Abstract
The concept of fuzzy set was initialized by Zadeh via membership function in 1965. In order to measure the chance of a fuzzy event occurs, Zadeh proposed the concepts of possibility measure and necessity measure. It is proved that both possibility measure and necessity measure satisfy the properties of normality, nonnegativity and monotonicity. However, neither of them is self-dual. Since the duality is intuitive and important in both theory and practice, Liu and Liu defined a credibility measure as the average value of possibility measure and necessity measure, which was redefined by Li and Liu as a set function satisfying the normality, monotonicity, duality, and maximality axioms. Nowadays, Credibility theory has become a branch of axiomatic mathematics for modeling fuzziness. This chapter mainly introduces some basic concepts and important theorems including credibility measure, fuzzy variable, credibility function, independence, identical distribution, credibility subadditivity theorem, credibility semicontinuous theorem, credibility extension theorem, product credibility theorem, credibility inversion theorem, Zadeh extension theorem, and so on.
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Li, X. (2013). Credibility Theory. In: Credibilistic Programming. Uncertainty and Operations Research. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36376-4_1
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DOI: https://doi.org/10.1007/978-3-642-36376-4_1
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