Dear Massimo

I refer to your sentence "I'm searching for real datasets of AHP pairwise"

I wonder since when intuitive data sets are real, especially coming from pair-wise comparisons, a method that has been criticized since its utilization in AHP. Please, consider that I am not talking about the method itself, since it is perfectly logical to make comparisons between two criteria. What is irrational is to assign a quantitative value to that preference.

As an example, you can rightly say that for you, love is more important than tenderness, it is your valid opinion and it is as good as any other, but to put a value to that preference is absurd, the same as if you compare criteria Number of ambulances in a city to Environmental contamination in the same city, in a five year urban plan.

Making surveys among interested people, is the right procedure, however, not asking people to make comparisons. When you make questions in a questionnaire, you have three issues.

First, most people might not have the faintest idea about the project and technicalities.

Second, people get tired of questions and answer anything to finish as soon as possible.

Third, the most important, you might get answers to your questions, but not about what the people want, wish or suffer.

I bear witness of survey done years ago following a questionnaire, AHP style. From there the City Hall, decided to go ahead with alternative B, that consisted in building a highway at land level.

It was published, and days later an old lady asked to talk about this selection among A and C. A was an elevated highway and C, part elevated and part at land level

She was introduced to the project manager, and said "as you know, our city is tucked between mountains and the sea. When Summer comes water from ice melting comes down and discharges into the sea through many natural ways and without problem, because there is no impediment. Now, if you build the highway at terrain level, did you realize that the city will be flooded?"

Needles to say, the project was cancelled

This is real, not a blackboard example

Dear Nolberto,

I totally agree with you: identifying, quantifying, and handling the "intrinsic fuzziness" of pairwise comparisons is exactly what I'm working on. And that's why I need this kind of datasets with multiple decision makers.

Dear Massimo

Thank you for your response, which in principle looks reasonable. It seems that your aim is to determine the mean value of a normal distribution of answers.

However, in my opinion, this is not applicable in this case for the following reasons:

1- Possibly there are thousands of these data base since the procedure has been used for years and still is very popular. However, those thousands of cases, correspond to different MCDM problems, with different number of criteria and addressing a multitude of different actions.

Therefore, they can be considered at random and so, following the Gauss Distribution.

I wonder which is the gain in getting that result? Can you apply that knowledge to a specific case?

2- There is another problem applicable to every pair-wise comparison, where, by intuition you get a value of preference, which as you know, is only a conceptual result, and as that. it can change with a new DM it will be his conceptual opinion, and not applicable to real-life scenarios.

Now, consider for instance, that you have 10 criteria, it means that the DM must compare each one with the other nine. Yes, it can be done, only if the DM is an expert in engineering, environment, financing, disposable income, and who knows what other discipline covered by the 10 criteria.

I am sure that you agree with me that this situation is highly improbable.

Nonetheless. you can have 10 experts, one for each criterion. Fine, only that each expert, whatever his/her expertise, must discuss for instance, with a medicine doctor about disposable income, when none of them know about the other, thus, how two people can argue and reach an agreement when both speak ‘different languages’, a typical example of maximum entropy, with zero information.

3- There is another issue, and it is why two persons, or a group of persons can vote or decide for thousands, and without being nominated for that job? I have also discussed back in 2017 this same subject with Dr. Saaty. The Arrow Impossibility Theorem calls it a dictatorship, and it is.

I of course, respect your work as that of any other scientist, but I cannot help to ask a question:

For what? Wouldn’t it be better to use an objective weight?

Of course, the contribution of the DM and opinions are fundamental and mandatory to consider, no question about that, but his judgment must bag applied at the right time, which certainly is not at the construction of the decision matrix.

Please, consider that the DM can get a clean, mathematical result, reflecting real-life data exact or not; on that, the DM can certainly work and apply his expertise.

That is, as is done in most industries, for instance in airplanes design, car designs, or in determining site location, or in applying certain government policies, or in railways, etc.

My theory calls for leaving the MCDM software do its job, which is to select the best alternative subject to a real data base and considering of course uncertainties. When the result obtained, it is based on mathematics (albeit not in most MCDM methods, based on assumptions without foundations), that is, it is mathematically realistic.

On that firm base the DM can analyze, research, consult, determine on what external factors depends the result, and analyze it thoroughly. At that moment the DM may introduce his ideas, estimates, values, or even reject the result. This is his job, not to arbitrarily modify the initial data. Does this procedure make sense?

