The complete set of efficient vectors for a reciprocal matrix

TL;DR

This paper introduces a novel method to generate all efficient vectors for any pairwise comparison matrix using graph theory and numerical techniques, with explicit results for 4x4 matrices.

Contribution

It provides the first known method to produce the complete set of efficient vectors for reciprocal matrices, expanding the tools for ranking and decision-making models.

Findings

All efficient vectors can be generated inductively for any matrix.

The set of efficient vectors is piecewise linearly connected.

Explicit efficient vectors are determined for 4x4 matrices.

Abstract

Efficient vectors are the natural set from which to choose a cardinal ranking vector for a pairwise comparison matrix. Such vectors are the key to certain business project selection models. Many ways to construct specific efficient vectors have been proposed. Yet, no previous method to produce all efficient vectors was known. Here, using some graph theoretic ideas, as well as a numerical extension technique, we show how to generate inductively all efficient vectors for any given pairwise comparison matrix. We apply this method to give a matricial proof of the fact that the set of efficient vectors and other related sets are piecewise linearly connected. In addition, we determine explicitly all efficient vectors for a 4-by-4 pairwise comparison matrix. Several examples are provided.