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.
Contents
- Definition 1
- Definition 2
- Representation theorem of Type 1 OWA operators
- Programming problems for Type 1 OWA operators
- Alpha level approach to Type 1 OWA operation
- Special cases
- Generalizations
- References
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
Definition 1
Let
Given n linguistic weights
such that
where
Definition 2
Using the alpha-cuts of fuzzy sets:
Given the n linguistic weights
where
Representation theorem of Type-1 OWA operators
Given the n linguistic weights
where
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
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:
- Let
i 0 = 1 ; - If
ρ α + i 0 ≥ A α + σ ( i 0 ) ρ α + i 0 -
i 0 ← i 0 + 1 , go to Step 2.1-2.
- Let
i 0 = 1 ; - If
ρ α − i 0 ≥ A α − σ ( i 0 ) ρ α − i 0 -
i 0 ← i 0 + 1 , go to step Step 2.2-2.
Special cases
Generalizations
Type-2 OWA operators have been suggested to aggregate the type-2 fuzzy sets for soft decision making.