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| The [[Ordered weighted averaging aggregation operator|Yager's OWA (ordered weighted averaging) operators]]<ref name="yagerOWA">{{cite journal|last=Yager|first=R.R|title=On ordered weighted averaging aggregation operators in multi-criteria decision making|journal=IEEE Transactions on Systems, Man and Cybernetics|year=1988|volume=18|pages=183–190}}</ref> have been widely used to aggregate the crisp values in decision making schemes (such as multi-criteria decision making, multi-expert decisin making, multi-criteria multi-expert decision making).<ref>{{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}}</ref><ref>{{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}}</ref> It is widely accepted that fuzzy sets<ref>{{cite journal|last=Zadeh|first=L.A|title=Fuzzy sets|journal=Information and Control |year=1965|volume=8 |pages=338–353|doi=10.1016/S0019-9958(65)90241-X}}</ref> are more suitable for representing preferences of criteria in decision making. But fuzzy sets are not crisp values, how can we aggregate fuzzy sets in OWA mechanism?
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| The type-1 OWA operators<ref name="fssT1OWA">{{cite journal|last=Zhou|first=S. M.|coauthors=F. Chiclana, R. I. John and 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}}</ref><ref name="kdeT1OWA">{{cite journal|last=Zhou|first=S. M.|coauthors=F. Chiclana, R. I. John and 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}}</ref> have been proposed for this purpose. So the type-1 OWA operators provides us with a new 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.
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| First, there are two definitions for type-1 OWA operators, one is based on Zadeh's Extension Principle, the other is based on <math>\alpha</math>-cuts of fuzzy sets. The two definitions lead to equivalent results.
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| ==Definitions==
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| '''Definition 1.<ref name="fssT1OWA" /> '''
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| Let <math>F(X)</math> be the set of fuzzy sets with domain of discourse <math>X</math>, a type-1 OWA operator is defined as follows:
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| Given n linguistic weights <math>\left\{ {W^i} \right\}_{i = 1}^n </math> in the form of fuzzy sets defined on the domain of discourse <math>U = [0,1]</math>, a type-1 OWA operator is a mapping, <math>\Phi</math>,
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| :<math>\Phi \colon F(X)\times \cdots \times F(X) \longrightarrow F(X)</math>
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| :<math>(A^1 , \cdots ,A^n) \mapsto Y</math>
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| such that
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| :<math>\mu _{Y} (y) =\displaystyle \sup_{\displaystyle \sum_{k =1}^n \bar {w}_i a_{\sigma (i)} = y }\left({\begin{array}{*{1}l}\mu _{W^1 } (w_1 )\wedge \cdots \wedge \mu_{W^n } (w_n )\wedge \mu _{A^1 } (a_1 )\wedge \cdots \wedge \mu _{A^n } (a_n )\end{array}}\right)</math>
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| where <math>\bar {w}_i = \frac{w_i }{\sum_{i = 1}^n {w_i } }</math>,and <math>\sigma \colon \{1, \cdots ,n\} \longrightarrow \{1, \cdots ,n\}</math> is a permutation function such that <math>a_{\sigma (i)} \geq a_{\sigma (i + 1)},\ \forall i = 1, \cdots ,n - 1</math>, i.e., <math>a_{\sigma(i)} </math> is the <math>i</math>th highest element in the set <math>\left\{ {a_1 , \cdots ,a_n } \right\}</math>.
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| '''Definition 2.<ref name="kdeT1OWA" /> '''
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| The definition below is based on the alpha-cuts of fuzzy sets:
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| Given the n linguistic weights <math>\left\{ {W^i} \right\}_{i =1}^n </math> in the form of fuzzy sets defined on the domain of discourse <math>U = [0,\;\;1]</math>, then for each <math>\alpha \in [0,\;1]</math>, an <math>\alpha </math>-level type-1 OWA operator with <math>\alpha </math>-level sets <math>\left\{ {W_\alpha ^i } \right\}_{i = 1}^n </math> to aggregate the <math>\alpha </math>-cuts of fuzzy sets <math>\left\{ {A^i} \right\}_{i =1}^n </math> is given as
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| : <math>
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| \Phi_\alpha \left( {A_\alpha ^1 , \ldots ,A_\alpha ^n } \right) =\left\{ {\frac{\sum\limits_{i = 1}^n {w_i a_{\sigma (i)} } }{\sum\limits_{i = 1}^n {w_i } }\left| {w_i \in W_\alpha ^i ,\;a_i } \right. \in A_\alpha ^i ,\;i = 1, \ldots ,n} \right\}</math>
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| where <math>W_\alpha ^i= \{w| \mu_{W_i }(w) \geq \alpha \}, A_\alpha ^i=\{ x| \mu _{A_i }(x)\geq \alpha \}</math>, and <math>\sigma :\{\;1, \cdots ,n\;\} \to \{\;1, \cdots ,n\;\}</math> is a permutation function such that <math>a_{\sigma (i)} \ge a_{\sigma (i + 1)} ,\;\forall \;i = 1, \cdots ,n - 1</math>, i.e., <math>a_{\sigma (i)} </math> is the <math>i</math>th largest
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| element in the set <math>\left\{ {a_1 , \cdots ,a_n } \right\}</math>.
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| == Representation theorem of Type-1 OWA operators<ref name="kdeT1OWA" />==
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| Given the ''n'' linguistic weights <math>\left\{ {W^i} \right\}_{i =1}^n </math> in the form of fuzzy sets defined on the domain of discourse <math>U = [0,\;\;1]</math>, and the fuzzy sets <math>A^1, \cdots ,A^n</math>, then we have that<ref name="kdeT1OWA" />
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| :<math>Y=G</math>
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| where <math>Y</math> is the aggregation result obtained by Definition 1, and <math>G</math> is the result obtained by in Definition 2.
