Yes, it is possible b ut not acceptable

You should modify your comparison.

Md Tanweer Ahmad

Why, because a formula says that it should be less than 01?

So, the DM decisions are subject to what a formula says?

This is one of the absurdities of AHP

Nolberto Munier,

Agreed your point, but, what I mean to say that if I got CI greater than 0.10. In this case, we have to modify our comparison matrix from start to finish.

Dear Md.

As my understanding, and according to AHP theory a value less thatn 10 % is correct.

Thus, what are you arguing about?

If the CI is less that 0.10 you have nothing to modify, if you accept this theory.

Thi s value was determined by Saaty using simulation. The Eigen Value Method can't be applied for CI>0.10

Pair-wise com parison makes sense when it is comparing real values as in Promethee, Electre anfd other methods, not based on experts values

It is possible for sure. It happens when decision-makers respond to the questionnaire carelessly. You can ask them to modify the comparisons.

Mir

So, you will ask experts to modify their earlier estimates?

Why should they do that? Based on what?

Because C does not coincide on what A and B say? And who guaranties that A and B are correct and C wrong or viceversa? By intuition?

And if you change the team of experts and have another team that gives different results from the former?

Who is right and who is wrong?

Do you see the incongruence?

Pairwise comparison methods have some limitations. You can use compensatory aggregation-based methods or any outranking methods for the solution.

Dear Rajesh

As far as I know outranking methods like PROMETHEE and ELECTRE do use weights from pair-wise comparisons, unless, as in other methods, they use objective weights

Dear Prof. Nolberto,

Compromise ranking and compensatory aggregation methods like VIKOR and TOPSIS or outranking methods like PROMETHEE or ELECTRE use input weights, which can be assumed based on the decision maker's requirements. It can be decided based on expert opinions, experiments, or by using a Delphi method. In my understanding, there is no mandatory condition that the weights need to be obtained using pairwise comparisons. But, pairwise comparison-based methods, such as AHP or ANP work on the eigenvectors and consistency and later by calculating the local priorities of alternatives.

Dear Rajesh

Decision's makers requirements? That would be logic, but remember that the main source of subjective weights is AHP, where the weights are based on DM intuitions, good perhaps for personal or trivial problems, but not for serious scenarios, like selecting a location to install an industry. It is absurd, that the DM can decide by intuitiuon when comparing for instance, 'Local workkers skill' with 'Exports facility', that one criterion is 2 or 3 times more important than the other.

The DELPHI method is biased since an agreement is reached after the decisions are 'adjusted' , in succesive iterations and according to precedence values and opinions.

No, there is no mandatory condition that the werights must be obtained by pair-wise comparaison. They can come from experts asssessing an individual criterion individually, by reasoning, experience and know-how. This is valid and makes sense.

Consistency in AHP and ANP is a pretext to be able to use Eigen Values, that needs some sort of transitivity.

The Eigen Value method works fine, but if using weighted from pair-wise comparison they work with false inputs. Garbage IN, garbage OUT.

Dear Prof. Nolberto,

Thanks for your answer. I was mentioning the alternative ways, the student can solve the problem (referring to the question posted). I am not in favor/ against any pairwise comparison-based methods. But, if a researcher got stuck with a consistency ratio in AHP/ANP-based methods, he can assume weights based on any of the suggested means and can continue with the selection problem using any compromise ranking or outranking methods. This is what I tried to answer.

Dear Nolberto,

It's true,

Thanks

Dear Abdul,

it is possible, but for larger matrices, CR could be handled more flexibly so the threshold could be extended to 0.2. 5x5 PCM might belong to the group of larger matrices.

However, in case the evaluations are very inconsistent (over 0.2) it might be beneficial to restructure the decision hierarchy or reevaluate the selected pattern of respondents or the surveying process itself.

Dear Szabolcs

• You say 'it might be beneficial to restructure the decision hierarchy or reevaluate the selected pattern of respondents or the surveying process itself'

Regarding de first bolded sentence, does it mean that the DM has to MODIFY the hyerarchical structure of a problem, seldom linear, to match his/her needs?

