On Z‐valuations using Zadeh's Z‐numbers

We first recall the concept of Z‐numbers introduced by Zadeh. These objects consist of an ordered pair (A, B) of fuzzy numbers. We then use these Z‐numbers to provide information about an uncertain variable V in the form of a Z‐valuation, which expresses the knowledge that the probability that V is A is equal to B. We show that these Z‐valuations essentially induce a possibility distribution over probability distributions associated with V. We provide a simple illustration of a Z‐valuation. We show how we can use this representation to make decisions and answer questions. We show how to manipulate and combine multiple Z‐valuations. We show the relationship between Z‐numbers and linguistic summaries. Finally, we provide for a representation of Z‐valuations in terms of Dempster–Shafer belief structures, which makes use of type‐2 fuzzy sets. © 2012 Wiley Periodicals, Inc.

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