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Efficiency Analysis of Simple Perturbed Pairwise Comparison Matrices

100%

Ábele-Nagy K. , Bozóki S.

Fundamenta Informaticae

2016

tom Vol. 144, nr 3/4

279--289

Efficiency, the basic concept of multi-objective optimization is investigated for the class of pairwise comparison matrices. A weight vector is called efficient if no alternative weight vector exists such that every pairwise ratio of the latter’s components is at least as close to the corresponding element of the pairwise comparison matrix as the one of the former’s components is, and the latter’s approximation is strictly better in at least one position. A pairwise comparison matrix is called simple perturbed if it differs from a consistent pairwise comparison matrix in one element and its reciprocal. One of the classical weighting methods, the eigenvector method is analyzed. It is shown in the paper that the principal right eigenvector of a simple perturbed pairwise comparison matrix is efficient. An open problem is exposed: the search for a necessary and sufficient condition of that the principal right eigenvector is efficient.

Balancing Pairwise Comparison Matrices by Transitive Matrices

100%

Farkas A.

Fundamenta Informaticae

2016

tom Vol. 144, nr 3/4

397--417

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.

Eigenvector Method and Rank Reversal in Group Decision Making Revisited

88%

Csató L.

Fundamenta Informaticae

2017

tom Vol. 156, nr 2

169--178

It has been shown recently that the Eigenvector Method may lead to strong rank reversal in group decision making, that is, the alternative with the highest priority according to all individual vectors may lose its position when evaluations are derived from the aggregated group comparison matrix. We give a minimal counterexample and prove that this negative result is a consequence of the difference of the rankings induced by the right and inverse left eigenvectors.

A Hierarchical Decision Model Based on Pairwise Comparisons

75%

Hou F.

Fundamenta Informaticae

2016

tom Vol. 144, nr 3/4

333--348

Pairwise comparisons (PC) method is an efficient technique and hierarchical analysis is a popular means coping with complex decision problems. Based on two proposed theorems, this paper shows that the PC-based hierarchical decision models stem from the weighted average methods (including the arithmetic form and the geometric form). Some issues (including the rank reversal, the criterion for acceptable consistency and the method for deriving priorities, etc) associated with the current PC-based hierarchical models (including the AHP, the multiplicative AHP and the FPR-AHP) are investigated. Another PC-based hierarchical decision model, which is different from the Saaty’s AHP, is introduced for applications by virtue of its desirable traits (such as the rank preservation, the isomorphic correspondence, etc).