Regrading the second, does it mean that the DM has to ALTER the data from a survey obtained by statistics, just to get transitivity? And who told him that the real problrem has transitive conditions?

I will very much appreciate your answer

Dear Nolberto,

thank you so much for your questions.

In case of high inconsistency, it is possible that the decision structure (created in the first step of the survey process) is misleading or false. Generally, this structure is determined by some experts who are not participating in the survey process itself and are most likely not involved in the decision problem itself (except for doing research, consultation, etc.). So it is possible that the initial structure is flawed. There might be missing decision elements, overgeneralization, etc.

Regarding your second question, in my opinion, most of the real decision problems are transitive (I think we already had this conversation some years before). If I prefer A over B and B over C, I prefer A over C. There might be some rare exceptions but it is much more general that a survey participant fills out the questionnaire almost randomly (just to get quickly to the end). These pointless evaluations are filtered out in AHP and for a reason, because they provide non-transitive responses so the assumption that this evaluated questionnaire was filled out randomly stands. This falsely evaluated questionnaire is considered in other methods (not only MCDM but take a look at DCM types of methods) but not in AHP. This is a strong advantage of the AHP and should not be wasted just because there are some special non-transitive cases for preferences.

I am always grateful for your comments, one of them gave us the inspiration to create a PROMETHEE type model for public transport mode choice with one of my Ph.D. students. Thank you so much for this conversation.

Dear Szabolcs

Thank you for your answer. I do appreciate it very much because you are one of the very few AHP supporters that answered my frequent request by challenging my comments, with reasons, with which I I may or may not agree. Thank you, friend, and also for your last paragraph. I feel rewarded to receive such comment from a scholar like you.

Now, coming back to your initial comments.

1- ‘In case of high inconsistency, it is possible that the decision structure (created in the first step of the survey process) is misleading or false’.

Yes, it can happen, but it is difficult to accept it because the structure of a problem is fixed by the problem not by the experts.

Suppose that you have two independent criteria C1 and C2. It may happen that a sub criterion from C2 is connected, by technical reasons of the problem, to C1, thus, the hierarchy is non longer lineal. You can’t modify it, and you can’t apply AHP because C1 and C2 are not independent (Saaty rightly dixit), and that condition is established for instance from an expert from the engineering area of the company.

You are right in saying that the expert is not involved in the decision problem. Exactly, normally he is not, but what he is expressing is reality. He is letting the DM know what reality is, which is not related with his knowledge of MCDM, and probably he is not interested on it. He describes reality and thus, giving the necessary input for the problem. What the DM does with it, is not his concern.

Agreed, there could be missing elements, but that is not for the DM to decide. He can’t be proficient in engineering, accounting, finances, social aspects, etc. at the same time.

From my humble point of view, I have always had the idea that in many cases, the developers of MCDM methods, mainly mathematicians, do not have experience in real issues, which is understandable because it is not their field; they don’t have hand-on experience in industry, commerce, construction, sustainability or environment, and apply mathematics to solve problems that don’t understand well, like industries location, supply chain, manufacturing, etc.

In my opinion, for this reason, none of the MCDM methods is designed to model aspects like using different times intervals, values minimum and maximum for the same criterion, precedence, inclusive and exclusive alternatives, etc., which are very often found in real scenarios.

This is one of my main critics on AHP and ANP, as well as many other methods. They TRY to solve a problem without a complete knowledge of its characteristics. For instance, they do not consider if the problem is feasible, or that there is precedence between alternatives.

Do you want an example?

Assume a construction company that has 5 projects of different characteristics. The DM applies MCDM to select which are the most important projects, subject, of course, to a set of criteria. Suppose that a MCDM method indicates that out of 5, only 3 projects can be built, because there are not enough funds to initiate all of them at the same time, WHICH IS WHAT ALL MCDM METHODS ASSUME, except Linear Programming and SIMUS by extension. Therefore, the whole scenario is unfeasible, unrealistic, because lack of enough funds, equipment, personnel, raw materials, etc.

Most methods, with the exception of PROMETHEE, assume that resources are infinite. How can AHP and many other methods select the best alternative, if they don’t not know if there are enough funding or manpower or raw materials?

