**Title:** Analyzing AHP-matrices by Robust Regression

**Authors:** Pertti Laininen and Raimo P. Hämäläinen

**Date:** April, 1999

**Status:** 5th International Symposium on the Analytic Hierarchy Process (ISAHP'99), August 12-14, 1999, Kobe, Japan.

**Key Words:** Analytical Hierarchy Process, eigenvector, regression, robust regression.

In the analytic hierarchy process (AHP) the decision maker makes comparisons between pairs of entities of interest. The comparisons can be thought of being influenced by random errors, and then the values of the ratios of the weights of the entities are values of random variables. Sometimes the ratio may be exceptionally different from the corresponding consistent value. Then the statement is called an outlier. In this paper we study the influence of the outlier on the estimates of the weights calculated by the eigenvector method and by the regression technique. It can be seen that outliers can have a significant influence on the weight estimates given by the eigenvector method and the logarithmic least squares regression. Here, we present the method of the logarithmic robust regression, which is robust in the presence of the outliers. We show by illustrative simulations how the solution of the logarithmic robust regression remains stable under random occurrences of outliers.