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.


Fuzzy Number Exponential Distribution Fuzzy Subset Possibility Distribution Ordered Weight Average 
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  1. 1.
    Zadeh, L.A.: A note on Z-numbers. Information Science 181, 2923–2932 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Yager, R.R.: A new approach to the summarization of data. Information Sciences 28, 69–86 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Yager, R.R.: On linguistic summaries of data. In: Piatetsky-Shapiro, G., Frawley, B. (eds.) Knowledge Discovery in Databases, pp. 347–363. MIT Press, Cambridge (1991)Google Scholar
  4. 4.
    Zadeh, L.A.: Probability measures of fuzzy events. Journal of Mathematical Analysis and Applications 10, 421–427 (1968)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ross, S.M.: Introduction to Probability and Statistics for Engineers and Scientists. Harcourt, San Diego (2000)zbMATHGoogle Scholar
  6. 6.
    Zadeh, L.A.: Computing with Words-Principal Concepts and Idea. Springer, Berlin (to appear)Google Scholar
  7. 7.
    Yager, R.R.: On ordered weighted averaging aggregation operators in multi-criteria decision making. IEEE Transactions on Systems, Man and Cybernetics 18, 183–190 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Yager, R.R., Kacprzyk, J., Beliakov, G.: Recent Developments in the Ordered Weighted Averaging Operators: Theory and Practice. Springer, Berlin (2011)CrossRefGoogle Scholar
  9. 9.
    Papoulis, A.: Probability, Random Variables and Stochastic Processes. McGraw-Hill, New York (1965)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Ronald R. Yager
    • 1
  1. 1.Machine Intelligence InstituteIona College New RochelleUSA

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