International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(4): 378-386
Published online December 25, 2024
https://doi.org/10.5391/IJFIS.2024.24.4.378
© The Korean Institute of Intelligent Systems
Masresha Wassie Woldie1, Jejaw Demamu Mebrat2, and Mihret Alamneh Taye1
1Department of Mathematics, Bahar Dar University, Bahir Dar, Ethiopia
2Department of Mathematics, Debark University, Gondar, Ethiopia
Correspondence to :
Masresha Wassie Woldie (masreshawassie28@gmail.com)
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.
An interval-valued fuzzy soft set theory is a strong instrument that can offer the uncertain data processing capacity in an imprecise environment. In this study, a generalized interval-valued fuzzy soft code (GIVFSC), which combines an interval-valued fuzzy soft set and a fuzzy code, is proposed. The GIVFSC framework supports its operation and decision-making processes. Various features of the GIVFSC are examined to improve the combination of interval-valued fuzzy soft sets and fuzzy codes.
Keywords: Fuzzy decoding, Fuzzy soft code, Interval-valued fuzzy soft set, Generalized interval-valued fuzzy soft set, Decision-Making, Generalized interval-valued fuzzy soft code
Molodtsov [1] established soft set theories and models as new mathematical tools for handling uncertainties that cannot be controlled by existing mathematical tools. Amudhambigai and Neeraja [2] studied the fuzzy codes and its application. Hong and Qin [3] discussed algebraic structures of fuzzy soft sets. Roy and Maji [4] presented a fuzzy soft theoretical approach to decision-making problems and shared several results on the application of fuzzy soft sets to these problems. Lal [5] applied fuzzy algebra to coding theory. Smarandache [6] initiated soft sets using a rough set of codes to create soft codes (soft linear codes). Ali et al. [7] developed and designed novel type of linear algebraic codes known as “soft linear algebraic codes” by employing soft sets. Alkhazaleh [8] introduced the concept of effective fuzzy soft sets, explored their operations, and investigated several of their properties. Kong et al. [9] examined the use of soft sets with fuzzy properties in grey theory-based decision-making problems. Gogoi et al. [10] designed a fuzzy softset theory applied to regular problems. Prade [11] proposed an interval-valued fuzzy set. Bustince [12] explored interval-valued fuzzy sets in soft computing. Majumdar and Samanta [13] introduced generalized fuzzy soft sets. Yang et al. [14] introduced the concept of interval-valued soft sets with fuzzy values by merging the notions of soft set models with interval-valued fuzzy sets. Chetia and Das [15] developed an application of interval-valued fuzzy software for medical diagnosis. Alkhazaleh and Salleh [16] introduced the concept of the generalized interval-valued fuzzy soft set (GIVFSs). A generalized Z-fuzzy soft covering-based rough matrix was defined by Sivaprakasam and Angamuthu [17], and its various algebraic properties were investigated. They used generalized Z-fuzzy soft covering-based rough matrices to offer a unique multi-attribute group decision-making problem model. Das and Granados [18] proposed a method to address group decision-making problems using fuzzy parameterized intuitionistic multi-fuzzy N-soft sets. They introduced an induced fuzzy parameterized hesitant N-soft set as an extension of multi-fuzzy N-soft sets, tailored for group decision-making problems. Mukherjee [19] introduced the concept of the intuitionistic fuzzy soft rough sets. Jiang et al. [20] proposed the interval-valued intuitionistic fuzzy soft set theory. Hayat and Mahmood [21] introduced an insight into a bipolar soft set in the union of two isomorphic hemirings. They characterized type-2 soft graphs as the underlying subgroups of simple graphs. Hayat et. al. [22] discussed the aggregation concepts of the selection of design parameter values by merging the acceptable and satisfactory level requirements of customers. Yang, et al. [23] developed the basic Dombi operational laws for spherical fuzzy soft numbers.
Alkhazaleh and Salleh [16] introduced the concept of a GIVFSs and its application to their characteristics. In [2] and [24], the authors defined fuzzy codes by applying the concepts of classical fuzzy sets and binary codes.
Based on these concepts, we introduce the concept of a generalized interval-valued fuzzy soft code (GIVFSC). The parameterization of fuzzy sets is related to the degree of the GIVFSs concept. However, in our study, for the GIVFSC, the degree was attached to the parameterization of the fuzzy code. This study aims to develop new ideas by integrating the definition of a GIVFSs and fuzzy codes, from which a novel fuzzy soft set model can be derived based on the GIVFSC and its operations. The novelty of this study is that it develops a combination of interval-valued fuzzy soft sets and fuzzy codes to generate an interval-valued fuzzy soft code. GIVFSCs are strong tools for processing uncertain data and are utilized in numerous different applications, such as ranking alternatives, processing uncertain data, and solving multigroup decision-making problems.
