Ordered Weighted Averaging (OWA) Aggregation Operators
Encyclopedia
Introduced by Ronald R. Yager
, the Ordered Weighted Averaging operators, commonly called OWA operators, provide a parameter
ized class of mean type aggregation operators. 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.
operator of dimension 
is a mapping 
that has an associated collection of weights 
lying in the unit interval and summing to one and with
where 
is the jth largest of the 
By choosing different W we can implement different aggregation operators. The OWA operator is a non-linear operator as a result of the process of determining the bj.
, monotonic, symmetric, and idempotent, as defined below.
if 
and 
for 
if 
and 
for 
This is defined as
It is known that
.
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
An alternative definition is
The dispersion characterizes how uniformly the arguments are being used
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.
The type-1 OWA operator is defined according to the alpha-cuts of fuzzy sets as follows:
Given the n linguistic weights
in the form of fuzzy sets defined on the domain of discourse 
, then for each 
, an 
-level type-1 OWA operator with 
-level sets 
to aggregate the 
-cuts of fuzzy sets 
is given as
where
, and 
is a permutation function such that 
, i.e., 
is the 
th largest
element in the set
.
The computation of the type-1 OWA output is implemented by computing the left end-points and right end-points of the intervals
:
and 
where
. Then membership function of resulting aggregation fuzzy set is:
For the left end-points, we we need to solve the following programming problem:
while for the right end-points, we need to solve the following programming problem:
The paper has presented a fast method to solve two programming problem so that the type-1 OWA aggregation operation can be performed efficiently.
Ronald R. Yager
Ronald Robert Yager is an American researcher in computational intelligence, decision making under uncertainty and fuzzy logic. He is currently Director of the Machine Intelligence Institute and Professor of Information Systems at Iona College.- Biography :Ronald R...
, the Ordered Weighted Averaging operators, commonly called OWA operators, provide a parameter
Parameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....
ized class of mean type aggregation operators. 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
Computational intelligence
Computational intelligence is a set of Nature-inspired computational methodologies and approaches to address complex problems of the real world applications to which traditional methodologies and approaches are ineffective or infeasible. It primarily includes Fuzzy logic systems, Neural Networks...
because of their ability to model linguistically expressed aggregation instructions.
Definition
Formally an operator of dimension is a mapping that has an associated collection of weights lying in the unit interval and summing to one and with where is the jth largest of the By choosing different W we can implement different aggregation operators. The OWA operator is a non-linear operator as a result of the process of determining the bj.
Properties
The OWA operator is a mean operator. It is boundedBounded operator
In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all non-zero vectors v in X...
, monotonic, symmetric, and idempotent, as defined below.
| Bounded Bounded operator In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all non-zero vectors v in X... |
 |
| Monotonic |  if  for  |
| Symmetric |  if  is a permutation map |
| Idempotent |  if all  |
Notable OWA Operators
if and for if and for Characterizing Features
Two features have been used to characterize the OWA operators. The first is the attudinal character(orness).This is defined as
It is known that
.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
An alternative definition is
The dispersion characterizes how uniformly the arguments are being used
Type-1 OWA Aggregation Operators
The above Yager's OWA operators are used to aggregate the crisp values. Can we aggregate fuzzy sets in the OWA mechanism ? TheType-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.
The type-1 OWA operator is defined according to the alpha-cuts of fuzzy sets as follows:
Given the n linguistic weights
in the form of fuzzy sets defined on the domain of discourse , then for each , an -level type-1 OWA operator with -level sets to aggregate the -cuts of fuzzy sets is given aswhere
, and is a permutation function such that , i.e., is the th largestelement in the set
.The computation of the type-1 OWA output is implemented by computing the left end-points and right end-points of the intervals
: and where
. Then membership function of resulting aggregation fuzzy set is:For the left end-points, we we need to solve the following programming problem:
while for the right end-points, we need to solve the following programming problem:
The paper has presented a fast method to solve two programming problem so that the type-1 OWA aggregation operation can be performed efficiently.