System reliability, as a quality index, is the capability to complete the specified functionsaccurately in mutually harmonious manner under the specified conditions within specified time period. The vague feature is intrinsic and inherent to the system reliability and inevitably engaging fuzzy mathematics. Fuzzy mathematics initiated by Zadeh (1965) facilitated a foundation dealing with vague phenomena in reliability modeling. However, the fuzzy mathematical foundation initiated by Zadeh (1965, 1978) is membership function and possibility measure based and widely used. However, possibility measure, which was originally expected to play the role of probability measure in probability theory but did not because possibility measure does not possess self-duality property as that in probability theory. To resolve this dilemma, Liu (2004) proposed an axiomatic foundation for modeling fuzzy phenomena, named as credibility theory. The credibility measure possesses self-duality property and is able to play the role of that in probability theory. In this paper, we will explore the concept of credibility measure, its axiomatic foundation, concept of fuzzy variable and its credibility distribution in the sense based on the credibility theoretical foundation. Furthermore, we propose the concept of credibility copula for characterizing the fuzzy dependence between fuzzy variables. Finally, we explore the credibility reliability evaluation based on the fuzzy load-strength concept.
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