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In the context of some multi-criteria decision-making methods, such as the Analytic Hierarchy Process, an expert is required to compare entities, e.g. alternatives and criteria. However, often, for various reasons, the expert cannot provide judgments on all pairs of entities. For these cases, several completion methods have been proposed in the literature to estimate the missing values of pairwise comparison matrices. In this paper, we study the similarity of eleven completion methods on the basis of numerical simulations and hierarchical clustering. We perform simulations for matrices of different orders considering various numbers of missing comparisons. Finally, the results suggest the existence of a cluster of five extremely similar methods, and a method significantly dissimilar from all the others.

Pairwise comparison matrices play a prominent role in multiple-criteria decision-making, particularly in the analytic hierarchy process (AHP). Another form of preference modeling, called an incomplete pairwise comparison matrix, is considered when one or more elements are missing. In this paper, an algorithm is proposed for the optimal completion of an incomplete matrix. Our intention is to numerically minimize a maximum eigenvalue function, which is difficult to write explicitly in terms of variables, subject to interval constraints. Numerical simulations are carried out in order to examine the performance of the algorithm. The results of our simulations show that the proposed algorithm has the ability to solve the minimization of the constrained eigenvalue problem. We provided illustrative examples to show the simplex procedures obtained by the proposed algorithm, and how well it fills in the given incomplete matrices.