The Cycle Inconsistency Index in Pairwise Comparisons Matrices

AbstractThe method of pairwise comparisons is widely applied in the decision making process. The inconsistency of data may significantly affect the final result. Since the notion of consistency is based on triads or cycles, there is a great need for defining the measure of a triad or cycle inconsistency.In the paper a set of properties of a good cycle inconsistency index is proposed. Two construction methods of a cycle-based inconsistency index for a pairwise comparisons matrix are introduced. All is supported by the examples

Inconsistency evaluation in pairwise comparison using norm-based distances

AbstractThis paper studies the properties of an inconsistency index of a pairwise comparison matrix under the assumption that the index is defined as a norm-induced distance from the nearest consistent matrix. Under additive representation of preferences, it is proved that an inconsistency index defined in this way is a seminorm in the linear space of skew-symmetric matrices and several relevant properties hold. In particular, this linear space can be partitioned into equivalence classes, where each class is an affine subspace and all the matrices in the same class share a common value of the inconsistency index. The paper extends in a more general framework some results due, respectively, to Crawford and to Barzilai. It is also proved that norm-based inconsistency indices satisfy a set of six characterizing properties previously introduced, as well as an upper bound property for group preference aggregation

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

Some Notes on the Properties of Inconsistency Indices in Pairwise Comparisons

Pairwise comparisons are an important tool of modern (multiple criteria) decision making. Since human judgments are often inconsistent, many studies have focused on the means of expressing and measuring this inconsistency, and several inconsistency indices have been proposed as an alternative to Saaty's inconsistency index, CI, and consistency ratio, CR, for reciprocal pairwise comparison matrices. The aims of this paper are threefold: firstly, a row inconsistency index (RIC) is proposed and the properties of this index are examined. Secondly, a comparison of selected inconsistency indices for a corner pairwise comparison matrix is provided. Last, but not least, another axiom about the upper bound on the value of an inconsistency index is postulated, and a set of selected inconsistency indices is examined with respect to this axiom. Numerical examples complete the paper. (original abstract

Comparing inconsistency of pairwise comparison matrices depending on entries

Pairwise comparisons have been a long-standing technique for comparing alternatives/criteria and their role has been pivotal in the development of modern decision-making methods. Since several types of pairwise comparison matrices (e.g., multiplicative, additive, fuzzy) are proposed in literature, in this paper, we investigate, for which type of matrix, decision-makers are more coherent when they express their subjective preferences. By performing an experiment, we found that the additive approach provides the worst level of coherence

A Local Adjustment Method to Improve Multiplicative Consistency of Fuzzy Reciprocal Preference Relations

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.Preferences that verify the transitivity property are usually referred to as rational or consistent preferences. Existent methods to improve the consistency of inconsistent fuzzy reciprocal preference relations (FPRs) fail to retain the original preference values because they always derive a new FPR. This article presents a new inconsistency identification and modification (IIM) method to detect and rectify only the most inconsistent elements of an inconsistent FPR. As such, the proposed IIM can be considered a local adjustment method to improve multiplicative consistency (MC) of FPRs. The case of inconsistent FPRs with missing values, i.e., incomplete FPRs, is addressed with the estimation of the missing preferences with a constrained nonlinear optimization model by the application of the IIM method. The implementation process of the proposed algorithms is illustrated with numerical examples. Simulation experiments and comparisons with existent methods are also included to show that the new method requires fewer iterations than existent methods to improve the MC of FPRs and achieves better MC level, while preserving the original preference information as much as possible than the existent methods. Thus, the results presented in this article demonstrate the correctness, effectiveness, and robustness of the proposed method