A new position weight correlation coefficient for consensus ranking process without ties

Abstract

Preference data represent a particular type of ranking data where a group of people gives their preferences over a set of alternatives. The traditional metrics between rankings do not take into account the importance of swapping elements similar among them (element weights) or elements belonging to the top (or to the bottom) of an ordering (position weights). Following the structure of the τx proposed by Emond and Mason and the class of weighted Kemeny–Snell distances, a proper rank correlation coefficient is defined for measuring the correlation among weighted position rankings without ties. The one‐to‐one correspondence between the weighted distance and the rank correlation coefficient holds, analytically speaking, using both equal and decreasing weights. In order to determine the consensus ranking among rankings, related to a set of subjects, the new coefficient is maximized modifying suitably a branch‐and‐bound algorithm proposed in the literature.