Your comments will be greatly appreciated

Dear Nolberto, thanks for your reply. I'll comment on each point.

1. I do not need "different MCDM problems, with different number of criteria" but pairwise comparisons for the same problem with the same criteria, collected by surveys on a relatively large sample, like in this paper Article Prioritizing factors determining environmental responsibilit...

By the way, I'm not planning to use normal distribution.

2. This is true since we only think about technical problems and use AHP to find the best alternative. But AHP can also be used to prioritize factors (without alternatives) on social and environmental issues too.

3. As in point 2, this doesn't apply if AHP (or any other MCDM) is used to analyze the prioritization of factors instead of choosing a technical/practical alternative.

4. An "objective weight" is not what I'm looking for. I want to analyze people's perceptions on a certain issue and, despite its drawbacks, I think AHP can be a valuable mathematical tool.

Dear Nolberto, thanks for your reply. I'll comment on each point.

1. I do not need "different MCDM problems, with different number of criteria" but pairwise comparisons for the same problem with the same criteria, collected by surveys on a relatively large sample, like in this paper

Article Prioritizing factors determining environmental responsibility...

Dear Massimo

As I understand it, you have a problem, use pair-wise comparisons and get results. Then, start again the same problem with different set of pairwise comparisons and repeat this operation many times. Is this correct?

What is your aim with this exercise? In you repeat the operation say 100 times, it is equivalent to have 100 DMs working individually. Fine, but what for?

The article cannot be accessed, you must request it. Not very practical indeed in this circumstance

However, I could read the abstract, and in my opinion, the result is correct, awareness is the most important, albeit that what was done in the past, appears in last place, something that I doubt , since that awareness is a consequence of actions taken in the past to develop it.

By the way, I'm not planning to use normal distribution.

Fine

2. This is true since we only think about technical problems and use AHP to find the best alternative. But AHP can also be used to prioritize factors (without alternatives) on social and environmental issues too.

What is a factor for you? In AHP we only have alternatives, criteria and performance values. Do you mean to prioritize criteria?

As I understand it, AHP has two procedures: The first one is to prioritize criteria, and the second, using those weights prioritize alternatives. Well, what I said is applicable to any problem, not necessarily technical, this is the first time that I see this discrimination between technical and environmental and social issues, which are usually linked.

The same procedure of one expert comparing many different things apply. Where is the difference? You may have a criterion relative to income, like disposable income (a very important metric in social studies), to be compared with the production of NOx from a factory, that belongs to environment

Of course, criteria refer to everything including social and environmental issues. Sorry, but I do not understand your explanation

3. As in point 2, this doesn't apply if AHP (or any other MCDM) is used to analyze the prioritization of factors instead of choosing a technical/practical alternative.

Are you sure? If a DM decides that criterion ‘inconvenience’ is less important than criterion ‘benefits’, you may be deciding for thousands and even for millions of people. Want an example? Find what happens when more than a million people were forced to abandon their land to build the Three Gorge Dam in China, or what happens when building the Itaipú Dam in Brazil? Do you have an idea of a social disaster that it provoked?

4. An "objective weight" is not what I'm looking for. I want to analyze people's perceptions on a certain issue and, despite its drawbacks, I think AHP can be a valuable mathematical tool.

In my opinion, in that case, a well-structured survey will be more than enough. Just by the number of answers you can measure that perception.

If you rely on the results obtained by 100 people in a desk, probably not even knowing the country that is being ‘analyzed’, without a knowledge of their problems and their social environment, what can you expect?

Dear Nolberto,

1. Why are you talking about "repeating the operation"? I need pairwise comparisons gathered with a survey on a relatively large sample (say from 200 to 5000 statistical units). Why do you think this is a "repetition"?

2. Yes, I meant criteria

3. I don't understand. Are you saying that, in your opinion, the whole AHP framework is not reliable at all?

4. What I'm going to do is exactly analyze the results of a survey, where questions are pairwise comparisons of an AHP, with the new framework I'm working on, to take account of uncertainty too.

Dear Massimo

1- When I said 'repeating' I mean that each new pair-wise comparisons and "weightings" is repeated 100 times by different actors

3- Yes, that is exactly what I mean. I have proved it many times, with reasons and examples, not with void words, and even published the reason by which I am against that method. Curiously, along maybe 8 or 9 years in RG, NOBODY has rebutted me, even when I asked for that.

