Calculating Maximum Eigenvalues in Pairwise Comparison Matrices for the Analytic Hierarchy Process

This paper focuses on a numerical method for calculating the maximum eigenvalue of a pairwise comparison matrix in the analytic hierarchy process. Two contributions are made: First, the first and second differentials of the characteristic polynomial of a pairwise comparison matrix are demonstrated to be always positive in the region larger than the maximum eigenvalue. This is proven by effectively utilizing the Gauss–Lucas theorem, which is a mathematical principle that helps in analyzing polynomials. By leveraging this fact, both Newton’s method and the secant method are shown to generate sequences converging to the maximum eigenvalue, bringing a new perspective to the numerical computation of the maximum eigenvalue. The second contribution is the judicious selection of initial points. Using an upper bound for the maximum eigenvalue as initial points, Newton’s method and the secant method generate a decreasing sequence that is bounded below. This allows generalizing our results for lower-order pairwise comparison matrices independent of the matrix order. The positivity of the first and second differentials of the characteristic polynomial also guarantees that Newton’s method and the secant method have quadratic and super-linear convergence, respectively. Numerical simulations confirm the superiority of Newton’s method in terms of convergence.