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Type-1 OWA operators : ウィキペディア英語版
Type-1 OWA operators

The Yager's OWA (ordered weighted averaging) operators have been widely used to aggregate the crisp values in decision making schemes (such as multi-criteria decision making, multi-expert decisin making, multi-criteria multi-expert decision making). It is widely accepted that fuzzy sets are more suitable for representing preferences of criteria in decision making. But fuzzy sets are not crisp values, how can we aggregate fuzzy sets in OWA mechanism?
The type-1 OWA operators have been proposed for this purpose. So the type-1 OWA operators provides us with a new technique for directly aggregating uncertain information with uncertain weights via OWA mechanism in soft decision making and data mining, where these uncertain objects are modelled by fuzzy sets.
First, there are two definitions for type-1 OWA operators, one is based on Zadeh's Extension Principle, the other is based on \alpha-cuts of fuzzy sets. The two definitions lead to equivalent results.
==Definitions==

Definition 1.〔
Let F(X) be the set of fuzzy sets with domain of discourse X, a type-1 OWA operator is defined as follows:
Given n linguistic weights \left\_^n in the form of fuzzy sets defined on the domain of discourse U = (), a type-1 OWA operator is a mapping, \Phi,
:\Phi \colon F(X)\times \cdots \times F(X) \longrightarrow F(X)
:(A^1 , \cdots ,A^n) \mapsto Y
such that
:\mu _ (y) =\displaystyle \sup__i a_ = y }\left(l}\mu _ (w_1 )\wedge \cdots \wedge \mu_ (w_n )\wedge \mu _ (a_1 )\wedge \cdots \wedge \mu _ (a_n )\end}\right)
where \bar _i = \frac },and \sigma \colon \ \longrightarrow \ is a permutation function such that a_ \geq a_,\ \forall i = 1, \cdots ,n - 1, i.e., a_ is the ith highest element in the set \left\.
Definition 2.〔
The definition below is based on the alpha-cuts of fuzzy sets:
Given the n linguistic weights \left\_^n in the form of fuzzy sets defined on the domain of discourse U = (), then for each \alpha \in (), an \alpha -level type-1 OWA operator with \alpha -level sets \left\_^n to aggregate the \alpha -cuts of fuzzy sets \left\_^n is given as
:
\Phi_\alpha \left( \right) =\left\ } } }\left| \right. \in A_\alpha ^i ,\;i = 1, \ldots ,n} \right\}
where W_\alpha ^i= \, A_\alpha ^i=\, and \sigma :\ \to \ is a permutation function such that a_ \ge a_ ,\;\forall \;i = 1, \cdots ,n - 1, i.e., a_ is the ith largest
element in the set \left\.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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