#147

Dear Jean Dezert, Andrii Shekhovtsov, Wojciech Sałabun, Albena Tchamova

I read your paper:

On Optimal Solution of the Compromise Ranking Problem

My comments:

1- It is for me the first time I read the expression Compromise Ranking. I guess that you refer to a ranking, that is a balance among the different alternatives conciliating the opinions of all intervening parties, or a compromise solution.

2- You say “Including a new alternative in the existing set of alternatives must not impact the preference ordering of the existing alternatives”

Since long time ago I agree with Arrow’s theorem and its axioms, except D4, where it appears entering in the rank reversal (RR) phenomenon. In my opinion, when adding or deleting an alternative you enter in a different spatial dimension, where ranking invariance cannot be guaranteed, because it depends on factors that have no relationship with former rankings. In reality, it is a geometrical issue.

That is, every time you delete or add an alternative you modify the existing object or structure, since there is not isomorphism. In other words, preserving rankings or invariance is something at random, and thus, it may or may not exist.

I agree with Arrow when he talks about dictatorship, as IT IS, when a DM establishes that C5 is for instance, three times more important than C7.

This is the reason, among others, by which I am very critic of AHP. The DM cannot vote for others, this is for me the essence of the Arrow’s Theorem. For this very subject I held a brief interchange of ideas with Tom Saaty on February 2017.

3- In page 1 you say “The search for an optimal solution in agreement with the common sense…..”

We cannot look for optimal solutions in multicriteria, because it is impossible to get the minimum cost and the maximum benefit at the same time. It is one or the other, or a balanced mix, or compromise solution, as shown by MCDM methods.

4- In page 2 “because the information about the intensity (or weights) of individuals’ preferences for the choice is rarely available, or very difficult to obtain precisely”

As well as useless. Weights are trade-offs, not good for evaluating alternatives, since they have no relation with the content of each criterion. Shannon explained this very well in developing his entropy theory.

Of course, in general, criteria set has different degrees of importance in each criterion, but this is not something that is determined by intuition. Criteria importance depends on its quality to evaluate alternatives, and this is given by entropy.

5-Page 3 “Our proposed method is mainly based on Kemeny’s and Frobenius distances between preference orderings, and we express the CRP as a least squares optimization problem in a discrete search space”

Fine, but I do not understand that the method is based on two different types of measures. One is Condorcet, while the other demands orthogonal matrices. And you do not even discuss ties breaking

I am not saying that your approach is wrong, only that it should be explained.

6-Page 3 “At this stage we already anticipate the difficulty of finding the best compromise ranking solution in more general (bigger) problems involving many alternatives and many members of the society under concern”

Agreed

7- Page 6 “Clearly, we see that the KOA and FOA solutions are not acceptable because with Kemeny’s distance we get multiple solutions and we cannot infer a priori which one is correct, and with Frobenius distance the FOA solution does not fit with the expected solution (A ≡ B) ≻ C”

Agreed

8- Page 6 “In fact there is no guarantee that ranking aggregation based on these distances and LS approach will give an acceptable solution in general even if it is optimal in the LS sense. Therefore, a better optimal approach must be sought to obtain more acceptable aggregated ranking solution”

Agreed

My personal opinion: This is an excellent research paper, however, as you mention, no solution has been found

If you allow me, I offer one, admittedly cumbersome, but that in my opinion, might work.

If we have, say five alternatives or options, and get 400 answers, with different rankings, I would commence by establishing a high correlation threshold, say 90 %, and group the set of rankings with correlation greater than 90%

Say for instance, that this set involves 45 rankings that meet this condition, disregard the others

Build a matrix where the criteria set is composed by the 45 selected rankings; forget weights, which anyway are equal for all criteria or ranking, because we do not have any reason to consider that one ranking is more important than other.

Run this system with an appropriate MCDM software, that will do the job for us by finding the thousands of interrelations among the 45 criteria, and will yield the alternative that best comply with it. Of course, the MCDM method must consider all criteria and alternatives simultaneously. All criteria must be subject to maximization, because we are looking for the maximum value of each ranking

Of course there could be ties, but they can be eliminated considering other factors, for instance, two different alternatives may have the same value; we can find for each one the marginal value of each criterion, and choose the alternative that corresponds to the maximum marginal value.

Unfortunatelly, as per my knowledge, this is only possible using Linear Programming and SIMUS, where all of this information is in last computer screen. Needless to say, the method is far from perfect, because we are considering only a fraction of the 400 answers, however, it is very exact in making the thousands of comparisons

These are my comments that I hope may help

Nolberto Munier