The complete set of efficient vectors for a reciprocal matrix

Efficient vectors are the natural set from which to choose a cardinal ranking vector for a reciprocal 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 all efficient vectors for any given reciprocal matrix. We apply this method to give explicitly all efficient vectors for a 4-by-4 reciprocal matrix. Several examples are provided.