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Abstract and Applied Analysis
Volume 2014, Article ID 349729, 12 pages
Research Article

Characterization of Consistent Completion of Reciprocal Comparison Matrices

1Instituto de Matemática Multidisciplinar (IMM), Universitat Politècnica de València, Camino de Vera S/N, 46022 Valencia, Spain
2Universitat Politècnica de València, Camino de Vera S/N, 46022 Valencia, Spain
3Fluing IMM, Universitat Politècnica de València, Camino de Vera S/N, Edificio 5C bajo, 46022 València, Spain

Received 22 October 2013; Revised 6 December 2013; Accepted 6 December 2013; Published 25 February 2014

Academic Editor: L. Jódar

Copyright © 2014 Julio Benítez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.