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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 = (bij)n×n, bij > 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.