Dear Zaher Sepehrian , Sahar Khoshfetrat , and Said Ebad

I read your paper

Approach for Generating Weights Using the Pairwise Comparison Matrix

My comments:

1- In page 1, you say” Several methods have been proposed in AHP-related articles for the determination of local weights based on the pairwise comparison matrices [9–18]. Each of these methods has specific advantages and disadvantages, and thus, none of them can be considered the best”

I can’t speak about the several methods you mention because you do not enumerate them, but in principle, using pair-wise comparisons is a technique that has been considered arbitrary and of dubious reliance , and without any mathematical support since the 80s. For this reason, I don’t understand why many papers are published using that procedure without analyzing if it is worth its use.

You use DEA and I wonder, which is the importance in determining efficiencies when data is invented? Honestly, it is time wasted for me.

You can weight criteria subjectively, one by one, using a simple scale. Why to consider a pair that requires to put a value of dominance that nobody knows and that can change with different DM? Where is the logic? Different is if you evaluate subjectively each criterion independently, where it is possible to use preferences, but more important, employing reasoning, analysis, research and common sense.

If once you have finished with the pair-wise comparison and present your result to the stakeholders, how are you going to answer if a stakeholder asks for the origin of that comparison value?

You will say that it comes by intuition? Do you think that it is a valid answer?

Instead, if you evaluate each criterion separately, you have a lot to talk, explain and justify the value or utility you assigned to each criterion. Or even better, why don’t you use objective weights than can also incorporate the DM justified preferences?

2- In page 1 “Although DEAHP can generate true weights for consistent pairwise comparison matrices, it generates illogical and meaningless weights when it comes to inconsistent pairwise comparison matrices”

Can generate true weights? Could you tell me what is the theorem or axiom that support that?

3- In page 2 “When a decision criterion or alternative evaluates its best weight, it also evaluates other decision criteria or alternatives.”

In my humble opinion, this is incorrect. What AHP determines are not weights but trade-offs that are useless to evaluate alternatives; even Saaty said that this equivalence was only an assumption

The only weights than can evaluate alternatives are those from entropy and from standard deviation, two well-established mathematical concepts, with no relation to the DM estimates.

4- On page “analysis of the most undesirable weight,”

And how do you determine them?

5- In page 2 “The principal condition for the analysis of the most undesirable weight is that, in addition to the most desirable weight, the most undesirable weight can also be assigned to every decision criterion and alternative”

Not in my opinion. When you compare two criteria weights, you do not take into account that in general all criteria are related, and thus, individually a criterion may be unimportant per se, however, its influence on other criteria may be paramount.

For instance, you may have a problem where one criterion is contamination, and another criterion is cost. You can say that contamination has a very low weight, and thus, it is not significant, however, it couldn’t be negligible because the company invested a lot of money in installing cost equipment to avoid contamination, like in a coal-fired powerhouse in installing gas cleaning equipment. If the company allows for a little higher contamination, for sure, it will produce a decrease in equipment costs and perhaps in operating costs.

In this way cost are linked to environment and vice versa, and this is real-life, not a school blackboard example

By the way, do you realize that Saayy clearly said that AHP is only applicable when criteria are independent, or not related, and you igored it?

6- In page 6 you say that equations 11 and 13 are non-linear? If they are the result of the Eigen Value procedure, I believe they are.

7- I don’t have any opinion on your method; however, I believe that by using par-wise comparisons and the AHP, its basis is very weak, and barely credible.

I hope that these comments can help

Nolberto Munier