Generalized and extended fuzzy sets with applications.

*(English)*Zbl 0649.94028In the paper different generalizations of fuzzy sets are proposed and studied. Fuzzy set \(\bar x\) defined in a finite space is described in terms of ordered pairs \(\bar x=(\#/x)\) where # stands for a number between 0 and 1 while x denotes a certain element of the universe of discourse. Two generalizations of fuzzy sets refer to (i) membership fuzzification (g-fuzzification), \(({\#}\bar{\;}/x),\) or equivalently ((#/#)/x) which lead to type-2 sets, and (ii) support fuzzification (s-fuzzification) \(({\#}/\bar x)\) (i.e. \(({\#}/({\#}/x))\) generating level-2 fuzzy sets. Next classes of generalized fuzzy sets are generated in a recursive manner which essence is to substitute any number # by \(\overline{\#}\) and/or to replace x by the fuzzy set \(\bar x\).

A reduction problem of generalized fuzzy sets is formulated and put into consideration. Three applicational examples dealing with decision-making, hierarchical analysis, expert systems within which generalized fuzzy sets arise are also provided.

A reduction problem of generalized fuzzy sets is formulated and put into consideration. Three applicational examples dealing with decision-making, hierarchical analysis, expert systems within which generalized fuzzy sets arise are also provided.

Reviewer: W.Pedrycz

##### MSC:

94D05 | Fuzzy sets and logic (in connection with information, communication, or circuits theory) |

03E72 | Theory of fuzzy sets, etc. |

##### Keywords:

mixed fuzzy sets; generalizations of fuzzy sets; decision-making; hierarchical analysis; expert systems
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\textit{J. J. Buckley}, Fuzzy Sets Syst. 25, No. 2, 159--174 (1988; Zbl 0649.94028)

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