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2002 | Buch

Aggregation Operators

New Trends and Applications

herausgegeben von: Professor Tomasa Calvo, Professor Gaspar Mayor, Professor Radko Mesiar

Verlag: Physica-Verlag HD

Buchreihe : Studies in Fuzziness and Soft Computing

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Über dieses Buch

1. The increasing number of research papers appeared in the last years that either make use of aggregation functions or contribute to its theoretieal study asses its growing importance in the field of Fuzzy Logie and in others where uncertainty and imprecision play a relevant role. Since these papers are pub­ lished in many journals, few books and several proceedings of conferences, books on aggregation are partieularly welcome. To my knowledge, "Agrega­ tion Operators. New Trends and Applications" is the first book aiming at generality , and I take it as a honour to write this Foreword in response to the gentle demand of its editors, Radko Mesiar, Tomasa Calvo and Gaspar Mayor. My pleasure also derives from the fact that twenty years aga I was one of the first Spaniards interested in the study of aggregation functions, and this book includes work by several Spanish authors. The book contains nice and relevant original papers, authored by some of the most outstanding researchers in the field, and since it can serve, as the editors point out in the Preface, as a small handbook on aggregation, the book is very useful for those entering the subject for the first time. The book also contains apart dealing with potential areas of application, so it can be helpful in gaining insight on the future developments.

Inhaltsverzeichnis

Frontmatter

Aggregation Operators: Basic Concepts, Issues and Properties

Frontmatter
Aggregation Operators: Properties, Classes and Construction Methods
Abstract
Aggregation (fusion) of several input values into a single output value is an indispensable tool not only of mathematics or physics, but of majority of engineering, economical, social and other sciences. The problems of aggregation are very broad and heterogeneous, in general. Therefore we restrict ourselves in this contribution to the specific topic of the aggregation of finite number of real inputs only. Closely related topics of aggregating infinitely many real inputs [23,109,64,52,43,42,44,99], of aggregating inputs from some ordinal scales [41,50], of aggregating complex inputs (such as probability distributions [107,114], fuzzy sets [143]), etc., are treated, among others, in the quoted papers, and we will not deal with them. In this spirit, if the number of input values is fixed, say n, an aggregation operator is a real function of n variables. This is still a too general topic. Therefore we restrict our considerations regarding inputs as well as outputs to some fixed interval (scale) I = [a, b] ⊑ [-∞, ∞]. It is a matter of rescaling to fix I = [0,1].
Tomasa Calvo, Anna Kolesárová, Magda Komorníková, Radko Mesiar

Theoretical Aspects of Aggregation Operators

Frontmatter
Aggregation Based on Integrals: Recent Results and Trends
Abstract
The paper presents an overview of recent results related to the use of non additive integrals (Choquet and Sugeno integrals) as aggregation operators. We essentially address two main issues, which are the treatment of negative numbers, and the case of ordinal information to aggregate. We do not explicitely presuppose any particular form of aggregation problem (e.g. multicriteria decision making), and remain at an abstract level, although decision making in general can be considered as the main motivation of the work.
Michel Grabisch
Associative Aggregation Operators
Abstract
An aggregation process occurs in many situations like in decision making or in statistical and economic measurement by aggregating expert’s opinions or by synthesizing judgements. So the typical situation is as follows:
Having n numerical values x 1,..., x n lying in an interval I of real numbers, the aggregation operator M defined on I n aggregates these numbers to a value of ℝ in an appropriate way so that the properties of M represent a model of the concrete situation.
W. Sander
Continuous WOWA Operators with Application to Defuzzification
Abstract
The basic operations for combining real values are the weighted mean (WM) and the Ordered Weighted Averaging (OWA) operator. The weighted mean allows the system to compute an aggregate value from the ones coming from several sources, taking into account the reliability of each information source. Alternatively, the OWA operator allows the user to weight the values supplied in relation to their alternative ordering. The Weighted OWA operator (WOWA) allows the user to consider both aspects using two sets of weights. In this chapter we describe an extension of the WOWA operator to the continuous case after arguing its convenience. Then, we analyze the use of this operator for defuzzification.
Vicenç Torra, Lluís Godo
Using Importances in Group Preference Aggregation to Block Strategic Manipulation
Abstract
We consider the problem of preference aggregation in group decision making. The role of the collaborative imperative used by the group in determining the form of the aggregation function is noted. We focus on one collaborative imperative, called the primal, deriving from participant autonomy, an agent doesn’t have to accept a decision by the group it doesn’t like. We showed that t-norms and more generally the class of uninorms having zero fixation provide appropriate aggregation operators to implement this primal collaborative imperative. We then discuss the possibility of an agent using a strategic manipulation of its preference information to get its preferred alternative. A mechanism based upon the use of an importance weighting is then suggested for modifying the construction of the group decision function to defend against strategic manipulation.
Ronald R. Yager

