Optimal filling in sequences for incomplete pairwise comparison matrices

Abstract

Pairwise comparisons form the basis of preference modelling and decision making. It is important from both theoretical and practical points of view to determine the number of questions and their arrangements to ask from the decision maker. We focus on incomplete pairwise comparison matrices, and provide the optimal filling in patterns, which result in the closest (LLSM) weight vectors on average to the complete case for at most six alternatives and for all possible number of comparisons, when the underlying representing graph is connected. These results are obtained by extensive numerical simulations with large sample sizes. Many of the optimal filling structures resulted in optimal filling in sequences, one optimal case can be reached by adding a comparison to a previous one, which are presented on GRAPH of graphs for the different number of alternatives. The star graph is revealed to be optimal among the cases with the same cardinality (spanning trees), while the optimal graphs are always close to bipartite ones. Regular graphs also correspond to optimal cases for the examined parameters. Regularity is important for all optimal graphs, as the degrees of the different vertices are always as close to each other as possible. Besides applying the optimal filling structure in given decision making problems, practitioners can utilize the optimal filling sequences in the cases, when the decision maker can abandon the problem at any period of the process. The effect of an additional comparison is also examined for the optimal cases, which can serve as a guide to determine the minimal sufficient number of comparisons in a given problem.