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Pairwise comparison matrix is a fundamental concept of the AHP. In this paper we provide recommendations for filling in patterns of incomplete pairwise comparison matrices, when we assume that the set of comparisons can be chosen and it is designed completely before the decision making process, without any further prior information. Our recommendations heavily rely on the graph representation of the incomplete pairwise comparison matrices, in fact, the suggested designs are represented by (quasi-)regular graphs with minimal diameter. One major contribution of our research is a list of proposed graphs for different parameter sets, where it is important to emphasize that the diameter of the graph has not been studied in the literature as a crucial property before. On the other hand, we validate the results with the help of extended numerical simulations, which show that these filling in patterns provide smaller errors compared to other well-known designs. Both theorists and practitioners can utilize the results, provided in several formats in our research: graph, adjacency matrix, list of edges. Also, a lot of other models based on pairwise comparisons can take advantage of our findings.

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.

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