A general unified framework for pairwise comparison matrices in multicriterial methods

In a Multicriteria Decision Making context, a pairwise comparison matrix $A=(a_{ij})$ is a helpful tool to determine the weighted ranking on a set $X$ of alternatives or criteria. The entry $a_{ij}$ of the matrix can assume different meanings: $a_{ij}$ can be a preference ratio (multiplicative case) or a preference difference (additive case) or $a_{ij}$ belongs to $[0,1]$ and measures the distance from the indifference that is expressed by 0.5 (fuzzy case). For the multiplicative case, a consistency index for the matrix $A$ has been provided by T.L. Saaty in terms of maximum eigenvalue. We consider pairwise comparison matrices over an abelian linearly ordered group and, in this way, we provide a general framework including the mentioned cases. By introducing a more general notion of metric, we provide a consistency index that has a natural meaning and it is easy to compute in the additive and multiplicative cases; in the other cases, it can be computed easily starting from a suitable additive or multiplicative matrix

Inconsistency and non-additive Choquet integration in the Analytic Hierarchy Process

We propose to extend the aggregation scheme of Saaty’s AHP, from the stan- dard weighted averaging to the more general Choquet integration. In our model, a measure of inconsistency between criteria is derived from the main pairwise comparison matrix and it is used to construct a non-additive capacity, whose associated Choquet integral reduces to the standard weighted mean in the con- sistency case. In the general inconsistency case, however, the new aggregation scheme based on Choquet integration tends to attenuate (resp. emphasize) the priority values of the criteria with higher (resp. lower) average inconsistency with the remaining criteria.Aggregation Functions, Multiple Criteria Analysis, AHP, Inconsintency, non-additive measures, Choquet integral, and Shapley values.

Approaches to improving consistency of interval fuzzy preference relations

This article introduces a consistency index for measuring the consistency level of an interval fuzzy preference relation (IFPR). An approach is then proposed to construct an additive consistent IFPR from a given inconsistent IFPR. By using a weighted averaging method combining the original IFPR and the constructed consistent IFPR, a formula is put forward to repair an inconsistent IFPR to generate an IFPR with acceptable consistency. An iterative algorithm is subsequently developed to rectify an inconsistent IFPR and derive one with acceptable consistency and weak transitivity. The proposed approaches can not only improve consistency of IFPRs but also preserve the initial interval uncertainty information as much as possible. Numerical examples are presented to illustrate how to apply the proposed approaches

A survey on pairwise comparison matrices over abelian linearly ordered groups

In this paper, we provide a survey of our results about the pairwise comparison matrices defined over abelian linearly ordered groups

An overview on managing additive consistency of reciprocal preference relations for consistency-driven decision making and Fusion: Taxonomy and future directions

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.The reciprocal preference relation (RPR) is a powerful tool to represent decision makers’ preferences in decision making problems. In recent years, various types of RPRs have been reported and investigated, some of them being the ‘classical’ RPRs, interval-valued RPRs and hesitant RPRs. Additive consistency is one of the most commonly used property to measure the consistency of RPRs, with many methods developed to manage additive consistency of RPRs. To provide a clear perspective on additive consistency issues of RPRs, this paper reviews the consistency measurements of the different types of RPRs. Then, consistency-driven decision making and information fusion methods are also reviewed and classified into four main types: consistency improving methods; consistency-based methods to manage incomplete RPRs; consistency control in consensus decision making methods; and consistency-driven linguistic decision making methods. Finally, with respect to insights gained from prior researches, further directions for the research are proposed

Algorithms to Detect and Rectify Multiplicative and Ordinal Inconsistencies of Fuzzy 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.Consistency, multiplicative and ordinal, of fuzzy preference relations (FPRs) is investigated. The geometric consistency index (GCI) approximated thresholds are extended to measure the degree of consistency for an FPR. For inconsistent FPRs, two algorithms are devised (1) to find the multiplicative inconsistent elements, and (2) to detect the ordinal inconsistent elements. An integrated algorithm is proposed to improve simultaneously the ordinal and multiplicative consistencies. Some examples, comparative analysis, and simulation experiments are provided to demonstrate the effectiveness of the proposed methods