Even for trivial problems like selecting a restaurant for dinner, if you don’t know if you can afford them because their prices, how can you select one?

In construction problems for instance, the correct way is for the DM to ask the planning and scheduling department of the company, about the dynamics of the of 5 projects scenarios. This is normally done[NM1] using Critical Path Method (CPM) for planning, and the Gantt Bar Chart for scheduling. These two methods can tell the DM about what year every project was scheduled to start, with different year advances in percent, which are different for each construction year, and establishing an annual budget that must not be exceeded.

With this information, the DM can model the scenario considering that not all the undertakings start and finish at the same time and also having different durations, and then getting the best project.

Do you think t hat AHP or ANP can solve this problem? Of course, not.

Believe me, I have worked many years as a Planning Engineer, in the construction industry for hydro dams, mining, chemicals, buildings construction, and bridges, and this is the way how the process goes in real life, using complex software to compute the CPM and the Gantt Chart.

Do you think that all of this can be done using intuition?

Regarding my second question, and according to many researchers, reality is not transitive. Suppose you must decide between building a bridge (A), a tunnel (B) or establishing a system of ferries (C) to cross a wide river. Do you think that if A > B and B >C, then A > C?

Not necessarily, because you can’t compare the three schemes from only a single point of view, say cost, where A cost more than B, but B, is less expensive than C. What about considering SIMULTANEOUSLY safety, maintenance, strong winds, floods, etc.?

Therefore, when the DM establishes for instance that A>B>C, that situation exists only in his mind.

What you say about the survey participant is very true. However, why their answers must be transitive? Is there an axiom does say that?

How do you know that they are false? It is as the DM saying “What comes from the survey is wrong, I will put it right”, or in other way he is saying that what thousands of people think is incorrect; it is only valid what he says. This thinking is wrong in two counts:

1- First, the DM, seated in his comfortable office perhaps at thousands of kilometers of the project size, without knowledge of the site, pretends that he is more knowledgeable than the people living there. This thinking violates Arrow’s Theorem, that says that a person cannot vote representing another, and that this situation is a ‘Dictatorship’.

2- It violates common sense

Please allow me to put a real example that happened years ago in a coastal city in Western USA. The project consisted in building a highway through a city according two alternatives: At high level, (on a viaduct), and at on-grade (i.e., at terrain level.) The DM used MCDM and recommended the second alternative, and thus, it was so published in newspapers.

About a week later, an old lady entered the offices of the company in charge of the project and asked to talk to the project manager.

In the meeting the lady said ‘You know that our city where you intent to build a highway, is between the mountains and the sea, and thus, in the raining season large amounts of water come from the mountains, and discharges into the sea.

In addition, she said to the project manager ‘Did your people realize that if the highway is built at city level it will act as dam’?

‘The water will not be able to reach the sea, and the whole area, including our houses, parks, commercial places and schools will be flooded’?

‘Do you know the effect that the highway will separate the city into two areas, generating lots of problems for our children going to school, or for going shopping, because the supermarkets are on the downtown area located between the highway and the sea?’

‘Do you have an idea of the problems that it can generate?’

Needless to say, the project manager was ashamed, and the project cancelled.

Years ago, in February 20l7 I had the privilege in corresponding with Tomas Saaty, about this same problem. I said to him the same thing that I am telling you. His answer was ‘They have to be taught’. Due to my respect for his memory, I prefer not to comment about this answer.

Szabolcs, please don’t interrupt this interchange of opinions. We, researchers, may be right in some aspects and grossly mistaken in others. The only way to settle differences is a sound and honest discussion and, in our field, supported by mathematics and examples taken from real life.

Best regards

Nolberto

Dear Nolberto,

thank you so much for your answer. As always, it motivates me very much. Most researchers take well-known methods for granted and apply them to several problems without even examining their applicability and produce tons of papers by this. But we are searching for the truth (if I may include myself in your company). The only difference between us is that I see some merits in AHP and ANP and would like to improve and fix their shortcomings but you think that both of these methods are originally flawed. You also have strong arguments, I admit.

