Comparing System Reliabilities with Ill-Known Probabilities
In reliability analysis, comparing system reliability is an essential task when designing safe systems. When the failure probabilities of the system components (assumed to be independent) are precisely known, this task is relatively simple to achieve, as
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bstract. In reliability analysis, comparing system reliability is an essential task when designing safe systems. When the failure probabilities of the system components (assumed to be independent) are precisely known, this task is relatively simple to achieve, as system reliabilities are precise numbers. When failure probabilities are ill-known (known to lie in an interval) and we want to have guaranteed comparisons (i.e., declare a system more reliable than another when it is for any possible probability value), there are different ways to compare system reliabilities. We explore the computational problems posed by such extensions, providing first insights about their pros and cons.
Keywords: System design probability · Comparison
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Reliability
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Introduction
Being able to compare system reliabilities is essential when designing systems. Provided the structure function mapping single component reliabilities to the overall system reliability is known, this step poses no particular problem (at least from a theoretical standpoint) when failure probabilities are precisely known. However, in practice, it may be difficult to provide precise assessments of such probabilities, for example because little data exist for the components (they may be issued from new technologies), or because they are given by expert opinions. This typically happens in early-stage phase design of new systems. In such a case, the problem of comparing system reliabilities become much more difficult, both conceptually and computationally speaking. In this paper, we explore what happens when the component probabilities of functioning are ill-known, that is are only known to lie in an interval. Several aspects of reliability analysis have been extended to the case of ill-known probabilities, such as importance indices [8], multi-state systems [4], common cause failure problems [9], . . . Yet, to our knowledge the problem of system reliability comparison remain to be formally studied within this setting. In Sect. 3, we extend usual system comparisons (recalled in Sect. 2) to interval-valued probabilities in two different ways, discussing the theoretical and c Springer International Publishing Switzerland 2016 J.P. Carvalho et al. (Eds.): IPMU 2016, Part II, CCIS 611, pp. 619–629, 2016. DOI: 10.1007/978-3-319-40581-0 50
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practical pros and cons of each extension. Section 4 provides a more complex examples than the very simple, illustrative ones provided along the paper. The necessary basics of reliability as well as notations are briefly recalled in Sect. 2.
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System Modelling and Comparison: Basics
In this paper, we assume that we want to compare the designs of K systems S1 , . . . , SK in terms of reliability, in order to choose (one of) the safest among them. The kth system will be composed of a set of rk components, and a given component can belong to one of T populations (types) of components, all components of a population being assumed to have the same stochastic behaviour (i.e., same failure rate).
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