Nonparametric predictive inference for system reliability with redundancy allocation

Abstract

This paper presents lower and upper probabilities for the reliability of k-out-of-m systems, which include series and parallel systems, and of series systems with independent k i -out-of-m i subsystems, for which optimal redundancy allocation is also presented in case of zero-failure testing. First, attention is restricted to k-out-of-m systems with exchangeable components. The lower and upper probabilities for successful functioning of the system are based on the nonparametric predictive inferential (NPI) approach for Bernoulli data. In this approach, it is assumed that test data are available on the components, and that the future components to be used in the system are exchangeable with these. Thereafter, systems are considered that consist of a series of independent subsystems, with subsystem i a k i -out-of-m i system consisting of exchangeable components. For such systems, an algorithm for optimal redundancy allocation after zero-failure testing is presented. A particularly attractive feature of NPI in reliability, with lower and upper probabilities, is that data containing zero failures can be dealt with in an attractive manner.

Nonparametric predictive inference for system reliability using the survival signature

“…It will also be of interest to consider possible system failure due to multiple failure modes [35], where the NPI approach provides interesting opportunities to consider unobserved or even unknown competing risks [22,32]. Topics of optimal system design in order to provide suitable levels of redundancy [3,10,14,23,31], possibly including costs, also pose interesting questions for which the use of the survival signature might provide new solutions. In the NPI framework some of such issues have been considered, but only for systems with relatively limited structures, for which the combinatorial aspects in computations already became quite complex [3].…”

“…Imprecise probabilities provide many exciting opportunities for reliability quantification [20,37,38]. The NPI method has already been used for system reliability [3,14,23,31], but only for systems with quite restricted structures. NPI for system reliability using the signature has also been presented, for systems consisting of only one type of components [4,5,15].…”

Nonparametric Predictive Inference for Binary Diagnostic Tests

“…These lower and upper probabilities follow from an assumed underlying latent variable representation together with Hill's assumption A (n) (Hill 1968), which has an explicitly predictive nature, and fit in the framework of nonparametric predictive inference (NPI) (Augustin and Coolen 2004;Coolen 2006). Several inferential problems involving Bernoulli data have been addressed using this NPI approach, for example comparisons of groups of Bernoulli data (Coolen and Coolen-Schrijner 2006;2007), acceptance sampling (Coolen and Elsaeiti 2009); and system reliability (Coolen-Schrijner et al 2008). We briefly summarize this approach; for more details and justification we refer to Coolen (1998) and Coolen-Schrijner (2006, 2007).…”

Nonparametric predictive inference for failure times of systems with exchangeable components

“…Applying this to systems with exchangeable components will be relatively straightforward and will generalize the results in reference [17]. In that paper a conjecture was formulated about optimal redundancy allocation, in line with the results in references [18] and [19] for different systems; analysis based on signatures might facilitate the proof of that conjecture.…”

“…Applying this to systems with exchangeable components will be relatively straightforward and will generalize the results in [17]. In that paper a conjecture was formulated about optimal redundancy allocation, in line with the results in [18] and [19] for different systems; There are major research challenges to the general theory of signatures, solutions to which may be of particular interest when working with lower and upper probabilities. For example, the fact that the theory of signatures [1] only applies to systems with exchangeable components is a very considerable restriction on the practical relevance of signatures and the related methods for reliability quantification.…”