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| ==Programming problems for Type-1 OWA operators==
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| According to the '''''Representation Theorem of Type-1 OWA Operators''''',a general type-1 OWA operator can be decomposed into a series of <math>\alpha</math>-level type-1 OWA operators. In practice, these series of <math>\alpha</math>-level type-1 OWA operators are 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 <math>\Phi _\alpha \left( {A_\alpha ^1 , \cdots ,A_\alpha ^n } \right)</math>. Then, the resulting aggregation fuzzy set is constructed with the membership function as follows:
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| :<math>\mu _{G} (x) = \mathop \vee \limits_{\alpha :x \in \Phi _\alpha \left( {A_\alpha ^1 , \cdots
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| ,A_\alpha ^n } \right)_\alpha } \alpha </math>
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| For the left end-points, we need to solve the following programming problem:
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| :<math> \Phi _\alpha \left( {A_\alpha ^1 , \cdots ,A_\alpha ^n } \right)_{-} = \mathop {\min }\limits_{\begin{array}{l} W_{\alpha - }^i \le w_i \le W_{\alpha + }^i A_{\alpha - }^i \le a_i \le A_{\alpha + }^i \end{array}} \sum\limits_{i = 1}^n {w_i a_{\sigma (i)} / \sum\limits_{i = 1}^n {w_i } } </math>
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| while for the right end-points, we need to solve the following programming problem:
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| :<math>\Phi _\alpha \left( {A_\alpha ^1 , \cdots , A_\alpha ^n } \right)_{+} = \mathop {\max }\limits_{\begin{array}{l} W_{\alpha - }^i \le w_i \le W_{\alpha + }^i A_{\alpha - }^i \le a_i \le A_{\alpha + }^i \end{array}} \sum\limits_{i = 1}^n {w_i a_{\sigma (i)} / \sum\limits_{i =
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| 1}^n {w_i } } </math>
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| 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.<ref name="kdeT1OWA" />
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| == Alpha-level approach to Type-1 OWA operation<ref name="kdeT1OWA" />==
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| * '''Step 1'''.To set up the <math>\alpha </math>- level resolution in [0, 1].
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| * '''Step 2'''. For each <math>\alpha \in [0,1]</math>,
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| ''Step 2.1.'' To calculate <math>\rho _{\alpha +} ^{i_0^\ast } </math>
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| # Let <math>i_0 = 1</math>;
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| # If <math>\rho _{\alpha +} ^{i_0 } \ge A_{\alpha + }^{\sigma (i_0 )} </math>, stop, <math>\rho _{\alpha +} ^{i_0 } </math> is the solution; otherwise go to ''Step 2.1-3''.
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| # <math>i_0 \leftarrow i_0 + 1</math>, go to ''Step 2.1-2''.
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| ''Step 2.2.'' To calculate<math>\rho _{\alpha -} ^{i_0^\ast } </math>
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| # Let <math>i_0 = 1</math>;
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| # If <math>\rho _{\alpha -} ^{i_0 } \ge A_{\alpha - }^{\sigma (i_0 )} </math>, stop, <math>\rho _{\alpha -} ^{i_0 } </math> is the solution; otherwise go to ''Step 2.2-3.''
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| #<math>i_0 \leftarrow i_0 + 1</math>, go to step ''Step 2.2-2.''
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| '''Step 3.'''To construct the aggregation resulting fuzzy set <math>G</math> based on all the available intervals <math>\left[ {\rho _{\alpha -} ^{i_0^\ast } ,\;\rho _{\alpha +} ^{i_0^\ast } } \right]</math>:
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| :<math>\mu _{G} (x) = \mathop \vee \limits_{\alpha :x \in \left[ {\rho _{\alpha -} ^{i_0^\ast } ,\;\rho _{\alpha +} ^{i_0^\ast } } \right]} \alpha </math> | |
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| ==Special cases of Type-1 OWA operators==
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| * Any OWA operators, like maximum, minimum, mean operators;<ref name="yagerOWA" />
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| * Join operators of (type-1) fuzzy sets,<ref name="MT">{{cite journal|last=Mizumoto|first=M.|coauthors=K. Tanaka|title=Some Properties of fuzzy sets of type 2|journal=Information and Control|year=1976|volume=31|pages=312–40}}</ref><ref name="zadehJ">{{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|pages=199–249}}</ref> i.e., fuzzy maximum operators;
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| * Meet operators of (type-1) fuzzy sets,<ref name="MT"/><ref name="zadehJ"/> i.e., fuzzy minimum operators;
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| * Join-like operators of (type-1) fuzzy sets;<ref name="kdeT1OWA"/><ref name="bookT1OWA">{{cite journal|last=Zhou|first=S. M.|coauthors=F. Chiclana, R. I. John, and 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}}</ref>
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| * Meet-like operators of (type-1) fuzzy sets.<ref name="kdeT1OWA"/><ref name="bookT1OWA"/>
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| ==Generalizations==
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| Type-2 OWA operators<ref>{{cite journal|last=Zhou|first=S.M.|coauthors=R. I. John, F. Chiclana and 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}}</ref> have been suggested to aggregate the [[Type-2 fuzzy sets and systems|type-2 fuzzy sets]] for soft decision making.
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| == References ==
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| {{reflist}}
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| [[Category:Artificial intelligence]]
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| [[Category:Logic in computer science]]
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| [[Category:Fuzzy logic]]
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| [[Category:Information retrieval]]
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