(T, S)-Based Single-Valued Neutrosophic Number Equivalence Matrix and Clustering Method
Abstract
:1. Introduction
2. Preliminaries
- FS: fuzzy set
- FEM: fuzzy equivalence matrix
- IFS: intuitionistic fuzzy set
- IVIFS: interval-valued intuitionistic fuzzy set
- IFEM: intuitionistic fuzzy equivalence matrix
- NS: neutrosophic set
- SVNS: single-valued neutrosophic set
- SVNN: single-valued neutrosophic number
- SVNNM: single-valued neutrosophic number matrix
- SVNNSM: single-valued neutrosophic number similarity matrix
- -SVNNEM: -based single-valued neutrosophic number equivalence matrix
- (1)
- Inclusion: if only if , , .
- (2)
- Complement: .
- (3)
- Union: .
- (4)
- Intersection: .
- (a)
- (Reflexivity) , for any ;
- (b)
- (Symmetry) , for any .
- (1)
- Reflexivity: ;
- (2)
- Symmetry: , i.e., .
- (1)
- , ;
- (2)
- , for any x and y;
- (3)
- , for any and z;
- (4)
- if , then .
- (1)
- , that is and ;
- (2)
- , that is and .
- (1)
- ;
- (2)
- .
- (1)
- Type I (min and max t-norm and t-conorm): , ;
- (2)
- Type II (Algebraic t-norm and t-conorm): , ;
- (3)
- Type III (Einstein t-norm and t-conorm): , ;
- (4)
- Type IV (Hamacher t-norm and t-conorm): , .
3. Main Results
3.1. Some Properties of Generalized Unions and Intersections
- (1)
- ;
- (2)
- ;
- (3)
- ;
- (4)
- ;
- (5)
- ;
- (6)
- .
- (1)
- ;
- (2)
- .
- (1)
- ;
- (2)
- .
3.2. A -Based Single-Valued Neutrosophic Number Composition Matrix and Its Properties
- (1)
- By Corollary 2, we know that is a SVNNM.
- (2)
- By Definition 10, for , we have
- (3)
- Since P is a SVNNSM, , i.e., . Then we have
3.3. -SVNNEM and -Cutting Matrix
- (1)
- Reflexivity: ;
- (2)
- Symmetry: , i.e. ;
- (3)
- -Transitivity: , i.e. , , .
- (1)
- Reflexivity: Since , we have for each SVNN . By Definition 13, we get .
- (2)
- Symmetry: By , we can easily get that for each SVNN .
- (1)
- Reflexivity: Since , we know that for each SVNN , that is and . Let , then .
- (2)
- Symmetry: If , then or or . Suppose , . Then . So , . , which is a contradiction. Therefore, .
4. A Algorithm for Single-Valued Neutrosophic Number Clustering
5. Illustrative Example and Comparative Analysis
5.1. Illustrative Example
5.2. Analysis of Comparative Results
- (1)
- For the -SVNNEM clustering algorithm, when the dual triangular modules of type I and II are chosen, they can be divided into five classifications with the same classification ability but different classification results. The reason is the min and max operators (type I) easily overlook the influence of other SVNN information on the whole.
- (2)
- Compare our method with literatures [18,19], can only be divided into three classifications by the FEM clustering algorithm in literature [18] and the IFEM clustering algorithm in literature [19]. The reason is that in the clustering process we use the SVNNM instead of the fuzzy matrix, which can better retain information. The classification results are more reasonable and comprehensive.
- (3)
- The method in literature [28] does not calculate the equivalence matrix based on the similarity matrix, but from the classification results, can only be divided into four classifications by SVNN orthogonal clustering algorithm in literature [28], while can be divided into five classifications by our clustering algorithm. For the example given in this paper, it shows that the -SVNNEM clustering algorithm classification result is more accurate than SVNN orthogonal clustering algorithm in literature [28].
- (4)
- Compared our method with the existing methods, as the value of changes, the result remains stable. That is, keeps the classification result within a certain range. Such as, when needs to be divided into three classifications, we have . The classification results remain unchanged in this interval, while in literature [18,28] it cannot be divided into three classifications.
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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(0.3,0.4,0.5) | (0.6,0.7,0.1) | (0.4,0.4,0.3) | (0.8,0.3,0.1) | (0.1,0.2,0.6) | (0.5,0.2,0.4) | |
(0.6,0.3,0.3) | (0.5,0.1,0.2) | (0.6,0.5,0.1) | (0.7,0.3,0.1) | (0.3,0.2,0.6) | (0.4,0.4,0.3) | |
(0.4,0.2,0.4) | (0.8,0.3,0.1) | (0.5,0.1,0.1) | (0.6,0.1,0.2) | (0.4,0.3,0.5) | (0.3,0.2,0.2) | |
(0.2,0.5,0.4) | (0.4,0.3,0.1) | (0.9,0.2,0.0) | (0.8,0.2,0.1) | (0.2,0.1,0.5) | (0.7,0.3,0.1) | |
(0.5,0.1,0.2) | (0.3,0.4,0.6) | (0.6,0.3,0.3) | (0.7,0.3,0.1) | (0.6,0.4,0.2) | (0.5,0.1,0.3) |
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Classification Results | |
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Mo, J.; Huang, H.-L. (T, S)-Based Single-Valued Neutrosophic Number Equivalence Matrix and Clustering Method. Mathematics 2019, 7, 36. https://doi.org/10.3390/math7010036
Mo J, Huang H-L. (T, S)-Based Single-Valued Neutrosophic Number Equivalence Matrix and Clustering Method. Mathematics. 2019; 7(1):36. https://doi.org/10.3390/math7010036
Chicago/Turabian StyleMo, Jiongmei, and Han-Liang Huang. 2019. "(T, S)-Based Single-Valued Neutrosophic Number Equivalence Matrix and Clustering Method" Mathematics 7, no. 1: 36. https://doi.org/10.3390/math7010036
APA StyleMo, J., & Huang, H.-L. (2019). (T, S)-Based Single-Valued Neutrosophic Number Equivalence Matrix and Clustering Method. Mathematics, 7(1), 36. https://doi.org/10.3390/math7010036