AHP is not reliable and it is deeply flawed. Now, If you want a technical discussion about it, I will be delighted to accept it.

4. Well, I wish you luck in your research. However, if you allow me, remember that I consider that pair-wise comparison invented by Franklin (1772), and developed by Thurstone (1927) is a legitimate tool.

What is INVALIDID is to extract quantitative values of the preferences. Three days ago in our interchange, I said the same.

You answered saying" I totally agree with you: identifying, quantifying, and handling the "intrinsic fuzziness" of pairwise comparisons is exactly what I'm working on."

It appears to be a contradiction in your own words.

You talk about the AHP framework, and it is based on assumptions, that possibly were partially correct when Saaty developed the method, based on the military hierarchy, but that changed in the 70s. In my opinion, the drawbacks of this framework are, among others:

1- The extraction of preference from pair-wise comparisons. No rational explanation

2- The need to work with only independent criteria. Not representing reality, albeit perhaps in personal and trivial problems

3- A formula that FORCES to modify, if necessary, the estimates of the DM. Where is the math here?

4- T he 'necessity' to determine a transitive data set. Why? Never explained

5- Assuming that the weights from Eigen Analysis are weights, when in reality they are trade-offs. The same Saaty used the word 'assumed'

6- The construction of a 'Fundamental Table', based on the Weber-Fetcher psycholocical law, when there is NO RELATION of this law working with incentives and responses, compared with preference values and responses

7- The fact that it is taken for granted that the weights can be applied to the real world. Who said that? Is there any theorem that justify this assertion?

I can continue, since more that 100 researched whose names I published, including me, have detected 29 drawbacks, also published in 2019.

Curiously, or perhaps not, NONE of these assumptions, after more than 11,000 accesses to this book (data from Springer), were rebutted or discussed, even when I publicly asked for it. Please, tell me what do you guess is the reason for that lack of response?

Dear Nolberto,

1. I disagree. I don't think it's repetition but sampling from a population. Surely, we could discuss the sampling method, but using a representative sample from a population is the base of inference. So, I don't understand why you don't agree with taking a sample of pairwise comparisons too.

I found your book. Interesting! I think I have access to it through my Institution, I just filed a request. So I'll take my time to read it before getting back to you and discuss all the drawbacks you highlighted.

For the question "tell me what do you guess is the reason for that lack of response?", I just don't know but I guess there are two options: nobody has the answers or nobody wants to invest time to find them (but, in both cases, I wonder why since there is also a biennial International Symposium on the Analytic Hierarchy Process).

Dear Massimo

Thank you for your answer

1- Maybe I did not express myself very well, for what you say is true, and there is no doubt of its validity and by determining the number of samples. What you suggest is to take 100 samples each one with the same questions, but in a different universe, and for me it is correct. The fact that you are using the same set of questions to 100 different universes, is what I, perhaps mistakenly, called ‘repetition’

I am delighted with the perspective to discuss with a scholar like you, about what I say on that little book and have maintained for decades. I remember that I learned AHP in my university, when taking a master course, decades ago, when I interrupted the professor several times, by pointing out some of the subjects that are in my book, especially those referring to quantifying preferences using pair-wise comparison. From the very beginning I found this process absurd, as well as some of my fellow students, all graduates in engineering

As an engineer, I am used to reasoning and thinking about technical issues and especially on real-world projects, not on blackboard examples. I couldn’t agree in following what somebody decided, without an explanation, unreasonable, just using intuition, very useful indeed in our life everyday, but not for deciding and giving a value among two criteria regarding real-life projects.

As you can see, I am not using mathematics, only common sense

Massimo, I like your answer about my question; you are probably right in both responses. You are the first person in maybe 8 years that suggested two reasons, for which I thank you. However, I believe that the true cause is the first one: There are no answers.

Then it poses a question:

Why some people continue using AHP? Most probably because it is easy, it does not demand any reasoning or thinking, and because they were educated by instructors that only knew this method, and in the final exam did not accept that a student t solving a problem with other method. This is not a supposition; I bear witness of this.

You question in turn: Why are there biennial conferences or ISAHP?

Simply, because it is a profitable legitimate business, and a site to meet people, like most conferences on MCDM around the world.

Dear Nolberto. Happy New Year! I've had access to your book and I'll begin to read it soon, but before I have to study for 5 exams... In the meantime, I'd like to start discussing with you some of the points you highlighted earlier.

