Incomplete pairwise comparison matrices in multi-attribute decision making

An extension of the pairwise comparison matrix is considered when some comparisons are missing. A generalization of the eigenvector method for the incomplete case is introduced and discussed as well as the Logarithmic Least Squares Method. The uniqueness problem regarding both weighting methods is studied through the graph representation of pairwise comparison matrices. It is shown that the optimal completion/solution is unique if and only if the graph associated with the incomplete pairwise comparison matrix is connected. An algorithm is proposed for solving the eigenvalue minimization problem related to the generalization of the eigenvector method in the incomplete case. Numerical examples are presented for illustration of the methods discussed in the paper.