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 -cuts of fuzzy sets. The two definitions lead to equivalent results.
Definitions
Definition 1
Let be the set of fuzzy sets with domain of discourse, a type-1 OWA operator is defined as follows:[2]
Given n linguistic weights in the form of fuzzy sets defined on the domain of discourse , a type-1 OWA operator is a mapping, ,
such that
where , and is a permutation function such that , i.e., is the th highest element in the set .
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:
where , and is a permutation function such that , i.e., is the th largest element in the set .
Representation theorem of Type-1 OWA operators
Given the n linguistic weights in the form of fuzzy sets defined on the domain of discourse , and the fuzzy sets , then we have that[2]
where is the aggregation result obtained by Definition 1, and is the result obtained by in Definition 2.
Programming problems for Type-1 OWA operators
According to the Representation Theorem of Type-1 OWA Operators, a general type-1 OWA operator can be decomposed into a series of -level type-1 OWA operators. In practice, this series of -level type-1 OWA operators is used to construct the resulting aggregation fuzzy set. So we only need to compute the left end-points and right end-points of the intervals . Then, the resulting aggregation fuzzy set is constructed with the
membership function
as follows:
For the left end-points, we need to solve the following programming problem:
while for the right end-points, we need to solve the following programming problem:
A fast method has been presented to solve two programming problem so that the type-1 OWA aggregation operation can be performed efficiently, for details, please see the paper.[2]
Step 1—To set up the - level resolution in [0, 1].
Step 2—For each ,
Step 2.1—To calculate
Let ;
If , stop, is the solution; otherwise go to Step 2.1-3.
, go to Step 2.1-2.
Step 2.2 To calculate
Let ;
If , stop, is the solution; otherwise go to Step 2.2-3.
, go to step Step 2.2-2.
Step 3—To construct the aggregation resulting fuzzy set based on all the available intervals :
Some Examples
The type-1 OWA operator with the weights shown in the top figure is used to aggregate the fuzzy sets (solide lines) in the bottom figure, and the dashed line is the aggregation result.
Special cases
Any OWA operators, like maximum, minimum, mean operators;[3]
Join operators of (type-1) fuzzy sets,[4] i.e., fuzzy maximum operators;
Meet operators of (type-1) fuzzy sets,[4][5] i.e., fuzzy minimum operators;