Characterization of Consistent Completion of Reciprocal Comparison Matrices

Analytic hierarchy process (AHP) is a leading multi-attribute decision-aiding model that is designed to help make better choices when faced with complex decisions involving several dimensions. AHP, which enables qualitative analysis using a combination of subjective and objective information, is a multiple criteria decision analysis approach that uses hierarchical structured pairwise comparisons. One of the drawbacks of AHP is that a pairwise comparison cannot be completed by an actor or stakeholder not fully familiar with all the aspects of the problem. The authors have developed a completion based on a process of linearization that minimizes the matrix distance defined in terms of the Frobenius norm (a strictly convex minimization problem). In this paper, we characterize when an incomplete, positive, and reciprocal matrix can be completed to become a consistent matrix. We show that this characterization reduces the problem to the solution of a linear system of equations—a straightforward procedure. Various properties of such a completion are also developed using graph theory, including explicit calculation formulas. In real decision-making processes, facilitators conducting the study could use these characterizations to accept an incomplete comparison body given by an actor or to encourage the actor to further develop the comparison for the sake of consistency.