Pairwise Comparisons Simplified

This study examines the notion of generators of a pairwise comparisons matrix. Such approach decreases the number of pairwise comparisons from $n\cdot (n-1)$ to $n-1$. An algorithm of reconstructing of the PC matrix from its set of generators is presented.Comment: 15 pages, two figure

Axiomatization of Inconsistency Indicators for Pairwise Comparisons

This study proposes revised axioms for defining inconsistency indicators in pairwise comparisons. It is based on the new findings that "PC submatrix cannot have a worse inconsistency indicator than the PC matrix containing it" and that there must be a PC submatrix with the same inconsistency as the given PC matrix. This study also provides better reasoning for the need of normalization. It is a revision of axiomatization by Koczkodaj and Szwarc, 2014 which proposed axioms expressed informally with some deficiencies addressed in this study.Comment: This paper should have been withdrawn by the first author a long time ago. The work has been finished with another researcher, I have been pushed out the projec

The Triads Geometric Consistency Index in AHP-Pairwise Comparison Matrices

The paper presents the Triads Geometric Consistency Index (T-GCI), a measure for evaluating the inconsistency of the pairwise comparison matrices employed in the Analytic Hierarchy Process (AHP). Based on the Saaty''s definition of consistency for AHP, the new measure works directly with triads of the initial judgements, without having to previously calculate the priority vector, and therefore is valid for any prioritisation procedure used in AHP. The T-GCI is an intuitive indicator defined as the average of the log quadratic deviations from the unit of the intensities of all the cycles of length three. Its value coincides with that of the Geometric Consistency Index (GCI) and this allows the utilisation of the inconsistency thresholds as well as the properties of the GCI when using the T-GCI. In addition, the decision tools developed for the GCI can be used when working with triads (T-GCI), especially the procedure for improving the inconsistency and the consistency stability intervals of the judgements used in group decision making. The paper further includes a study of the computational complexity of both measures (T-GCI and GCI) which allows selecting the most appropriate expression, depending on the size of the matrix. Finally, it is proved that the generalisation of the proposed measure to cycles of any length coincides with the T-GCI. It is not therefore necessary to consider cycles of length greater than three, as they are more complex to obtain and the calculation of their associated measure is more difficult

Incomplete pairwise comparative judgments: Recent developments and a proposed method

The current paper deals with incomplete Pairwise Comparisons (‘PWs’) when a large number of alternatives is evaluated. PWs are used to quantify decision maker's preferences, both ordinal and cardinal, in multi-criteria decision-making settings for eliciting the priorities of alternative options or weights of criteria. We use additive PWs with a different scale and show how 2-diagonal samples are used to deduce the implied weights thus prioritizing the alternatives. As a consequence, the number of PWs in incomplete judgment decision matrices is greatly reduced while preserving consistency and quality of the results. Computational results are provided and an example from the literature is applied to demonstrate the effectiveness of this method