Inconsistency in the ordinal pairwise comparisons method with and without ties

In the presented work, the maximal number of inconsistent triads in the set of ordinal pairwise comparisons with ties of arbitrary size is determined (formula 14). This, in turn, opens the possibility of effectively extending the Kendall and Babington Smith index to pairwise comparisons, where the result of an individual comparison can be: better, worse or equal. Hence, this effectively extends the use of this index to the Analytic Hierarchy Process and other quantitative methods based on comparing alternatives in pairs. The work also introduces the notions of a generalized tournament and a double tournament as graphs that model ordinal pairwise comparisons with ties and the maximally inconsistent set of pairwise comparisons with ties, respectively. The relationship between the most inconsistent set of pairwise comparisons with ties and the set cover problem is also shown.