Type-1 OWA operators are a set of aggregation operators that generalise the Yager's OWA (ordered weighted averaging) operators in the interest of aggregating fuzzy sets rather than crisp values in soft decision making and data mining. These operators provide a mathematical 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. The two definitions for type-1 OWA operators are based on Zadeh's Extension Principle and α {\displaystyle \alpha } -cuts of fuzzy sets. The two definitions lead to equivalent results.
Definitions
Definition 1 Let F ( X ) {\displaystyle F(X)} be the set of fuzzy sets with domain of discourse X {\displaystyle X} , a type-1 OWA operator is defined as follows: Given n linguistic weights { W i } i = 1 n {\displaystyle \left\{{W^{i}}\right\}_{i=1}^{n}} in the form of fuzzy sets defined on the domain of discourse U = [ 0 , 1 ] {\displaystyle U=[0,1]} , a type-1 OWA operator is a mapping, Φ {\displaystyle \Phi } ,
Φ : F ( X ) × ⋯ × F ( X ) ⟶ F ( X ) {\displaystyle \Phi \colon F(X)\times \cdots \times F(X)\longrightarrow F(X)}
( A 1 , ⋯ , A n ) ↦ Y {\displaystyle (A^{1},\cdots ,A^{n})\mapsto Y}
such that
μ Y ( y ) = sup ∑ k = 1 n w ¯ i a σ ( i ) = y ( μ W 1 ( w 1 ) ∧ ⋯ ∧ μ W n ( w n ) ∧ μ A 1 ( a 1 ) ∧ ⋯ ∧ μ A n ( a n ) ) {\displaystyle \mu _{Y}(y)=\displaystyle \sup _{\displaystyle \sum _{k=1}^{n}{\bar {w}}_{i}a_{\sigma (i)}=y}\left({\begin{array}{*{1}l}\mu _{W^{1}}(w_{1})\wedge \cdots \wedge \mu _{W^{n}}(w_{n})\wedge \mu _{A^{1}}(a_{1})\wedge \cdots \wedge \mu _{A^{n}}(a_{n})\end{array}}\right)}
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