Type-1 OWA operators

Type-1 OWA operators[1][2] are a set of aggregation operators that generalise the Yager's OWA (ordered weighted averaging) operators)[3] 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.

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,

is a permutation function such that

th highest element in the set

Using the alpha-cuts of fuzzy sets:[2] Given the n linguistic weights

in the form of fuzzy sets defined on the domain of discourse

-level type-1 OWA operator with

-cuts of fuzzy sets

is a permutation function such that

th largest element in the set

Given the n linguistic weights

in the form of fuzzy sets defined on the domain of discourse

, and the fuzzy sets

is the aggregation result obtained by Definition 1, and

is the result obtained by in Definition 2.

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] Three-step process:[2] Type-2 OWA operators[7] have been suggested to aggregate the type-2 fuzzy sets for soft decision making.

Type-1 OWA operators have been applied to different domains for soft decision making.