Multi-Granulation Graded Rough Intuitionistic Fuzzy Sets Models Based on Dominance Relation
Abstract
:1. Introduction
2. Preliminaries
3. GRIFS Model Based on Dominance Relation
3.1. GRIFS Model Based on Type-I Dominance Relation
3.2. GRIFS Model Based on Type-II Dominance Relation
4. Multi-Granulation GRIFS Models Based on Dominance Relation
4.1. GRIFS Model Based on Type-I Dominance Relation
4.2. GRIFS Model Based on Type-II Dominance Relation
5. Algorithm and Example Analysis
5.1. Algorithm
Algorithm 1. Computing multi-granulation GRIFS models based on dominance relation. |
Input:, , IFS , k is a natural number Output: Multi-granulation GRIFS models based on dominance relation 1:if ( and ) 2: if can build up GRS 3: if ( && i = 1 to m && ) 4: compute and , and , for each ; 5: then compute and of and , and and compute and of and ; 6: if 7: for (i = 1 to m ) 8: compute ; 9: end 10: compute 11: end 12: end 13: end 14:else return NULL 15:end |
5.2. An Illustrative Example
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
- Pawlak, Z. Rough sets. Int. J. Comput. Inf. Sci. 1982, 11, 341–356. [Google Scholar] [CrossRef]
- Yao, Y.Y.; Lin, T.Y. Generalization of rough sets using modal logics. Intell. Autom. Soft Comput. 1996, 2, 103–119. [Google Scholar] [CrossRef]
- Zhang, X.Y.; Mo, Z.W.; Xiong, F.; Cheng, W. Comparative study of variable precision rough set model and graded rough set model. Int. J. Approx. Reason. 2012, 53, 104–116. [Google Scholar] [CrossRef]
- Zadeh, L.A. Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems; World Scientific Publishing Corporation: Singapore, 1996; pp. 433–448. [Google Scholar]
- Qian, Y.H.; Liang, J.Y.; Yao, Y.Y.; Dang, C.Y. MGRS: A multi-granulation rough set. Inf. Sci. 2010, 180, 949–970. [Google Scholar] [CrossRef]
- Xu, W.H.; Wang, Q.R.; Zhang, X.T. Multi-granulation fuzzy rough sets in a fuzzy tolerance approximation space. Int. J. Fuzzy Syst. 2011, 13, 246–259. [Google Scholar]
- Lin, G.P.; Liang, J.Y.; Qian, Y.H. Multigranulation rough sets: From partition to covering. Inf. Sci. 2013, 241, 101–118. [Google Scholar] [CrossRef]
- Liu, C.H.; Pedrycz, W. Covering-based multi-granulation fuzzy rough sets. J. Intell. Fuzzy Syst. 2015, 30, 303–318. [Google Scholar] [CrossRef]
- Xue, Z.A.; Si, X.M.; Xue, T.Y.; Xin, X.W.; Yuan, Y.L. Multi-granulation covering rough intuitionistic fuzzy sets. J. Intell. Fuzzy Syst. 2017, 32, 899–911. [Google Scholar]
- Hu, Q.J. Extended Graded Rough Sets Models Based on Covering. Master’s Thesis, Shanxi Normal University, Linfen, China, 21 March 2016. (In Chinese). [Google Scholar]
- Wang, H.; Hu, Q.J. Multi-granulation graded covering rough sets. In Proceedings of the International Conference on Machine Learning and Cybernetics, Guangzhou, China, 12–15 July 2015; Institute of Electrical and Electronics Engineers Computer Society: New York, NY, USA, 2015. [Google Scholar]
- Wu, Z.Y.; Zhong, P.H.; Hu, J.G. Graded multi-granulation rough sets. Fuzzy Syst. Math. 2014, 28, 165–172. (In Chinese) [Google Scholar]
- Yu, J.H.; Zhang, X.Y.; Zhao, Z.H.; Xu, W.H. Uncertainty measures in multi-granulation with different grades rough set based on dominance relation. J. Intell. Fuzzy Syst. 2016, 31, 1133–1144. [Google Scholar] [CrossRef]
