In applied mathematics, specifically in fuzzy logic, the ordered weighted averaging (OWA) operators provide a parameterized class of mean type aggregation operators. They were introduced by Ronald R. Yager. Many notable mean operators such as the max, arithmetic average, median and min, are members of this class. They have been widely used in computational intelligence because of their ability to model linguistically expressed aggregation instructions.
Definition An OWA operator of dimension n {\displaystyle \ n} is a mapping F : R n → R {\displaystyle F:\mathbb {R} ^{n}\rightarrow \mathbb {R} } that has an associated collection of weights W = [ w 1 , … , w n ] {\displaystyle \ W=[w_{1},\ldots ,w_{n}]} lying in the unit interval and summing to one and with
F ( a 1 , … , a n ) = ∑ j = 1 n w j b j {\displaystyle F(a_{1},\ldots ,a_{n})=\sum _{j=1}^{n}w_{j}b_{j}}
where b j {\displaystyle b_{j}} is the jth largest of the a i {\displaystyle a_{i}} . By choosing different W one can implement different aggregation operators. The OWA operator is a non-linear operator as a result of the process of determining the bj.
Notable OWA operators
F ( a 1 , … , a n ) = max ( a 1 , … , a n ) {\displaystyle \ F(a_{1},\ldots ,a_{n})=\max(a_{1},\ldots ,a_{n})} if w 1 = 1 {\displaystyle \ w_{1}=1} and w j = 0 {\displaystyle \ w_{j}=0} for j ≠ 1 {\displaystyle j\neq 1}
F ( a 1 , … , a n ) = min ( a 1 , … , a n ) {\displaystyle \ F(a_{1},\ldots ,a_{n})=\min(a_{1},\ldots ,a_{n})} if w n = 1 {\displaystyle \ w_{n}=1} and w j = 0 {\displaystyle \ w_{j}=0} for j ≠ n {\displaystyle j\neq n}
F ( a 1 , … , a n ) = a v e r a g e ( a 1 , … , a n ) {\displaystyle \ F(a_{1},\ldots ,a_{n})=\mathrm {average} (a_{1},\ldots ,a_{n})} if w j = 1 n {\displaystyle \ w_{j}={\frac {1}{n}}} for all j ∈ [ 1 , n ] {\displaystyle j\in [1,n]}
Properties The OWA operator is a mean operator. It is bounded, monotonic, symmetric, and idempotent, as defined below.
Characterizing features Two features have been used to characterize the OWA operators. The first is the attitudinal character, also called orness. This is defined as
A − C ( W ) = 1 n − 1 ∑ j = 1 n ( n − j ) w j . {\displaystyle A-C(W)={\frac {1}{n-1}}\sum _{j=1}^{n}(n-j)w_{j}.}
It is known that A − C ( W ) ∈ [ 0 , 1 ] {\displaystyle A-C(W)\in [0,1]} . In addition A − C(max) = 1, A − C(ave) = A − C(med) = 0.5 and A − C(min) = 0. Thus the A − C goes from 1 to 0 as we go from Max to Min aggregation. The attitudinal character characterizes the similarity of aggregation to OR operation(OR is defined as the Max). The second feature is the dispersion. This defined as
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