The Consistency between Cross-Entropy and Distance Measures in Fuzzy Sets

Round 1

Reviewer 1 Report

Thank you for inviting me as a reviewer for manuscript titled The Consistency between Cross-entropy and Distance Measures in Fuzzy Sets. The paper presents the relation between the discrimination measure of fuzzy sets and distance measures.

General comments

The paper is really impressive for the efforts made from you to demonstrate the valence of your algorithm. The model is well explained. Methodology is clear. I must congratulate to the authors for very quality research and effort for presenting results. In my opinion the paper is almost ready for the publication in Symmetry.  The paper would be more exiting if you implement below improvements.

Specific comments

- Need to better highlight the novelty of study in the introduction.

- Introduction should be clearly stated research questions and targets first. Then answer several questions:

Why is the topic important (or why do you study on it)?

What are the research questions?

What are your contributions?

Why is to propose this particular methods?

- Page 2, line 36 – Authors write “Information measures are essential to decision-making in information processing…” but in literature review there is no decision making references. I agree with this statement and in my opinion authors should enrich literature review with references presented below:

A sensitivity analysis in MCDM problems: A statistical approach. Decision Making: Applications in Management and Engineering, 1 (2), 51-80. https://doi.org/10.31181/dmame1802050m.

A multicriteria model for the selection of the transport service provider: A single valued neutrosophic DEMATEL multicriteria model. Decision Making: Applications in Management and Engineering, 1 (2), 121-130. https://doi.org/10.31181/dmame1802128l.

- All abbreviations should be introduced in text.

- Remove typos from heading of sections.

- Authors presented impressive results, but without any detailed discussion and numerical examples.

- Focus should be on results section and discussion section. This should be main parts of your paper. Please add numerical examples and show comparisons with existing relations models.

- Conclusion should be rewritten and extended by highlighting the study novelty. More future directions should be presented. Show limitations of the proposed algorithms.

I will review the final version of the paper with pleasure. Once again, congrats to the Authors.

Author Response

List of Changes
Manuscript ID: symmetry-455332
Title: The Consistency between Cross-entropy and Distance Measures in Fuzzy Sets
Authors: Yameng Wang, Han Yang, Keyun Qin
E-mails: [email protected], [email protected], [email protected]
The authors are very 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. Need to better highlight the novelty of study in the introduction.
This comment is constructive. We have revised our manuscript according to the Reviewer’s
comment. The revised part in Introduction is as follows:
In the study, we found that both of the fuzzy discrimination proposed by Bhandari [8] and
improved fuzzy cross-entropy based on discrimination by Shang [9] have similar properties
corresponding to distance measure such as non-negativity, symmetry and if cross-entropy(distance)
between two fuzzy sets is 0 if and only if the two sets are coincide. Furthermore, the decision
principle of cross-entropy and distance applied to decision-making are the same. That is to say, in
the process of decision-making, among all the choices we finally choose the one with the smallest
cross-entropy(distance) from the ideal one. On the basis of above analysis we tend to study their
relationships between cross-entropy and distance measure. There is no research on their
relationships previously. So, we mainly proved that the fuzzy discrimination proposed by Bhandari
[8] and improved fuzzy cross-entropy defined by Shang [9] and neutrosophic cross-entropy
proposed by Ye [12] based on discrimination are distance measures in fact.
2. Introduction should be clearly stated research questions and targets first. Then answer several
questions:
Why is the topic important (or why do you study on it)?
Firstly, we found that the cross-entropy E(A,B) (A, B are fuzzy sets or neutrosophic sets) can be
applied to decision-making just because of:
(4) ( , ) ( , ).
(3) ( , ) ( , );
(2) ( , ) 0 ;
(1) ( , ) 0;
E A B E Ac Bc
E A B E B A
E A B A B
E A B


  

