Dear Vanessa, index of inconsistency has no relationship with number of parameters. You can find mathematical approval in Saaty book from 1996 (about ANP and AHP) concerning that issue.

Dear  Dra. Mirjana,

Does this method accept more than 30 parameters simultaneously? (using decision lens software)

Thank You for Your time and knowledge.

Best Regards,

Vanessa

Dear Vanessa,

Could you explain me your question? What do you mean "parameters"? Do you ask about consistency index CI, which is computing as diferrence between lambda max and n, and after that divided by (n-1)? parameters means criterions, subcriterions or alternatives?

Dear Dr.Wojciech,

First I Thank You for your time and knowledge and for your interest in understanding my problem.

For your explanation is precisely what I meant.

9 years ago I used the Decision Lens program and there was a consistency index that was calculated automatically.

I don’t have this program and I'm trying to set up an alternative in Excel.

The idea is to use 30 or more parameters (would be the “criteria”) to be-ranked (would be the “alternative”) combined simultaneously without sub-criteria.

In the software it seemed that when I used only 10 0u 15 parameters the index was

Dear Vanessa,

I am sure that it was a misunderstanding. Consistency index is calculated based on the formula (1). As we can see, it is a simple function, which is dependent on two parameters (2). One of these parameters is the number of assessment elements. Therefore, in my opinion, CI is related to the number of assessed elements. Lambda max, in some way, is related to the number of assessment elements. Lambda max is always equal or greater than the number n.

However, there is also consistency ratio (CR). It is obtained as the following quotient: CI / RI. CR should be less than 0.1 (in extremely situations 0.15 or 0.2).

Let's lambda max is equal "n+x" and n is equal 3, and then we have:

CI = (lambda_max -n)/(n-1) = (n+x-n)/(3-1)= x/2

CI / RI

Dear Vanessa

The consistency ratio is estimated from a comparison between consistency index and a mean value of random matrices of the same dimension. There are some references about regression equaction in order to estimate CI. One of them are Alonso & Lamata, work. This kind of equations let you estimate CI for any n-dimension matrix.

Alonso & Lamata do not use eigenvector but use geometric row mean and there are differences about CI estimation when you use different methods to synthetize weigths.

In any case I strongly believe that AHP Method has a deep root in psichological issues.and one of them is Miller Magic Number 7 +- 2 related with the capacity of the human mind to store and manipulate data. Because of that the original source speaks about small n random CI. Technically you must break down the hierarchy when the number of criteria, sub criteria, or alternatives are more than 9. This number is associated also with the original Saaty's Scale.

Conference Paper A Statistical Criterion of Consistency in the Analytic Hiera...

Article Consistency in the Analytic Hierarchy Process: a New Approach.

Dear Manuel,

Why do you advise approximation? We can very easy obtain the lambda max. So, we can compute CI.

If we have more than nine alternatives, we cannot reduce the dimensionality of an evaluation matrix by changed hierarchy. It will be work only for criteria or sub-criteria. Additionally, if two different hierarchies give us two different results, which one will be real? In my opinion, the hierarchy is dependent on expert knowledge and not on the number of elements (n).

Best Regards Wojciech Sałabun

Dear Wojciech,

Thank You for showing the mathematical formulation in detail (i'm reading your excellent works that are in RG).

Really I believe that  I am confusing the terms used and I appreciate if You can look at the table that I will send to You.

Thank You for Your time.

Best Regards,

Vanessa

Dear Dr. Manuel,

Your explanation is very useful. I read your articles and they also help a lot for me to understand how to deal with the AHP (exactly about the index).

Thank You very much for Your time and examples.

Best Regards,

Vanessa

Dear Vanessa,

Sorry, but I not sure that I understand it properly.

In columns, you have parameters (IEX, IDH, AAA and so on) (yes/no)

In rows, you have interviews (interview means a survey or a decision-maker or what?). Do you use pairwise comparison? I don't see pairwise comparison.

I see that your target is ranking of parameters. Do you need only ranking or weights of them and ranking?

Dear Wojciech ,

Yes, in columns are parameters.

Interviews represent the opinion of 16 experts on a scale from 1 to 9.

I desire to have classification of parameters and % of each one

I did a previous question and some RG colleagues and I explained how you try to mount something like "decision Lens (pairwise comparison)" in excel, but I haven' t done the test yet.

https://www.researchgate.net/post/Can_someone_suggest_free_computer_programs_based_on_the_analytic_hierarchy_process_AHP?_tpcectx=profile_questions

With these data can already estimate the consistency index or is necessary to make all the combinations?

