Spherical Linear Diophantine Fuzzy Graphs: Unleashing the Power of Fuzzy Logic for Uncertainty Modeling and Real-World Applications

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This manuscript explores the intricacies and nuances surrounding the regularity of spherical linear Diophantine fuzzy graphs. However, there are several concerns regarding the content:

 

1. In the abstract, the authors claim that the theory of spherical linear Diophantine fuzzy sets (SLDFS) surpasses other fuzzy set (FS) theories such as Picture fuzzy sets (PFS), spherical fuzzy sets (SFS), and T-spherical fuzzy sets (T-SFS). However, they neglect to mention the T-Spherical Linear Diophantine Fuzzy Sets (T-SLDFSs) proposed in a separate publication (https://www.aimspress.com/article/doi/10.3934/math.2023618). T-SLDFS is a generalization of the proposed method and addresses limitations related to the (RPs). Specifically, the sum of RPs provided by a decision-maker is often larger than one (a+b+c ≥ 1), which violates the restriction of SLDFS. 

T-SLDFS introduces the q-th power of RPs, which covers the space of existing structures and membership grades through the utilization of the q-th power of RPs.

 

2. The motivation behind the research is not clearly stated in the introduction section. It would be beneficial to provide a more comprehensive explanation of the reasons for undertaking this study and its potential contributions to the field.

 

3. The usage of capital letters in words that are not at the beginning of a sentence should be corrected to conform to standard writing conventions.

 

4. The comparative analysis section should be expanded to include additional discussions. This would enhance the depth of the analysis and provide a more thorough examination of the proposed method in relation to other existing methods.

 

5. It is important to address the challenges faced during the research process and discuss the limitations of the proposed method. Identifying and acknowledging potential limitations will contribute to a more comprehensive understanding of the research and its applicability.

 

6. The references list should be checked to ensure that all cited references are properly mentioned within the text. This will ensure the accuracy and completeness of the reference section.

Comments on the Quality of English Language

Minor editing of English language required

Author Response

This manuscript explores the intricacies and nuances surrounding the regularity of spherical linear Diophantine fuzzy graphs. However, there are several concerns regarding the content:

 

1. In the abstract, the authors claim that the theory of spherical linear Diophantine fuzzy sets (SLDFS) surpasses other fuzzy set (FS) theories such as Picture fuzzy sets (PFS), spherical fuzzy sets (SFS), and T-spherical fuzzy sets (T-SFS). However, they neglect to mention the T-Spherical Linear Diophantine Fuzzy Sets (T-SLDFSs) proposed in a separate publication (https://www.aimspress.com/article/doi/10.3934/math.2023618). T-SLDFS is a generalization of the proposed method and addresses limitations related to the (RPs). Specifically, the sum of RPs provided by a decision-maker is often larger than one (a+b+c ≥ 1), which violates the restriction of SLDFS. 

T-SLDFS introduces the q-th power of RPs, which covers the space of existing structures and membership grades through the utilization of the q-th power of RPs.

Author Response: Included the concept of T-SLDFSs in the introduction along with the reference paper.

2. The motivation behind the research is not clearly stated in the introduction section. It would be beneficial to provide a more comprehensive explanation of the reasons for undertaking this study and its potential contributions to the field.

 Author Response: Motivation and Objectives of the paper are included in the introduction.

3. The usage of capital letters in words that are not at the beginning of a sentence should be corrected to conform to standard writing conventions.

 Author Response: Changed in all the places and only in Diophantine D will be in capital as it is the correct form.  

4. The comparative analysis section should be expanded to include additional discussions. This would enhance the depth of the analysis and provide a more thorough examination of the proposed method in relation to other existing methods.

Author Response: Changed in all the places except Diophantine D will be in capital as it is the correct form. 

5. It is important to address the challenges faced during the research process and discuss the limitations of the proposed method. Identifying and acknowledging potential limitations will contribute to a more comprehensive understanding of the research and its applicability.

 Author Response: Limitations are included in table 10

6. The references list should be checked to ensure that all cited references are properly mentioned within the text. This will ensure the accuracy and completeness of the reference section.

 Author Response: References are checked and corrected 

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

Please open the attachment file

Comments for author File: Comments.pdf

Comments on the Quality of English Language

Author Response

Dear Editor,

The advantages of the theory of spherical linear Diophantine fuzzy sets over other theories -- Picture, spherical, and T-spherical fuzzy sets, for example-- are discussed in the study. Spherical linear Diophantine fuzzy sets can depict a greater variety of unclear and uncertain knowledge data sets because they have a broader portrayal space for acceptable triplets. This study investigates the regularity of spherical linear Diophantine fuzzy graphs, giving a geometrical interpretation in a spherical setting and establishing their basic ideas. Additionally, the authors provide the idea of a spherical linear Diophantine isomorphic fuzzy graph and use a social network scenario to illustrate its use. The paper demonstrates how different graph-theoretical problems can be studied using this implicated portrayal space.

