Dear Chiranjib

Very interesting your question. However allow me to correct what you say that any MCDM method gives ranking that vary with the normalization methods. Not with SIMUS. I have run differeent tests using four differente normalization process, namely

a) Dividing each performance value in a row by thye sum of all values in that row

b) Dividing each performance value in a row by the highest value in that row

c) Using the Euclidean formula

d) Using the Max/Min formula

The same problem is solved y SIMUS using any of the three first methods gives the same results, that is scores and rankings.

There is little difference with the fourth method, which by the way is very useful, in the sense thatr the the first methods give rankins with little discrimination, while the fourth, because its structure discriminates in a larger extend

This is useful when you get ties, that is alternatives with equal or very similar values. Try to use Max/Min and chances are that the ties disappear.

There are other methods of normalization with which I am not familiar and may be with one of them SIMUS will give different rankings. However the four methods mentioned are the most commonmly sedu

Thank you Adis Puška Sir and Nolberto Munier Sir for your elaborate discussion.

Suggest reviewing this paper for MCDM methods:

https://www.sciencedirect.com/science/article/abs/pii/S092583881732827X

It is a classical approach to keep the MCDM basic equation constant, which is the next step after the normalization stage, and to change the normalization techniques for pre-normalization (first decision matrix transformation). So you can also observe the interesting variation of the results. Normalization techniques sometimes do not affect the MCDM results at all, but sometimes they can turn the results inside out. Therefore, I think the innocence of these techniques should be questioned. Depending on the data type and problem type, the selection of normalization techniques may also vary. Even the number of alternatives and criteria affects a fair choice. They are such volatile and unresolved techniques that there have also been those who claim that they sometimes cause “rank reversal” for MCDM rankings. Nowadays, fixed/static normalization techniques are generally used in every scenario for MCDM methods, which is wrong. Because there is no normalization that fits every scenario. For example, “vector normalization” technique is used for TOPSIS. Whereas Max, Min-Max, Sum, Logarithmic etc all should be tested. The other issue is that Min-Max is often preferred for artificial intelligence and machine learning applications, which is also wrong. Let’s not forget that normalization homogenizes different units ams does not always guarantee a fair conversion. Sometimes the thing that transforms becomes far from the original and turns into something completely different. Sometimes things are good when the water is steaming, but sometimes the coal turns into ashes.

MB- It is a classical approach to keep the MCDM basic equation constant, which is the next step after the normalization stage, and to change the normalization techniques for pre-normalization (first decision matrix transformation). So, you can also observe the interesting variation of the results

NM- Sorry, I don’t understand. Could you please explain what you intend to say? Which is the classical approach? What is the basic equation and pre-normalization? Could you put an example?

MB- Normalization techniques sometimes do not affect the MCDM results at all, but sometimes they can turn the results inside out.

NM- All normalization techniques involve dividing each original aij value of a criterion by its norm or distance. Using different norms such as Manhattan, Infinite, Euclidean, etc. will produce different aij values in each criterion. However, if f or a certain criterion ‘i’ you use the same norm for that criterion, all aij values will change in the same sense and proportionally.

If it is true that using different norms you get different new aij, the original relationship, measured by the distance in percentages between two values, does not change. From here, you can deduct that normalization does not have any participation in giving different results according to the norm used, and has nothing to do either with RR. I can send you an example if you wish. The fact that effectively TOPSIS uses the Euclidean norm, which in my opinion is the best, because it is the shortest distance, does not mean that you cannot use another.

MB- Therefore, I think the innocence of these techniques should be questioned.

NM- ‘Innocence’? This is new for me.

MB- Depending on the data type and problem type, the selection of normalization techniques may also vary.

NM- Why? On what grounds?

MB- Even the number of alternatives and criteria affects a fair choice.

NM- Please tell me the source or the reasoning for that assertion.

MB- They are such volatile and unresolved techniques that there have also been those who claim that they sometimes cause “rank reversal” for MCDM rankings.

NM- ‘Unsolved techniques?’ There is nothing to solve with them. By the way, it reminds me that you are one of those researchers that support the idea that normalization produces RR.

MB- Nowadays, fixed/static normalization techniques are generally used in every scenario for MCDM methods, which is wrong. Because there is no normalization that fits every scenario.

NM- Why not? Please, don’t use words, rather explain. Why is it wrong?

MB- For example, “vector normalization” technique is used for TOPSIS. Whereas Max, Min-Max, Sum, Logarithmic etc. all should be tested.