In my opinion, the cross-entropy is used to express the discrimination between two objects, and
only these conditions are less convincing. However, the conditions of distance measure are more
convincing compared with cross-entropy, especially the following condition of distance D(A,B)(A,
B are fuzzy sets or neutrosophic sets) that is:
If A  B  C, then D(A,C)  D(A, B), D(A,C)  D(B,C).
So, we think that we can verify that the fuzzy discrimination and cross-entropy satisfy this
condition. If they meet this condition, it’s more persuasive in decision-making.
What are the research questions?
The research question is based on the relationships between cross-entropy and distance measure
in fuzzy sets and neutrosophic sets. In the study, we found that the fuzzy discrimination proposed
by Lin and improved fuzzy cross-entropy by Shang and neutrosophic cross-entropy proposed by Ye
all have similar properties with distance measure such as non-negativity, symmetry and if
cross-entropy(distance) between two fuzzy sets is 0 if and only if the two sets are coincide.
Furthermore, the decision principle(among all the choices we finally choose the one with the
smallest cross-entropy(distance) from the ideal one) of cross-entropy and distance applied in
decision-making are the same. On the basis of above analysis we tend to study the relationships
between cross-entropy and distance, and we mainly proved that the fuzzy discrimination proposed
by Bhandari [8] and improved fuzzy cross-entropy proposed by Shang [9] and neutrosophic
cross-entropy proposed by Ye [12] are also distance measures in our manuscript.
What are your contributions?
In the study, we firstly proved that the fuzzy discrimination proposed by Bhandari [8] satisfies all
the conditions of distance measure. In other words, it actually is a kind of distance measure. In the
next, we continue to prove that the fuzzy cross-entropy( formula (7) in our manuscript) improved
by the above fuzzy discrimination is a distance measure as well. Furthermore, the neutrosophic
cross-entropy proposed by Ye [12] also is a distance measure in the fact that it satisfies the
conditions of distance measure.
Why is to propose this particular methods?
Our research is based on the relationships between cross-entropy and distance measure in fuzzy
sets and neutrosophic sets, we mainly proved that the fuzzy discrimination proposed by Bhandari
[8] and improved fuzzy cross-entropy proposed by Shang [9] and neutrosophic cross-entropy
proposed by Ye [12] are also distance measures in our manuscript. Our research mainly focuses
on theoretical research and there is no new method proposed in our manuscript.
The Reviewer’s comment is very constructive. We have revised our manuscript according to the
Reviewer’s comment. And added some contents in introduction as follows:
The research question is based on the relationships between cross-entropy and distance measure
in fuzzy sets and neutrosophic sets. In the manuscript, we mainly proved that the fuzzy
discrimination proposed by Bhandari [8] and improved fuzzy cross-entropy proposed by Shang [9]
and neutrosophic cross-entropy proposed by Ye [12] are distance measures in the fact that they
satisfied all the conditions of distance measure. In section 2, we mainly introduced some relevant
knowledge, and we mainly proved that the fuzzy discrimination measure satisfies all the conditions
of distance measure. In other words, it actually is a kind of distance measure. In the section 3, we
mainly proved the fuzzy cross-entropy satisfies all the conditions of distance measure, and
cross-entropy in single value neutrosophic sets also is a distance. That is to say cross-entropy
measure is consistent with distance measures.
3. Page 2, line 36 – Authors write “Information measures are essential to decision-making in
information processing…” but in literature review there is no decision making references. I agree
with this statement and in my opinion authors should enrich literature review with references
presented below:
A sensitivity analysis in MCDM problems: A statistical approach. Decision Making: Applications
in Management and Engineering, 2018, 1(2), 51-80. https://doi.org/10.31181/dmame1802050m.
A multicriteria model for the selection of the transport service provider: A single valued
neutrosophic DEMATEL multicriteria model. Decision Making: Applications in Management
and Engineering, 1 (2), 121-130. https://doi.org/10.31181/dmame1802128l.
This comment is constructive. We added some references and introduced them in Introduction as
follows:
Such as Liu et al. [14] applied single value neutrosophic number to Decision-making Trial and
Evaluation Laboratory Method and presented the model of SVNN-DEMATEL. Irik et al. [12]
provided a analysis about some multi-criteria decision-making (MDM) methods and the finally
selection and presented a result consistency evaluation model. Tu et al. [13] introduced a
symmetry simplified neutrosophic measures and applied it to decision-making.
