András Farkas

Consistency adjustments for pairwise comparison matrices

This paper is concerned with the development of a ‘best’ rank one transitive approximation to a general paired comparison matrix in a least-squares sense. A direct attack on the non-linear problem is frequently replaced by a sub-optimal linear problem and, here, the best procedure of this kind is obtained. The Newton–Kantorovich method for the solution of the non-linear problem is also studied, including the role of the best linear approximation as a starting point for this method. Numerical examples are included. Copyright

Transitive matrices and their applications

Abstract Transitive matrices and symmetrically reciprocal (SR) matrices are defined and spectral properties of certain SR perturbations of transitive matrices are studied. The results are applied to a multicriteria decision making method, the analytic hierarchy process (AHP), which uses an SR matrix with positive entries. A proof is given that rank reversal is inherent in this method, if its input matrix is perturbed.

Data Perturbations of Matrices of Pairwise Comparisons

This paper deals with data perturbations of pairwise comparison matrices (PCM). Transitive and symmetrically reciprocal (SR) matrices are defined. Characteristic polynomials and spectral properties of certain SR perturbations of transitive matrices are presented. The principal eigenvector components of some of these PCMs are given in explicit form. Results are applied to PCMs occurring in various fields of interest, such as in the analytic hierarchy process (AHP) to the paired comparison matrix entries of which are positive numbers, in the dynamic input–output analysis to the matrix of economic growth elements of which might become both positive and negative and in vehicle system dynamics to the input spectral density matrix whose entries are complex numbers.

A recursive least-squares algorithm for pairwise comparison matrices

Pairwise comparison matrices are commonly used for setting priorities among competing objects. In a leading decision making method called the analytic hierarchy process the principal right eigenvector components represent the weights of the alternatives. The direct least-squares method extracts the weight vector by first finding a rank-one matrix which minimizes the Euclidean distance from the original ratio matrix. We develop a recursive least-squares algorithm and reveal a striking correspondence between these two approaches for these matrices. The recursion applies for merely positive matrices also. We prove that a convergent iteration leads to matrices by which the Perron-eigenvectors and the Perron approximation of the original matrix may be produced. We show that certain useful properties of the recursion advance the development of reliable measures of perturbations of transitive matrices. Numerical analysis is included for a macroeconomic problem taken from the literature.

Balancing Pairwise Comparison Matrices by Transitive Matrices

We discuss the development and use of a recursive rank-one residue iteration (triple R-I) to balancing pairwise comparison matrices (PCMs). This class of positive matrices is in the center of interest of a widely used multi-criteria decision making method called analytic hierarchy process (AHP). To find a series of the ’best’ transitive matrix approximations to the original PCM the Newton-Kantorovich (N-K) method is employed for the solution to the formulated nonlinear problem. Applying a useful choice for the update in the iteration, we show that the matrix balancing problem can be transformed to minimizing the Frobenius norm. Convergence proofs for this scaling algorithm are given. A comprehensive numerical example is included to illustrate the useful features to measuring and reducing perturbation errors and inconsistency of a PCM as a result of the respondents’ judgments on the pairwise comparisons.