Why we used to take Geometric mean instead of arithmetic mean in Analytic hierarchy process ?
What will be the difference in result if we take the arithmetic mean instead of Geometric mean in AHP or any other multi-criteria decision making method??
One important thing that has to be mentioned is that geometric mean is the only one that keep the first axiom of AHP alive (Axiom1: if A=nB then B=1/nA) keeping the matrix reciprocal. If you use the arithmetic mean the axiom do not hold, i.e. In the arithmetic mean mode the inverse of the aggregation of (2+5)/2=7/2 which inverse is 2/7, in the other side of the matrix we have (1/2+1/5)/2=7/20 which is different to 2/7 (the expected value to keep the matrix reciprocal). By the other hand, in geometric mean mode the the aggregation of 2 and 5 is: SQRT(2*5)=3.162, and in the other side of the matrix the inverse is SQRT(1/2)*SQRT(1/5) which is the same than 1/SQRT(2*5)=0.3162 which is the inverse of 3.162, (the expected value).
Thats why the geometric mean is the only aggregation form valid. (it keep "reciprocity" alive).
See also Saaty-Aczel paper.
I have recent papers in Int Jnl of Production Research and Supply Chain Managemnt: An international Journal that discuss the reason why.
But if we want to aggregate the priorities of criteria using the mean geometric we will have a problem that the sum of all criteria is not 1 and we should normalize the weights.
Is it possible to but Decimals instead of integer inside the AHP matrix or must I rounded of to integer number when I collected the results from geometric mean..
Thanks a lot for your valuable comments ,
Please which one is the better and consistent , the integer or the decimals numbers inside AHP matrix ?
Dear , Prof. dr.Theo K. Dijkstra
Thanks a lot for your valuable comments.
Dear Faez try to add something to valuable insights provided by Theo is challenging but I think it is valuable to say that Saaty's AHP has some pillars (seven according to Saaty himself). One of them is the ratio scales used to compare. Then when you use geometric mean instead of arithmetic mean you are preserving ratios instead of intervals (James, 1968 cited by Forman & Peniwati, 1998; p. 168)
Hi Manuel,
How it was? did you have lucky? (I mean, can you found it?)
Anyway, if you send me your email I can send you an extract.
Regards,
Claudio
Dear Chandra
For AHP some researchers suggested its creator, Dr. Saaty to use MAW that is Multiplicative Additive Weightings, instead as SAW or Simple Additive Weightings, in order to avoid Rank Reversal, that is the irregularity that may appear in a ranking when there are two alternatives with similar values, or when adding an alternative inferior to a non-optimal alternatives in the ranking
For multi criteria decision making, why cant we use Weighted average ( too simple) over AHP model( bit complex) ?
Mr. Abhinav,
I can think at least in 3 simple reasons:
1.- Weighted average is not a real model for any decision problem
2.- Weighted average use and mix raw numbers of different procedence
3.- There is no way to measure any kind of consistency of the DM's input
There are other technical reasons too (normalization, quality of measurement, combination of different DMs, and many more.
Of course, (As you say) WAverage is simple (temptingly simple), but with that simplicity comes also a big source of errors.
When a DM prepares an initial matrix in AHP, that matrix is composed by ratios, that is, all values are dependent from another
If those values were independent, then the arthritic mean could be used, but not when they are related,
Dear friends,
I think this paper is the best answer.
Krejčí, J., & Stoklasa, J. (2018). Aggregation in the analytic hierarchy process: Why weighted geometric mean should be used instead of weighted arithmetic mean. Expert Systems with Applications, 114, 97-106.
Chicago
Regards,
Sarbast Moslem many thanks.
I will refer this paper.
regards, Chandra
...really interesting discussion, thank you. I'll profit from all your contributions..
Fundermental differances are : GM is for relative value and AM for abloute value. For dependent relation GM will be used and for Independent value AM is used and many more differances....
Krejčí, J., & Stoklasa, J. (2018). Aggregation in the analytic hierarchy process: Why weighted geometric mean should be used instead of weighted arithmetic mean. Expert Systems with Applications, 114, 97-106.
Although no differences were observed among the methods, the find the AHP of every expert and then get their arithmetic mean is inefficient. Forth, the number of experts should be considered when decision makers are selecting the aggregation method; if the number of experts is large, a geometric mean is inappropriate, because it cannot be calculated; and thus the arithmetic mean is a better method in this situation. Variance should be considered, decision maker need to check the variance before dealing the AHP, if there is outlier in the
Opinion set, it should be considered to drop.
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