Irik, Mukhametzyanov; Dragan, Pamučar. A sensitivity analysis in MCDM problems: A statistical
approach. Decision Making: Applications in Management and Engineering, 2018, 1(2), 51-80.
Angyan, Tu; Ye, J.; Bing, Wang. Symmetry Measures of Simplified Neutrosophic Sets for Multiple
Attribute Decision-Making Problems. symmetry, 2018, 10, 144, doi:10.3390/sym10050144.
Feng, Liu; Guan, Aiwu; Vesko, Lukovac; Milena, Vukić. A multicriteria model for the selection of
the transport service provider: A single valued neutrosophic DEMATEL multicriteria model.
Decision Making:Applications in Management and Engineering, 2018, 1, 2, 121-130.
4. All abbreviations should be introduced in text, Remove typos from heading of sections.
We are sorry to make these mistakes. We checked whole manuscript and these mistakes have been
revised as follows:
“fuzzy discrimination” has been revised to “Fuzzy discrimination” in heading of sections 2,
“fuzzy cross-entropy” has been revised to “Fuzzy cross-entropy” in heading of sections 3, and
“neutrosophic cross-entropy” has been revised to “Neutrosophic cross-entropy” in heading of
sections 4.
5. Authors presented impressive results, but without any detailed discussion and numerical
examples.
This comment is constructive. Our manuscript is mainly focused on the theoretical research, and
our results are the proof of the fuzzy discrimination and fuzzy cross-entropy and neutrosophic
cross-entropy mentioned in the paper are distance measures, and the conclusions are exactly exist
in Section 2, Section 3 and Section 4. We think careful the review, and added some examples
behind the Theorem 2, 5 and 7 respectively as follows:
Example 1. Let X be a space of universe course, M, N,T FS(X ) which
M  { x,0.5 x X}, N  { x,0.7 x X},T  { x,0.9 x X}, it’s obvious that
M  N  T , and we can get: E(M, N)  0.1695, E(N,T)  0.2700, E(M,T)  0.8789
that is E(M,T)  E(M, N), E(M,T)  E(N,T).
Example 2. Let X be a space of universe course, M, N,T FS(X ), which
M  { x,0.5 x X}, N  { x,0.7 x X},T  { x,0.9 x X}, it’s obvious that
M  N  T , and we can get ( , ) 0.042, ( , ) 0.0648, ( , ) 0.2035, 2 2 2 E M N  E N T  E M T 
that is ( , ) ( , ), ( , ) ( , ). 2 2 2 2 E M T  E M N E M T E N T
Example 3. Let X be a space of universe course, M, N,T SVNS(X ), which
M  {x,0.5,0.3,0.7 x X}, N  {x,0.7,0.2,0.5 x X},T  {x,0.8,0.1,0.1 x X}, it’s
obvious that M  N  T , and we can get:
( , ) 0.042, ( , ) 0.0134, ( , ) 0.1013, 2 2 2 ET M N  ET N T  ET M T 
that is ( , ) ( , ), ( , ) ( , ). 2 2 2 2 ET M T  ET M N ET M T  ET N T
( , ) 0.0134, ( , ) 0.0199, ( , ) 0.0648, 2 2 2 EI M N  EI N T  EI M T 
that is ( , ) ( , ), ( , ) ( , ). 2 2 2 2 EI M T  EI M N EI M T  EI N T
( , ) 0.0420, ( , ) 0.2035, ( , ) 0.4101, 2 2 2 EF M N  EF N T  EF M T 
that is ( , ) ( , ), ( , ) ( , ). 2 2 2 2 EF M T  EF M N EF M T  EF N T
( , ) ( , ) ( , ) ( , ) 0.5762, ( , ) 0.0974, ( , ) 0.2368 3 2 2 2 3 3 E M T  ET M T  EI M T  EF M T  E M N  E N T 
that is ( , ) ( , ), ( , ) ( , ). 3 3 3 3 E M T  E M N E M T E N T
6. Focus should be on results section and discussion section. This should be main parts of your
paper. Please add numerical examples and show comparisons with existing relations models.
This comment is constructive. Our manuscript is mainly focused on the theoretical research, and
our results are the proof of the fuzzy discrimination and fuzzy cross-entropy and neutrosophic
cross-entropy mentioned in the manuscript are distance measures actually, and the conclusions
are exactly exist in Section 2, Section 3 and Section 4, so, there is no dedicated part to discuss our
results. There is no study about the relation between cross-entropy and distance measures before.
7. Conclusion should be rewritten and extended by highlighting the study novelty. More future
directions should be presented. Show limitations of the proposed algorithms.
This comment is constructive. Our article mainly focuses on theoretical research. There is no
algorithm is involved. We have revised our manuscript according to the Reviewer’s comment. The
revised Conclusion is as follows:
On account of these similar properties between distance measure and cross-entropy such as
non-negativity, symmetry and if cross-entropy(distance) between two fuzzy sets is 0 if and only if
the two sets are coincide. We tend to study their relationships, and the decision principle of
cross-entropy and distance applied in decision-making are the same. That is, among all the
choices we finally choose the one with the smallest cross-entropy(distance) from the ideal solution.
Based on above analysis, we mainly proved that the fuzzy discrimination and fuzzy cross-entropy
and neutrosophic cross-entropy improved by fuzzy discrimination are distance measures in fact.
That is to say the symmetry cross-entropy mentioned in the paper is consistent with distance
measure. In the next, we will try to simplify the formula and proposed a new improvement. It is
precisely because the cross-entropy formulas are composed of logarithmic functions that make the
calculation complicated.

