三维模糊集
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摘要
本文引入了一种特殊的L-模糊集:三维模糊集,它是Zadeh模糊集、区间值模糊集和直觉模糊集的推广.本文研究了三维模糊集的基础理论,如三维模糊集的截集、分解定理、表现定理和扩展原理;建立了三维模糊集的范畴;研究了模糊点与三维模糊集的邻属关系;讨论了基于三维模糊集的模糊决策问题.具体研究工作如下:
1.第二章研究了三维模糊集的基础理论.首先,给出三维模糊集的四类(共8种)截集的定义.我们将三维模糊集的截集定义为4-值模糊集,指出如此定义的截集与Zadeh模糊集的截集有完全一样的性质.其次,基于三维模糊集的截集和4-值集合套的概念,我们建立了三维模糊集的8种分解定理和8种表现定理,从而建立了三维模糊集与4-值模糊集的联系.最后,研究了三维模糊集的范畴.我们首先建立了4-值模糊集的范畴QFuz、4-值反序集合套的范畴QNS和三维模糊集的范畴TFuz,然后证明了范畴QFuz和范畴TFuz为弱topos,范畴QNS为topos,从而说明了三维模糊集的范畴与Zadeh模糊集的范畴有同样的topos性质.
2.第三章建立了4-值模糊集与Zadeh模糊集的联系.首先,给出了Zadeh模糊集的三维向量水平截集的概念.我们将Zadeh模糊集的三维向量水平截集定义成4-值模糊集,指出这种截集与Zadeh模糊集的λ-截集有完全一样的性质.其次,建立了Zadeh模糊集的基于三维向量水平截集的8种分解定理和8种表现定理.
3.第四章在引入了三维模糊关系概念的基础上,首先利用三维模糊集的表现定理刻画了三维模糊关系的合成、内合成、投影和内投影;其次建立了三维模糊集的极大扩展原理、极小扩展原理、极大多元扩展原理、极小多元扩展原理和广义扩展原理;最后给出了三维模糊集的两种模糊线性变换的定义,建立了三维模糊集与三维模糊关系的两种合成方法.
4.第五章研究了三维凸模糊子集和基于三维模糊集的决策问题.首先,给出了模糊点与三维模糊集的邻属关系的定义,在此基础上给出了(α,β)-三维凸模糊子集的定义,指出在这16种(α,β)-三维凸模糊子集中,有意义的是(∈,,∈)-三维凸模糊子集、(∈,∈∨q)-三维凸模糊子集和((?),(?)∨(?))-三维凸模糊子集.在这些讨论的基础上,给出了具有边界值的三维凸模糊子集的定义,并用模糊点与三维模糊集的邻属关系刻画了具有边界值的三维凸模糊子集.其次,建立了一种基于三维模糊集的TOPSIS决策方法.最后,通过引入基于三维向量的截函数的概念,给出了两个三维向量α与β比较可能度p(α≥β)的定义,并具体给出了两个三维向量比较可能度p(α≥β)在20种情况下的具体表达式.在此基础上建立了基于三维向量的一种模糊决策方法.
A new type of L—fuzzy set called 3-dimensional fuzzy set is introduced in this paper. It extends Zadeh fuzzy sets, interval-valued fuzzy sets and intuitionistic fuzzy sets. We consider the 3-dimensional fuzzy sets with their basical theories which include concepts of cut sets, decomposition theorems, representation theorems and extension principles. We construct categories of 3-dimensional fuzzy sets, present a membership-relation between a fuzzy point and a 3-dimensional fuzzy sets, and study decision making problem based on the 3-dimensional fuzzy sets. More specially we present the results as follows.
1. In section 2, we consider the 3-dimensional fuzzy sets with their basical theories. First, we give definitions of 4 types of cut sets which are 4-valued fuzzy sets. The cut sets defined in this way have the same properties as those of Zadeh fuzzy sets. Second, we develop 8 decomposition theorems and 8 representation theorems, based on the cut sets of 3-dimensional fuzzy sets and 4-valued nested sets. Therefore we establish relations between 3-dimensional fuzzy sets and 4-valued fuzzy sets. Last, we construct category QFuz of 4-valued fuzzy sets , category TPuz of 3-dimensional fuzzy sets and category QNS of 4-valued nested sets respectively. We prove that categories QFuz and TFuz are weak toposes while QNS is a topos. Therefore we conclude that the category of 3-dimensional fuzzy sets has the same topos properties as that of Zadeh fuzzy sets.