Interestingly, I am dealing with the Arrow theorem right now because the aggregation technique of AHP and ANP seems very dissatisfactory to me and I believe that a more efficient aggregation method can be found.

Friendly regards,

Szabolcs

Dear Szabolcs

Thank you for your letter. I am 100 % in agreement with your first paragraph and it is nice to see how a learned and AHP supporter colleague is not biased.

You are right, I support the idea, and you know that I am not alone in this opinion (I can give you the list of researchers that think similarly to me), that AHP and ANP are flawed on many counts. In one of my books, I listed 29 drawbacks in AHP.

For me is inconceivable a DM taking decisions on criteria importance based on his intuition and WITHOUT considering the alternatives those criteria are to evaluate.

It is not valid even for trivial problems.

It is also no scientific that a DM must modify his prior estimates based on what a formula says. If it is true that the 10 % limit is based on simulation, this forcefully is partial, because it is rather impossible to simulate all the characteristics of some problems. It is based on the assumption that decisions on criteria relationships are transitive. There is no axiom, theorem or common sense that support this idea.

I don’t want to abuse of your patience so the last serious drawback that I will comment is the assumption that the real world is transitive, and most important, that what is I n mind of the DM applies to this real world. Again, there is no proof about this, only an assumption from Saaty. Why should we accept that it is true?

If you are a father, think that in your mind you have the best ideas and purposes for your children, but it does not mean that they accept them (unfortunately!), they have their own. The same happens with AHP that directs what must be done, concerning lots of people. It does not mean that people must follow them.

AHP was conceived when Saaty worked for the military, where there is a lineal hierarchy, and from that point of view it is quite good, and thus, adopted by industries and companies. We can't blame Saaty or his method for it.

But when companies abandoned the lineal hierarchy replacing it by the matrix one, it was the beginning of the end for AHP. Most possible Saaty detected that and for that reason developed ANP, however, the drawbacks and the lack of logic and common sense in AHP permeated the new system, which also incorporated new concepts without any mathematical foundation like the concepts of Influence, to replace a simple precedence, and still, worst of all, the concept of feedback. Both of them were never explained by Saaty.

As you can appreciate, you and I have more than one difference.

In my opinion, if a MCDM can't justify every step of the process regarding assumptions, the method is flawed. Look at PROMETHEE, ELECTRE, TOPSIS and VIKOR. These methods have little imperfections, the main one is using AHP weights, and, if it is true that sometimes thresholds in the two first are subjective, they are based on analysis and reasoning, and thus, they have credit; they are not invented.

Same for TOPSIS, in which, other that using weights, its only weakness is the selection of the type of distance, and the same with VIKOR.

Methods like BWM and many others are based on subjective assumptions about which is the best or the worse alternative or criteria. Based on what?

But AHP and ANP have many unjustified assumptions in comparison with the methods above mentioned.

In general, for all MCDM methods, hundreds of pages, deep and complex mathematical reasoning using fuzzy, have been performed trying to improve data, which is a good thing. However, NONE of them consider the most important: Build an initial decision matrix reflecting reality as much as possible.

I wonder, and have formulated many times my concern about why we spent time improving data when it does not represent a problem. It leads to solving a problem with better data and thus, with more reliability; too bad that the problem solved is only a bad imitation of the real one. In other words, we are solving something fake.

Interesting that you are working with the Arrow Theorem. I never thought about a connection, however, I think that you are on the right track when researching about aggregation. I always found strange that AHP adds up priorities or trade-offs, meaning that if you increase say C6 in a 10%, all the other Cj will vary proportionally. In so doing, it means that AHP considers that priorities are in a lineal relationship, which is very debatable.

That is, AHP decides that they are lineally comparable when in reality they are not, for instance comparing criteria Price and Demand.

Thus, at first sight it appears that this violates Arrow

Curiously; Saaty said that in AHP criteria MUST be independent (and with reason), so, how can this dichotomy be explained?

Warmest regards my friend

Nolberto

Dear Nolberto,

thank you so much for your response. Let me just react to the preference transitivity issue (I hope our conversation is interesting to others too, for us it definitely is).

