Jaroslav Ramík: Incomplete pairwise comparison matrix and its 
application to rating of alternatives.

Abstract:
Pairwise comparison is a popular method for solving decision making
problems of finding the best alternative(s) among a finite number of
ones. The method is based on the psychological observation that the
human brain can compare only 5-9 independent values in one moment. For
a human being it is much easier to compare only two elements, a pair.
Then for all pairs we obtain a pairwise comparison matrix. The core of
the rating method is how to aggregate the results into a final
prioritization, i.e. ranking or rating of the given alternatives.
In this presentation we deal with some properties of such pairwise
comparison matrices, particularly reciprocity, consistency and
transitivity. We show how to measure the grade of consistency and/or
transitivity. We propose a new method for measuring of inconsistency
based on Saaty´s principal eigenvector method. Finally, we deal with
the problem of the incomplete fuzzy pairwise comparison matrix, where
some elements are missing. We propose a special method for dealing
with such problems. Some illustrating examples will be presented to
clarify the proposed theory.