## AHP and the Magical Number Seven Plus or Minus Two

In the analytic hierarchy process you define a set of criteria and sub-criteria arranged in a hierarchy, to do pairwise comparisons and find the weights of criteria or decision alternatives. In my AHP excel template the number of criteria is limited to ten, in my AHP online software to 15. Still sometimes I am asked to extend and allow for more criteria.

### Why the number of criteria should not exceed the magical number seven plus or minus two?

There are three reasons not to exceed the number of 9 criteria in any AHP project. Two of them are quite clear and published in the literature:

• The first has to do with the human limits on our capacity for processing information, and was published by George A Miller, as well as in the context of AHP by Saaty and Ozdemir.
• The second is related to the first one. The number of pairwise comparisons increases with the number of criteria, it is (n2n)/2. For example, 9 criteria require 36 comparisons. For a  high number of comparisons easily logical inconsistencies occur, and the consistency ratio CR exceeds values of 10% to 20%, making the basic assumption of near consistent matrices invalid and the AHP results questionable.

The third reason is not so obvious and not so well known. It is based on the limited 1 to 9 AHP ratio scale for the judgment. The maximum preference you can give to one criterion is 9, i.e. this criterion is 9 times more important than all other criteria. Assume, you have only two criteria, then – if you fully prefer one over the other – the preferred one will result in a weight of 90%, the other gets a weight of 10%. The weights depend on the number of criteria, the maximum weight or maximum priority wmax is always

wmax = M/(n + M – 1)

with M = 9, the maximum of the AHP scale and n the number of criteria. The below diagram shows wmax as a function of the number of criteria.

Clearly you can see that for 10 criteria the maximum possible weight reduces to 50%, or in other words, although you give full preference to one criterion, it only gets a weight of 50%! For more than ten criteria the weight will be below 50%. This is the reason, why the number of criteria should never exceed the magical number seven plus or minus two.

## AHP Online Calculator

The AHP online calculator is part of BPMSG’s free web-based AHP online system AHP-OS. If you need to handle a complete decision hierarchy, group inputs and alternative evaluation, use AHP-OS.

Calculate priorities from pairwise comparisons using the analytic hierarchy process (AHP) with eigen vector method. Input the number of criteria  between 2 and 20 1) and a name for each criterion. Next, do a pairwise comparison: Which of the criterion in each pair is more important, and how many times more, on a one to nine scale. With Check consistency you will then get the resulting priorities, their ranking, and a consistency ratio CR2) (ideally < 10%). Calculation is done using the fundamental 1 to 9 AHP ratio scale.

The three judgments with highest inconsistency will be highlighted, with the last column showing the recommended judgment for lowest consistency ratio. Slightly modify your comparisons, if you want to improve consistency, and recalculate the result, or download the result as a csv file.

Example of inconsistent pair-wise comparisons. The most inconsistent judgment no 2 is marked in red (Color or Delivery); the consistent judgment would be 3 (B) and is highlighted in light green. Decision makers can decide to adjust some of their original judgments to improve consistency.

 Here the link: AHP priority calculator

Kindly rate the software from 1 star (poor) to 5 stars (excellent) at the bottom of this post.

### Format of the csv file

Fields are separated by tabs:

• Line 1: Date (yyyy-mm-dd)  Time (hh:mm:ss) Title (text)
• Line 2: Number of criteria n
• Line 3: Criteria
• Line 4 to 4+n: Decison matrix
• Second last line:  Priority vector
• Last line: eigenvalue and consistency ratio CR

### References

1) Though the maximum number of criteria is 15, you should always try to structure your decision problem in a way that the number of criteria is in the range 5 to 9.

2) Alonso, Lamata, (2006). Consistency in the analytic hierarchy process: a new approach. International Journal of Uncertainty, Fuzziness and Knowledge based systems, Vol 14, No 4, 445-459.

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