International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(4): 360-377
Published online December 25, 2024
https://doi.org/10.5391/IJFIS.2024.24.4.360
© The Korean Institute of Intelligent Systems
Misbah Rasheed1, Muntazim Abbas Hashmi1, Muhammad Kamran2, Aamir Hussain Khan2, Lakhdar Ragoub3, Mohammad Mahtab Alam4, and Umber Rana1
1Institute of Mathematics, Khwaja Fareed University of Engineering & Information Technology,Rahim Yar Khan, Pakistan
2Department of Mathematics, Thal University Bhakkar, Pakistan
3Mathematics Department, Prince Mugrin University, Al Madinah, Saudi Arabia
4Department of Basic Medical Sciences, College of Applied Medical Science, King Khalid University, Abha, Saudi Arabia
Correspondence to :
Lakhdar Ragoub (l.ragoub@upm.edu.sa)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
The recent development of a two-way structure for fuzzy concept lattices, based on intuitionistic fuzzy sets, aims to characterize the ambiguity and vagueness in specific data sets by accounting for acceptance, opposition, and uncertainty. This paper leverages the graphical properties of interval-valued intuitionistic fuzzy sets (IVIFSs) to address the issue of partial ignorance in a two-way fuzzy framework. It demonstrates how an IVIFS-based environment reflects hidden patterns in concept lattices, such as formal concepts. We propose a novel strategy for generating interval-valued formal concepts by integrating the properties of concept lattices, interval-valued intuitionistic fuzzy graphs, and IVIFSs. Additionally, we introduce a method to enhance specific patterns using interval-valued intuitionistic fuzzy formal principles, defined via (α, β)-cut for their respective truth (α-cut) and falsity (β-cut) membership values. Examples are provided to illustrate the proposed approaches, demonstrating their practical applicability and effectiveness.
Keywords: Intuitionistic fuzzy set, Lattice features, Fuzzy graph, Decision-making
No potential conflict of interest relevant to this article was reported.
International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(4): 360-377
Published online December 25, 2024 https://doi.org/10.5391/IJFIS.2024.24.4.360
Copyright © The Korean Institute of Intelligent Systems.
Misbah Rasheed1, Muntazim Abbas Hashmi1, Muhammad Kamran2, Aamir Hussain Khan2, Lakhdar Ragoub3, Mohammad Mahtab Alam4, and Umber Rana1
1Institute of Mathematics, Khwaja Fareed University of Engineering & Information Technology,Rahim Yar Khan, Pakistan
2Department of Mathematics, Thal University Bhakkar, Pakistan
3Mathematics Department, Prince Mugrin University, Al Madinah, Saudi Arabia
4Department of Basic Medical Sciences, College of Applied Medical Science, King Khalid University, Abha, Saudi Arabia
Correspondence to:Lakhdar Ragoub (l.ragoub@upm.edu.sa)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
The recent development of a two-way structure for fuzzy concept lattices, based on intuitionistic fuzzy sets, aims to characterize the ambiguity and vagueness in specific data sets by accounting for acceptance, opposition, and uncertainty. This paper leverages the graphical properties of interval-valued intuitionistic fuzzy sets (IVIFSs) to address the issue of partial ignorance in a two-way fuzzy framework. It demonstrates how an IVIFS-based environment reflects hidden patterns in concept lattices, such as formal concepts. We propose a novel strategy for generating interval-valued formal concepts by integrating the properties of concept lattices, interval-valued intuitionistic fuzzy graphs, and IVIFSs. Additionally, we introduce a method to enhance specific patterns using interval-valued intuitionistic fuzzy formal principles, defined via (α, β)-cut for their respective truth (α-cut) and falsity (β-cut) membership values. Examples are provided to illustrate the proposed approaches, demonstrating their practical applicability and effectiveness.
Keywords: Intuitionistic fuzzy set, Lattice features, Fuzzy graph, Decision-making
IVIF relationship network.
IVIF concept lattice.
Table 3 . An IVIF relations for §3.
⋎1 | ⋎2 | ⋎3 | |
---|---|---|---|
0.34 | 0.56 | 0.43 | |
0.65 | 0.65 | 0.56 | |
0.34 | 0.34 | 0.78 | |
0.43 | 0.43 | 0.90 |
Table 4 . An IVIF relationship by §4.
⋎1 | ⋎2 | ⋎3 | |
---|---|---|---|
0.78 | 0.65 | 0.65 | |
0.87 | 0.78 | 0.78 | |
0.12 | 0.12 | 0.87 | |
0.21 | 0.34 | 0.90 |
Table 5 . IVIF representation for Tables 1 to
⋎1 | ⋎2 | ⋎3 | |
---|---|---|---|
§1 | [(0.43, 0.56), (0.34, 0.43)] | [(0.43, 0.65), (0.21, 0.43)] | [(0.78, 0.90), (0.43, 0.56)] |
§2 | [(0.65, 0.78), (0.21, 0.34)] | [(0.65, 0.78), (0.21, 0.34)] | [(0.34, 0.65), (0.87, 0.90)] |
§3 | [(0.34, 0.65), (0.34, 0.43)] | [(0.56, 0.65), (0.34, 0.43)] | [(0.45, 0.56), (0.78, 0.90)] |
§4 | [(0.78, 0.87), (0.12, 0.21)] | [(0.65, 0.78), (0.12, 0.34)] | [(0.65, 0.78), (0.87, 0.90)] |
Table 6 . A three-way fuzzy context representation using intuitionistic fuzzy sets.
⋎1 | ⋎2 | ⋎3 | |
---|---|---|---|
§1 | (0.43, 0.34) | (0.43, 0.34) | (0.21, 0.56) |
§2 | (0.65, 0.21) | (0.65, 0.21) | (0.56, 0.21) |
§3 | (0.34, 0.34) | (0.56, 0.34) | (0.56, 0.21) |
§4 | (0.78, 0.12) | (0.65, 0.21) | (0.43, 0.21) |
Algorithm 1. Proposed algorithm to develop the IVIF concept.
An IVIF envirnment ![]() ![]() | |
Set of IVIFFCs | |
1 | Determine the subsets |
2 | |
3 | Determine the value of the acceptance membership = ([1.00, 1.00], [0.00, 0.00]) for each subsets |
4 | Use the scrolling operator to locate the encompassing object set |
5 | The down operator (↓) provides the following object set: |
6 | The calculation of the membership value for the acquired set of objects can be determined in the following manner: |
7 | Determine the attribute set that covers the specified object set using the upper operator, denoted as ↑ (UP). |
8 | Now, calculate the membership value for the obtained attribute set using the following procedure: |
9 | The formal concepts ( |
10 | |
11 | In a similar way, more concepts can be developed with the other subsets. |
12 | Build the IVIF concept lattice. |
Algorithm 2. Suggested algorithm for breaking down the IF environment into
An IF environment ![]() ![]() | |
The set of decomposed environment | |
1 | Let us assume an IF environment ![]() |
2 | Define the granulation for the TMV and FMV i.e., ( |
3 | Now the decomposed the given environment |
4 | |
5 | |
6 | The decomposed environment follows the equality: |
7 | The decomposed binary environment write for the user-defined granulation. |
8 | Derive the knowledge from the decomposed environment |
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|@|~(^,^)~|@|IVIF concept lattice.