Journal
IEEE TRANSACTIONS ON FUZZY SYSTEMS
Volume 21, Issue 5, Pages 937-949Publisher
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TFUZZ.2012.2236843
Keywords
Fuzziness; fuzzy set; gradual number; shadowed set
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The existing methods of determining an alpha-cut of a fuzzy set to construct its underlying shadowed set do not fully comply with the concept of shadowed sets, namely, a retention of the total amount of fuzziness and its localized redistribution throughout a universe of discourse. Moreover, no closed formula to calculate the corresponding alpha-cut is available. This paper proposes analytical formulas to calculate threshold values required in the construction of shadowed sets. We introduce a new algorithm to design a shadowed set from a given fuzzy set. The proposed algorithm, which adheres to the main premise of shadowed sets of capturing the essence of fuzzy sets, helps localize fuzziness present in a given fuzzy set. We represent the fuzziness of a fuzzy set as a gradual number. Through defuzzification of the gradual number of fuzziness, we determine the required threshold (i.e., some alpha-cut) used in the formation of the shadowed set. We show that the shadowed set obtained in this way comes with a measure of fuzziness that is equal to the one characterizing the original fuzzy set.
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