An algorithm of judgment matrix consistency measurement

Abstract

The consistency of judgment matrix in Analytic Hierarchy Process is analyzed. By using the concepts of partial relation and chain, this paper proves that a judgment matrix is an ordinal consistent matrix if and only if its elements can be sorted as a chain under a partial order. An “ordering” algorithm has been presented to sort all elements in the order of importance to form a chain. A simple method has been proposed to examine ordinal consistency of a judgment matrix, which is based on a new operation called “simplification” on matrices. It has been shown that a judgment matrix is ordinal consistent if and only if it can be simplified into a matrix of order two. The operations of “elementary transformation” is used to show that a judgment matrix is ordinal consistent if and only if its elements can be arranged to form a matrix B = (b ij )n × n, b ij > 1, j = i, i + 1,… n and bij < 1 for i < j, that is, all of the upper triangle elements of matrix B are bigger than one by a series of elementary transformations. Then, an efficient algorithm has been proposed to examine the ordinal consistency of a judgment matrix, which has an Ο(n3) time complexity. Finally, an algorithm has been designed to examine and modify the ordinal consistency of a judgment matrix.