- Yu, J.H.; Xu, W.H. Multigranulation with different grades rough set in ordered information system. In Proceedings of the International Conference on Fuzzy Systems and Knowledge Discovery, Zhangjiajie, China, 15–17 August 2015; Institute of Electrical and Electronics Engineers Incorporated: New York, NY, USA, 2016. [Google Scholar]
- Wang, X.Y.; Shen, J.Y.; Shen, J.L.; Shen, Y.X. Graded multi-granulation rough set based on weighting granulations and dominance relation. J. Shandong Univ. 2017, 52, 97–104. (In Chinese) [Google Scholar]
- Zheng, Y. Graded multi-granularity rough sets based on covering. J. Shanxi Normal Univ. 2017, 1, 5–9. (In Chinese) [Google Scholar]
- Shen, J.R.; Wang, X.Y.; Shen, Y.X. Variable grade multi-granulation rough set. J. Chin. Comput. Syst. 2016, 37, 1012–1016. (In Chinese) [Google Scholar]
- Atanassov, K.T.; Rangasamy, P. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986, 20, 87–96. [Google Scholar] [CrossRef]
- Zadeh, L.A. Fuzzy sets. Inf. Control 1965, 8, 338–353. [Google Scholar] [CrossRef] [Green Version]
- Huang, B.; Zhuang, Y.L.; Li, H.X. Information granulation and uncertainty measures in interval-valued intuitionistic fuzzy information systems. Eur. J. Oper. Res. 2013, 231, 162–170. [Google Scholar] [CrossRef]
- Slowinski, R.; Vanderpooten, D. A generalized definition of rough approximations based on similarity. IEEE Trans. Knowl. Data Eng. 1996, 12, 331–336. [Google Scholar] [CrossRef]
- Chang, K.H.; Cheng, C.H. A risk assessment methodology using intuitionistic fuzzy set in FMEA. Int. J. Syst. Sci. 2010, 41, 1457–1471. [Google Scholar] [CrossRef]
- Zhang, X.H. Fuzzy anti-grouped filters and fuzzy normal filters in pseudo-BCI algebras. J. Intell. Fuzzy Syst. 2017, 33, 1767–1774. [Google Scholar] [CrossRef]
- Gong, Z.T.; Zhang, X.X. The further investigation of variable precision intuitionistic fuzzy rough set model. Int. J. Mach. Learn. Cybern. 2016, 8, 1565–1584. [Google Scholar] [CrossRef]
- He, Y.P.; Xiong, L.L. Generalized inter-valued intuitionistic fuzzy soft rough set and its application. J. Comput. Anal. Appl. 2017, 23, 1070–1088. [Google Scholar]
- Zhang, X.H.; Bo, C.X.; Smarandache, F.; Dai, J.H. New inclusion relation of neutrosophic sets with applications and related lattice structure. Int. J. Mach. Learn. Cybern. 2018, 9, 1753–1763. [Google Scholar] [CrossRef]
- Zhang, X.H.; Smarandache, F.; Liang, X.L. Neutrosophic duplet semi-group and cancellable neutrosophic triplet groups. Symmetry 2017, 9, 275. [Google Scholar] [CrossRef]
- Huang, B.; Guo, C.X.; Li, H.X.; Feng, G.F.; Zhou, X.Z. An intuitionistic fuzzy graded covering rough set. Knowl.-Based Syst. 2016, 107, 155–178. [Google Scholar] [CrossRef]
- Tiwari, A.K.; Shreevastava, S.; Som, T. Tolerance-based intuitionistic fuzzy-rough set approach for attribute reduction. Expert Syst. Appl. 2018, 101, 205–212. [Google Scholar] [CrossRef]
- Guo, Q.; Ming, Y.; Wu, L. Dominance relation and reduction in intuitionistic fuzzy information system. Syst. Eng. Electron. 2014, 36, 2239–2243. [Google Scholar]
- Ai, A.H.; Xu, Z.S. Line integral of intuitionistic fuzzy calculus and their properties. IEEE Trans. Fuzzy Syst. 2018, 26, 1435–1446. [Google Scholar] [CrossRef]
- Rizvi, S.; Naqvi, H.J.; Nadeem, D. Rough intuitionistic fuzzy sets. In Proceedings of the 6th Joint Conference on Information Sciences, Research Triangle Park, NC, USA, 8–13 March 2002; Duke University/Association for Intelligent Machinery: Durham, NC, USA, 2002. [Google Scholar]