Author Response File: Author Response.pdf

Reviewer 2 Report

There is no numerical example confirming the correctness of the thesis.

It is necessary to present an example, as it is done in most of the works cited by the authors of this article, for example as in item 13 of the References.

Author Response

List of Changes
Manuscript ID: symmetry-455332
Title: The Consistency between Cross-entropy and Distance Measures in Fuzzy Sets
Authors: Yameng Wang, Han Yang, Keyun Qin
E-mails: [email protected], [email protected], [email protected]
Reviewer #2:
There is no numerical example confirming the correctness of the thesis. It is necessary to present
an example, as it is done in most of the works cited by the authors of this article, for example as in
item 13 of the References.
This comment is constructive. Our manuscript is mainly focused on the theoretical research and
we think careful the review, added some examples behind the Theorem 2, 5 and 7 respectively as
follows:
Example 1. Let X be a space of universe course, M, N,T FS(X ), which
M  { x,0.5 x X}, N  { x,0.7 x X},T  { x,0.9 x X}, it’s obvious that
M  N  T , and we can get E(M, N)  0.1695, E(N,T)  0.2669, E(M,T)  0.8789
that is E(M,T)  E(M, N), E(M,T)  E(N,T).
Example 2. Let X be a space of universe course, M, N,T FS(X ), which
M  { x,0.5 x X}, N  { x,0.7 x X},T  { x,0.9 x X}, it’s obvious that
M  N  T , and we can get ( , ) 0.042, ( , ) 0.0648, ( , ) 0.2068, 2 2 2 E M N  E N T  E M T 
that is ( , ) ( , ), ( , ) ( , ). 2 2 2 2 E M T  E M N E M T E N T
Example 3. Let X be a space of universe course, M, N,T SVNS(X ), which
M  {x,0.5,0.3,0.7 x X}, N  {x,0.7,0.2,0.5 x X},T  {x,0.8,0.1,0.1 x X}, it’s
obvious that M  N  T , and we can get:
( , ) 0.042, ( , ) 0.0134, ( , ) 0.1013, 2 2 2 ET M N  ET N T  ET M T 
that is ( , ) ( , ), ( , ) ( , ). 2 2 2 2 ET M T  ET M N ET M T  ET N T
( , ) 0.0134, ( , ) 0.0199, ( , ) 0.0648, 2 2 2 EI M N  EI N T  EI M T 
that is ( , ) ( , ), ( , ) ( , ). 2 2 2 2 EI M T  EI M N EI M T  EI N T
( , ) 0.0420, ( , ) 0.2035, ( , ) 0.4101, 2 2 2 EF M N  EF N T  EF M T 
that is ( , ) ( , ), ( , ) ( , ). 2 2 2 2 EF M T  EF M N EF M T  EF N T
( , ) ( , ) ( , ) ( , ) 0.5762, ( , ) 0.0974, ( , ) 0.2368 3 2 2 2 3 3 E M T  ET M T  EI M T  EF M T  E M N  E N T 
that is ( , ) ( , ), ( , ) ( , ). 3 3 3 3 E M T  E M N E M T E N T

