Michele Fedrizzi

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.

Axiomatic properties of inconsistency indices for pairwise comparisons

Pairwise comparisons are a well-known method for the representation of the subjective preferences of a decision maker. Evaluating their inconsistency has been a widely studied and discussed topic and several indices have been proposed in the literature to perform this task. As an acceptable level of consistency is closely related to the reliability of preferences, a suitable choice of an inconsistency index is a crucial phase in decision-making processes. The use of different methods for measuring consistency must be carefully evaluated, as it can affect the decision outcome in practical applications. In this paper, we present five axioms aimed at characterizing inconsistency indices. In addition, we prove that some of the indices proposed in the literature satisfy these axioms, whereas others do not, and therefore, in our view, they may fail to correctly evaluate inconsistency.

Inconsistency indices for pairwise comparison matrices: a numerical study

Evaluating the level of inconsistency of pairwise comparisons is often a crucial step in multi criteria decision analysis. Several inconsistency indices have been proposed in the literature to estimate the deviation of expert’s judgments from a situation of full consistency. This paper surveys and analyzes ten indices from the numerical point of view. Specifically, we investigate degrees of agreement between them to check how similar they are. Results show a wide range of behaviors, ranging from very strong to very weak degrees of agreement.

Soft consensus and network dynamics in group decision making

We propose a dynamical network model for consensus reaching in group decision making. The model combines the minimization of a soft measure of collective dissensus and an individual inertial mechanism which emulates opinion changing aversion. Both components of the dynamics are nonlinear. The collective consensual trend corresponds to a process of anisotropic diffusion among the various individual preference structures. The anisotropy is designed so as to outline and enhance the natural group segmentation into homogeneous preference subgroups (weak consensus). The individual inertial mechanism, on the other hand, opposes changes from the original preferences and provides an appropriate framework to deal with preference outliers. We examine in detail the simple case in which each decision maker must choose between only two alternatives. Finally we comment on the possibility of incorporating in the dynamics a form of transitivity constraint regarding the group segmentation. ©1999 John Wiley & Sons, Inc.

On the priority vector associated with a reciprocal relation and a pairwise comparison matrix

We propose two straightforward methods for deriving the priority vector associated with a reciprocal relation, by some authors called fuzzy preference relation. Then, using transformations between pairwise comparison matrices and reciprocal relations, we study the relationships between the priority vectors associated with these two types of preference relations. Eventually, we show a brief example involving the newly introduced characterizations.

A Fuzzy Approach to Social Network Analysis

Adjacency relations for social network analysis have usually been tackled in their bidimensional form, in the sense that relations are computed over pairs of objects. Nevertheless, this paper considers the bidimensional case as restrictive and it proposes an approach where the dimension of the analysis is not limited to binary relations. With the aid of fuzzy logic and OWA operators, it is showed that the interpretation of m-ary adjacency relations is the same of binary relations and therefore they can consistently be employed in social network analysis and some novel results be derived. Besides justifying the use of m-ary relations, the paper proposes a way to characterize them and, eventually, it will provide the reader with an example section.

On a Consensus Measure in a Group MCDM Problem

A method for consensus measuring in a group decision problem is presented for the multiple criteria case. The decision process is supposed to be carried out according to Saaty’s Analytic Hierarchy Process, and hence using pairwise comparison among the alternatives. Using a suitable distance between the experts’ judgements, a scale transformation is proposed which allows a fuzzy interpretation of the problem and the definition of a consensus measure by means of fuzzy tools as linguistic quantifiers. Sufficient conditions on the expert’s judgements are finally presented, which guarantee any a priori fixed consensus level to be reached.

Boundary properties of the inconsistency of pairwise comparisons in group decisions

This paper proposes an analysis of the effects of consensus and preference aggregation on the consistency of pairwise comparisons. We define some boundary properties for the inconsistency of group preferences and investigate their relation with different inconsistency indices. Some results are presented on more general dependencies between properties of inconsistency indices and the satisfaction of boundary properties. In the end, given three boundary properties and nine indices among the most relevant ones, we will be able to present a complete analysis of what indices satisfy what properties and offer a reflection on the interpretation of the inconsistency of group preferences.

Towards fuzzy modelling in economics

Abstract The general methodological framework of fuzzy modelling is considered. Particular emphasis is set on the analytical approach to building fuzzy models, especially linear fuzzy models. Such models are obtained as a fuzzy extension of usual linear models. Apart from the known extension principle of Zadeh and the so called fuzzy parameter extension, a new model of fuzzy combination is also considered.

On the issue of consistency in dynamical consensual aggregation

In this paper we propose to use the consistency of preferences in order to endogenously assign different weights to decision makers in a consensual dynamics process. For this purpose, we first define a consistency index for preferences expressed by means of fuzzy preference relations. Then we introduce this index in an iterative law for updating the individual preferences. The updating law is formulated in the spirit of some previous papers on consensual dynamics (see for instance [5]) and, as a result, both the initially declared preferences and the present ones (in the ongoing process) act with a strength determined on the basis of their consistency.