2. In section 3, we consider the relations between 4-valued fuzzy sets and Zadeh fuzzy sets. First, we put forward 3-dimensional vector level cut sets of Zadeh fuzzy sets which are 4-valued fuzzy sets. The cut sets defined in this way have the same properties as A—cut sets of Zadeh fuzzy sets. Second, we develop 8 decomposition theorems and 8 representation theorems based on the 3-dimensional vector level cut sets. Therefore we establish relations between Zadeh fuzzy sets and 4-valued fuzzy sets.
3. In section 4, we first introduce the 3-dimensional fuzzy relations, and describe their composition, inner-composition, projection, and inner-projection by the representation theorems. We then develop the extension principles of the 3-dimensional fuzzy sets which include the maximum extension principle, the minimum extension principle, the maximum multiple extension principle, the minimum multiple extension principle and generalized extension principle. Last we put forward two kinds of fuzzy linear transformation of 3-dimensional fuzzy sets, and present two composition methods of a 3-dimensional fuzzy relation and a 3-dimensional fuzzy set.
4. In section 5, we study convex 3-dimensional fuzzy sets and fuzzy decision making problem based on 3-dimensional fuzzy sets. First, we give definition of membership-relation between a fuzzy point and a 3-dimensional fuzzy set, and present definition of (α,β)—convex 3-dimensional fuzzy sets. we show that, in 16 kinds of (α,β)—convex 3-dimensional fuzzy sets, the significant are (∈,,∈)—,(∈,∈∨q)—and ((∈|-), (∈|-)∨(q|-))—convex 3-dimensional fuzzy sets. Furthermore, we give definitions of convex 3-dimensional fuzzy sets with thresholds and describe them using the membership-relation between a fuzzy point and a 3-dimensional fuzzy set. Second, we construct a TOPOSIS method on fuzzy decision making problem based on 3-dimensional fuzzy sets. Last, by introducing cut functions based on 3-dimensional vectors, we give the definition of comparision possibility degree that compares one 3-dimensional vector with another and its expressions for 20 cases. Relating to these, we develop a fuzzy decision making method based on 3-dimensional vectors.
1.第二章研究了三维模糊集的基础理论.首先,给出三维模糊集的四类(共8种)截集的定义.我们将三维模糊集的截集定义为4-值模糊集,指出如此定义的截集与Zadeh模糊集的截集有完全一样的性质.其次,基于三维模糊集的截集和4-值集合套的概念,我们建立了三维模糊集的8种分解定理和8种表现定理,从而建立了三维模糊集与4-值模糊集的联系.最后,研究了三维模糊集的范畴.我们首先建立了4-值模糊集的范畴QFuz、4-值反序集合套的范畴QNS和三维模糊集的范畴TFuz,然后证明了范畴QFuz和范畴TFuz为弱topos,范畴QNS为topos,从而说明了三维模糊集的范畴与Zadeh模糊集的范畴有同样的topos性质.
2.第三章建立了4-值模糊集与Zadeh模糊集的联系.首先,给出了Zadeh模糊集的三维向量水平截集的概念.我们将Zadeh模糊集的三维向量水平截集定义成4-值模糊集,指出这种截集与Zadeh模糊集的λ-截集有完全一样的性质.其次,建立了Zadeh模糊集的基于三维向量水平截集的8种分解定理和8种表现定理.
3.第四章在引入了三维模糊关系概念的基础上,首先利用三维模糊集的表现定理刻画了三维模糊关系的合成、内合成、投影和内投影;其次建立了三维模糊集的极大扩展原理、极小扩展原理、极大多元扩展原理、极小多元扩展原理和广义扩展原理;最后给出了三维模糊集的两种模糊线性变换的定义,建立了三维模糊集与三维模糊关系的两种合成方法.
