Ricardo Alberto Marques Pereira

Aggregation with generalized mixture operators using weighting functions

This paper regards weighted aggregation operators in multiple attribute decision making and its main goal is to investigate ways in which weights can depend on the satisfaction degrees of the various attributes (criteria). We propose and discuss two types of weighting functions that penalize poorly satisfied attributes and reward well-satisfied attributes. We discuss in detail the characteristics and properties of both functions. Moreover, we present an illustrative example to clarify the use and behaviour of such weighting functions, comparing the results with those of standard weighted averaging operators.

Generalized mixture operators using weighting functions: A comparative study with WA and OWA

Abstract In the context of multiple attribute decision making, we present an aggregation scheme based on generalized mixture operators using weighting functions and we compare it with two standard aggregation methods: weighted averaging and ordered weighted averaging. Specifically, we consider linear and quadratic weight generating functions that penalize bad attribute performances and reward good attribute performances. An illustrative example, borrowed from the literature, is used to perform the operators’ comparison. We believe that this comparative study will highlight the potential and flexibility of generalized mixture operators using weighting functions that depend on attribute performances.

The self-dual core and the anti-self-dual remainder of an aggregation operator

In most decisional models based on pairwise comparison between alternatives, the reciprocity of the individual preference representations expresses a natural assumption of rationality. In those models self-dual aggregation operators play a central role, in so far as they preserve the reciprocity of the preference representations in the aggregation mechanism from individual to collective preferences. In this paper we propose a simple method by which one can associate a self-dual aggregation operator to any aggregation operator on the unit interval. The resulting aggregation operator is said to be the self-dual core of the original one, and inherits most of its properties. Our method constitutes thus a new characterization of self-duality, with some technical advantages relatively to the traditional symmetric sums method due to Silvert. In our framework, moreover, every aggregation operator can be written as a sum of a self-dual core and an anti-self-dual remainder which, in some cases, seems to give some indication on the dispersion of the variables. In order to illustrate the method proposed, we apply it to two important classes of continuous aggregation operators with the properties of idempotency, symmetry, and stability for translations: the OWA operators and the exponential quasiarithmetic means.

Classical inequality indices, welfare and illfare functions, and the dual decomposition

Abstract In the traditional framework, social welfare functions depend on the mean income and on the income inequality. An alternative illfare framework has been developed to take into account the disutility of unfavorable variables. The illfare level is assumed to increase with the inequality of the distribution. In some social and economic fields, such as those related to employment, health, education, or deprivation, the characteristics of the individuals in the population are represented by bounded variables, which encode either achievements or shortfalls. Accordingly, both the social welfare and the social illfare levels may be assessed depending on the framework we focus on. In this paper we propose a unified dual framework in which welfare and illfare levels can both be investigated and analyzed in a natural way. The dual framework leads to the consistent measurement of achievements and shortfalls, thereby overcoming one important difficulty of the traditional approach, in which the focus on achievements or shortfalls often leads to different inequality rankings. A number of welfare functions associated with inequality indices are OWA operators. Specifically this paper considers the welfare functions associated with the classical inequality measures due to Gini, Bonferroni, and De Vergottini. These three indices incorporate different value judgments in the measurement of inequality, leading to different behavior under income transfers between individuals in the population. In the bounded variables representation, we examine the dual decomposition and the orness degree of the three classical welfare/illfare functions in the standard framework of aggregation functions on the [ 0 , 1 ] n domain. The dual decomposition of each welfare/illfare function into a self-dual central index and an anti-self-dual inequality index leads to the consistent measurement of achievements and shortfalls.

A class of poverty measures induced by the dual decomposition of aggregation functions

In this paper we introduce a new family of poverty measures for comparing and ordering social situations. The aggregation scheme of these poverty measures is based on the one-parameter family of exponential means. The poverty measures introduced satisfy interesting properties and the dual decomposition of the underlying exponential means induces a natural decomposition of the proposed poverty indices themselves into three underlying factors: incidence, intensity, and inequality among the poor.

