To apply fuzzy methods to real-life problems having exact values is to introduce uncertainties yourself into the system. According to me, in many exact values possessing situations, a researcher should look for the appropriate case of need for fuzzy values to be implemented. If it is found therein, we should proceed; otherwise, skip to another application genre where you can find the role of uncertain information to be processed adequately. Applying the proposed fuzzy algorithm to normal cases having exact values, according to me, is an injustice to the system as well as to the fuzzy environment in which you are working. So, we should carefully look for uncertain information first before implementing a decision-making approach onto it.

Fuzzy methods are principally too complex compared to crisp ones. Such undenied fact can be seen in every computation method. Take for example the Machine Learning classifiers and their obvious complexity in the preparing data and the learning phase before getting a mapping model. Such complexity would be smaller in the crisp method. Not only the complexity but also the running speed, which is higher in the crisp than in the fuzzy-based methods.

No doubt that nothing makes fuzzy methods are necessary need in solving the real problems that their data are of crisp nature.

I think, as an approach, we can also make some confirmatory judgments based on the data as an alternative (or additional support) to purely theoretical logic and causation. While the MCDM input algorithm has a solid foundation in theory, real-life ranking results based on output scores are also important for validation. Logical algorithms may seem solid in theory. But when their hypothetical strength meets the realities of real life, it turns out that they can actually be very weak (and vice versa). What is the best solution. In my opinion, there are some critical factors as to which MCDM method is more successful for a well-structured problem. These affect ultimate success. One of them should be about what type of data you are using. Fuzzy is recommended if there is missing value or no crisp value. On the other hand, fuzzy can also improve crisp data. It is not recommended to use fuzzy numbers when there is crisp value in terms of information loss. I do not think so. Sometimes conversion from crisp value to fuzzy numbers can be more useful. This is because the relationship between real life and MCDM results is not always linear. That is, the transformed fuzzy numbers interpretation rather than the crisp value itself can sometimes be a critical factor.

Dear Mahmut

I agree with you. If we have crisp values like a consolidated price, or the degree of contamination produced by a process, why do we need fuzzy?

I don't understand your sentence; "Fuzzy is recommended if there is missing value or no crisp value. On the other hand, fuzzy can also improve crisp data".

Could you please explain the bold sentence, or was it a mistake?

Regarding using fuzzy when there is loss of information, I agree with you, because it depends on several factors, that ca'nt be improved using fuzzy.

You say "Sometimes conversion from crisp value to fuzzy numbers can be more useful" Could you put an example regarding this assertion?

It is curious; we spend a lot of efforts trying to improve some values using fuzzy, which is correct, however, some researchers happily accept that invented weights born in the mind of a DM, can be directly applied to the real world, and that it is transitive.

Considering this, I don't think it makes sense using fuzzy in some scenarios

Everybody knows that AHP is a completely arbitrary method to determinning

criteria weights, what is a contradiction, and yet, many researchers apply fuzzy, even when Saaty specifically stated that it is not correct, because AHP and ANP are already fuzzy in their conception.

Dear Nolberto,

I know that you are sensitive about AHP, alert to the subject and eager to discuss it. But on the contrary, I do not want to discuss this issue, and this is not our topic anyway. We are not talking about AHP or other subjective weighting methods. Our topic is about the use of “fuzzy” or “crisp” values ​​for the first decision matrix. For MCDM problems, it is usually clear when fuzzy is used: Its use is more recommended when secondary data (crisp value) is not available or there is missing data. Some researchers convert "crisp value" data into "fuzzy numbers". Some argue that this type of conversion causes information loss (So it is an unnecessary conversion). I stressed that this may not be the case. Because what “crisp value” means to you is also important. For example, a firm's Altman-Z score is 2.00. This value is considered as the "grey zone" for listed manufacturing companies. If this value was 0.2 last year for the same company, this year's 2 value would be more valuable. If the industry average was 1.5, these 2 values ​​would be more meaningful. Or let's think about it this way, let this score be 5 for this company, the industry average is 15, and last year's value is 10. Now compare the two situations. Is 2 better or 5 worse? We can easily interpret these nonlinear situations with linguistic based fuzzy numbers.

In short, "2 is not just 2".

Dear Mahmut

Thank you for your comments.

Yes, I am eager to discuss AHP and I have said it many times, but it was not my intention to discuss it with you.

Regarding the first part of your message, I am afraid that you did not interpret the meaning of my reference, or perhaps, I did not explain myself very well.

Regarding what you say about crisp and fuzzy, and refer to not available crisp data or missing, may I remind you that missing data means also data values that do not supply complete information? If you use fuzzy with data that are invented or assumed or highly unrerliable, from any origin, what is the purpose of adjustimng it? This is what I am aiming to.

The DM is concerned with analyzing uising fuzzy a part of the data and not assuming that it can be useless. He can't see the forest (the whole scenario), because the trees (the unreliable data). Wouldn't it be better to make sure that the data depicts reality and is as reliable and true as possible, before refining it?

What I tried to convey, is that the general trend is using fuzzy to get more reliable values in case of uncertainty, that is, addressing a factor that is indeed very important, but closing the eye when there are values totally uncertain when using, AS AN EXAMPLE, the values given by AHP. I can say the same for other methods like Best and Worse, and others, where is up to the DM, to establish criteria importance.

As you can see, I am just making a comparison between two kinds of uncertainties

I like the example you put on Altman Z-score, but in this case, you are incorporating an element, - I believe fundamental in finance - but not common in most MCDM problems, as is its relation to time.

Your example is very illustrative and useful, and for those readers that are not in the finance activity, it is convenient to explain that the Altman -Z is a probabilistic metric based on 5 different financial indicators, which measures the risk for a company heading for bankruptcy. I presume that in this analysis, not considered in the metric, are other factors that the DM must take into account and evaluate. As you say “2 is not just 2”.

As you know my knowledge in finance is very limited, so I am not in condition of answering your last question, but it appears to me that in certain degree it depends on the analyst plus other aspects, to evaluate the risk portrayed by this metric

Nolberto