Type-1 OWA operators

The Yager's OWA (ordered weighted averaging) operators are used to aggregate the crisp values in decision making schemes (such as multi-criteria decision making, multi-expert decision making and multi-criteria/multi-expert decision making). It is widely accepted that Fuzzy sets are more suitable for representing preferences of criteria in decision making.
The type-1 OWA operators have been proposed for this purpose. The type-1 OWA operators provides a 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 formula_1-cuts of fuzzy sets. The two definitions lead to equivalent results.
Definitions.
Definition 1.
Let formula_2 be the set of fuzzy sets with domain of discourse formula_3, a type-1 OWA operator is defined as follows:
Given n linguistic weights formula_4 in the form of fuzzy sets defined on the domain of discourse formula_5, a type-1 OWA operator is a mapping, formula_6,
such that
where formula_10,and formula_11 is a permutation function such that formula_12, i.e., formula_13 is the formula_14th highest element in the set formula_15.
Definition 2.
Using the alpha-cuts of fuzzy sets:
Given the n linguistic weights formula_16 in the form of fuzzy sets defined on the domain of discourse formula_17, then for each formula_18, an formula_19-level type-1 OWA operator with formula_19-level sets formula_21 to aggregate the formula_19-cuts of fuzzy sets formula_23 is:
where formula_25, and formula_26 is a permutation function such that formula_27, i.e., formula_28 is the formula_14th largest
element in the set formula_15.
Representation theorem of Type-1 OWA operators.
Given the "n" linguistic weights formula_16 in the form of fuzzy sets defined on the domain of discourse formula_17, and the fuzzy sets formula_33, then we have that
where formula_35 is the aggregation result obtained by Definition 1, and formula_36 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 formula_1-level type-1 OWA operators. In practice, this series of formula_1-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 formula_39. 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.
Alpha-level approach to Type-1 OWA operation.
Three-step process:
Generalizations.
Type-2 OWA operators have been suggested to aggregate the type-2 fuzzy sets for soft decision making.