An algorithm of judgment matrix consistency measurement

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 b_ij < 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 O(n³) time complexity. Finally, an algorithm has been designed to examine and modify the ordinal consistency of a judgment matrix.