Section 2 presents the fundamental definitions and properties of the preliminary concepts. Section 3 presents the GIVFSC. In Section 4, the basic operations of GIVFSC are presented. In Section 5, the operations of GIVFSC and its applications are discussed.
In this section, we discuss some basic concepts used to interpret our findings.
In particular, the soft set is a parametrized family of the set
If
A GIVFSs over the soft universe (
In this section, we introduce the GIVFSC, which combines a GIVFSs with a fuzzy code.
where
Subsequently,
Over the soft universe (
For every
Then,
In matrix notation,
Let
Consider another GIVFSC over (
Hence,
(i)
(ii)
Here,
Proof. As
This is complete the proof.
such that
such that
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
Proof. (iii) Assume
From the definition, we have,
However,
Now,
Then,
Hence,
The remaining proofs follow directly from the definition.
(i) (
(ii) (
Proof. (i)
The proof of (ii) follows from (i).
(i)
(ii)
Proof. (i) For all
and also,
(ii) Similar to Proof (i).
The concepts of AND and OR operations on GIVFSCs are presented in this section, along with an example of their application to a decision-making problem.
where
Let two GIVFSCs (
and
Determine the “∧” between the two GIVFSCs over (
Now, for each
The outcome is displayed in Tables 1 and 2.
Let
We now record the best possible scores in numbers, denoted by parentheses, in each row, with the exception of the final row, which represents the machine’s grade of belongingness in relation to each set of parameters (refer to Table 2 for details). Currently, all of these numerical grades’ items combined with the matching value of
Therefore, machine
where
Proof. (i) Considering two GIVFSCs (
Here,
Hence, we have
Assume that
Then we have
Therefore, (
Consequently, based on our discussion, we have a result.
The same method is applied to demonstrate the second.
To illustrate the significance of our work in the decision-making process, we compared the GIVFSs first described by Alkhazaleh and Salleh [6]. Both GIVFSCs and GIVFSs are used to solve real-world issues and identify the best options based on candidate grades. While our model GIVFSCs yield the parameterized set as the set of fuzzy codes—that is, after fuzzifying the universal set, we parameterized it—GIVFSs yield the parameterized set as the universal set/objects.
In this study, we present a GIVFSC and investigate some of its properties using an interval-valued fuzzy soft set and fuzzy code. The complement, intersection, union, AND, and OR operations were defined using the GIVFSC. Decisions regarding the study were made. The limitation of this study is the integration of a generalized interval-valued fuzzy set and fuzzy codes, because this is a new formulation. Future researchers in this discipline will find this study as an introduction to related fields. Further studies could investigate generalized interval-valued fuzzy soft cyclic codes, generalized interval-valued fuzzy soft linear codes, and other relevant areas.
No potential conflict of interest relevant to this article was reported.
The author(s) did not declare any conflicts of interest or shared interests.
No potential conflict of interest relevant to this article was reported.
The author(s) did not declare any conflicts of interest or shared interests.
Table 1. Statistical grades of GIVFSCs.
( | [0.1, 0.2] | [0.2, 0.3] | [0.3, 0.4] | |
( | [0.1, 0.2] | [0.5, 0.6] | [0.3, 0.4] | 0.67 |
( | [0.1, 0.2] | [0.5, 0.6] | [0.3, 0.4] | 0.82 |
( | [0, 0.1] | [0.2, 0.3] | [0.4, 0.5] | 0.53 |
( | [0, 0.1] | [0.2, 0.3] | [0.4, 0.5] | 0.53 |
( | [0, 0.1] | [0.2, 0.3] | [0.3, 0.4] | 0.53 |
( | [0.3, 0.4] | [0.2, 0.3] | [0.4, 0.5] | 0.54 |
( | [0.4, 0.5] | [0.5, 0.6] | [0.7, 0.8] | 0.54 |
( | [0.1, 0.2] | [0.5, 0.6] | [0.3, 0.4] | 0.54 |
Table 2. Numeral rating
−0.6 | 0 | (0.6) 0.6 | ||
−1.2 | (1.2) | 0 | 0.67 | |
−1.2 | (1.2) | 0 | 0.82 | |
−1.2 | 0 | (1.2) | 0.53 | |
−1.2 | 0 | (1.2) | 0.53 | |
−1 | (0.2) | (0.8) | 0.53 | |
0 | −0.6 | (0.6) | 0.54 | |
−0.8 | −0.2 | (1) | 0.54 | |
−1.2 | (1.2) | 0 | 0.54 |
E-mail: masreshawassie28@gmail.com
E-mail: jejaw@yahoo.com
E-mail: mihretmahlet@yahoo.com
International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(4): 378-386
Published online December 25, 2024 https://doi.org/10.5391/IJFIS.2024.24.4.378
Copyright © The Korean Institute of Intelligent Systems.
