Enumerating all spanning trees for pairwise comparisons

Determining the most suitable prioritization method for pairwise comparisons remains an open problem. This paper proposes a method based on the generation of all possible preferences from a set of judgements in pairwise comparisons. A concept of pivotal combination is introduced using a graph-theoretic approach. A set of preferences is generated by enumerating all spanning trees. The mean of these preferences is proposed as a final priority vector, and variance gives a measure of inconsistency. The proposed method provides a way of ordering objects according to a voting scheme. The proposed method is also applicable to incomplete sets of pairwise comparisons without modification, unlike other popular methods which require intermediate steps to estimate missing judgements.