The basic properties of some typical systems’ reliability in interval form

A proper mathematical representation of uncertainties is indispensable for reliability analysis of a practical engineering structural system. A general uncertainty analysis approach is probability bounds analysis (PBA), which propagates constraints on a distribution function through mathematical operations. The uncertainty about a probability distribution is represented by the set of cumulative distribution functions lying entirely within a pair of bounding distribution functions, which is called a P-box. Interval analysis as a special case of PBA is useful when there is no or less probabilistic information. It is common sense that great efforts must be paid to get enough probabilistic information used for probabilistic analysis of large and complex engineering structural systems. Even if there is no or less probabilistic information; the interval of possible values of probability of an event can be easily specified, such as the interval value of each element’s reliability of an engineering structural system.

This paper aims to introduce the concept of system reliability and its relationship to the reliability of its individual elements in an interval form. In terms of extension principle, interval arithmetic and possibility degree formula (PDF) for ranking interval numbers, basic properties of system reliability in interval form are investigated. The conclusion is that relationships between point reliability (point reliability used to describe a precise value of probability reliability is distinct with interval reliability) of some typical systems, such as series system, parallel system, series–parallel system, parallel–series system and r/n(G) system, etc., and point reliability of their individual elements are maintained in their interval forms. This is called quasi-consistency in this paper. A simple review of order relations of interval numbers, which will play an important role in interval reliability analysis, is given. The proposed quasi-consistency establishes the foundations for interval reliability analysis of a complex engineering structural system.