First-Level Transitivity Rule Method for Filling in Incomplete Pair-Wise Comparison Matrices in the Analytic Hierarchy Process

The paper discusses the problem of performing the prioritization of decision elements within the multicriteria optimization method, analytic hierarchy process (AHP), with incomplete information. An approach is proposed on how to fill in the gap in the pair-wise comparison matrix generated within an AHP standard procedure; that is, to reproduce one missing judgment of the decision maker while assuring the reproduced judgment belongs to the same ratio scale used while other judgments are elicited. The first-level transitivity rule (FLTR) approach is proposed based on screening matrix entries in the neighborhood of a missing one. Scaling (where necessary) and geometric averaging of screened entries allows filling of the gap in the matrix and later prioritization of involved decision elements by the eigenvector, or any other known method. Illustrative examples are provided to compare the proposed method with the other two known methods also aimed to fill-in gaps in AHP matrices. The results indicate some similarities in attaining consistency. However, unlike other methods, the FLTR assures coherency of the generating process in a sense that all numeric values in a matrix (original entries, plus one generated) come from the same ratio scale and have correct element-wise semantic equivalents.