Fuzzy Domination Graphs in Decision Support Tasks

Round 1

Reviewer 1 Report

This paper lacks a clear presentation. The terms used in the paper have not been defined before used. My recommendations are to reconsider it after major revision with a clear presentation of the article. An article should contain the all necessary material to understand.

Please write in a clear and easy way and avoid long and difficult sentences.

Author Response

Dear Reviewer,
Thank you for your attention to our article. 
The text of the article has been significantly improved with the necessary explanations and definitions. Tables with data are given. There is a link to GitHub with all the data and calculations. Additional graphs are displayed. Existing graphs are redesigned. 
Some sentences are simplified and split into several sentences. 
Sincerely yours, the Authors

Reviewer 2 Report

The article proposes a method of decision support  which allows to rank the alternatives by taking into account the fuzzy judgments of the DM. The proposed method looks complicated and it is not presented explicitly so that to be easily understood by the reader. Further, the development of the method is not accompanied by a practical application illustrating it and making it more friendly to the reader. In my opinion the manuscript needs a major revision according to these lines. 

Author Response

Dear Reviewer,
Thank you for your attention to our article. 
The methods section has been significantly improved. Additional definitions and explanations are given. The algorithm of the method is modified. It now contains the formulas used in calculations.
A brief description of a practical example in which the method can be applied is given. And the practical usefulness in some applications is described. Two new synthetic examples are given in the results section. The synthetic rather than real data are deliberately chosen. The article adds the explanation that the use of synthetic data is due to the fact that we needed to estimate the accuracy of the solution obtained. The real DM estimates will not allow us to do this, as he himself has difficulty saying whether there are errors in the system's sentences. And in modeling, we ourselves know what kind of error we embedded in the judgments so we can check how the proposed method deals with it.
We have added some additional illustrative graphs. 
Sincerely yours, the Authors

Reviewer 3 Report

1. This manuscript does not have a good framework structure. Moreover, it is suggested that some new proposals should be provided by means of definitions or theorems.

2. In order for the reader to better understand the work of this manuscript, the examples presented in Figures 1 and 2 should be elaborated in the text.

3. The introduction section and the literature review section do not provide a more in-depth and comprehensive discussion of the existing relevant research. More importantly, the innovations and contributions of this manuscript seem to be insufficient.

4. The data for computational experiments should be provided.

5. The sample of 10 alternatives is too small. It is recommended to add more samples to perform computational experiments.

6. Compared with the existing studies, the advantages of the work are not prominent.

• The language quality needs to be improved.

Author Response

Dear Reviewer,
Thank you for your attention to our article. 
1. The text of the article has been significantly improved with the necessary explanations and definitions. The article does not contain theorems for several reasons: 1) Some formulas are based on theorems given in sources 2) Some formulas and algorithm are obtained by simple constructive inference 3)
2.    New figures and tables have been introduced in the text, explaining the sequence of operations. Figures 1 and 2 now have numbers 3 and 4. They are explained. 
3. The literature review has been expanded. The resulting fuzzy ranks are not always easy to use in practice. It will be shown next that the defuzzification of fuzzy dominance relations used in the works studied can lead to an incoherent graph and to non-transitive judgments. The novelty lies in the consistent derivation of Kemeny medians from fuzzy rankings. It allows us to derive a transitive dominance graph or a final generalized fuzzy ranking from them. 
4.    The data is partially embedded in the article. A link to GitHub with all the data and calculations is given.
5.    Calculations for 50 alternatives have been added to the results and software section. Dimensions greater than 50 were not analyzed because they would require a lot of time and powerful computing resources. This is because finding the Kemeny median is an NP-hard problem.
6.    We hope that after the corrections made, the benefits will be seen in the work. 
The English translation has been rechecked and simplified. No serious errors were found. The translation was done by a professional translator and checked by the authors. If you still have the comment "Extensive editing of English language required", please indicate in which sentences mistakes were found.
Sincerely yours, the Authors

Round 2

Reviewer 2 Report

I agree with the authors claim, expressed in their letter of response to my first review report, that the methods section in the revised article has been significantly improved. The authors attempted also to sketch, in a very general and abstract way, how their DM method COULD be applied in a practical paradigm. They add, however, in their letter the following paragraph: "The synthetic rather than real data are deliberately chosen. The article adds the explanation that the use of synthetic data is due to the fact that we needed to estimate the accuracy of the solution obtained. The real DM estimates will not allow us to do this, as he himself [WHO the D-maker?] has difficulty saying whether there are errors in the system's sentences". In other words, the authors admit that it is practically impossible to check the validity of their method in case of real data!

I really appreciate the authors'  hard effort to present their DM method in the best possible way. In my opinion, however, a DM method has some value, if it can be easily applied in practice. Therefore, even if the present method is mathematically correct (it is too complicated to allow me to be sure about this), its practical value is highly questionable. 

 

Author Response

Dear Reviewer,
Thank you for your time in reviewing our article.
The Decision Support System is designed to help the decision maker (DM) in situations where he or she is having difficulty making a decision. For this reason, the DM has difficulty saying whether there are errors in the system's sentences. That is, the difficulties are not caused by the proposed method, but by the nature (essence) of the problem to be solved. With further practical use of the method its evaluation in a real environment is quite possible. For this purpose, it is necessary that it was used by several DM in different situations and after some time on a set of precedents DM can qualitatively estimate its work and estimate consequences of decision-making on its basis. Work on the approbation of the proposed method is planned in the future.
We have made these explanations in the article (lines 356-364).
Sincerely yours, the Authors

Reviewer 3 Report

Accept

No

Author Response

Dear Reviewer,

Thank you for your time in reviewing our article.

Sincerely yours, the Authors