- Zhang, X.X.; Chen, D.G.; Tsang, E.C.C. Generalized dominance rough set models for the dominance intuitionistic fuzzy information systems. Inf. Sci. 2017, 378, 1339–1351. [Google Scholar] [CrossRef]
- Zhang, Y.Q.; Yang, X.B. An intuitionistic fuzzy dominance-based rough set. In Proceedings of the 7th International Conference on Intelligent Computing, Zhengzhou, China, 11–14 August 2011; Springer: Berlin, Germany, 2011. [Google Scholar]
- Huang, B.; Zhuang, Y.L.; Li, H.X.; Wei, D.K. A dominance intuitionistic fuzzy-rough set approach and its applications. Appl. Math. Model. 2013, 37, 7128–7141. [Google Scholar] [CrossRef]
- Zhang, W.X.; Wu, W.Z.; Liang, J.Y.; Li, D.Y. Theory and Method of Rough Sets; Science Press: Beijing, China, 2001. (In Chinese) [Google Scholar]
- Wen, X.J. Uncertainty measurement for intuitionistic fuzzy ordered information system. Master’s Thesis, Shanxi Normal University, Linfen, China, 21 March 2015. (In Chinese). [Google Scholar]
- Lezanski, P.; Pilacinska, M. The dominance-based rough set approach to cylindrical plunge grinding process diagnosis. J. Intell. Manuf. 2018, 29, 989–1004. [Google Scholar] [CrossRef]
- Huang, B.; Guo, C.X.; Zhang, Y.L.; Li, H.X.; Zhou, X.Z. Intuitionistic fuzzy multi-granulation rough sets. Inf. Sci. 2014, 277, 299–320. [Google Scholar] [CrossRef]
- Greco, S.; Matarazzo, B.; Slowinski, R. An algorithm for induction decision rules consistent with the dominance principle. In Proceedings of the 2nd International Conference on Rough Sets and Current Trends in Computing, Banff, AB, Canada, 16–19 October 2000; Springer: Berlin, Germany, 2011. [Google Scholar]
x | |||
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x | |||||||||
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0.9 | 0.8 | 0.65 | 0.85 | 0.95 | 0.7 | 0.5 | 0.87 | 0.75 | |
0 | 0.1 | 0.3 | 0.1 | 0.05 | 0.3 | 0.2 | 0.1 | 0.2 | |
0.78 | 0.78 | 0.65 | 0.78 | 0.78 | 0.7 | 0.5 | 0.72 | 0.75 | |
0.22 | 0.22 | 0.3 | 0.22 | 0.22 | 0.3 | 0.22 | 0.22 | 0.22 | |
0.9 | 0.8 | 0.78 | 0.85 | 0.95 | 0.78 | 0.78 | 0.78 | 0.78 | |
0 | 0.1 | 0.22 | 0.1 | 0.05 | 0.22 | 0.2 | 0.1 | 0.2 | |
0.67 | 0.67 | 0.65 | 0.67 | 0.67 | 0.67 | 0.5 | 0.6 | 0.67 | |
0.33 | 0.33 | 0.33 | 0.33 | 0.33 | 0.33 | 0.33 | 0.33 | 0.33 | |
0.89 | 0.8 | 0.67 | 0.85 | 0.95 | 0.7 | 0.67 | 0.67 | 0.75 | |
0.11 | 0.1 | 0.3 | 0.1 | 0.05 | 0.3 | 0.2 | 0.1 | 0.2 | |
0.9 | 0.8 | 0.65 | 0.85 | 0.89 | 0.7 | 0.5 | 0.87 | 0.75 | |
0 | 0.11 | 0.3 | 0.11 | 0.11 | 0.3 | 0.2 | 0.11 | 0.2 | |
0.9 | 0.89 | 0.89 | 0.89 | 0.95 | 0.89 | 0.89 | 0.89 | 0.89 | |
0 | 0.1 | 0.11 | 0.1 | 0.05 | 0.11 | 0.11 | 0.1 | 0.11 |
x | |||
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x | |||
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Share and Cite
Xue, Z.-a.; Lv, M.-j.; Han, D.-j.; Xin, X.-w. Multi-Granulation Graded Rough Intuitionistic Fuzzy Sets Models Based on Dominance Relation. Symmetry 2018, 10, 446. https://doi.org/10.3390/sym10100446
Xue Z-a, Lv M-j, Han D-j, Xin X-w. Multi-Granulation Graded Rough Intuitionistic Fuzzy Sets Models Based on Dominance Relation. Symmetry. 2018; 10(10):446. https://doi.org/10.3390/sym10100446
Chicago/Turabian StyleXue, Zhan-ao, Min-jie Lv, Dan-jie Han, and Xian-wei Xin. 2018. "Multi-Granulation Graded Rough Intuitionistic Fuzzy Sets Models Based on Dominance Relation" Symmetry 10, no. 10: 446. https://doi.org/10.3390/sym10100446