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 pairs for each criterion and wC2 pairs for the set of criteria, and the total number of them becomes up tp C-) X M + C2. 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, 51. 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. 1

A NEW APPROACH OF REVISING UNSTABLE DATA IN ANP BY BAYES THEOREM

Abstract In the typical type of ANP with a matrix U evaluating alternatives by criteria and a matrix W evaluating criteria by alternatives in the so-called supermatrix S, W is often said to be unstable. Here, we propose a method to revise W into a stable ˆ W and to calculate the weights of criteria and alternatives at the same time under the revised supermatrix ˆ S. Our method is formulated as an optimization problem based on Bayes Theorem which T.L. Saaty claimed to be included in ANP scheme. Concurrent Convergence method developed by Kinoshita and Nakanishi also intends to be correct W, but this method includes some contradictions. We prove that our method has no such contradiction. We introduce some eigenvalue problems, which give a lower bound of our optimal value and their special cases coincide with our problem. Furthermore, we clear what perturbations of W preserve weights of criteria and alternatives to be invariant under the concept of inactive alternatives