Masresha Wassie Woldie1, Jejaw Demamu Mebrat2, and Mihret Alamneh Taye1
1Department of Mathematics, Bahar Dar University, Bahir Dar, Ethiopia
2Department of Mathematics, Debark University, Gondar, Ethiopia
Correspondence to:Masresha Wassie Woldie (masreshawassie28@gmail.com)
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.
An interval-valued fuzzy soft set theory is a strong instrument that can offer the uncertain data processing capacity in an imprecise environment. In this study, a generalized interval-valued fuzzy soft code (GIVFSC), which combines an interval-valued fuzzy soft set and a fuzzy code, is proposed. The GIVFSC framework supports its operation and decision-making processes. Various features of the GIVFSC are examined to improve the combination of interval-valued fuzzy soft sets and fuzzy codes.
Keywords: Fuzzy decoding, Fuzzy soft code, Interval-valued fuzzy soft set, Generalized interval-valued fuzzy soft set, Decision-Making, Generalized interval-valued fuzzy soft code
Molodtsov [1] established soft set theories and models as new mathematical tools for handling uncertainties that cannot be controlled by existing mathematical tools. Amudhambigai and Neeraja [2] studied the fuzzy codes and its application. Hong and Qin [3] discussed algebraic structures of fuzzy soft sets. Roy and Maji [4] presented a fuzzy soft theoretical approach to decision-making problems and shared several results on the application of fuzzy soft sets to these problems. Lal [5] applied fuzzy algebra to coding theory. Smarandache [6] initiated soft sets using a rough set of codes to create soft codes (soft linear codes). Ali et al. [7] developed and designed novel type of linear algebraic codes known as “soft linear algebraic codes” by employing soft sets. Alkhazaleh [8] introduced the concept of effective fuzzy soft sets, explored their operations, and investigated several of their properties. Kong et al. [9] examined the use of soft sets with fuzzy properties in grey theory-based decision-making problems. Gogoi et al. [10] designed a fuzzy softset theory applied to regular problems. Prade [11] proposed an interval-valued fuzzy set. Bustince [12] explored interval-valued fuzzy sets in soft computing. Majumdar and Samanta [13] introduced generalized fuzzy soft sets. Yang et al. [14] introduced the concept of interval-valued soft sets with fuzzy values by merging the notions of soft set models with interval-valued fuzzy sets. Chetia and Das [15] developed an application of interval-valued fuzzy software for medical diagnosis. Alkhazaleh and Salleh [16] introduced the concept of the generalized interval-valued fuzzy soft set (GIVFSs). A generalized Z-fuzzy soft covering-based rough matrix was defined by Sivaprakasam and Angamuthu [17], and its various algebraic properties were investigated. They used generalized Z-fuzzy soft covering-based rough matrices to offer a unique multi-attribute group decision-making problem model. Das and Granados [18] proposed a method to address group decision-making problems using fuzzy parameterized intuitionistic multi-fuzzy N-soft sets. They introduced an induced fuzzy parameterized hesitant N-soft set as an extension of multi-fuzzy N-soft sets, tailored for group decision-making problems. Mukherjee [19] introduced the concept of the intuitionistic fuzzy soft rough sets. Jiang et al. [20] proposed the interval-valued intuitionistic fuzzy soft set theory. Hayat and Mahmood [21] introduced an insight into a bipolar soft set in the union of two isomorphic hemirings. They characterized type-2 soft graphs as the underlying subgroups of simple graphs. Hayat et. al. [22] discussed the aggregation concepts of the selection of design parameter values by merging the acceptable and satisfactory level requirements of customers. Yang, et al. [23] developed the basic Dombi operational laws for spherical fuzzy soft numbers.
Alkhazaleh and Salleh [16] introduced the concept of a GIVFSs and its application to their characteristics. In [2] and [24], the authors defined fuzzy codes by applying the concepts of classical fuzzy sets and binary codes.
Based on these concepts, we introduce the concept of a generalized interval-valued fuzzy soft code (GIVFSC). The parameterization of fuzzy sets is related to the degree of the GIVFSs concept. However, in our study, for the GIVFSC, the degree was attached to the parameterization of the fuzzy code. This study aims to develop new ideas by integrating the definition of a GIVFSs and fuzzy codes, from which a novel fuzzy soft set model can be derived based on the GIVFSC and its operations. The novelty of this study is that it develops a combination of interval-valued fuzzy soft sets and fuzzy codes to generate an interval-valued fuzzy soft code. GIVFSCs are strong tools for processing uncertain data and are utilized in numerous different applications, such as ranking alternatives, processing uncertain data, and solving multigroup decision-making problems.
