Incomplete pairwise comparison matrices are increasingly employed to save resources and reduce cognitive load by collecting only a subset of all possible pairwise comparisons. We present their graph representation and some completion algorithms, including the incomplete eigenvector and incomplete logarithmic least squares methods, as well as a lexicographical minimisation of triad inconsistencies. The issue of ordinal violations is discussed for matrices generated by directed acyclic graphs and the best--worst method. We also show a reasonable approach to generalise the inconsistency threshold based on the dominant eigenvalue to the incomplete case, and state recent results on the optimal order of obtaining pairwise comparisons. The benefits of using incomplete pairwise comparisons are highlighted by several applications.
The (logarithmic) least squares optimality of the arithmetic (geometric) mean of weight vectors calculated from all spanning trees for incomplete additive (multiplicative) pairwise comparison matrices
A DEMATEL-Based Completion Method for Incomplete Pairwise Comparison Matrix in AHP
Southwest University（西南大学）；Jinan University（暨南大学）；Vanderbilt University（范德堡大学）；The Hong Kong Polytechnic University（香港理工大学）；University of Portsmouth（朴茨茅斯大学）
Inconsistency thresholds for incomplete pairwise comparison matrices
Corvinus University of Budapest (BCE)（布达佩斯考文纽斯大学）；Institute for Computer Science and Control (SZTAKI)（计算机科学与控制研究所）；Eötvös Loránd Research Network (ELKH)（厄特沃什·罗兰研究网络）
Refined thresholds for inconsistency: The effect of the graph associated with incomplete pairwise comparisons
Corvinus University of Budapest (BCE), Institute of Operations and Decision Sciences, Department of Operations Research and Actuarial Sciences（布达佩斯考文纽斯大学）；Institute for Computer Science and Control (SZTAKI), Hungarian Research Network (HUN-REN)（匈牙利国家研究网络计算机科学与控制研究所）
Size independence of consistency index for pairwise comparison matrices in analytic hierarchy process
Faculty of Science and Engineering, Otemon Gakuin University（帝塚山学院大学理工学部）；Faculty of Applied Information Science, Hiroshima Institute of Technology（广岛工业大学应用信息科学部）
