Review Reports - New Similarity and Entropy Measures of Interval Neutrosophic Sets with Applications in Multi-Attribute Decision-Making

Round 1

Reviewer 1 Report

New Similarity And Entropy Measures of Interval Neutrosophic Sets With Applications in Multi-attribute Decision Making:

The abstract is very detailed, the authors should point out only the main results of their study and not the organisation of the paper. This is usually provided in the last part of the Introduction section.

Definition 8 must be accompanied by the corresponding citation, because it is not the work of the authors. Also, after Definition 8, there are enumerated several similarity measures, but no citation is given, are they proposed by the authors? Or they exist in the specialised literature and the authors just remind them?

The Theorems 3 and 4 are logical consequences of all the theory presented in the previous section, I do not understand the need of a new section for them, that is the need of Section 4.

The numerical example is very weak, it seems to be written in rush. Please consider the existing similarity measures and then the authors should apply their method in order to illustrate the advantages of their method.

The following phrases should be rewritten:

“There is an originally inclusion relationship between INS”

and

“We note that in the original inclusion relationship …”

The authors could evidence the originality of their paper using phrases of the following form: “in this study we propose … at this moment, the inclusion relationship existing in the literature … “

A sentence can not begin with an “And” as in “And the existing similarity measure is mainly designed …”

There are two “and”s in the first part of the Introduction section.

There are some problems with grammatical agreement such as: “the neutrosophic sets has”.

There are some spaces needed, after punctuation marks or before the citation marks, etc.

Author Response

List of Changes

Manuscript ID: symmetry-452676

Type of manuscript: Article

Title: New Similarity And Entropy Measures of Interval Neutrosophic Sets With Applications in Multi-attribute Decision Making

Authors: Han Yang *, Xiaoman Wang, Keyun Qin *

Received: 9 February 2019

E-mails: yanghan@163.net, xiaomanwang@aliyun.com, keyunqin@263.net

The authors are grateful to the editor and referee for his/her constructive suggestions on our paper. We have revised our paper according to the referee’s comments. The concrete revision is as follows.

Reviewer #1:

1. The abstract is very detailed, the authors should point out only the main results of their study and not the organization of the paper. This is usually provided in the last part of the Introduction section.

This comment is constructive. We have revised our manuscript according to the Reviewer’s comment. The revised Abstract is as follows:
Information measures play an important role in the interval neutrosophic sets (INS) theory. The main purpose of this paper is to study the similarity and entropy of INS and its application in multi-attribute decision making. We propose a new inclusion relation between interval neutrosophic sets where the importance of the three membership functions may be different. Then, we propose the axiomatic definitions of the similarity measure and entropy of the interval neutrosophic set (INS) based on the new inclusion relation. Based on the Hamming distance, cosine function and cotangent function, some new similarity measures and entropies of INS are constructed. Finally, based on the new similarity and entropy, we propose a multi-attribute decision making method and illustrate that these new similarity and entropy are reasonable and effective.

2. Definition 8 must be accompanied by the corresponding citation, because it is not the work of the authors. Also, after Definition 8, there are enumerated several similarity measures, but no citation is given, are they proposed by the authors? Or they exist in the specialised literature and the authors just remind them?

We are sorry to make these mistakes. We checked whole manuscript and these mistakes have been revised, and added the following paper in the References and cited it.

Ye, J. Some distances, similarity and entropy measures for interval-valued neutrosophic sets and their relationship. International Journal of Machine Learning and Cybernetics. 2019, 10(2), 347-355.

3. The Theorems 3 and 4 are logical consequences of all the theory presented in the previous section, I do not understand the need of a new section for them, that is the need of Section 4.

The Reviewer’s comment is correct and constructive. The Theorems 3 and 4 are logical consequences of Theorems 1 and 2 respectively, So we have merged the Section 3 and Section 4 together, and added comments before Theorems 3 as follows:

By aggregating the similarities and entropies of interval neutrosophic values, we have the following similarity and entropy of interval neutrosophic sets.

4. The numerical example is very weak, it seems to be written in rush. Please consider the existing similarity measures and then the authors should apply their method in order to illustrate the advantages of their method.

This comment is constructive. We added a subsection (Subsection 4.1), and made a comparison of the existing methods in Ye[11] and Sahin[19]. Our conclusion is different form [11], due to the differences in inclusion relation and in the best choice, we have different conclusions. Moreover, in [11], the distances between INSs are first calculated and any difference is then amplified in the results using criteria weights, which cause a distortion in the similarity between an alternative and the ideal alternative. While in [19], Sahin defined the interval neutrosophic cross-entropy in two different ways, which are based on extension of fuzzy cross-entropy and single-valued neutrosophic cross-entropy. Additionally, two multi-criteria decision-making methods using the interval neutrosophic cross-entropy between an alternative and the ideal alternative are developed in order to determine the order of the alternatives and choose most preferred one(s). And Sahin ranked the alternatives as A_{4}>A_{1}>A_{2}>A_{3}. For this example, by using the new similarity measure proposed in this paper, we obtained the same ranking order of alternatives as in [19]. It shows that the new similarity measures proposed in this paper are effective and efficient.

