05C50 Online Seminar: Susana Furtado

An n-by-n matrix A=[a_{ij}] is said to be a pairwise comparison matrix (PC matrix) or a reciprocal matrix if it is positive and a_ij=1/a_ji, for all i,j=1,...,n. If, in addition, a_ika_kj=a_ij for all i,j,k, the matrix is said to be consistent. There is a natural consistent matrix associated to any positive vector.

PC matrices play an important role in decision making, namely in models for ranking alternatives, as the Analytic Hierarchy Process, proposed by Saaty (1977). In these models, a PC matrix represents independent, pairwise, ratio comparisons among n alternatives and a cardinal ranking vector should be obtained from it. The consistent matrix constructed from this vector should be a good approximation of the PC matrix. So, it is desirable to choose a ranking vector from the set of efficient vectors for the PC matrix, as, otherwise, there would be a positive vector such that the consistent matrix obtained from it better approximates the PC matrix in at least one entry and is not worse in all other entries.

In this talk we give two descriptions of the set of efficient vectors for a PC matrix and discuss some of its consequences. Saaty proposed the right Perron vector of a PC matrix as the ranking vector. We discuss the efficiency of this vector and, in addition, we provide vectors constructed from the matrix whose efficiency is universal.

This is a joint work with Charles Johnson.