# 167

Dear Silvia Carpitella , Václav Kratochvíl , Miroslav Pištěk

I read your paper

Multi-criteria decision making beyond consistency: An alternative to AHP for real-world industrial problems

My comments:

1- On page 1 you say, “The Analytic Hierarchy Process (AHP) relies on consistent pairwise comparisons,”

Why? Because it is convenient for AHP? Which are the technical reasons for that?

There are none.

2- “The method is validated through practical examples and …”

No MCDM method can be validated because for that you need to compare your result with the real one, something that you do not have. If you did, that is, if you know the solution, what would you need MCDM for?

3- Page 1 “However, what if the pairwise comparison matrix is inconsistent but still valid? This situation can occur, especially when the entered alternatives cannot be linearly arranged based on their importance. This is well illustrated by the following example, inspired by the game of “intransitive dice”, see, e.g., Butler and Blavatskyy (2020).

“This is well illustrated by the following example, inspired by the game of “intransitive dice”, see, e.g., Butler and Blavatskyy (2020).wo probabilities ?

Suppose that there are three statistically independent assets. Asset D yields 9% with a probability of 1/3 and does not yield any return otherwise. Asset E yields a negative return of −1% with a probability of 1/3 and 5. % otherwise. A risk-free asset F yields 3%; note that the expected returns of both assets D and E are also 3%. Now, given that a fund manager’s preference is to choose an asset that is more likely to yield a higher return, she would prefer asset D over E, asset E over F, and finally asset F over D.1

I am afraid that your presentation of the example is confusing. What does ‘otherwise’ mean here? Of course, my question does not refer to the meaning of the word but to what it means here, in this financial content

As I understand, for alternative D there is a probability of 33% that an IRR of 9% can be achieved

For alternative E the probability in 33 % indicates that the IRR is 1% negative, meaning that the IRR is less than the discount rate. I understand that there is certainty or 100% probability that the IRR reach a value of 5%

For alternative F, it appears that there is certainty or 100% probability that an IRR reaches a value of 30%

The problem appears to be feasible, because all conditions can be met, although probabilities of 100% are dubious. Anyway, I solved this problem with SIMUS and the results are D: 0.31, E:1.06 and F:1.98

Therefore, the best alternative is F

This demonstrates that transitivity or consistency does not matter

4- Indeed, an inconsistent input is inherently considered flawed, and the decision-maker internally ambivalent and irrational”

With due respect, I believe exactly the opposite, and your example shows it. Assume for instance that a DM is comparing C1 and C3 and he finds that C1 =2C3. If he now compares C1 with C6, it could very well be that C6 is say, 4 C1. Is the DM an irrational by expressing his preferences or must adjust to what a formula says, as in AHP? In my opinion, this is irrational and without any support,

Suppose that you compare two criteria, quality and price in three restaurants, you can prefer quality over price, and for this you select restaurant A over restaurant B. Now, he compares B and restaurant C and finds that they are equal. When comparing A and C he thinks that we prefer to pay less in C than in B, with only a little difference in quality between C and A

What is wrong with this?

5- “Moreover, the consistency assumption imposes limitations, as it may force decision-makers to artificially adjust their preferences to fit a mathematical framework rather than reflect their true, sometimes inconsistent, judgments. As a result, AHP’s reliance on consistency may hinder its ability to capture the full complexity of subjective preferences, making it less applicable in practical scenarios where inconsistencies are common”

I agree 100% with you, especially when you say that it may force to artificially adjust preferences. What I do not understand is why a method that is flawed on many aspects, not only in what you mention, is being used, and so many efforts wasted with the subject of consistency, when it is completely irrelevant.

6- “Allowing for inconsistency can deliver rational and effective outcomes by reflecting more realistic and adaptable human judgment patterns (Szczypińska and Piotrowski, 2009, Tavana et al., 2023)”

True, without a doubt

These are may comments, hope they can help

Nolberto Munier