Recommendation

My strong opinion about the thesis is the following.

1. The contribution of the candidate in the thesis is original.
2. Introduction part in the paper is appreciable.
3. The sections are arranged suitably.
4. The contents of the paper will definitely motivate the young researchers in the field of the paper.

 Author Response: We thank the reviewer for the positive comments

 

In view of the above evaluation report, I recommend the paper for publication with the following minor comments:

Comments:

1. I suggest adding a summary at the beginning of each section that summarizes its results

 Author Response: Summary included in each section

2. The preliminaries section is very long. I suggest shortening it by pointing out that some concepts are present in certain references. The preliminaries section is very long. I suggest shortening it by pointing out that some concepts are present in certain references.

 Author Response: Preliminary shortened by giving appropriate references

3. Recent references should be cited in the paper.

 Author Response: Recent references included in the paper

 

Author Response File: Author Response.docx

Reviewer 3 Report

Comments and Suggestions for Authors

The authors of the paper describe their proposed approach for Spherical Linear Diophantine Fuzzy Graphs for Unleashing the Power of Fuzzy Logic for Uncertainty Modeling and Real-World Applications. The topic is interesting and with possible applicability, but now is mostly theoretical. However, the paper needs several improvements:

1) the main contribution and originality should be explained in more detail, which part of the proposal is really new?

2) the motivation of the approach with Spherical Linear Diophantine Fuzzy Graphs needs further clarification, why not another fuzzy approach?

3) discussion of related work in fuzzy logic for decision making should be expanded with more recent work

4) Minor grammar and syntax issues need correction to enhance readability

5) more simulation results and formal comparison of results are needed with real problem data (right now is only data for examples)

6) the conclusions should be extended with more discussion of possible future works

7) More references to related fuzzy decision making papers should be included, like for example: Metaverse integration alternatives of connected autonomous vehicles with self-powered sensors using fuzzy decision making model, An integrated fuzzy multi-measurement decision-making model for selecting optimization techniques of semiconductor materials, A novel spherical decision-making model for measuring the separateness of preferences for drivers' behavior factors associated with road traffic accidents, and similar ones

Comments on the Quality of English Language

Minor English grammar issues should be fixed

Author Response

Spherical Linear Diophantine Fuzzy Graphs for Unleashing the Power of Fuzzy Logic for Uncertainty Modeling and Real-World Applications.

We would like to appreciate the reviewer comments and they are valuable to strengthen the paper.

Comment: The main contribution and originality should be explained in more detail, which part of the proposal is really new?

Response: Main contribution is explained in the updated file.

Comment: the motivation of the approach with Spherical Linear Diophantine Fuzzy Graphs needs further clarification, why not another fuzzy approach?

Response: Motivation of the work is improved in the updated file.

Comment: Discussion of related work in fuzzy logic for decision making should be expanded with more recent work

Response: Discussion of related work is included in the updated file.

Comment: Minor grammar and syntax issues need correction to enhance readability

Response: Grammar and syntax corrections are carried out.

Comment: more simulation results and formal comparison of results are needed with real problem data (right now is only data for examples)

Response: It is true, and simulation results and formal comparison of results with real problem data will be included in our future research which needs more time and effort to prepare.

Comment: the conclusions should be extended with more discussion of possible future works

Response: Future work mentioned in the updated file.

Comment: More references to related fuzzy decision making papers should be included, like for example: Metaverse integration alternatives of connected autonomous vehicles with self-powered sensors using fuzzy decision making model, An integrated fuzzy multi-measurement decision-making model for selecting optimization techniques of semiconductor materials, A novel spherical decision-making model for measuring the separateness of preferences for drivers' behavior factors associated with road traffic accidents, and similar ones

Response: reference related to this study are included in the updated file.

 

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

 Authors were able to make the requested corrections effectively.  It seems that the paper is suitable for publication in Axioms in its current form.

Comments on the Quality of English Language

Minor editing of English language required

Reviewer 3 Report

Comments and Suggestions for Authors

The authors appear to have improved the paper, but the changes are not highlighted in the manuscript (making it difficult to verify all things), for this reason I suggest minor revision.

Comments on the Quality of English Language

Minor errors that could be fixed easily.