NM- Please, illustrate me why you can’t you use another normalization type in TOPSIS?

In an example, I have 3 alternatives and 10 criteria using both Euclidean or vector norm, and Sum of values (Manhatan norm). Naturally the normalized results are different, but the proportion is strictly maintained between normalized values. If you want, I can send it to you.

MB- The other issue is that Min-Max is often preferred for artificial intelligence and machine learning applications, which is also wrong. Let’s not forget that normalization homogenizes different units and does not always guarantee a fair conversion. Sometimes the thing that transforms becomes far from the original and turns into something completely different. Sometimes things are good when the water is steaming, but sometimes the coal turns into ashes.

NM – I don’t know on AI and ML, but in MCDM, normalization does no homogenize the values in each criterion, it only converts them in 0-1 scale. Please, demonstrate what you say.

Interesting your quotation. Is it a Turkish idiom? And what does it mean?

My friend, I enjoy your comments, but usually you make many assertions as if were truth but you never demonstrate anything. In so doing you could make mistakes, the same as me, but this is a technical discussion, and thus, please document what you say. If you don’t, there is nothing we can discuss because you don’t give any explanation. Again, I also probably make mistakes, have wrong assumptions and faulty conclusions, but at least I try to find some answers.

Dear Munier,

We've actually discussed these issues in depth with you in a way that hasn't been concluded many times before. I thank you for this. Below are some references that support my claims. Of course, some of my claims are unique to me, and they will perhaps be proven with a publication one day. But I've invited you to explain before, and I'll say it again: Why do normalization techniques sometimes produce very different sequences in an MCDM? For example, why don't Vector based TOPSIS, Sum based TOPSIS, Max based TOPSIS and Min-Max based TOPSIS produce 100% same ranking? Sometimes zero and negative value paralyze some normalization techniques. But even if we ensure that there are no zero and negative values ​​in a data set, the results may still not be 100% the same. According to you, the difference is very small and is valid after the comma. However, for some datasets, our findings also show that the difference may be too large. So how and with what do you explain this unfair data conversion? A question for you: Can outliers be effective in this?

The following article may be eye-opening:

Article Elimination of the Domains’ Displacement of the Normalized V...

Dear Munier,

MB- We've actually discussed these issues in depth with you in a way that hasn't been concluded many times before. I thank you for this.

NM- We never discussed these issues in depth, because I never got straight answers from you. This is not a scientific discussion, rather a tee gathering

MB- Below are some references that support my claims. Of course, some of my claims are unique to me, and they will perhaps be proven with a publication one day.

NM – Where are those references? You only mention procedures. You speak as proving what you say perhaps in publications one day. Why don’t you prove them NOW? This is the way a scientific discussion is held.

It is legitimate for you to make claims, but another thing is to develop them. You express your ideas and feelings, as is your right, but never support them with rational arguments. You just write your ideas, and that is all. How can we discuss if you do not provide arguments? If you don’t support your claims?

MB- But I've invited you to explain before, and I'll say it again: Why do normalization techniques sometimes produce very different sequences in an MCDM? For example, why don't Vector based TOPSIS, Sum based TOPSIS, Max based TOPSIS and Min-Max based TOPSIS produce 100% same ranking?

NM-That is true, you ask, and sometimes there are very interesting questions. But it does not mean that I must know the answers. I am not the Delphi Oracle.

OK, here is my answer. Normalization is only a ratio between two values, that is, the value to be normalized and the norm of the criterion it belongs to. I explained this yesterday, Consequently, if you divide the components of each criterion using the same norm, the values will change, but mathematics says that is it does not alter the proportion between values. In my opinion, rankings change, no for the ratios but because the different subjective assumptions made in most methods.

Are you satisfied with my explanation? It can be wrong, sure, but it is at least an explanation. Which is yours?

Now, if you use normalization by min/max, chances are that the ranking changes because in this method you have a ratio of differences, not of numbers.

MB- Sometimes zero and negative value paralyze some normalization techniques.

NM – They shouldn’t

MB- But even if we ensure that there are no zero and negative values ​​in a data set,

NM- Where did you get that? You can have zero values as well as negatives. For instance, installing a domestic waste incinerator, produces negative effects in the neighborhood, and thus, they must be negative. The fact that most MCDM methods cannot handle them, is not proof that they are inexistent.

Regarding zeros, if you are evaluating alternatives like fossil fuel plants, and PV farms, and there is a criterion like ‘Unitary emissions of NOx”, there will be a positive value for the fossil plant and zero for PV plant.