4.第五章研究了三维凸模糊子集和基于三维模糊集的决策问题.首先,给出了模糊点与三维模糊集的邻属关系的定义,在此基础上给出了(α,β)-三维凸模糊子集的定义,指出在这16种(α,β)-三维凸模糊子集中,有意义的是(∈,,∈)-三维凸模糊子集、(∈,∈∨q)-三维凸模糊子集和((?),(?)∨(?))-三维凸模糊子集.在这些讨论的基础上,给出了具有边界值的三维凸模糊子集的定义,并用模糊点与三维模糊集的邻属关系刻画了具有边界值的三维凸模糊子集.其次,建立了一种基于三维模糊集的TOPSIS决策方法.最后,通过引入基于三维向量的截函数的概念,给出了两个三维向量α与β比较可能度p(α≥β)的定义,并具体给出了两个三维向量比较可能度p(α≥β)在20种情况下的具体表达式.在此基础上建立了基于三维向量的一种模糊决策方法.
A new type of L—fuzzy set called 3-dimensional fuzzy set is introduced in this paper. It extends Zadeh fuzzy sets, interval-valued fuzzy sets and intuitionistic fuzzy sets. We consider the 3-dimensional fuzzy sets with their basical theories which include concepts of cut sets, decomposition theorems, representation theorems and extension principles. We construct categories of 3-dimensional fuzzy sets, present a membership-relation between a fuzzy point and a 3-dimensional fuzzy sets, and study decision making problem based on the 3-dimensional fuzzy sets. More specially we present the results as follows.
1. In section 2, we consider the 3-dimensional fuzzy sets with their basical theories. First, we give definitions of 4 types of cut sets which are 4-valued fuzzy sets. The cut sets defined in this way have the same properties as those of Zadeh fuzzy sets. Second, we develop 8 decomposition theorems and 8 representation theorems, based on the cut sets of 3-dimensional fuzzy sets and 4-valued nested sets. Therefore we establish relations between 3-dimensional fuzzy sets and 4-valued fuzzy sets. Last, we construct category QFuz of 4-valued fuzzy sets , category TPuz of 3-dimensional fuzzy sets and category QNS of 4-valued nested sets respectively. We prove that categories QFuz and TFuz are weak toposes while QNS is a topos. Therefore we conclude that the category of 3-dimensional fuzzy sets has the same topos properties as that of Zadeh fuzzy sets.
2. In section 3, we consider the relations between 4-valued fuzzy sets and Zadeh fuzzy sets. First, we put forward 3-dimensional vector level cut sets of Zadeh fuzzy sets which are 4-valued fuzzy sets. The cut sets defined in this way have the same properties as A—cut sets of Zadeh fuzzy sets. Second, we develop 8 decomposition theorems and 8 representation theorems based on the 3-dimensional vector level cut sets. Therefore we establish relations between Zadeh fuzzy sets and 4-valued fuzzy sets.
3. In section 4, we first introduce the 3-dimensional fuzzy relations, and describe their composition, inner-composition, projection, and inner-projection by the representation theorems. We then develop the extension principles of the 3-dimensional fuzzy sets which include the maximum extension principle, the minimum extension principle, the maximum multiple extension principle, the minimum multiple extension principle and generalized extension principle. Last we put forward two kinds of fuzzy linear transformation of 3-dimensional fuzzy sets, and present two composition methods of a 3-dimensional fuzzy relation and a 3-dimensional fuzzy set.
4. In section 5, we study convex 3-dimensional fuzzy sets and fuzzy decision making problem based on 3-dimensional fuzzy sets. First, we give definition of membership-relation between a fuzzy point and a 3-dimensional fuzzy set, and present definition of (α,β)—convex 3-dimensional fuzzy sets. we show that, in 16 kinds of (α,β)—convex 3-dimensional fuzzy sets, the significant are (∈,,∈)—,(∈,∈∨q)—and ((∈|-), (∈|-)∨(q|-))—convex 3-dimensional fuzzy sets. Furthermore, we give definitions of convex 3-dimensional fuzzy sets with thresholds and describe them using the membership-relation between a fuzzy point and a 3-dimensional fuzzy set. Second, we construct a TOPOSIS method on fuzzy decision making problem based on 3-dimensional fuzzy sets. Last, by introducing cut functions based on 3-dimensional vectors, we give the definition of comparision possibility degree that compares one 3-dimensional vector with another and its expressions for 20 cases. Relating to these, we develop a fuzzy decision making method based on 3-dimensional vectors.
引文
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