Multi-Granulation Covering Rough Intuitionistic Fuzzy Sets Based on Maximal Description
Abstract
1. Introduction
2. Literature Review
3. Basic Concepts
3.1. Rough Sets
- . Let be the degree of rough membership of x in R. is the cardinality of a set, and the equivalence class of x with respect to relation R is denoted as .
3.2. Covering Approximation Spaces
3.3. Multi-Granulation Rough Sets
3.4. Intuitionistic Fuzzy Sets
4. Methodology
4.1. New Kind of Covering Rough Intuitionistic Fuzzy Sets Based on Maximal Description
- (1)
- ,
- (2)
- ,
- (3)
- ,
- (4)
- If , then
4.2. Multi-Granulation Covering Rough Intuitionistic Fuzzy Sets Based on Maximal Description
4.2.1. Optimistic and Pessimistic Multi-Granulation Covering Rough Intuitionistic Fuzzy Sets
- (1)
- ,
- (2)
- ,
- (3)
- ,
- (4)
- .
- (1)
- ,
- (2)
- ,
- (3)
- ,
- (4)
- ,
- (5)
- ,
- (6)
- ,
- (7)
- If , then
- (8)
- If , then ,
4.2.2. Neutral Multi-Granulation Covering Rough Intuitionistic Fuzzy Sets
- (1)
- ,
- (2)
- ,
- (3)
- ,
- (4)
- If , then ,
4.3. Comparative Analysis
5. An Application Example
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Notations | Explanations |
---|---|
U | Nonempty and finite universe of discourse |
R | Equivalence relation |
C | Covering relation |
X | |
The degree of rough membership of x in R | |
The lower and upper approximations of X in the rough sets | |
The maximal description of x | |
The lower and upper approximations of A with optimistic multi-granulation rough sets | |
The lower and upper approximations of A with pessimistic multi-granulation rough sets | |
A | The intuitionistic fuzzy set (IFS) |
The membership and non-membership degrees of the element in x | |
The intuitionistic fuzzy covering rough membership and non-membership degrees of x based on maximal description | |
The lower and upper approximations of A with covering rough intuitionistic fuzzy sets | |
The lower and upper approximation of A with optimistic multi-granulation covering rough intuitionistic fuzzy sets | |
The lower and upper approximation of A with pessimistic multi-granulation covering rough intuitionistic fuzzy sets | |
The maximal description based on multi-granulation of x | |
The multi-granulation intuitionistic fuzzy covering rough membership and non-membership degrees of x based on maximal description | |
The lower and upper approximation of A with neutral multi-granulation covering rough intuitionistic fuzzy sets |
U | C1 | C2 | C3 |
---|---|---|---|
x1 | High | High | High |
x2 | High | High | High |
x3 | High/Moderate | High/Moderate | Moderate |
x4 | High/Moderate | Moderate | Moderate/Low |
x5 | Moderate/Low | Moderate | Moderate/Low |
x6 | Moderate/Low | Moderate/Low | Low |
x7 | Low | Low | Low |
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Si, X.-M.; Xue, Z.-A. Multi-Granulation Covering Rough Intuitionistic Fuzzy Sets Based on Maximal Description. Symmetry 2025, 17, 1217. https://doi.org/10.3390/sym17081217
Si X-M, Xue Z-A. Multi-Granulation Covering Rough Intuitionistic Fuzzy Sets Based on Maximal Description. Symmetry. 2025; 17(8):1217. https://doi.org/10.3390/sym17081217
Chicago/Turabian StyleSi, Xiao-Meng, and Zhan-Ao Xue. 2025. "Multi-Granulation Covering Rough Intuitionistic Fuzzy Sets Based on Maximal Description" Symmetry 17, no. 8: 1217. https://doi.org/10.3390/sym17081217
APA StyleSi, X.-M., & Xue, Z.-A. (2025). Multi-Granulation Covering Rough Intuitionistic Fuzzy Sets Based on Maximal Description. Symmetry, 17(8), 1217. https://doi.org/10.3390/sym17081217