# 203

Dear Andrii Shekhovtsov, Amirkia Rafiei, Wojciech Sałabun

I read you paper

Comparison of Monolithic and Structural Decision Models Using the Hamming Distance

My comments

1- In the abstract you speak about a structured division of the model

In my opinion you can’t divide a model, it is OK for analyzing it, but not for solving, because an initial decision matrix is a system, where everything is normally interrelated. I think that you can’t divide, calculate and add-up, because you must consider intersections not additions. In addition, customer preferences, especially in selecting a car, generally is more that one, as maximum comfort and minimum price, and that is impossible to get. It is necessary to compromise, and this is reached by giving and taking, looking for a balance, as in a normal negotiation.

2- Page2 -you say “However, there is no simple way to compare COMET models or predict how much their results might differ, creating a gap in the tools avail able for decision-making analysis:”

The cause, in my opinion, is that comparing different results from different COMET decision models is a waste of time, since not all of them work on the same basis

3- In page 2 you use the Hamming distance, which is great to measure two strings with equal distance, but does it happen in MEJ? No, that I know

4- “This highlights the utility of comparing results, but also points to the limitations of relying solely on final rankings, because it does not offer a true model comparison, which should focus on comparing the entire structure of the models rather than discrete points or outputs”

In my opinion there is no utility in comparing results, but agree when you say “because it does not offer a true model comparison “, since what is the gain in comparing results when you have to compare criteria considering objectives and targets, something that you do not do?

In using a rational MCDM you should get a unique result, that is the consequence of considering the initial matrix as a whole. It is a fact that maybe 99% of MCDM methods work using addition of partial results, and thus, based neither in mathematics not in reality, but in assuming that the whole is the sum of its components, when in real-life projects it maybe not so. It is like analyzing in a car, INDIVIDUALLY, the engine, the electrical system, the transmission etc., when it is evident that the engine is a FUNCTION of aerodynamics of the car, fuel consumption, type of tires, comfort offered, etc.; you can’t optimize each component and add them up.

Thermodynamically an engine can have a high efficiency in the test bench, and completely different when in operation, because it is subject to wind speed, to power the air conditioning unit, charge the battery, plus all the lighting system of the car. This can be seen clearly in the wind tunnel that car factories have, to study the performance of a mock-up (a wooden model of the car) and acquire knowledge about how other elements affect the engine.

Your theory may have a high academic value, but if it does not take into account the facts and characteristics of a project, what is then its utility? It is based on pair-wise comparison which is useless to quantify criteria, as is to demanding transitivity in the DM estimates, or assuming, against all logic, that what is in the mind of the DM can be applied to the real environs. Did you realize that none of them have a mathematical or common-sense support?

5- Page 3 “They tested the impact of varying the number of alternatives and criteria on the final rankings and then compared the results using correlation coefficients (Shekhovtsov et al., 2021). Their findings indicated that the rank ings were very similar and that an increase in alter natives and criteria improved this similarity.”

In here, I think that there is a mixed-up, because when varying criteria, what you do is to put new values, and keeping the number of criteria, and keeping the same dimensional space. That is, is the same problem with other data, very used in simulation. But when you vary alternatives, you change the problem and in reality, solving a new one. Remember that in this case a new alternative also means a vector that modifies all criteria. What is the senesce in comparing problems that are different? This is the reason why rank reversal may appear, because there is no certainty that the ranking you had originally, will be preserved when you are solving a new problem.

6- Page 4 “The Characteristic Objects Method (COMET) distinguishes itself from other Multiple Criteria Decision Analysis (MCDA) methods by being completely immune to the ranking reversal phenomenon”

On what grounds do you assert that COMET is immune to RR? I have been investigating RR extensively and I can assure you that there no MCDM immune to RR, whatever its structure. According to my research RR is not a phenomenon, but a logical consequence derived from space geometry.

Let’s analyze briefly my statement

Dimensions are not for criteria as said in the paper; they refer to the umber of alternatives, i.e., two alternatives are in 2D and can be represented by 2 Cartesian axes. That is, the axes represent alternatives or variables x1 and x2. If in this Cartesian space we draw the two lines representing criteria C1 and C2, which are linear inequations, since we want to compare them, there will be two spaces that overlap forming a geometric irregular shape.