MB- the results may still not be 100% the same. According to you, the difference is very small and is valid after the comma.

NM- When did I say that?

MB- However, for some datasets, our findings also show that the difference may be too large. So how and with what do you explain this unfair data conversion? A question for you: Can outliers be effective in this?

NM- Can you describe your findings?

Rewarding outliers, I am not aware that they exist in MCDM problems, they pertain to regression analysis. Probable you refer to extreme values in a criterion. Their action is to have a high value of quality of information, because the dispersion, and thus, giving a high significance to said criterion. But in this case, it affects equally all alternatives.

I suggest you to put an example, send it to me and we together try to find what happens. What do you think?

MB- The following article may be eye-opening:

Article Elimination of the Domains’ Displacement of the Normalized V...

Possibly, if I can read it in full. I already asked for a copy

Dear Nolberto,

( according to your)Normalization can be a simple ratio for Sum, Max, Min-Max. But I guess things are a bit more complicated for Z-Score, Logarithmic, Vector normalization.

Well, I asked you why sometimes results and ranking differ when techniques such as min-max, max, sum or vector normalization are used for the same MCDM method. According to you, the rankings differ because of the different subjective assumptions made in most methods. If so, make your claim scientific and prove it! For example, what is the relationship between the assumptions in TOPSIS and the results of the normalization technique?

Actually, things are a little different and more complicated.. You can check the following reference:

https://dmame-journal.org/index.php/dmame/article/view/493

According to Mukhametzyanov, the answer is: “The displacement of the domains of the normalized values ​​of various attributes relative to each other and the local priorities of the alternatives are the main factors that change the rating when using various normalization methods”. According to the author, Normalization techniques are actually a transformative. Sometimes these converters cannot preserve the tendencies of the natural values ​​of the criteria. they cannot transfer the contribution of the criteria to the final performance indicator in a way that preserves the original.

Dear Nolberto,

MB - (according to your) Normalization can be a simple ratio for Sum, Max, Min-Max. But I guess things are a bit more complicated for Z-Score, Logarithmic, Vector normalization.

NM - Z-score is the ratio between a difference (value and mean) and the standard deviation. Logarithmic normalization is the ratio between two logarithms. Vector normalization, is in my opinion, a misnomer, for it is a ratio between a value and a norm, which is the Pythagorean distance, or vector, or Euclidean.

MB - (Well, I asked you why sometimes results and ranking differ when techniques such as min-max, max, sum or vector normalization are used for the same MCDM method. According to you, the rankings differ because of the different subjective assumptions made in most methods. If so, make your claim scientific and prove it! For example, what is the relationship between the assumptions in TOPSIS and the results of the normalization technique?

NM- TOPSIS uses normally what you call vectorial normalization, perhaps because it looks for s shortest distance to the positive ideal point (as the crow flies). This normalization complies with that requirement, because the minimum distance between two points is given by a straight line, which happen to be the hypothenuse of two orthogonal values.

In TOPSIS you can also use another distance like Manhattan, which is the sum of the distances assuming there are similar as a taxi cab in Manhattan Island has to drive (hence the name), between two points. Therefore, the type of distance may affect the scores.

Kizielewicz, BartlomiejWięckowski, JakubWątrobski, Jarosław, in their paper “ A Study of Different Distance Metrics in the TOPSIS Method, and after comparing Euclidean and Manhattan, state that:

“The Euclidean and Manhattan distances were compared, while obtained rankings were compared with the similarity coefficients to check their correlation. It shows that used distance metric has an impact on the results and they are significantly different”:

In addition, TOPSIS uses compensation. These are my answers.

MB - (Actually, things are a little different and more complicated. You can check the following reference:

According to Mukhametzyanov, the answer is: “The displacement of the domains of the normalized values ​​of various attributes relative to each other and the local priorities of the alternatives are the main factors that change the rating when using various normalization methods”.

According to the author, Normalization techniques are actually a transformative. Sometimes these converters cannot preserve the tendencies of the natural values ​​of the criteria. they cannot transfer the contribution of the criteria to the final performance indicator in a way that preserves the original.