The binomial Gini inequality indices and the binomial decomposition of welfare functions

In the context of Social Welfare and Choquet integration, we briefly review, on the one hand, the generalized Gini welfare functions and inequality indices for populations of n>=2 individuals, and on the other hand, the Mobius representation framework for Choquet integration, particularly in the case of k-additive symmetric capacities. We recall the binomial decomposition of OWA functions due to Calvo and De Baets [14] and we examine it in the restricted context of generalized Gini welfare functions, with the addition of appropriate S-concavity conditions. We show that the original expression of the binomial decomposition can be formulated in terms of two equivalent functional bases, the binomial Gini welfare functions and the Atkinson-Kolm-Sen (AKS) associated binomial Gini inequality indices, according to Blackorby and Donaldsons correspondence formula. The binomial Gini pairs of welfare functions and inequality indices are described by a parameter j = 1,...,n, associated with the distributional judgements involved. The j-th generalized Gini pair focuses on the (n - j + 1)/n poorest fraction of the population and is insensitive to income transfers within the complementary richest fraction of the population.

The Dynamics of Consensus in Group Decision Making: Investigating the Pairwise Interactions between Fuzzy Preferences

In this paper we present an overview of the soft consensus model in group decision making and we investigate the dynamical patterns generated by the fundamental pairwise preference interactions on which the model is based. The dynamical mechanism of the soft consensus model is driven by the minimization of a cost function combining a collective measure of dissensus with an individual mechanism of opinion changing aversion. The dissensus measure plays a key role in the model and induces a network of pairwise interactions between the individual preferences. The structure of fuzzy relations is present at both the individual and the collective levels of description of the soft consensus model: pairwise preference intensities between alternatives at the individual level, and pairwise interaction coefficients between decision makers at the collective level. The collective measure of dissensus is based on non linear scaling functions of the linguistic quantifier type and expresses the degree to which most of the decision makers disagree with respect to their preferences regarding the most relevant alternatives. The graded notion of consensus underlying the dissensus measure is central to the dynamical unfolding of the model. The original formulation of the soft consensus model in terms of standard numerical preferences has been recently extended in order to allow decision makers to express their preferences by means of triangular fuzzy numbers. An appropriate notion of distance between triangular fuzzy numbers has been chosen for the construction of the collective dissensus measure. In the extended formulation of the soft consensus model the extra degrees of freedom associated with the triangular fuzzy preferences, combined with non linear nature of the pairwise preference interactions, generate various interesting and suggestive dynamical patterns. In the present paper we investigate these dynamical patterns which are illustrated by means of a number of computer simulations.

Inconsistency and non-additive capacities: The Analytic Hierarchy Process in the framework of Choquet integration

We examine the AHP in the framework of Choquet integration and we propose an extension of the standard AHP aggregation scheme on the basis of the Shapley values associated with the criteria. In our model a measure of dominance inconsistency between criteria is defined in terms of the totally inconsistent matrix associated with the main pairwise comparison matrix of the AHP. The measure of dominance inconsistency is then used to construct a non-additive capacity whose associated Shapley values reduce to the standard AHP priority weights in the consistency case. In the general inconsistency case, however, the extended aggregation scheme based on the Shapley weighted mean tends to attenuate (resp. emphasize) the priority weights of the criteria with higher (resp. lower) average dominance inconsistency with respect to the other criteria.

The dual decomposition of aggregation functions and its application in welfare economics

In this paper, we review the role of self-duality in the theory of aggregation functions, the dual decomposition of aggregation functions into a self-dual core and an anti-self-dual remainder, and some applications to welfare, inequality, and poverty measures.

A decision making approach to relevance feedback in information retrieval : A model based on soft consensus dynamics

Information retrieval (IR) can be regarded as a natural instance of multicriteria decision making (MCDM). Queries are formulated as selection criteria aggregated by means of appropriate operators. Retrieval is then performed as a MCDM process by evaluating the degrees of satisfaction of the criteria by each document, and then aggregating them. Another decisional instance in IR concerns the problem of improving retrieval performance by taking into account user indications on documents relevance. Relevance feedback mechanisms exploit user‐system interaction in order to improve retrieval results by means of an iterative process of query refinement. In this process the main decisional issue is that of finding new concepts (terms) with which to expand–modify the initial query so that it better reflects the users information needs. In this paper we introduce a relevance feedback mechanism based on a dynamical consensus model originally proposed in the framework of group decision making. In the relevance feedback context the consensual interaction highlights associations among the most significant terms in the relevant retrieved documents selected by the user. The resulting associative structure can then be used to expand the original query by including new terms which result strongly associated with those in the query. ©1999 John Wiley & Sons, Inc.