1- The extraction of preference from pair-wise comparisons. No rational explanation. What do you mean? If you mean the extraction of priorities from pairwise comparison matrices, mathematically it makes perfect sense and it's easy to prove it.

2- The need to work with only independent criteria. Not representing reality, albeit perhaps in personal and trivial problems. I agree. This is an issue. Many researchers are working on it though, with AHP extensions.

3- A formula that FORCES to modify, if necessary, the estimates of the DM. Where is the math here? I agree. In my opinion, inconsistent matrices should be discarded and/or considered unreliable (according to their Consistency Ratio). Adjusting inconsistent matrices would be like taking a survey from a sample and then adjusting the responses (or asking the respondents to adjust their responses) if we "don't like" the results...

4- The 'necessity' to determine a transitive data set. Why? Never explained. What do you mean? Are you talking about (in)transitive matrices and (in)consistency?

5- Assuming that the weights from Eigen Analysis are weights, when in reality they are trade-offs. The same Saaty used the word 'assumed'. I agree. The priorities of AHP should be, in my opinion, interpreted as the result of respondents' preferences (based on their knowledge, experience, etc) but not as objective weights.

6- The construction of a 'Fundamental Table', based on the Weber-Fechner psychological law, when there is NO RELATION of this law working with incentives and responses, compared with preference values and responses. I'm not enough of an expert on the topic to be able to discuss it.

7- The fact that it is taken for granted that the weights can be applied to the real world. Who said that? Is there any theorem that justify this assertion? It would be interesting to compare the results of AHP with other techniques on the same topic. Is there any research like this in the literature? I never searched.

Massimo Pierini added an answer

7 hours ago

Dear Nolberto. Happy New Year! I've had access to your book and I'll begin to read it soon, but before I have to study for 5 exams... In the meantime, I'd like to start discussing with you some of the points you highlighted earlier.

1- The extraction of preference from pair-wise comparisons. No rational explanation. What do you mean? If you mean the extraction of priorities from pairwise comparison matrices, mathematically it makes perfect sense and it's easy to prove it.

2- The need to work with only independent criteria. Not representing reality, albeit perhaps in personal and trivial problems. I agree. This is an issue. Many researchers are working on it though, with AHP extensions.

3- A formula that FORCES to modify, if necessary, the estimates of the DM. Where is the math here? I agree. In my opinion, inconsistent matrices should be discarded and/or considered unreliable (according to their Consistency Ratio). Adjusting inconsistent matrices would be like taking a survey from a sample and then adjusting the responses (or asking the respondents to adjust their responses) if we "don't like" the results...

4- The 'necessity' to determine a transitive data set. Why? Never explained. What do you mean? Are you talking about (in)transitive matrices and (in)consistency?

5- Assuming that the weights from Eigen Analysis are weights, when in reality they are trade-offs. The same Saaty used the word 'assumed'. I agree. The priorities of AHP should be, in my opinion, interpreted as the result of respondents' preferences (based on their knowledge, experience, etc) but not as objective weights.

6- The construction of a 'Fundamental Table', based on the Weber-Fechner psychological law, when there is NO RELATION of this law working with incentives and responses, compared with preference values and responses. I'm not enough of an expert on the topic to be able to discuss it.

7- The fact that it is taken for granted that the weights can be applied to the real world. Who said that? Is there any theorem that justify this assertion? It would be interesting to compare the results of AHP with other techniques on the same topic. Is there any research like this in the literature? I never searched.

Dear Massimo

Thank you for your wishes for the new year, that I heartly reciprocate.

I am glad that you got access to my book. With it, will be easy for us to maintain a friendly technical discussion and especially for me to be conversing with a person like you, that do not hesitate in researching, especially if you are curious on what the literature on MCDM claims.

It is not a matter of doubting on what an author writes, but in finding the truth behind those claims, and of course, this includes me. I am not against AHP; what I am against is students, practitioners, professors, that do not investigate, and if they do, say nothing, and continue using a method (any), that they suppose may hint as diffuse or not clears, and it applies to all methods, including SIMUS. I would love to have some negative comments on any aspect of the method from anybody, to correct it if necessary. You, my friend, are excerption

Now. Concerning your last response.

1- The extraction of preferences, as you call it, does not have any mathematical support, and not even analyzed from the common-sense point of view, unless the problem is a personal one, like deciding a restaurant for dinner or a movie to watch in Netflix

Now, in my turn for making questions: Could you please tell me where are the mathematics in pair-wise comparisons?

