Derivatives of the Perron root of a positive reciprocal matrix: with application to the analytic hierarchy process (Q1089415)

This paper presents a refinement of the classical results on the derivatives of the Perron root of an essentially nonnegative matrix 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.