One of the vertexes of this shape is the optimal solution, which values may be seen when projecting the vertexes on the two axes, and forming a ranking, like x1 > x2. Observe t hat this ranking is not related to preferences, it is only a geometric construction, built with the data from the problem, and thus, real. When adding an alternative, we are in 3D, and the original shape is now a polyhedron in three dimensions in a Cartesian system of 3 axes

If we continue adding alternatives we can have linear equations in 4 dimensions like a tesseract, or i n5, 7 or 20 abstract dimensions, that we cannot visualize in our 3D world.

Therefore, each new addition involves an object with a higher dimensionality and most possibly with a different format, depending on the vectors added. Therefore, the two vertexes found in 2D may or may not coincide with the 3 vertexes in 3D. If they do, there is ranking invariance. since the two original values are preserved. If not, new optimal vertexes appear that determine a new ranking, with possible disappearance of precedent rankings.

This is the reason for RR, and applies to all MCDM methods in a lesser or greater degree according to their algebraic structure. Thus, RR is unavoidable, at random, because it depends son the vectors added. The same of course, for decreasing dimensions.

7- Pag. 4 “The algorithm assumes that if the expert judgments are consistent and free from errors, the evaluations of characteristic objects should form a transitive relationship.”

Well, this is a strong assertion, but v very difficult to prove

For instance:

Why the expert judgement MUST be consistent? Assume that a DM compares criteria C2 and C3 and C4, on quality/price, and decides that for him quality is 2 times more important than price, ALLWAYS.

In selecting a restaurant for dinner, the DM knows very well three of them (R1, R2 and R3). Comparing R1 with R2, and following his preference, he selects R1, more expensive but with better food quality.

Now, the DM compares R1 to R3, and finds thar R3 has very good quality too, but lightly inferior to R1, and at a substantial lower price, because it is not located in an upscale neighborhood as R1, and thus the DM reasons “Why to select R1 over R3 when I may have similar quality in R3 at a lower price, and he selects R3 on R1”

Therefore, the DM is inconsistent, but he is expressing a legitimate change in preferences. In addition, the DM is forced to admit that he was wrong, because he is consistent or transitive, obviously absurd.

R2 is the same as R3, therefore, the ranking is R1 > R2 = R3 where R3 > R1. You can see that this necessity of alleged consistency, may not work in real-life. Do you see any wrong here? If not, the premise that the estimates from the DM must be consistent does not have any support neither mathematically nor from the common-sense point of view.

Why then the insistence of many papers regarding consistency or transitivity.? The only reason is that a method like AHP needs to use consistent matrices to apply the Eigen Value algebraic analysis. As a bottom line, it is established for convenience, to make the method appear to be mathematically supported.

8- Page 4 “Finally, the Structural COMET approach offers a solution by breaking down a complex decision problem into several smaller sub models, aggregated to form the final decision model.”

As I mentioned and explained it, that breaking- down a system is not a good practice. This is against Systems Theory and opposed to the opinion of many researchers. I think that my little example is illustrative enough. I invite you to look in the Web the definition of System Theory.

9- Page 6 “year of the start of the production, approximated or estimated price, range of the car, technical parameters of the engine, tank capacity, acceleration and maximum speed.”

You put a good example here, but these different criteria cannot be considered independent since some heavily influences others. For instance:

Price depends of the year of fabrication. Therefore, a more modern car surely will cost more.

At the same time a modern car will have less fuel consumption, more comfort, higher acceleration, and son on, therefore, how can you consider say three different criteria and add them up? See the dependency? Can you break them down and add results?

Range of cars depend on engine fuel consumption, maximum speed, tank capacity and technical parameters. It is rational to consider them one by one?

10- Page 7 “Usage of the structural approach requires only 423 pair wise comparisons “

I must confess that I was shocked when I read this paragraph. ONLY 423 Pair-Wise comparisons?

Sorry for ``lack of answer, but I do not think that comments are necessary. It speaks by itself

These are my comments. I hope they can help

Nolberto Munier