The Saaty rating scale is rather nonlinear, but aggregation approach is definitely linear. Is the AHP a linear or a nonlinear method? I think it is a linear method (e.g. Zarghami and Szidarovszky).
Zarghami M. and Szidarovszky F. (2011). Multicriteria Analysis, Springer, pp. 33-39.
The AHP uses a rational Scale to compare in pairwise the alternatives/criteria, so, I believe it is a nonlinear method.
When you use the mean, you have to use the geometric mean.
If we use the geometric mean, we use multiplicative AHP. In my opinion, both version (additive and multiplicative ) are linear. The main reason is a global character of significance of each criterion. The all priorities are identical for any set of alternatives. Finally, those values are used to aggregation partial ranks as linear combination. The final ranking of alternatives lie in a common plane.
AHP is a Multilinear method and, as any multilinear function, is not linear function.
The only case when an AHP model can be represented by a linear function is when the hierarchy has only one level of criteria (without subcriteria).
Indeed, as Dr. Flavio Autran point out, AHP deals with linear model (the Hierarchy). But, it is important to clarify that AHP is NOT a linear function like Sum(AxB)i, instead is a multilinear one. Thus, we can simulate a non linear process through several linear steps (the hierarchy), just like a Fourier serie can simulate a non linear function. When its feedback present, then is better to use a Network model like ANP, (the mathematic generalization of AHP), which of course is not linear in essence.
When you are dealing with many linear steps (different one of another), then, mathematically speaking, you are dealing with a non linear function but a multilinear one. Just like any step-linear function (i.e. Fourier serie) commonly used to approximate non linear functions, and that of course do not correspond to the definition of linearity (at less the definition of linearity have been changed lastly).
Thats right, it is a non linear method, with several linear steps which means multilinear.
It is a multilinear form because of the way that composition works. Multilinear forms are the foundation of solutions of polynomial equations and so the conclusion is a non-linear method. You can look the following link for more details: http://www.google.gr/books?id=rhhlQ0FyBTkC&lpg=PT13&ots=SbCWA1M8VP&dq=principia%20mathematica%20%20saaty&lr&pg=PT145#v=snippet&q=multilinear&f=false
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