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Characterization of the consistent completion of analytic hierarchy process comparison matrices using graph theory
Author(s) -
Benítez Julio,
Carpitella Silvia,
Certa Antonella,
Izquierdo Joaquín
Publication year - 2018
Publication title -
journal of multi‐criteria decision analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.462
H-Index - 47
eISSN - 1099-1360
pISSN - 1057-9214
DOI - 10.1002/mcda.1652
Subject(s) - pairwise comparison , uniqueness , analytic hierarchy process , judgement , invertible matrix , characterization (materials science) , computer science , consistency (knowledge bases) , hierarchy , mathematics , matrix (chemical analysis) , decision problem , theoretical computer science , mathematical optimization , algorithm , mathematical economics , artificial intelligence , mathematical analysis , materials science , political science , economics , pure mathematics , law , market economy , composite material , nanotechnology
Decision‐making is frequently affected by uncertainty and/or incomplete information, which turn decision‐making into a complex task. It is often the case that some of the actors involved in decision‐making are not sufficiently familiar with all of the issues to make the appropriate decisions. In this paper, we are concerned about missing information. Specifically, we deal with the problem of consistently completing an analytic hierarchy process comparison matrix and make use of graph theory to characterize such a completion. The characterization includes the degree of freedom of the set of solutions and a linear manifold and, in particular, characterizes the uniqueness of the solution, a result already known in the literature, for which we provide a completely independent proof. Additionally, in the case of nonuniqueness, we reduce the problem to the solution of nonsingular linear systems. In addition to obtaining the priority vector, our investigation also focuses on building the complete pairwise comparison matrix, a crucial step in the necessary process (between synthetic consistency and personal judgement) with the experts. The performance of the obtained results is confirmed.