2- Glad that you agree, but what are the AHP extensions? Please don not tell me that using fuzzy solves that invention of criteria weights. Fuzzy will effectively find a reasonable average. On what?

On the DM coherence, but this does not mean that said coherence can be applied to real-life problems, since mathematics, linking mind to reality, has not been invented yet

3- Your opinion is very respectable, but did you ask yourself why there must be transitivity or consistency in the estimates of the DM. As a simple example. Assume that Peter must decide among three restaurants A, B and C for dinner, based on quality and price in each one, and that is constant, according to AHP.

Peter always say that for him, quality ifs more important than price, and for that reason selects A on B even when A is expensive, or A>B

Now he compares B and C and finds that C has a little less quality than A but at a lower price, and then, he prefers C > A or price on quality, or hat price is more important than quality. Obviously, there is inconsistency here, but is it wrong?

Why inconsistent matrices should be discarded? Don’t you think that is going against the DM estimates? If we do as you say the result is biased because you are violating at will what the DM says. But mainly Massimo, on what authority can you do that? The Arrow Inconsistency Theorem calls that a dictatorship’, and it is.

AHP looks for consistent matrices because the principal eigenvector of an inconsistent positive reciprocal matrix is a weak priority vector, which is not what the method looks for, while in a consistence matrix it is a strong priority vector (this I from AI, not from me). In other words, this is, in my opinion, the reason for the only mathematical application of AHP

4- Yes, I am talking about them

5- Exactly

6- Saaty himself says that his Table is based on this law. Now, incentives imply normally an increase on for instance a wage for a job well done, and that originates a non-proportional answer, because the person will put more emphasis in improving his performance, i.e., a log answer, that is, when an incentive may be lineal. but the response is represented by an ascending curve. Saaty Table tries to mimic this relationship and thus assuming that a number that indicates a preference may originate a log response.

The fact that an incentive is not the same as a preference seems not to raise any concern, and happily accepted as a truth.

I am neither a psychologist nor a mathematician, simply somebody that is always asking, observing if a mathematic procedure is correctly applied to a problem, by reasoning, researching, asking WHY and HOW, which is a constant feature in all my books

7- In my opinion it is better to ask why AHP applies that concept. Saaty never explained it.

As far as I know other techniques using AHP weights are also in the same wavelength, and I wonder why they are not using objective weights like entropy.

I don’t know if there are more research on this, but I guess that t here is no reason to be considered the first one.

Nolberto

Dear Nolberto, I'll start with points 1, 2, and 3. I'll add some pictures for the math since it is very difficult to write math in RG (it's a pity that an excellent platform like this doesn't have LaTeX support yet...).

1. Let's suppose we know the truth, i.e. we know that for n criteria there exist π priorities, as in eq(1.1). We can build a perfectly consistent pairwise comparison matrix, as in eq(1.2). Since it is consistent, its maximum eigenvalue λmax is equal to the rank, i.e. λmax=n and each column, normalized to sum to 1, is the associated eigenvector. So, normalizing each column we can compute the original priorities π, eq (1.3).

2. I was thinking about ANP. Yes, it is no longer a hierarchical model, but it's AHP-derived, so I think it can be considered (maybe improperly?) an AHP extension. What do you think about it?

3. Sorry Nolberto, maybe you forgot to write a comparison in your example. If you have a nxn matrix, you need (n)(n-1)/2 value to fulfil the matrix with reciprocals, so in this case 3(3-1)/2=3 comparison, but you wrote 2 comparisons only (maybe a typo?). Using (n-1) comparisons (in this case, 2) you can fulfil the matrix anyhow, but using all the consistency rules; so it would be perfectly consistent. Anyhow, how could you distinguish these cases you're talking about from cases where the inconsistency is due to boredom, lack of understanding, etc? If we always accept inconsistent matrices because in some cases they come from oversimplified frameworks (like, I guess, in your example), we accept to incorporate "intrinsic uncertainty errors" in our priorities estimates too, and it would lead to much more biased results then discarding too much inconsistent matrices or considering them as reliable as their CR. Or... try to estimate the amount of this "uncertainty" and take the real "preference" (or, better, its estimate). What do you think about this?

Dear Massimo

I agree with you regarding the difficulty in writing math formulas in RG. It is a pity, considering that it is by far, the best platform between scientists, if not the unique.