Applications of Aggregation Operators

Frontmatter
Aggregation Operators in Engineering Design
Abstract
Engineering design is conducted with incomplete and imperfect information, and in this paper there are presented some of the tools for decision making under risk and uncertainty and the application of these tools to engineering design. First it is presented an axiomatization of engineering design based on von Neumannn and Morgenstern axiomatization. Then it is given a general definition of decision making problem which enables to apply also fuzzy systems and non-additive measures. Special attention is taken on different aggregation operators which can model the decision making in engineering design. A procedure for finding the global maximum as well as some procedures for identification of non-additive measure are presented.
Endre Pap
Aggregation of Interacting Criteria by Means of the Discrete Choquet Integral
Abstract
The most often used operator to aggregate criteria in decision making problems is the classical weighted arithmetic mean. In many problems however, the criteria considered interact, and a substitute to the weighted arithmetic mean has to be adopted. Under rather natural conditions, the discrete Choquet integral is proved to be an adequate aggregation operator that extends the weighted arithmetic mean by the taking into consideration of the interaction among criteria. The axiomatic that supports the Choquet integral is presented and some subfamilies are studied.
Jean-Luc Marichal
Representation Models for Aggregating Linguistic Information: Issues and Analysis
Abstract
The linguistic information has been used successfully in many areas. The aggregation of linguistic information is a crucial aspect. In the literature we can find different linguistic computational models that present linguistic aggregation operators as: (i) The computational model based on the Extension Principle, which operates over the fuzzy numbers that supports the semantics of the linguistic labels. (ii) The symbolic one makes the computations directly over the order index of the linguistic labels. And, (iii) the model based on the linguistic 2-tuple representation, which uses the symbolic translation to make the linguistic computations.
F. Herrera, E. Herrera-Viedma, L. Martinez
Aggregation Techniques for Statistical Confidentiality
Abstract
This chapter describes microaggregation, a technique for statistical confidentiality that uses aggregation operators. We describe the goals of statistical confidentiality and its application to continuous and categorical data. We show the application of the method to a small publicly available data set. The chapter finishes by reviewing some of the practical problems of the application of microaggregation to statistical disclosure control.
Josep Domingo-Ferrer, Vicenç Torra
Quantifier Guided Aggregation of Fuzzy Criteria with Associated Importances
Abstract
The evaluation of quantified sentences of the form “Q of D are A” is recognized as a suitable tool for quantifier guided aggregation of fuzzy criteria with associated importances. In this paper we discuss the properties any good evaluation method should verify. We study a new method to evaluate quantified sentences. Our new method is shown to be an extension of the quantified aggregation via the Choquet integral when all the criteria are equally important and a monotone increasing quantifier is employed.
I. Blanco, M. Delgado, M. J. Martín-Bautista, D. Sánchez, M. A. Vila

Fuzzy Quantities and Their Aggregation

Frontmatter
Verbally Generated Fuzzy Quantities and Their Aggregation
Abstract
The processing of vague data recently becomes one of attractive topics in the fuzzy set theory and its applications. As the vagueness is usually represented by some verbal expressions, this branch of the fuzzy sets is frequently called “computing with words”. Seemingly, but only seemingly, it could be understood as computational processing of fuzzy numbers or fuzzy quantities in the already classical sense. Other authors understand the computing with words rather as a fuzzy logical discipline being near to fuzzy reasoning methods and other related branches. Both approaches are rational and fully acceptable but, in the matter of facts, none of them appears to be complete. Their parallel existence offers a conclusion that the fair approach to computing with words can consist in some kind of their combination. Computing with words has two faces — quantitative and qualitative one — and each of them would be somehow reflected. The fuzzy set theoretical model of verbal variables, their generating and processing suggested in this contribution and in some of the referred papers is intended to offer such combined view on the quantitative — qualitative dualism existing in the “computing with words” and to develop at least elementary methods for manipulation with such dualistic verbal data.
Milan Mareš, Radko Mesiar
Backmatter
Metadaten
Titel
Aggregation Operators
herausgegeben von
Professor Tomasa Calvo
Professor Gaspar Mayor
Professor Radko Mesiar
Copyright-Jahr
2002
Verlag
Physica-Verlag HD
Electronic ISBN
978-3-7908-1787-4
Print ISBN
978-3-662-00319-0
DOI
https://doi.org/10.1007/978-3-7908-1787-4