Combinatorial analysis for probabilistic assessment of dependent failures in systems and portfolios

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Abstract

System reliability is usefully applied to assess the performance of individual structures and portfolios of structures. In many instances, one is interested in knowing the probability that m failures have occurred among n components in a system, or that at least m failures have occurred among n components. Examples include structural failure modes within a single infrastructure or building, buildings and structures within a portfolio, or components of infrastructure systems such as transportation, water, power and communications. In cases where failure events in different components are statistically independent and their probability is identical, the number of component failures follows the binomial distribution. However, in most situations, these conditions do not hold. In this paper, we present a combinatorial formulation for computing the probability of m failures out of n system components for the general case, based on an extension of the inclusion–exclusion​ principle for computing the probability of a union. The results have been verified by an existing mathematical approach previously available in a mathematical textbook on combinatorics. The current derivation presents a closed-form accounting scheme derived for the inclusion/exclusion problem of multiple building failures in a portfolio, but also applicable to multiple modes of failure in a structural system. An example application to a portfolio of buildings is presented.

Introduction

In structural reliability considerations, one is often interested in the number (or ratio) of buildings that exceed a particular damage limit state, and are thus defined as “failed”. This could be important in assessing the vulnerability of a community, wherein the number of damaged buildings of particular type (such as hospitals, residences, schools, etc.) may be crucial.

Let Fi denote the event of failure of component i, and M the random variable describing the total number of component failures. To compute the probability of the union of all failure events, PrF1F2Fn=PrM1, corresponding to the failure probability of a series system, the inclusion/exclusion principle (e.g., [1]), can be invoked. It is: PrM1=1inPrFi1i<jnPrFiFj+1i<j<knPrFiFjFk+1n1PrF1F2Fn=k=1n1k11i1<<iknPrFi1Fik

Note that the k=1 to n summation terms are over all possible subsets of k failures, the number of which is given by the binomial coefficient nk.

In this paper we present a generalization of the inclusion–exclusion principle to compute the probability of more than m events, and the probability of exactly m events, where m is any particular value of M between zero and n. This generalization is useful for computing the reliability of systems that fail as soon as k components out of n fail (k out of n systems), but also for computing the distribution of the number of failures or damages in a portfolio. The proposed generalization is related to alternative system reliability approaches, such as Ditlevsen bounds [2], Daniels systems [3], system reliability based on linear programming concepts proposed by Song and Der Kiureghian [4], bounds by partial expansion [5], or other methods [6].

Section snippets

A generalization of the inclusion-exclusion principle for system reliability

The inclusion–exclusion principle can be extended to compute the probability of two or more failure events as follows: Pr(M2)=1i<jnPrFiFj21i<j<knPrFiFjFk+31i<j<k<lnPrFiFjFkFl+1n1n1PrF1F2Fn=k=2n1kk11i1<<iknPrFi1Fik and to compute the probability of m or more failure events as follows: PrMm=m1m11i1<<imnPrFi1Fimmm11i1<<im+1nPrFi1Fim+1+m+1m11i1<<im+2nPrFi1Fim+2+1nmn1m1PrF1F2Fn=k=mn1kmk1m11i1<<iknPrFi1Fik

Note that the

Special Cases

A common special case occurs when all marginal failure probabilities are identical and all failure events have the same statistical dependence, i.e.: PrFi=P1for1in,Pr(FiFj)=P2for1i<jn,Pr(FiFjFk)=P3for1i<j<kn, and so on, in which Pr has been introduced for the probability of the intersection of r events. In this special case, Eq. (6) reduces to: PrM=m=k=mn1kmkmnkPk In a more general setting, it is often possible to identify groups of components with identical marginal failure

Conclusions

Realistic safety assessment of systems is vitally important, including those with for which a large number of failure modes exhibit significant probabilities (i.e., those not dominated by a small number of modes). This is true for multiple modes of failure in a single structure, as well as performance criteria for a portfolio of systems within a community. For many such realistic systems those modes exhibit correlation and differing marginal probabilities. A straightforward understanding of the

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

The authors gratefully acknowledge the support of the National Science Foundation, USA , grant number CMMI 1063790. The opinions expressed in this paper are those of the authors, and do not necessarily reflect the views or policies of NSF.

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