Now, answering your points:

1- Yes, coming from the roots of the polynomial, but bult on invented estimates. What is the utility of that? Garbage in garbage out, and remember that those priorities, since the matrix is transitive, only represents what is in the mind of the DM, no more than that

2- In reality ANP is the generalization of AHP. In my opinion, AHP is a method within that generalization. By the way it is not difficult to guess the reason by which Saaty developed ANP after AHP.

I presume it was because he realized the inconvenience to work in real-life projects with independent criteria

3- Really, I never worried about reciprocals. Therefore, I did not forget, simply I do not care about them. I know that in AHP they are important but not for me or in real projects. Again, using common sense, why assuming Peter selecting a restaurant for dinner, need to have in consideration reciprocals?

He is not a mathematician, only a guy who reasons. If we want to apply MCDM methods, we need to think as a normal person purposes, otherwise the method is not very useful.

Remember that the only methods, as far as I know, that use reciprocals is AHP and ANP, all of the most of other 200+ methods do not care about that mathematical nuance. Does it mean that all of them are wrong? Nevertheless, considering your point, I built the same example, and the result is the same, the matrix is inconsistent

The inconsistency may very well be due to boredom. Tiring or whatever other circumstance.

Do you imagine the nightmare that would be making a series of tests in a problem where the criteria values changed in each run? Years ago, I participated in an European agri- project that involved 120 runs, each one different, of course not using AHP or ANP

Yes, my example is very simplistic, but I do not understand your reasoning. As per AHP, inconsistency matrices must never be accepted, but it has no concern in accepting forced or fraudulent matrices. A question, what happens if the DM refuses to correct his estimates? He cannot continue, it is like ‘it is this or nothing’, very similar at what the Arrows Impossibility Theorem say

What happens if we have an inconsistent matrix and you apply the geometric mean instead f the Eigen Value analysis?

Jiří Mazurek et al. / Procedia Computer Science 207 (2022) 504–513

“Ordinal rankings’ similarity of EVM and GMM

From the previous example, it can be seen that priority vectors obtained by different methods are different. In general, priority vectors derived using EV and GM methods are identical in the case of a consistent pairwise comparison matrix [5]. Also, in the case of n = 3, priority vectors for both methods coincide even when the PC matrix is inconsistent”

Intrinsic uncertainty errors? To interpret an error, we need to have a value to compare to, something that we do not have.

Now, for you, forcing a consistency reduces uncertainty? And what about the uncertainty of forcing consistency?

I often perform an exercise. For a problem, I find the entropy weight, make the ranking and then compare it to the ranking from AHP. Never found even more or less coincidences, which is understandable. Entropy is science, AHP is lottery. Entropy is always the same, whoever is the DM, while AHP weights may change according to a number of different DMs.

I do not understand this sentence: “we accept to incorporate "intrinsic uncertainty errors" in our priorities estimates too, and it would lead to much more biased results then discarding too much inconsistent matrices or considering them as reliable as their CR

It appears that you are saying that accepting uncertainties is more biased than accepting

Inconsistent matrices? I agree, are less biases due to estimates from inconsistent matrices than with consistent.

The first, even considering that are intuitions, are at least what the DM wishes. In the second, the same intuitions exist, but aggravated by the forcing to consistency

By the way, remember that AHP considers matrices with up to 10% on inconsistence and I wonder how EV reacts to that

Dear Nolberto

1. "what is in the mind of the DM" is exactly what I care about in my research. As I said, and you agreed to, I interpret AHP as "DM preferences" not as "objective weights". I want to use AHP to analyze DMs opinions, not to use the results for decision-making. I know, it may be a paradigm shift in AHP framework, but it's what I'm going to do.

