“Type-1 OWA operators”的意思、由来-开放百科全书

The Yager's OWA (ordered weighted averaging) operators{{R|yagerOWA}} 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).{{R|Yager|YagerBeliakov}} It is widely accepted that Fuzzy sets{{R|Zadeh}} are more suitable for representing preferences of criteria in decision making.

The type-1 OWA operators{{R|fssT1OWA|kdeT1OWA}} 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.

Definitions

Definition 1

Let

be the set of fuzzy sets with domain of discourse

, a type-1 OWA operator is defined as follows:{{R|kdeT1OWA}}

Given n linguistic weights

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

, a type-1 OWA operator is a mapping,

where

,and

is a permutation function such that

, i.e.,

is the

th highest element in the set

Definition 2

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:

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{{R|kdeT1OWA}}

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:

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.{{R|kdeT1OWA}}

Alpha-level approach to Type-1 OWA operation

Special cases

Generalizations

Type-2 OWA operators{{R|Zhou}} have been suggested to aggregate the type-2 fuzzy sets for soft decision making.

References

1. ^{{cite book|last=Yager|first=R. R. and Kacprzyk, J|title=The Ordered Weighted Averaging Operators: Theory and Applications|year=1997|publisher=Kluwer: Norwell, MA}}
2. ^{{cite book|last=Yager|first=R.R, Kacprzyk, J. and Beliakov, G|title=Recent Developments in the Ordered Weighted Averaging Operators-Theory and Practice|year=2011|publisher=Springer}}
3. ^{{cite journal|last=Zadeh|first=L.A|title=Fuzzy sets|journal=Information and Control |year=1965|volume=8 |issue=3|pages=338–353|doi=10.1016/S0019-9958(65)90241-X}}
4. ^{{cite journal|last=Zhou|first=S. M. |author2=F. Chiclana |author3=R. I. John |author4=J. M. Garibaldi|title=Alpha-level aggregation: a practical approach to type-1 OWA operation for aggregating uncertain information with applications to breast cancer treatments|journal=IEEE Transactions on Knowledge and Data Engineering|year=2011|volume=23|issue=10|pages=1455–1468|doi=10.1109/TKDE.2010.191}}
5. ^{{cite journal|last=Zhou|first=S. M. |author2=F. Chiclana |author3=R. I. John |author4=J. M. Garibaldi|title=Type-1 OWA operators for aggregating uncertain information with uncertain weights induced by type-2 linguistic quantifiers|journal=Fuzzy Sets and Systems|year=2008|volume=159|issue=24|pages=3281–3296|doi=10.1016/j.fss.2008.06.018}}
6. ^{{cite journal|last=Mizumoto|first=M.|author2=K. Tanaka |title=Some Properties of fuzzy sets of type 2|journal=Information and Control|year=1976|volume=31|issue=4|pages=312–40|doi=10.1016/s0019-9958(76)80011-3}}
7. ^{{cite journal|last=Zadeh|first=L. A.|title=The concept of a linguistic variable and its application to approximate reasoning-1|journal=Information Sciences|year=1975|volume=8|issue=3|pages=199–249|doi=10.1016/0020-0255(75)90036-5}}
8. ^{{cite book|last=Zhou|first=S. M.|author2=F. Chiclana |author3=R. I. John |author4=J. M. Garibaldi |title=Fuzzificcation of the OWA Operators in Aggregating Uncertain Information|journal=R. R. Yager, J. Kacprzyk and G. Beliakov (ed): Recent Developments in the Ordered Weighted Averaging Operators-Theory and Practice|year=2011|volume=Springer|pages=91–109|doi=10.1007/978-3-642-17910-5_5|series=Studies in Fuzziness and Soft Computing|isbn=978-3-642-17909-9}}
9. ^{{cite journal|last=Zhou|first=S.M. |author2=R. I. John |author3=F. Chiclana |author4=J. M. Garibaldi|title=On aggregating uncertain information by type-2 OWA operators for soft decision making|journal=International Journal of Intelligent Systems|year=2010|volume=25|issue=6|pages=540–558|doi=10.1002/int.20420}}