How to select paired comparisons in AHP of incomplete information - Strongly regular graph design

Abstract

It is said that paired comparison is the essence of AHP. But if there are N alternatives and M criteria in a standard AHP, we must compare MC2 pairs for each criterion and NC2 pairs for the set of criteria, and the total number of them becomes up tp NC2 × M + MC2. So for rather large M and N it takes much cost and time to get paired comparison data. But even if we have not the whole set Sn of nC2 pairs (let such a case be called incomplete information case), we can estimate the weights based on comparison data in an appropriate subset of Sn by Harker method or Two-stage method [4, 5]. We can use LLS (logarithmic least square) method in AHP analysis, by which we can analyze AHP for incomplete information case. So we can reduce the number of paired comparisons by using incomplete information case. The problem is how to select pairs to be compared in Sn, that is, a design to get data. We propose the strongly regular (SR) design based on strongly regular graphs, and by numerical simulation show that the errors of the estimations by SR designs are smaller than any random designs for almost all cases. Since SR graphs are rather difficult to be constructed, we generalize them to quasi-strongly regular (quasi-SR) graphs, and propose quasi-SR design based on quasi-SR graphs. By simulation we show that quasi-SR designs also give the same good results as the SR designs.