Thank you for your understanding and patience. I apologize for the English and the lack of clarity in explaining the question.

Thank You very much.

Vanessa

Dear Amaury,

I thank You for Your suggestion. I’ll read about this method.

Thank You very much!

Best Regards,

Vanessa

Dear Wojciech

I appreciate so much your comment because it let me to clarify better my opinions.

I'm not intending to promote using any kind of approximation. My intention was to suggest that others before Vanessa have dealed with big matrices (and in a seriously fashion as I believe). In that case the spanish research group that I referred use some statistical regresion in order to get a faster estimate.

I agree with you that if you need to choose between more than nine alternatives there are no simple way to split them. There are a possibility if we have sound knowledge about the problem and the alternatives: We can cluster the alternatives by means of some attribute. In this case the cluster-criteria will be another level in the hierarchy.

I tried to emphasize in psychological aspects of AHP instead of mathematical aspects. From abstract-mathematical point of view there are no problem at all with the dimensionality of the matrices, specially when we realize about the computational power available to lay people this days. But when you realize that Saaty intention was to put forward to managers a simple but robust tool to decision making and also you acknowledge that our rationality is limited, taking in consideration psychological aspects are welcome. That doesn't mean that we can't deal with bigger problems but suggest that our knowledge about any problem is better when that problem not overwhelm ourselves.

Dear Vanessa

Do you like to send me more information about your actual application?

Dear Manuel,

Continuing the work I did in the period 2003-07 using Decision Lens, I am working with urban indicators (28 so far)that I want to classify (ranking) according to their performance in cities from assigning weights 1-9 attributed by experts in area.

Thank you for your interest.

Best Regards,

Vanessa

Dear Manuel,

I agree with you that every situation is a different, and the solution needs more data about the problem. I believe that nine criteria is a lot of. In my opinion, very often only few of them have sense (to solve our problem). We can check their significance level, and then we can reduce dimensionality of our problem. This is only my general approach. In any case we must know what is the main aim to decide about using a specific method or approach.

Alas, I cannot agree with you that AHP is a robust tool. Why? Because, a robust tool is always verified. AHP has a few strong shortcomings. E.g. rank reversal, Saaty's scale and so forth. We can only belive that it works well.

Best Regards WS

Dear Vanessa

Why dont you try BWM? its a new method which is presented in 2015 by Dr Jafar Rezaei from Delft university. You can solve your problem easier with this method. Its totally better than AHP and still based on pairwise comparison. 

Dear Sarfaraz,

I'm looking for more information about Dr. Jafar and his work in this page:

http://www.tbm.tudelft.nl/en/about-faculty/departments/engineering-systems-and-services/tlo-section/staff/jafar-rezaei/publications-jafar-rezaei/#c104903

and I read this paper:

Rezaei, J., 2015. Best-Worst Multi-Criteria Decision-Making Method, Omega,Vol. 53, pp. 49–57.

Do you know where i can geet more information about this new method (ifIs it a free program or i have to develop it)?

Thank You very much,

Best wishes,

Vanessa

Dear Vanessa

Its a useful and easy method. If you couldnt learn the method, send me an email. I have some problems still in learning the method but I generally learn the method through this article. [email protected]

Thank You very much Dear Sarfaraz for Your konwledge and kindness.

Vanessa

Dear Wojciech and Vanessa

I like very much Wojciech interesting reply. In fact I was mistaken explaining my advice about breaking the hierarchy when you need to evaluate more than 9 alternatives because it could raise a misinterpretation

When we need to breakdown the hierarchy to simplify the pairwise comparison in order to attend human beings' psychological constraints, it comes with the cost of expanding the vertical direction of the hierarchy. Then we never have two independent hierarchies but only one.

I do not really recommend approximate consistency value in larger matrices, because I believe that a large matrix is an inappropriate use of AHP not regarding the mathematical issue but regarding the psychological frame of the decision.

As you mention any matrix could be solved easy with actual computing power but when at the cost of needing a very exhausting questionnaire to get experts' opinions. Instead if you cluster the issues using an overarching criteria o subcriteria it lets you to get a experts-clustering as well (i.e., you will ask for lesser pairwise comparisons to different experts).

My intention pointing Vanessa to Alonso & Lamata's paper was twofold. First to show that the consistency test could be performed on matrices of any dimensionality and also because in that research one can see how the consistency index grows with dimensionality

I hope this extension could make more clear my previous answer

Dear Manuel,

Thank You very much. You all are experts in multi-criteria and I only can thank you for sharing your knowledge with me.

I am crawling but learning from all the explanations.

Best wishes,

Vanessa