Author Response File: Author Response.pdf

Reviewer 3 Report

The paper is well written and presented. The scientific part is relevant.

Author Response

Thank you very much for your review and affirmation

Reviewer 4 Report

The study focuses on The Consistency between Cross-entropy and Distance Measures in Fuzzy Set.  The paper shows good matechmatical background.  

My main attention to the paper is that: 

1. Space between word and reference <Verma[17,18]...> 

2. Page 2, line 72. ...2. fuzzy discrimination please change to 2. Fuzzy discrimination...

3. Page 3, line 97 new sentence..... We can redef.....

4. Page 4, please connect line 112 with 113. 

5. Page 4, please connect line 133 with 134. 

6. Page 4, line 135 new sentence..... when n.....

7. Page 5, please connect line 155 with 156. 

8. Page 5, line 157 new sentence..... when m.....

9. Space between word and reference <Smarandache[4,5]...> line 243

10. The paper needs additional formatting in some places. 

Author Response

List of Changes

List of Changes
Manuscript ID: symmetry-455332
Title: The Consistency between Cross-entropy and Distance Measures in Fuzzy Sets
Authors: Yameng Wang, Han Yang, Keyun Qin
E-mails: [email protected], [email protected], [email protected]
Reviewer #3:
1. Space between word and reference <Verma[17,18]...>
2. Page 2, line 72. ...2. fuzzy discrimination please change to 2. Fuzzy discrimination...
3. Page 3, line 97 new sentence..... We can redef.....
4. Page 4, please connect line 112 with 113.
5. Page 4, please connect line 133 with 134.
6. Page 4, line 135 new sentence..... when n.....
7. Page 5, please connect line 155 with 156.
8. Page 5, line 157 new sentence..... when m.....
9. Space between word and reference <Smarandache[4,5]...> line 243
10. The paper needs additional formatting in some places.
We are sorry to make these mistakes. We checked whole paper carefully and revised these mistakes
as follows:.
1. Added a space in front of reference.
2. “fuzzy discrimination” has been revised to “Fuzzy discrimination”.
3. “we” has been revised to “We”.
4. We have connected the line 112 with 113 with “,”.
5. “since...” has been revised to “Since...” in line 134 .
6. “when...” has been revised to “When...” in line 135 .
7. “since...” has been revised to “Since...” in line 156 .
8. “when...” has been revised to “When...” in line 157 .
9. Added a space between word and reference.
10. “fuzzy cross-entropy” has been revised to “Fuzzy cross-entropy”, and “neutrosophic
cross-entropy” has been revised to “Neutrosophic cross-entropy”.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Authors have  improved the previous version of this paper.

Reviewer 2 Report

The authors took into account all the imperfections of the previous version of the manuscript and the article is suitable for publication.