2. I agree.

3. It is not a matter of "worrying" or "nor worrying" about reciprocals: to fulfil a matrix you need reciprocals, i.e. in order to perform all needed calculations you need a complete comparison matrix because in AHP you never ask for n2-n comparisons (otherwise the questionnaire would be too long). I'll explain. Say we have three criteria: A, B, C. If the AHP questionnaire has only (n-1) = (3-1) = 2 questions, we're missing n(n-1)-(n-1) = 3(3-1) - (3-1) = 4 elements, as in matrix M1. To complete the matrix we need to take the reciprocals of the two known elements and use transitivity for the last two elements, as in M1*. In this case, λmax = rank and we CANNOT have any inconsistency at all. Notice that, for the sake of simplicity, here I used elements on the same row but it's trivial to prove that this holds for any non-symmetrical elements. If we instead add a third question (provided that it's not symmetrical to the other two) so that we have n(n-1)/2 = 3(3-1)/2 = 3 elements, as in M2, we need to complete the matrix using reciprocals only, as in M2* (we cannot use transitivity here). In this case, we CAN have inconsistency if the third element is not transitive. Notice that again, for the sake of simplicity, here I used the elements of the upper triangular matrix but it's trivial to prove that this holds for any non-symmetrical elements. I'm not saying that the DM should care about reciprocals (he/she should not care about them at all...!), I'm only saying that this is how AHP matrices work. If a matrix is inconsistent, as I said earlier, we should not try to adjust it but understand WHY it is inconsistent and adjust (or completely change, if needed) the method because there's something wrong in the method not in the DM, who only expressed its preferences as requested. If the matrix resulting from its preferences is inconsistent, what is wrong is not the DM but the method we used (too simplified, too complicated, too long questionnaire, non-independent criteria, etc, etc). The DM (and his/her response) is never wrong because he/she only did what was requested: express an opinion. Regarding "uncertainty", well this is the core of my research and of the assumptions I'm going to verify (or falsify, if I'm wrong) with experimental data. So, maybe, we should talk extensively about it once I'll have verified/falsified my assumptions...

Dear Massimo

1- If your objective is determining what is the mind of the DM, it is, in my opinion, not an area of MCDM but pertaining to psychology, as apparently was Saaty aimed to, since he chose the descriptive process instead of the normative one, followed by 99% of MCDM methods, or the European School.

I follow this school because I believe that there is no room for psychology in real-life MCDM projects. These are not being built by wishes, opinions and intuitions. They are ruled by needs, facts and numbers, by economics, health and environment, real things, and because of that they are the field of applied mathematics. The problem with AHP and many other methods is that they work with assumptions that are not demonstrated, and from this point of view, anything is OK.

There are multiples real examples, when people say that transmission of words at large distance, was impossible; Marconi showed that it was not true. Or people with high standing saying that nothing heavier than air could fly from Lord Kelvin (nonetheless) in 1890, or that cars will never replace horses, from a prominent American banker in the 1880s. Those were coming from the minds of very illustrated people for that time, and science showed different.

2-

You can legitimately express your preferences in many personal decision, as we do every day, but not to decide a location site, or for selection of renewable sources for electric generation. This is the reason I work with objectives.

3- Of course, reciprocals is not either a matter of wishes, but as you say, AHP never asks for n2-n comparisons due to the work load. Consequently, in your questioners are incomplete, and you develop the reciprocals on your own. Is that legitimate?

4- As I understand λmax is the highest value root of the polynomial and is used as maximum criterion, which is logical, but it is not a rank because this word involves considering the balance of roots.

You say: “The DM (and his/her response) is never wrong because he/she only did what was requested: express an opinion. Regarding "uncertainty”

I am surprised with this assertion, because nobody requested the opinion of the DM about uncertainty; he only was requested to reveal his intuition about which of two criteria is more important. Curiously, if he is never wrong why he may be corrected by a formula?

Is it possible to know your assumptions about uncertainty?

Dear Nolberto,

3. When you ask n(n-1)/2 non-symmetrical comparisons (e.g. the upper triangular matrix) and then use the reciprocals for the remaining n(n-1)/2, you're simply assuming that, for example, if "A is strongly more important than B" then "B is strongly less important than A" so you simply complete the matrix in order to do calculations, you're not building up comparisons out of nothing. If we asked to compare A and B, we do not need to compare B and A, it would be redundant. If you got less than n(n-1)/2 comparisons and you use transitivity too then you're building up comparisons, and this is not legitimate.

4. I'm not saying that the DM should care or answer about uncertainty, I'm saying that uncertainty is intrinsic in the DMs responses and that we should care about it.

Is it possible to know your assumptions about uncertainty? We should talk about it in private. I will launch a survey to collect data and don't want to affect the results by publicly expressing the assumptions (that any possible respondent could read).

Massimo

3 and 4 - I agree with you and with your reasoning. I mentioned that before because you said that using the complete matrix for questioning, would be too much a workload

Regarding your last point it is perfectly understandable that you want to keep private. If you wish and think that it could be useful, my email address is [email protected]

Dear Nolberto,

Sure it can be useful and, if you'll share the concepts, we could also work together on it. I've sent you an email.