models play a central role in the decision-making process, as they provide analysts with crucial insights on how to achieve a given system performance. Several studies have demonstrated that components do not contribute to system performance in the same way (Lambert, 1975, Butler, 1977, El-Neweihi, 1980, Vesely et al., 1990, Cheok et al., 1998, Borgonovo and Apostolakis, 2001, Zio and Podofillini, 2006, Lu and Jiang, 2007, Gao et al., 2007). Thus, it is essential for analysts “to identify …the critical components (Gao et al., 2007, p. 282).” This information is delivered by importance measures. The recent works of Zio and Podofillini, 2006, Lu and Jiang, 2007, Gao et al., 2007, Do Van et al., 2008 have renewed interest towards the study and utilization of importance measures. In particular, they have extended the joint and differential importance measures (Hong and Lie, 1993, Borgonovo and Apostolakis, 2001) so as to capture higher-order interactions. An examination of the literature (see Section 2) shows that the research on these importance measures has sofar proceeded on two parallel but almost independent tracks, with works addressing the properties of either one of the importance measures. The lack of a comprehensive framework limits the insights analysts can obtain from the decision-support model. In fact, as we are to show, it is the simultaneous utilization of the two importance measures that allows analysts to exploit the reliability model information at best.

Our purpose is to build a unified framework for the utilization of the joint and differential reliability importance measures. In this respect, we observe that the distinction between coherent or non-coherent system marks a quite net partition in the literature. For instance, the works of Barlow and Proschan, 1976, Birnbaum, 1969, Lambert, 1975, Agrawal and Barlow, 1984, El-Neweihi, 1980, Ball and Provan, 1988, Hong and Lie, 1993, Armstrong, 1995, Cheok et al., 1998, Borgonovo and Apostolakis, 2001, Giglio and Wynn, 2004, Zio and Podofillini, 2006, Borgonovo, 2007, Gao et al., 2007, Do Van et al., 2008 assume coherent systems, while Inagacki and Henley, 1980, Andrews and Beeson, 2003, Beeson and Andrews, 2003, Lu and Jiang, 2007 address non-coherent systems. Thus, we set up our analysis so that our findings hold independently of the system type. For each finding, however, we discuss whether/how the finding is affected by the coherency (or non-coherency) of the system. We begin with the properties of multilinear functions. In particular, we address the coincidence of a multilinear function with its Maclaurin and Bernstein polynomials. By showing that the reliability function of any coherent and non-coherent system is multilinear, we prove that the Maclaurin (or Taylor) expansion of any system reliability function is exact and can be arrested at a finite order T, where TN (see Table 1 for notation and symbols used in this work).

We discuss how this finding relates to classical reliability results (in particular Theorem 3.2 in El-Neweihi (1980)) and observe that it provides an answer to the research question opened by Do Van et al. (2008) concerning the determination of “the bounds of Maclaurin series to find the minimal k for which the differential importance of order k can provide the true importance ranking (Do Van et al., 2008, p. 7).” It is then possible to prove that, if the rare event approximation applies, then for both coherent and non-coherent systems: a) the joint reliability importance of components i1,i2,,ik is equal to unity, if they are a prime implicant; and b a group of components has null joint reliability importance if and only if it is not contained in any prime implicant. We also offer an interpretation of these results in terms of the probability of group i1,i2,,ik being critical to the system.

The fact that the Taylor expansion of any system reliability function can be arrested at an order T has the following relevant consequence: any finite change in reliability that follows a discrete change in component failure probabilities is exactly apportioned at an order of at most T. We then introduce a new importance indicator, the total order importance measure, denoted by DT. DT includes the joint differential importance (Ji1,i2,,ikk) of all groups of components (k=1,2,,T). It is shown that DT is the exact fraction of the change in system reliability caused by generic changes in component reliabilities/unreliabilities, both for coherent and non-coherent systems, and both in the presence/absence of the rare event approximation. We study the limiting properties of DT, proving that it ends to the differential importance (D), when changes become small. Furthermore, DT tends to the Birnbaum importance measure (B), if uniform changes are assumed. Results show that, however, DT differs from lower orders Dk and from D (or B) significantly as interaction effects become relevant – i.e., when changes are finite.

The remainder of the paper is organized as follows. Section 2 provides a literature review and the definitions of Birnbaum, differential and joint importance measures. Section 3 lays out the mathematical framework of the work. Section 4 presents a general result for reliability functions of coherent and non-coherent systems. Section 5 introduces the total order importance measure. An algorithm for the numerical estimation of DT is presented in Section 6. Section 7 offers conclusions.

Section snippets

Birnbaum, differential and joint reliability importance measures

This section investigates the definitions and relationships between the Birnbaum (B), Joint (J) and Differential (D) reliability importance measures.

The concept of reliability importance stems from the seminal work of Birnbaum (1969) (Table 2).

In Birnbaum (1969) a system of N components is considered. The Birnbaum importance of component i(Bi) is defined as the probability of component i being critical to the system. Letting p be the (vector of) component success probabilities, and h(p):[0,1]NR

Multilinear functions and Maclaurin (Taylor) expansion

Reliability theory originates with the seminal works of Barlow and Proschan, 1976, Birnbaum, 1969, Barlow and Proschan, 1975, Lambert, 1975 (see also Block, 2001). Core of the theory is the representation of a system as a set of N components, each of which can be in two possible states (Birnbaum, 1969, Ball and Provan, 1988, Boros et al., 2000, Giglio and Wynn, 2004, Khachiyan et al., 2007). In these instances, multilinear functions play a crucial role and many reliability results are a

Finite changes in reliability functions of coherent and non-coherent systems

In this section, we show that the reliability (unreliability) function of any coherent and non-coherent system is multilinear.

Consider a system of N components (notations and symbols are listed in Table 1). Each component can be working or failed. Let Zi=10 denote the state variable of component i,Z=(Z1,Z2,,ZN) the state vector, and ϕ(Z) the structure function of the system. ϕ(Z) is a Boolean function and ϕ(Z)=1 denotes system success/failure. A structure function is coherent if ϕ(Z2)>ϕ(Z1)

The total order importance measure

This Section introduces a new importance measure. The new indicator provides the relationship between joint and differential reliability importance of all orders.

Consider a generic system, coherent or non-coherent. Let G(x) be the system reliability/unreliability function.

Definition 1

Let T be the highest order of the Taylor expansion in Eq. (32). LetΔThlBlΔxl+k=2Ti1<i2,,<ikli1<i2,,<ikJi1,i2,,ikk(x0)s=1kΔxis,denote the sum of the terms in Eq. (32) containing a contribution from xl. Then, we callDlT:=

Algorithmic computation of DT

In this Section, we address the numerical estimation of DT. In complex systems the number of components (N) can be of the order of 103 or more. A brute-force estimation of DT, i.e., an estimation procedure based on Eq. (48), would entail the computation of all mixed partial derivatives up to T. The cost of such an algorithm grows more than linearly with N. For instance, when N=2, the number of model evaluations necessary to estimate all first and second order partial derivatives is equal to N+N2

Conclusions

The recent works of Lu and Jiang, 2007, Gao et al., 2007 have renewed interest in reliability importance measures. A literature review, however, reveals that the studies of the properties of the joint and differential importance measures have developed on parallel tracks, an a common background is missing. In this work, we have proposed a unified framework for the utilization of the differential and joint reliability importance measures.

We have first addressed the question of the order T at

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