In my opinion, if we have merely one aspect, the preferences have to be transitive. If you want to buy a car and the only aspect (criterion) is the price and A is cheaper than B (so A is preferred) and B is cheaper than C (so B is preferred) then A has to be preferred over C. The same is true for more subjective criteria like esthatics, perceived safety, etc.

The merit of AHP is that it examines only one aspect at a time (even if there are definitely multiple criteria in the decision hierarchy).

Let us assume that you want to buy a car and consider five different criteria with three alternatives (car options). Thus we have two levels in the hierarchy, a criteria level and below an alternative level.

What happens in AHP? First you have to evaluate pairwisely the criteria from the aspect of their importance in the decision. So we have only one aspect (importance), thus, the pairwise comparisons have to be transitive.

Then you have to evaluate the alternatives from the aspect of each criterion. So the aspect is always sole, thus the evaluations have to be transitive.

Finally you multiply the alternative scores by the criteria weights.

Consequently, all phases of the AHP deals with a single aspect in the evalautions at a time so it is valid to check the transitivity of these preferences.

I agree with you in the possible influence of the criteria and the rare cases of the strict hierarchy. But I have seen several cases in surveying public opinion by Discrete Choice Models that do not consider the possible biased and random evaluations and led to false consequencies due to that. These fasle consequencies ended up in non-sustainable transport developments. AHP is a better alternative because of the consistency check.

My main concern about the aggregation is the mean-based approach. You cannot characterize a group by calculating merely their mean. How about standard deviation? It is definitely not considered not only in AHP but in many other decision support methods.

Friendly regards,

Szabolcs

Dear Szabolcs

- Well, simulation could be a good solution, if we could simulate jointly all the characteristics of a problem and their interactions, however, it is difficult for me to understand how we could have a larger pattern that reality, since we don’t know it.

That is, it is impossible for us to consider all the components of reality; we know some, but probably not all of them. In addition, in real world problems, matrices are normally much larger. I have worked (using SIMUS), with matrices of 120 projects and 12 criteria. Since there are not ‘adjustments’ to get transitivity, it takes less than 3 minutes to solve this problem.

If we make say 5000 simulations most probably, we can get a convergence of results, but what does it mean? What is it good for?

If we know the true result, it is a different problem, because we have something to use as a yardstick. As you know, this is the procedure used in supervised AI.

I understand that more than 2-9 restrictions are a convenience based on the psychologist George Miller hypothesis, that refer to capacity limits in information processing, suggesting that it is limited to about seven units plus or minus 2 units. Not too practical indeed.

2- Yes, the Saaty fundamental scale is very well-known, but also criticized, not for me, since I never discussed it. This is the first time that I do.

As you know, it is based on two psychophysical laws from Weber and Fechner, addressing the relationship between stimuli and response. In reality, the purpose of the Saaty scale is to approximate a ratio between two criteria - and thus, involving two values - in an absolute integer given by the Saaty fundamental scale. I recognize that it is an ingenuous ploy.

I have three questions:

1- Is it valid to apply these laws when the ratio is completely arbitrary, reflecting the appraisal of the DM regarding importance, and taking the dominated value as unit?

2- Is it valid to apply the Weber -Fechner equations to convert stimuli in responses?

I don’t think that saying that quality if preferred to price is a stimulus. It is simple a value reflecting an intuition.

Saaty equates the pair-wise comparisons with the neuronal system. However, in here, the stimulus is an emotion produced by something objective, like the birth of a son, or subjective, as observing a beautiful sunrise, and that fires a neuron, which response is pleasure.

In pair-wise comparisons it is a number got from nowhere; it does not have any effect in your body or spirit.

Saaty, in Chapter 2 of his paper ‘Fundamentals of the Analytic Hierarchy Process’, says “Weassume that the stimuli arise in making pairwise comparisons of relatively comparable activities”. It is only an assumption, and by translating it using Weber-Fechner, he tries to convey the notion that his scale is scientifically supported.

3- There is another problem related with the number 9. Saaty explains that it is not a limit, but in cases of large stimuli, the responses are incorrect.

