Desirable Properties of Weighting Vector in Pairwise Comparisons Matrix

The aim of this paper is to show a new categorization of inconsistent pairwise comparisons matrices (PCM) proposed with respect to a satisfaction/violation of selected PCM properties, such as the fundamental selection (FS) condition, preservation of order preference (POP) condition, preservation of order of intensity of preference (POIP) condition, and reliability priority condition (RP). A new non-linear optimization problem for finding the weights (i.e. priority vector) satisfying the aforementioned conditions is proposed such that the distance between the given PCM matrix and the ratio matrix composed of the weights function is minimized. Moreover, we presented important examples of alo-groups where the above optimization problem for finding the desirables weights can be solved by standard optimization methods.