Derivatives of the Perron root of a positive reciprocal matrix: With application to the analytic hierarchy process

摘要

The derivatives of the Perron root of an essentially nonnegative matrix have been thoroughly studied in the literature. This paper presents a refinement of these results to a special class of essentially nonnegative matrices, namely, positive reciprocal matrices. This class of matrices arises in Saaty's analytic hierarchy process, a decision-analysis method which is based upon pairwise comparisons among a set of alternatives over various criteria. Formulas for the first and second partial derivatives of the Perron root of a positive reciprocal matrix are developed, and the use of these formulas in conjunction with Saaty's method is illustrated.