NM- You propose to read this paper but unfortunately, we only have the abstract of this publication

https://dmame-journal.org/index.php/dmame/article/view/493

As I could gather from the abstract, it speaks about the transformation of a domain X to a domain Y. Since de domain is the set of values in X space, it says that “the displacement of the domains of the normalized values of various attributes relative to each other and the local priorities of the alternatives are the main factors that change the rating when using various normalization methods”

By ‘displacement’, I guess that the author refers to changing values in X by normalizing, and getting a set of values that depend on the normalization method used. This is of course true.However, when using say Euclidean normalization for value a11 and that is for instance, 9, and if because the normalization it is now 12, that is an increase of 3 point. These same three points are now in domain Y, Therefore, if for the original X set is normalized to Y, it does not matter which method you use, Y will reflect that change whatever the method. This indicates that the different rankings are not related to normalization methods.

Look at the following exercise: Normalize criterion C1 where x1 = 3, x2 =5, x3 = 8

Euclidean: x1 = 0.30, x2 = 0.50, x3= 0.8, Ranking: x3 > x2 > x1

Sum of values: x1 = 0.19, x2 = 0.31, x3= 0.5, Ranking: x3 > x2 > x1

Maximum value: x1 = 0.37, x2 = 0.62, x3= 1, Ranking: x3 > x2 > x1

Min/Max: x1 = 0, x2 = 0.40, x3= 1, Ranking: x3 > x2 > x1

Log: x1 = 0, x2 = 1.66, x3= 2.66, Ranking: x3 > x2 > x1

Coincidence?

The author of the paper you sent me says” The displacement of the domains of the normalized values of various attributes relative to each other and the local priorities of the alternatives are the main factors that change the rating when using various normalization methods.

Obviously, the above exercise says the opposite.

Since I don’t have the whole paper I don’t know which are his arguments to defend that position.

https://dmame-journal.org/index.php/dmame/article/view/493

(One of the methods of accessing the Full text of the open access reference I suggested earlier: You can access the full text by clicking on Full text (pdf) in the middle right part of the page.)

Thank you Mahmut, I already did that and perused the paper.

It is the best publication I have seen on this subject

Yes, it is one of the good publications I have seen on this subject. But the technical jargon in the article makes it difficult to gain insights. I'm curious about your comments on this article. I would be happy if you share it with us.

Dear Mahmut

This is one of thee papers to be saved for reference.

One of the good things of this paper is that the author is very precise in his terms.

For instance, he clearly separates conceptes like attrinutes and criteria, very rarely seen in other papers that considers the twothings the same. As you know I have been saying the same in RG during years.

Also, he analyzes the normalization without considering criteria weights, which is very reasonable for an unbiased comparison.

In several places, he confirms what I was saying latelly, in the sense that normalized matrices or normaliazed domains are equivlanet to original domains.

What he does is to analyze relationships between two consecutive performance values of each criterion, and this is, in my opinion, the key to understand why the same MCDM method, like TOPSIS, using different normalization processes give different ranking. From this point of view this author is by far more advanced and knowledgeable that me.

Since normally I base my analysis in reasoning and examples, rather that in mathematics, there is something that calls my attention and it is why Linear Programming, using a complely different procedure to all other MCDM methods, always give the same results using different normalization methods, except, sometimes when using min-max, and which for me has an explanation, that also I shared in RG.

That is, if we agree in the beginning about normalization equivalence, and I get equal results using LP or SIMUS, it is obvious that the fact that produces different rankings, is not suibjectvity, as I initially suggested, but something in the middle on the two ends, and that in my opinion is the algebraic structure of each MCDM method.

My reasoning, that I gladly share with you, is that the problem lies in comparing performance values WITHIN each criterion. Since LP does not follows this procedure and is apparently immune to getting different results, it appears that said comparison could be the reason. Please think about it. I would very much like to read your comments

Dear Nolberto,

I agree with what you said. I compared the initial decision matrix created with raw data and the decision matrices obtained from normalization transformations. For each criterion (for example, for the C1 raw and C1 transformed column), I found a correlation of around 100 percent between the raw data and the transformed data. So, at first glance, the original data and the transformed version appear to be the same. But let's come to the second stage, where we introduce the transforming criteria into the MCDM equation, I think the data shifts from the original in some cases. In other words, when different units are homogenized and algebraic operations are performed, the total performance score we get can differ. The author emphasizes here that normalization changes some features in the original data. (Meanwhile, the Kurtosis and Skewness values ​​did not change). But if I'm not mistaken, I think the author means that with normalization, the relations between the criteria change a little. The author also claims that the priority of alternatives has changed. I can't be too sure and I want to think a little more about the issue, do some tests and clarify my final opinion later. If you have any questions or have a hypothesis, please share it and I would be happy.

I will, do not doubt about it!

It will be a pleasure working with you