On optimal completion of incomplete pairwise comparison matrices

Authors:
SáNdor BozóKi;JáNos FüLöP;Lajos RóNyai
Affiliations:
Laboratory on Engineering and Management Intelligence, Research Group of Operations Research and Decision Systems, Computer and Automation Research Institute, Hungarian Academy of Sciences, 1518 B ...;Laboratory on Engineering and Management Intelligence, Research Group of Operations Research and Decision Systems, Computer and Automation Research Institute, Hungarian Academy of Sciences, 1518 B ...;Informatics Laboratory, Computer and Automation Research Institute, Hungarian Academy of Sciences, Hungary and Institute of Mathematics, Budapest University of Technology and Economics, Hungary
Venue:
Mathematical and Computer Modelling: An International Journal
Year:
2010

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Abstract

An important variant of a key problem for multi-attribute decision making is considered. We study the extension of the pairwise comparison matrix to the case when only partial information is available: for some pairs no comparison is given. It is natural to define the inconsistency of a partially filled matrix as the inconsistency of its best, completely filled completion. Here we study the uniqueness problem of the best completion for two weighting methods, the Eigenvector Method and the Logarithmic Least Squares Method. In both settings we obtain the same simple graph theoretic characterization of the uniqueness. The optimal completion will be unique if and only if the graph associated with the partially defined matrix is connected. Some numerical examples are discussed at the end of the paper.