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It is well known that Medal Fields Prize is intended for excellent research of mathematicians under forty years old because many mathematicians think that the main contributions in the life of the researchers are obtained when they are younger than forty. I do not believe so. It is true, by common experience, that the students of Mathematics, which are constantly in interaction at the same time, with several (and sometimes, very different) subjects, develop a high degree of good ideas which inspire them and lead them to obtain new and interesting results. This interaction between different branches is expected to remain (more or less consciently) up to forty years old. By the same reason, if necessary, whoever researcher, independently of his/her age, may return to study the different mathematical matters and create new important contributions, even in his/her very definite area of research. Furthemore, it may help to overcome a blockade. It is incredible the fact that when one studies again different matters it inspires you, and combined with your experience and knowledge, you see the contents of these different subjects with new perspective, often helping in your area of research creating new knowledge and solving problems. This is the motive why I believe that the career of each mathematician is always worthly and continuous independently of his/her age as demonstrated by most senior mathematicians in all the areas of research who are living examples for us.

What is your opinion on the relationship between the age of a researcher and the quality of his/her contributions?

Thank you very much beforehand.

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Greetings and respect! If you have an article or book about the dual product of two Banach spaces, please cooperate.

# 169

Dear Ravipudi Venkata Rao

I read your paper

BHARAT: A simple and effective multi-criteria decision-making method that

does not need fuzzy logic, Part-2: Role in multi- and many-objective.

My comments:

1- In your abstract you say, “Three case studies are presented to illustrate and validate the proposed BHARAT method.”

I am afraid that you are mistaken since no MCDM method can be validated because there is not a yardstick you can use for comparison.

2- “The objective functions f1 and f2 and the constraints are described in terms of the decision variables x1, x2, x3 and x4. Both the constraints and the objective functions can generally be either linear or non-linear. The best option should meet the decisionmaker's preferences and constraints.”

An objective function and a criterion or constraint, as you call it, are the same algebraic entity, both are represented by straight lines, but with different purposes. The difference is that the function z = f(x,y) is indefinite while the criteria (Ci), that are also objectives, but not functions, are finite and constrained by a resource or by a target. Normally, an objective function says, ‘Minimize production cost’, but does not establish a value for it, while the objective ‘cost’, to be minimized sets a value, for instance, that the cost cannot be less that the production cost. Thus, as an objective function, the cost may be any, but as an objective, it must be for instance, equal a greater than the production cost, say 3.25 /unit. This is Linear Programming Theory (LP

)

Genetic algorithm can indeed address this problem, but offering a lot of different solutions, however there is a MCDM algorithm, SIMUS, based on LP, which gives a unique solution

3- “Recently, a few researchers have started using the fuzzy logic for expressing the values of decision variables and the objective functions. However, these may be called as heedless attempts to apply fuzzy techniques anywhere there are numerical values, without scrutinizing the legitimacyof the methodology. Certain procedures view all numbers as amenable to fuzzy logic.

I have read hundreds of papers on MCDM along the years an arrived exactly to the same conclusion. The aim is to publish; it does not matter if a method is appropriate or not. I have been saying this for years in RG, without any rebuttal, and yours is the first paper that dares to say so.

Glad to have a learned ally.

Regarding reviewers, just reading a few initial paragraphs of papers addressing for instance AHP, and I would say that in a 90% they should have been rejected. The most common is not even respecting the method’s algebra. In AHP Saaty warned, and he was right, that his method should not be used when criteria are interrelated, but neither author not reviewers, who should know better, happily accepted papers with highly related criteria using that very much used method.

The problem, as I see it, is that people do not think, they do not analyze which is the most adequate method for a certain problem, and if the method is not adequate, it does not matter; nobody will notice.

For instance, what is the sense of applying fuzzy to invented weights from AHP? Again, Saaty clearly wrote that fuzzy should not be applied to AHP, because the method was already fuzzy, right again. People just ignored this and created FAHP….Do editors and reviewers reacted? May be some, but most just went blind, or do not know the subject.

The important is to be in Publons stating that they were reviewers for a journal.

4- “Therefore, the best solution can be defined as the one that achieves the best compromise between the objectives.”

Exactly. This is the best and valid definition, long time ago annunciated by Zeleny (1974)

5- “The pertinent objectives include both beneficial and non-beneficial objectives. The

This is a trivial definition, simple because they are not linked to the objectives. You cannot ask for maximum benefits for a product if you do not establish a limit, which is tied to your capacities or the resources, be them money, workers or raw materials. to produce it. Pls. remember what I wrote above when a minimum cost is related to a minimum than cannot be lowered.

Because the no consideration of resources, these methods cannot represent real-life.

5- “To determine the objectives’ weights wi (for i=1, 2,..., m), order the objectives according to the decision-maker's assessment of their significance in terms of 1, 2, 3, 4, and so on.”

Sorry, I do not think that is realistic to trust in the intuition of a DM to fix criteria relative significance. This is the most possible arbitrary procedure, because there can be different weights for the same problem if more than a DM intervene, even using a group. The reasoned, studied, investigated weight from a DM must be used to weight criteria, but not based on pair-wise comparison (absurd!), Delphi (biased) or intuition (irrelevant).

These well-founded opinions MUST be considered but not affecting initial data. Remember that criteria weights are trade-offs and have NO ROLE in alternatives evaluation.

6- “Four different MADM methods are used on the same case study to study the efficacy of the suggested BHARAT,”

And what the DM learn from this comparison? Nothing.

There is no mathematical support, only intuition, that if there are quasi coincidences, then the solution is correct. This is an assumption very much used.

These are my comments, hope they can help.

Nolberto Munier

Is there an analytical or experimental approach to measuring fault normal and shear stiffness? In both cases, a shear zone is present, and when the fault is tight. I need to calculate the normal and shear stiffness for numerical analysis.