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

Incomplete pairwise comparison and consistency optimization

This paper proposes a new method for calculating the missing elements of an incomplete matrix of pairwise comparison values for a decision problem. The matrix is completed by minimizing a measure of global inconsistency, thus obtaining a matrix which is optimal from the point of view of consistency with respect to the available judgements. The optimal values are obtained by solving a linear system and unicity of the solution is proved under general assumptions. Some other methods proposed in the literature are discussed and a numerical example is presented.consistency, pairwise comparison matrices

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

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.

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

A multi-step goal programming approach for group decision making with incomplete interval additive reciprocal comparison matrices

This article presents a goal programming framework to solve group decision making problems where decision-makers’ judgments are provided as incomplete interval additive reciprocal comparison matrices (IARCMs). New properties of multiplicative consistent IARCMs are put forward and used to define consistent incomplete IARCMs. A two-step goal programming method is developed to estimate missing values for an incomplete IARCM. The first step minimizes the inconsistency of the completed IARCMs and controls uncertainty ratios of the estimated judgments within an acceptable threshold, and the second step finds the most appropriate estimated missing values among the optimal solutions obtained from the previous step. A weighted geometric mean approach is proposed to aggregate individual IARCMs into a group IARCM by employing the lower bounds of the interval additive reciprocal judgments. A two-step procedure consisting of two goal programming models is established to derive interval weights from the group IARCM. The first model is devised to minimize the absolute difference between the logarithm of the group preference and that of the constructed multiplicative consistent judgment. The second model is developed to generate an interval-valued priority vector by maximizing the uncertainty ratio of the constructed consistent IARCM and incorporating the optimal objective value of the first model as a constraint. Two numerical examples are furnished to demonstrate validity and applicability of the proposed approach