Efficient vectors for simple perturbed consistent matrices

In the Analytic Hierarchy process, a method used in Decision Making, it may be important to approximate a pairwise comparison matrix (PC matrix) by a consistent one. In this context, the notion of efficient vector for a PC matrix arises. In this paper we describe all efficient vectors for an n×n comparison pairwise matrix obtained from a consistent one by perturbing one entry above the main diagonal, and the corresponding reciprocal entry. As a consequence, we give a simple proof of the result obtained by K. Ábele-Nagy and S. Bozóki (2016) that states that the principal eigenvector of a simple perturbed consistent matrix is efficient. In addition, we consider a set of non-efficient vectors associated with the simple perturbed consistent matrix and describe all the efficient vectors that dominate each vector in that set.