Section 2 presents the fundamental definitions and properties of the preliminary concepts. Section 3 presents the GIVFSC. In Section 4, the basic operations of GIVFSC are presented. In Section 5, the operations of GIVFSC and its applications are discussed.
In this section, we discuss some basic concepts used to interpret our findings.
In particular, the soft set is a parametrized family of the set
If
A GIVFSs over the soft universe (
In this section, we introduce the GIVFSC, which combines a GIVFSs with a fuzzy code.
where
Subsequently,
Over the soft universe (
For every
Then,
In matrix notation,
Let
Consider another GIVFSC over (
Hence,
(i)
(ii)
Here,
Proof. As
This is complete the proof.
such that
such that
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
Proof. (iii) Assume
From the definition, we have,
However,
Now,
Then,
Hence,
The remaining proofs follow directly from the definition.
(i) (
(ii) (
Proof. (i)
The proof of (ii) follows from (i).
(i)
(ii)
Proof. (i) For all
and also,
(ii) Similar to Proof (i).
The concepts of AND and OR operations on GIVFSCs are presented in this section, along with an example of their application to a decision-making problem.
where
Let two GIVFSCs (
and
Determine the “∧” between the two GIVFSCs over (
Now, for each
The outcome is displayed in Tables 1 and 2.
Let
We now record the best possible scores in numbers, denoted by parentheses, in each row, with the exception of the final row, which represents the machine’s grade of belongingness in relation to each set of parameters (refer to Table 2 for details). Currently, all of these numerical grades’ items combined with the matching value of
Therefore, machine
where
Proof. (i) Considering two GIVFSCs (
Here,
Hence, we have
Assume that
Then we have
Therefore, (
Consequently, based on our discussion, we have a result.
The same method is applied to demonstrate the second.
To illustrate the significance of our work in the decision-making process, we compared the GIVFSs first described by Alkhazaleh and Salleh [6]. Both GIVFSCs and GIVFSs are used to solve real-world issues and identify the best options based on candidate grades. While our model GIVFSCs yield the parameterized set as the set of fuzzy codes—that is, after fuzzifying the universal set, we parameterized it—GIVFSs yield the parameterized set as the universal set/objects.
In this study, we present a GIVFSC and investigate some of its properties using an interval-valued fuzzy soft set and fuzzy code. The complement, intersection, union, AND, and OR operations were defined using the GIVFSC. Decisions regarding the study were made. The limitation of this study is the integration of a generalized interval-valued fuzzy set and fuzzy codes, because this is a new formulation. Future researchers in this discipline will find this study as an introduction to related fields. Further studies could investigate generalized interval-valued fuzzy soft cyclic codes, generalized interval-valued fuzzy soft linear codes, and other relevant areas.
No potential conflict of interest relevant to this article was reported.
The author(s) did not declare any conflicts of interest or shared interests.
Table 1 . Statistical grades of GIVFSCs.
( | [0.1, 0.2] | [0.2, 0.3] | [0.3, 0.4] | |
( | [0.1, 0.2] | [0.5, 0.6] | [0.3, 0.4] | 0.67 |
( | [0.1, 0.2] | [0.5, 0.6] | [0.3, 0.4] | 0.82 |
( | [0, 0.1] | [0.2, 0.3] | [0.4, 0.5] | 0.53 |
( | [0, 0.1] | [0.2, 0.3] | [0.4, 0.5] | 0.53 |
( | [0, 0.1] | [0.2, 0.3] | [0.3, 0.4] | 0.53 |
( | [0.3, 0.4] | [0.2, 0.3] | [0.4, 0.5] | 0.54 |
( | [0.4, 0.5] | [0.5, 0.6] | [0.7, 0.8] | 0.54 |
( | [0.1, 0.2] | [0.5, 0.6] | [0.3, 0.4] | 0.54 |
Table 2 . Numeral rating
−0.6 | 0 | (0.6) 0.6 | ||
−1.2 | (1.2) | 0 | 0.67 | |
−1.2 | (1.2) | 0 | 0.82 | |
−1.2 | 0 | (1.2) | 0.53 | |
−1.2 | 0 | (1.2) | 0.53 | |
−1 | (0.2) | (0.8) | 0.53 | |
0 | −0.6 | (0.6) | 0.54 | |
−0.8 | −0.2 | (1) | 0.54 | |
−1.2 | (1.2) | 0 | 0.54 |