The Yager's OWA operators are used to aggregate the crisp values in decision makingschemes. 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 -cuts of fuzzy sets. The two definitions lead to equivalent results.
Let be the set of fuzzy sets with domain of discourse, a type-1 OWA operator is defined as follows: 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.
Definition 2
Using the alpha-cuts of fuzzy sets: 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.
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 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 tocompute 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.
Alpha-level approach to Type-1 OWA operation
Three-step process:
Step 1—To set up the - level resolution in .
Step 2—For each,
Let ;
If, stop, is the solution; otherwise go to Step 2.1-3.
, go to Step 2.1-2.
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 :
Special cases
Any OWA operators, like maximum, minimum, mean operators;
Join operators of fuzzy sets, i.e., fuzzy maximum operators;
Meet operators of fuzzy sets, i.e., fuzzy minimum operators;
Join-like operators of fuzzy sets;
Meet-like operators of fuzzy sets.
Generalizations
Type-2 OWA operators have been suggested to aggregate the type-2 fuzzy sets for soft decision making.