5. The following phrases should be rewritten: “There is an originally inclusion relationship between INS” and “We note that in the original inclusion relationship …”The authors could evidence the originality of their paper using phrases of the following form: “in this study we propose … at this moment, the inclusion relationship existing in the literature … “

A sentence can not begin with an “And” as in “And the existing similarity measure is mainly designed …” There are two “and”s in the first part of the Introduction section.

There are some problems with grammatical agreement such as: “the neutrosophic sets has”.

There are some spaces needed, after punctuation marks or before the citation marks, etc.

We are sorry to make these mistakes. We checked whole manuscript and these mistakes have been revised.

Author Response File: Author Response.pdf

Reviewer 2 Report

See enclosed file

Comments for author File: Comments.pdf

Author Response

List of Changes

Manuscript ID: symmetry-452676

Type of manuscript: Article

Title: New Similarity And Entropy Measures of Interval Neutrosophic Sets With Applications in Multi-attribute Decision Making

Authors: Han Yang *, Xiaoman Wang, Keyun Qin *

Received: 9 February 2019

E-mails: yanghan@163.net, xiaomanwang@aliyun.com, keyunqin@263.net

Reviewer #2:

1. The text in lines 117-118 is unclear. Thesymbol at the same time means interval neutrosophic value or its element. At the same time the symboldenotes the family of all interval neutrosophic values and a single interval neutrosophic value. The whole fragment should be intensively rewritten. Only then its validity will be able to be assessed.

In the existing literature, symbols the interval neutrosophic value as well as element in the universe. It is distinguished according to the context. This notation is followed in this paper. In addition, it is clearly pointed out that is the set of all interval neutrosophic values.

2. The Definition 10 in the lines 191-198 should be discussed.

There, the distinct set is not defined. Based on the semantic analysis, I suppose that it is following interval neutrosophic values

Then the conditions (N1) and (N2) are contradictory because of the interval neutrosophic values

is not a distinct set.

Moreover, the authors should check extension principles:

a) If interval neutrosophic values is a fuzzy set i.e.

then the Definition 10 is equivalent the definition of entropy measure given by de Luca and Termini [15].

b) If interval neutrosophic values is an intuitionistic fuzzy set i.e.

then the Definition 10 is equivalent the definition of entropy measure given in [ Piasecki K., (2017) Some remarks on axiomatic definition of entropy measure, Journal of Intelligent and Fuzzy Systems 33(3),https://doi.org/10.3233/JIFS-15364 ].

The results of this discussion may impact on the entire content of the article.

We are sorry to make these mistakes. An interval neutrosophic value is distinct if or and or . We revised (N1) according to this definition. In addition, (N2) should be “if and only if .” The revised definition is as follows:

Definition 10: Let , the real function E is an entropy of interval neutrosophic value, if E satisfies the following conditions:

(N1)E(x)=0 if and only if or and or ;

(N2)E(x)=1 if and only if ;

(N3)E(x)=E(x^{c});

(N4)Let ,, then , , that is, x is more ambiguous than y, if , when, or if , when .

Furthermore, the comments a) and b) are correct. We added a) and b) after Definition 10. Also we added the following paper in the References and cited it.

Piasecki K., (2017) Some remarks on axiomatic definition of entropy measure, Journal of Intelligent and Fuzzy Systems 33(3),https://doi.org/10.3233/JIFS-15364.

3. In my opinion, all proofs may by simplified with use of functional analysis in the Banach space.

Actually, all proofs can be indeed presented by using the notations of Banach space. However, if the proofs are presented by using the notations of Banach space, we need introduce firstly the theory of Banach space. It will dilute the main content of this article. So we have not used the notations of Banach space.

Moreover, the article contains a lot of linguistic, stylistic, punctuation and editorial errors.

In line 129: S_4(x,y) is defined incorrectly. Instead of the second and fourth "+", commas should be used.

In lines 133 and 134 S_2(x,y), S_2(y,z) and S_2(x,z) are calculated incorrectly.

Proofs of Theorem 1 and Theorem 2 are difficult to read. They should be shorted and simplified.

We are sorry to make these mistakes. We checked whole References and these mistakes have been revised.

In line 197 the symbol <=_2 should be explained.

This comment is constructive. We explained the symbol <=_2 in Definition 7:

Let and be the interval neutrosophic values. x<=< span="">_2 y if and only if one of the following three conditions is true:

(1) and;

(2) and;

(3) and and .

There is no proof or at least comments of Theorem 3 and Theorem 4.

By aggregating the similarities and entropies of interval neutrosophic values, we have the following similarity and entropy of interval neutrosophic sets.

Notations A_11, A_12,...used in lines 265-267 should be explained.

This comment is constructive. We explain Aij indicates the neutrosophic value in line i column j.

In line 267, the Authors refer to the formula (3), which is not in the article.

We are sorry to make these mistakes. We checked whole References and these mistakes have been revised.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The present form of the article can be published in the Symmetry journal.

Reviewer 2 Report

All comments from the reviewer have been included. I accept the article in the presented form.