When calculating idealized priorities, Saaty (2008) defines that: “To obtain the idealised priorities, normalise by dividing by the largest of the priorities. The idealised priorities are always used for ratings.”
In my case, I am using the AHP for developing an environmental risk indicator. In this sense, the “ideal rating” (1.0) would be the one that presents the lowest risk. Therefore, when applying Saaty’s definition of idealized priorities, the highest risk would obtain the ideal rating of 1.0.
Would it be correct if for obtaining the idealized priorities, I normalise by dividing by the smallest value?
Thanks!
Saaty, T. L. (2008). Decision making with the analytic hierarchy process. International Journal of Services Sciences, 1, 83. https://doi.org/10.1504/IJSSCI.2008.017590
If I have understood correctly 1.0 - is the best value. However, values can be both more than 1 and under 1? If Yes, then idealized priorities will be normalized by e^(-(a-1)^2), where a - values of indicator, e - EXP.
If 1.0 - is the minimum value and the best one, then you have 1.0 divide by a.
Also, you can substract each value to highest value. Then apply Saaty normally, and the highest risk will obtain the ideal rating of 1.0 or any close value.
My best regards.
Dear Alejandro
What is the meaning of 'idealized priorities'?
Certainly they are not the priorities derived from the AHP method
In my interpretation of what Saaty says is that you can normalize by dividing each value by the highest value.
I have never seen that normalization is executed by dividing by the lowest value
What happens if the minimum value is zero? You can't divide by zero
Dear Artiom Volkov, Andrey Vinajera Zamora and Nolberto Munier
Thanks for your message. My point is that when calculating priorities for a nxn matrix, the resulting eigenvector distributes weights (100%) among the criteria. Now, as I cited in my previous message, Saaty (2008) explains that "idealised priorities" are those used for the ratings, meaning this the lowest level of the hierarchy (e.g. for criterion groundwater pollution, ratings could be high, medium and low).
In my environmental risk indicator the higher the aggregated score, the larger the risk.
The problem when applying the formula of “idealized priorities” Saaty (2008), is that when “dividing by the largest of the priorities”, the rating high for groundwater pollution, based on the example, would be 100% and the others (medium and low) would be a percentage of this. As Saaty (2008) says: “the effect is to make this alternative the ideal one with the others getting their proportionate value”. It is important to have in mind that in this case the eigenvalues do not sum up 100%.
Now, in my indicator the "idealised priorities" of 100% will always be assigned to the rating high. In my opinion, this does not make sense, since the ideal value should be assigned to the rating low. In the example, the ideal priority of 100% should be assigned to low groundwater pollution. When doing this (dividing by the smallest value instead of the largest) the rating low would be 100%. I checked the proportionalities among the ratings (high, medium and low) with both methods and it is satisfied. Unfortunately, I have not found any of this online and of course makes me doubt if it is accepted.
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