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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Type-1 OWA operators ： ウィキペディア英語版 Type-1 OWA operators The Yager's OWA (ordered weighted averaging) operators 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). It is widely accepted that fuzzy sets 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? The type-1 OWA operators 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. First, there are two definitions for type-1 OWA operators, one is based on Zadeh's Extension Principle, the other is based on $\backslash alpha$-cuts of fuzzy sets. The two definitions lead to equivalent results. ==Definitions== Definition 1.〔 Let $F(X)$ be the set of fuzzy sets with domain of discourse $X$, a type-1 OWA operator is defined as follows: Given n linguistic weights $\backslash left\backslash \_^n$ in the form of fuzzy sets defined on the domain of discourse $U\; =\; ()$, a type-1 OWA operator is a mapping, $\backslash Phi$, :$\backslash Phi\; \backslash colon\; F(X)\backslash times\; \backslash cdots\; \backslash times\; F(X)\; \backslash longrightarrow\; F(X)$ :$(A^1\; ,\; \backslash cdots\; ,A^n)\; \backslash mapsto\; Y$ such that :$\backslash mu\; \_\; (y)\; =\backslash displaystyle\; \backslash sup\_\_i\; a\_\; =\; y\; \}\backslash left(l\}\backslash mu\; \_\; (w\_1\; )\backslash wedge\; \backslash cdots\; \backslash wedge\; \backslash mu\_\; (w\_n\; )\backslash wedge\; \backslash mu\; \_\; (a\_1\; )\backslash wedge\; \backslash cdots\; \backslash wedge\; \backslash mu\; \_\; (a\_n\; )\backslash end\}\backslash right)$ where $\backslash bar\; \_i\; =\; \backslash frac\; \}$,and $\backslash sigma\; \backslash colon\; \backslash \; \backslash longrightarrow\; \backslash$ is a permutation function such that $a\_\; \backslash geq\; a\_,\backslash \; \backslash forall\; i\; =\; 1,\; \backslash cdots\; ,n\; -\; 1$, i.e., $a\_$ is the $i$th highest element in the set $\backslash left\backslash$. Definition 2.〔 The definition below is based on the alpha-cuts of fuzzy sets: Given the n linguistic weights $\backslash left\backslash \_^n$ in the form of fuzzy sets defined on the domain of discourse $U\; =\; ()$, then for each $\backslash alpha\; \backslash in\; ()$, an $\backslash alpha$-level type-1 OWA operator with $\backslash alpha$-level sets $\backslash left\backslash \_^n$ to aggregate the $\backslash alpha$-cuts of fuzzy sets $\backslash left\backslash \_^n$ is given as : $$ \Phi_\alpha \left( \right) =\left\ } } }\left| \right. \in A_\alpha ^i ,\;i = 1, \ldots ,n} \right\} where $W\_\backslash alpha\; ^i=\; \backslash ,\; A\_\backslash alpha\; ^i=\backslash$, and $\backslash sigma\; :\backslash \; \backslash to\; \backslash$ is a permutation function such that $a\_\; \backslash ge\; a\_\; ,\backslash ;\backslash forall\; \backslash ;i\; =\; 1,\; \backslash cdots\; ,n\; -\; 1$, i.e., $a\_$ is the $i$th largest element in the set $\backslash left\backslash$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Type-1 OWA operators」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.