Therefore, I doubt that this scale is a good chance for valid simulation

4- Regarding what you say of how a model behaves, I don’t think than that is the point. We are trying to get reliable solutions independently of how a model behaves, and in my opinion, it is impossible for us, because we don’t know which is the ‘true’ result.

5- Regarding the efficiency of a technique, indeed it could be a coincidence. True, but it is precisely what we are looking for, albeit I would be uneasy and doubtful if there is a perfect correlation, or even 0.9.

I would say that a correlation of about 0.75 would be a good indicator, and I would use Kendal Tau Correlation Coefficient, instead of Spearman.

Regards

Nolberto

Dear Nolberto,

please read my paper published in Expert Systems with Applications journal: Duleba, S., Szádoczki, Zs: Comparing aggregation methods in large-scale group AHP: Time for the shift to distance-based aggregation. Expert Systems with Applications, 196, 116667.

This paper gives the response to all your questions hopefully.

Szabolcs

Dear Szabolcs

Thank you for giving me access to your paper.

From the title I realized that it is not related with the aspects that we have been discussing, let alone to answer my questions. Notwithstanding, I read it, although not in depth.

I am not trying to avoid this important issue, simply I don't see where is the relation with what we have discussed.

A little digression:

When I developed SIMUS I also allowed it to work with groups.

However, I don't need to aggregate anything. In SIMUS, each criterion is analyzed by each member of the group, and each one is free to express his approval or disapproval, regarding the attributes of each one. The criterion under study is then modified according to the different opinions considered simultaneously,for instance changing some values, adding up a new criterion, considering for the same criterion both, lower and upper levels, also simultaneously, etc., that is, modifying the initial decision matrix. To make it clearer, the new matrix contains the comments of all members of the group, as well as the silent of those that don’t see anything wrong in the original matrix.

As you know, in Linear Programming, if the problem is feasible, there is always an optimal solution, not only showing the scores of the alternatives but the OPTIMAL value of the whole system. SIMUS takes advantage of this. This optimal value, which is the sum of the products between the objective factors and the corresponding scores of the alternatives, is identified as Zj.

Once all experts agree that their observations have been registered in the new matrix, the software is run again, and yields, or not, a new solution that contemplated all opinions simultaneously, not by their sum, but considering the influence of each one on the problem.

This Zj is then compared with the former, that is, before the opinions (Zj-1).

If Zj > Zj-1 when the criterion calls for maximization, all opinions are accepted. If the criterion calls for minimization, the new solution is accepted if Zj < Zj-1.

As you can see, after the ‘n’ experts freely express their opinion, the system computes the answer.

When this is done, the second criterion is examined in the same way, and the same for the others, until all of them have been analyzed.

A very important issue to remember, is that the Zj, whatever their results are optimal, according to the Lineal Programming Theorem, of course, if the project is feasible.

As you can see, the process is completely different as is being done with other MCDM methods.

If you are interested, I can send you the section of one of my books where this is explained, step-by-step in a real problem.

Coming back to your article, I don’t see, how, as you hope, it answers my questions. As a matter of fact, it does not answer any, since it is not related with our debate. I don’t really understand how you thought that your article is related with our discussions.

Best regards

Nolberto

Dear Nolberto,

we were discussing the possible role of simulation in AHP. If you read the paper in depth, you could see that it contains simulations of entries as I mentioned before. You are insisting that simulations have no meaning in AHP because they cannot reflect reality so the comparison is always pointless (as the original question sounds in this topic).

As you can read in the paper, simulation is a great tool to reach thousands or millions of possible cases of evaluations. Currently, we are examining over 7 million cases of pairwise comparison matrices filled by the Saaty-scale and with CR less than 0.1. They are not reflecting reality but demonstrate possible evaluations by the Saaty-scale with tolerable inconsistency so that possible responses to decision problems.

If you read the paper thoroughly, you can see that the comparison of the outcomes makes sense because they examine the vector compatibility and rank correlation of the individual vectors and the aggregated global preference vector.

Friendly regards,

Szabolcs

Dear Nolberto,

I am very curious about your response to the preference transitivity issue. If we consider only one aspect at a time (as AHP does), do you think the preferences are transitive? If so you accept a merit of AHP.

Friendly regards,

Szabolcs