Dear Nolberto Munier,

When I look at the questions that you have listed, I see that you critique in depth and in depth every stage of the method, and this is really inspiring for me. In a sense, I am ashamed of myself. In many studies that utilize MCDM in decision problems, we witness a casual application of the formula as a combination of the formula with the numbers of the problem, without even caring whether the method is appropriate for that problem. I hope all our authors can be as aware of what they are doing as you are.

You have already questioned every issue that can be examined about the AHP method. Let's also include the questioning of the 7 +/- 2 rule, which is one of the basic principles of the method.

Best Regards.

Dear Hamit

Thank you for your answer

Regarding what you say about my criticism, I am afraid that I am not done, I have many other aspects that in my opinion must be examining in AHP.

You rightly mention the 7 +- rule. Good point. As per my knowledge, Saaty took that from the work of a psychologist (Miller).

As Saaty at al (2003) assert “In 1956, Miller conjectured that there is an upper limit on our capacity to process information on simultaneously interacting elements with reliable accuracy and with validity”

For me it makes sense, but in AHP you work with one pair of criteria at a time, not with 7 pairs at the same time, what is the purpose of the 7 limits, when there is not simultaneity. Another unexplained assumption?

In a study I did in 2021I detected 29 strange aspects in AHP, that further were published in a book dedicated to AHP limitations, therefore, I still have plenty of room for criticism. In that book I replicated verbatim the opinions of 104 researchers, with their names of course

From that number, no less than 101 researchers, criticized different aspects of the AHP method, while only three, including of course Saaty, where supporting it.

Since Springer, the publisher of my books, publishes every day the metrics of ebooks, I learn than at present 8300 persons have read it. Do you know how many comments or rebuttals I receive? None; what does it suggests to you?

Your sentence "we witness a casual application of the formula as a combination of the formula with the numbers of the problem, without even caring whether the method is appropriate for that problem"

Is absolutely correct.

I would suggest a little experiment that I already did: Go to the Web, or Scopus, and read the abstracts of papers on projects where AHP were used. Observe the criteria of the problems and you will see that many openly violate AHP claim for independency of AHP criteria.

For instance, a fabrication cost is considered independent of the unit cost of components or environmental issue, or manpower.

It is exactly as you say. They took the AHP method as a simple combination of formula and numbers, and on this we cannot blame the method, because Saaty was very clear when he said that AHP works with independent criteria.

Years ago, I wrote to an author mentioning this fact in his article. His answer: “Yes, we knew this, but we decided that it is not important”

However, even with this flagrant violation that forbids using AHP when criteria, as in most real-life cases are related, why these papers were accepted for publication?

Because the reviewers, the ‘experts’, approved it.

The Editors, of course happy, because it meant that they can collect those absurd and exorbitant fees, the so-called publication fees.

I conclude that the authors are responsible, but also the reviewers in a larger extent, because you can understand than authors may ignore these facts, but a reviewer is assumed to know it. I prefer not to ask for the reasons for which they approve something that shouldn’t have been accepted.

Do you want another example: How many times have you read in an article that is has been validated. I accept that the author ignorance, but it unacceptable that reviewers accept that sentence when he/she must be aware, that not validation is possible in MCDM, for any method, since as the true result is not known, no comparison es possible.

Years ago, I proposed to the Chairman of the International Society on Multi Criteria, that there was a need to establish a quality control on MCDM publishing as in many other activates since decades ago. Answer: None.

However, I believe that people like you and another very few colleagues in RG, should start a movement to demand quality control on MCDM methods, on ALL of them, and decidedly not recommending their utilization, if they do not meet certain standards that must be established, as for instance a through mathematical explanation of each single step of a proposed method. Of course, in this, I am including my own MCDM method: SIMUS.

It would be an honor if SIMUS is selected as the first method to be analyzed and results made known publicly. I would be very happy if I receive criticism on any aspect of it.

The quality control must be extended to reviewers, which even if they do not have any economic reward - another absurd - they gain in their names to be published in certain data base as Publons.

Thank you for your answer.

@The Analytic Hierarchy Process (AHP) is founded on mathematical principles, specifically the eigenvector method and pairwise comparisons. While it's widely used and respected, "proof" in the strictest mathematical sense might be elusive due to the complexity of real-world decision-making scenarios and subjective judgments involved. However, there are numerous studies validating its effectiveness and demonstrating its consistency with rational decision-making principles.

Dear Mr. Sing

Thank you for your answer

If I don't remember wrong, I mentioned that EV and Geometric mean are the only mathematical tools used by AHP.

Could you please tell me where is the mathematics in pair-wise comparisons? I fail to see any, other than comparison of criteria and assigning an intuitive preference to it.

Is this mathematical for you?

It is true that it is widely used, by respected? By who?

Not precisely for more that 100 scientists. I have also published a list of them with their opinions, could you please do the same with people supporting AHP?

You are right that proof may be elusive not only for AHP but also for all MCDM methods, since there is not a yardstick linked with reality that we can use as a reference.

However, don't you think that MCDM methods, that reach results based on reasoning, discussions, research, experience, must perform AHP's results based on intuitions and feelings?

You responded your own question when saying "subjective judgements involved".

"Numerous studies validating"? Could you mention at least one, based of course, on mathematics?

You speak of consistency, and of course, you realize that said consistency is applied to the DM estimates, nothing wrong with that.

Now I ask you two things

1- Why there should be consistency?

Do you think that an estimate done in good faith by a human being must be corrected by a formula? Too bad that there are not a formula for MCDM problms

2- Why the estimates from a person, must apply to the real-world?

Which is the mathematical axiom or theorem that supports that assumption? What happens if another DM has different estimates? Does it mean that there are two or more realities?

Not even the Quantum Theory or the Theory of Relativity propose that.

Well perhaps, if we adhere to the hypothesis of the parallel worlds