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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

Enhancing Complex Relationships between Interval-Valued Intuitionistic Fuzzy Graphs and Concept Lattice Exploration

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)

Received: January 31, 2024; Revised: June 10, 2024; Accepted: September 30, 2024

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.

Misbah Rasheed is a Ph.D. scholar at the Institute of Mathematics, Khwaja Fareed University of Engineering and Information Technology, Rahim Yar Khan, Pakistan. She specialized in computational mathematics during her master’s studies. Her research focuses on fuzzy logic, decision-making, and graph-based approaches.

Muntazim Abbas Hashmi is an assistant Professor of Mathematics at the Institute of Mathematics, Khwaja Fareed University of Engineering and Information Technology, Rahim Yar Khan 64200, Pakistan. His research interests include mathematical models and differential equations, cryptography, steganography, watermarking, statistical models and simulation, Monte Carlo methods, fuzzy hypothesis testing, Bayesian hypothesis testing, estimation theory, information theory, credibility theory, uncertainty theory, distribution theory, data analysis (environmental, agricultural, and medical), system analysis (environmental water pollution), computer technologies, mathematical statistics, and computational mathematics.

Muhammad Kamran is a lecturer in Mathematics at Thal University, Bhakkar, Pakistan. His research interests include fuzzy logic, decision-making, uncertainty theory, fuzzy hypothesis testing, system analysis, computer technologies, mathematical statistics, computational mathematics, and graph theory.

Aamir Hussain Khan is a visiting lecturer in Mathematics at Thal University, Bhakkar, Pakistan. His research focuses on fuzzy logic, numerical methods, and graph theory.

Lakhdar Ragoub is a professor in the Mathematics Department at Prince Mugrin University, Al Madinah, Saudi Arabia. His research interests include mathematical statistics and computational mathematics.

Mohammad Mahtab Alam is a Professor with the Department of Basic Medical Sciences, College of Applied Medical Science, King Khalid University, Abha, Saudi Arabia. His research focuses on computational mathematics and its applications.

Umber Rana is an assistant professor of Mathematics at the Institute of Mathematics, Khwaja Fareed University of Engineering and Information Technology, Rahim Yar Khan, Pakistan. Her research interests include mathematical models and differential equations, cryptography, steganography, watermarking, statistical models and simulation, Monte Carlo methods, fuzzy hypothesis testing, Bayesian hypothesis testing, estimation theory, information theory, credibility theory, uncertainty theory, distribution theory, data analysis (environmental, agricultural, and medical), system analysis (environmental water pollution), computer technologies, mathematical statistics, and computational mathematics.

Article

Original Article

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.

Enhancing Complex Relationships between Interval-Valued Intuitionistic Fuzzy Graphs and Concept Lattice Exploration

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)

Received: January 31, 2024; Revised: June 10, 2024; Accepted: September 30, 2024

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.

Abstract

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

Fig 1.

Figure 1.

IVIF relationship network.

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 360-377https://doi.org/10.5391/IJFIS.2024.24.4.360

Fig 2.

Figure 2.

IVIF concept lattice.

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 360-377https://doi.org/10.5391/IJFIS.2024.24.4.360

Table 1 .1) involves IVIF relationship shown in Figure 1.

123
§1L'0.430.430.78
§1I'0.560.650.90
Ω§1L'0.430.210.43
Ω§1I'0.430.430.56

Table 2 . IVIF relation for §2, shown in Figure 1.

123
§1L'0.650.650.34
§1I'0.780.780.65
Ω§1L'0.210.210.87
Ω§1I'0.340.340.90

Table 3 . An IVIF relations for §3.

123
§3L'0.340.560.43
§3I'0.650.650.56
Ω§3L'0.340.340.78
Ω§3I'0.430.430.90

Table 4 . An IVIF relationship by §4.

123
§4L'0.780.650.65
§4I'0.870.780.78
Ω§4L'0.120.120.87
Ω§4I'0.210.340.90

Table 5 . IVIF representation for Tables 1 to 4.

123
§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.

123
§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)

Table 7 . A decomposition shown in Table 6.

123
§1(0, 1)(0, 1)(0, 1)
§2(0, 1)(0, 1)(0, 1)
§3(0, 1)(0, 1)(0, 1)
§4(1, 0)(0, 1)(0, 1)

Table 8 . An analysis of the background presented in Table 6.

123
§1(0, 1)(0, 1)(0, 1)
§2(1, 0)(1, 0)(0, 1)
§3(0, 1)(0, 1)(0, 1)
§4(1, 0)(1, 0)(0, 1)

Table 9 . Context breakdown of Table 6 (0.12, 0.21).

123
§1(0, 1)(0, 1)(1, 1)
§2(1, 1)(1, 1)(0, 1)
§3(0, 1)(0, 1)(0, 1)
§4(1, 0)(1, 1)(1, 1)

Algorithm 1. Proposed algorithm to develop the IVIF concept.

Input:An IVIF envirnment , whereas , and ℜ̃ express IVIF relation.
Output:Set of IVIFFCs
Extent:{§,[A(§)L',A(§)I'],[ΩA(§)L',ΩA(§)I']},
Intent:{,[B(yj)L',B()I'],[ΩB(yj)L',ΩB()I']}, where ⋎ ≤ n and ⋌ ≤ m.
1Determine the subsets S of given IV IF attribues 2m.
2for ⋌ = 1 to 2m.
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 ():{,[BSL'(),BSI'()],[ΩBSL'(),ΩBSI'()]}.
5 The down operator (↓) provides the following object set: {§,[A(§)L',A(§)I'],[ΩA(§)L',ΩA(§)I']}
6 The calculation of the membership value for the acquired set of objects can be determined in the following manner:ASL'=minBSL'[μL'˜(§,)],ASI'=minBSL'[μI'˜(§,)],ΩASL'=maxΩBSL'[μΩL'˜(§,)],ΩASI'=maxΩBSI'[μΩI'˜(§,)].
7 Determine the attribute set that covers the specified object set using the upper operator, denoted as ↑ (UP). {§,[A(§)L',A(§)I'],[ΩA(§)L',ΩA(§)I']}.
8 Now, calculate the membership value for the obtained attribute set using the following procedure:BSL'=maxASL'[μL'˜(§,)],BSI'=minASI'[μI'˜(§,)],ΩBSL'=maxΩASL'[μΩL'˜(§,)],ΩBSI'=maxΩASI'[μΩI'˜(§,)].
9 The formal concepts (AS, BS) are derived from the selected subset.
10end for
11In a similar way, more concepts can be developed with the other subsets.
12Build the IVIF concept lattice.

Algorithm 2. Suggested algorithm for breaking down the IF environment into α and β.

Input:An IF environment where , and (ℝ̃ = (℧ℜ̃(§, ⋎), Ωℜ̃(§, ⋎))
Output:The set of decomposed environment Kα, β
1Let us assume an IF environment .
2Define the granulation for the TMV and FMV i.e., (α, β).
3Now the decomposed the given environment Kα, β as follows:
4if {℧ℜ̃(§, ⋎)|μ℧ℜ̃(§,⋎)α}then represent 1.00 at the place of TMV,else 0.00 at the place of TMV.
5ifℜ̃(§, ⋎)|μΩℜ̃(§,⋎)β}then represent 0.00 at the place of FMV,else 1.00 at the place of FMV.
6The decomposed environment follows the equality:K = ∪α, βKα, β.
7The decomposed binary environment write for the user-defined granulation.
8Derive the knowledge from the decomposed environment Kα, β.

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