Michel Grabisch

THE APPLICATION OF FUZZY INTEGRALS IN MULTICRITERIA DECISION MAKING

Abstract This paper presents a synthesis on the application of fuzzy integrals as an innovative tool for criteria aggregation in decision problems. The main point is that fuzzy integrals are able to model interaction between criteria in a flexible way. The methodology has been elaborated mainly in Japan, and has been applied there successfully in various fields such as design, reliability, evaluation of goods, etc. It seems however that this technique is still very little known in Europe. It is one of the aims of this review to disseminate this emerging technology in many industrial fields.

k -order additive discrete fuzzy measures and their representation

Abstract In order to face with the complexity of discrete fuzzy measures, we propose the concept of k -orderadditive fuzzy measure, including usual additive measures and fuzzy measures. Every discrete fuzzy measure is a k -order additive fuzzy measure for a unique k . A related topic of the paper is to introduce an alternative representation of fuzzy measures, called the interaction representation, which sets and extends in a common framework the Shapley value and the interaction index proposed by Murofushi and Soneda.

Fuzzy integral in multicriteria decision making

Abstract This paper presents a synthesis on the use of fuzzy integral as an aggregation operator in multicriteria decision making. Definitions, essential properties are given, and compared to those of usual aggregation operators. A quick survey on applications is given.

Fundamentals of Uncertainty Calculi with Applications to Fuzzy Inference

From the Publisher: This decade has witnessed increasing interest in fuzzy technology both from academia and industry. It is often said that fuzzy theory is easy and simple so that engineers can progress quickly to real applications. However, the lack of knowledge of design methodologies and the theoretical results of fuzzy theory have often caused problems for design engineers. The aim of this book is to provide a rigorous background for uncertainty calculi, with an emphasis on fuzziness. Fundamentals of Uncertainty Calculi with Applications to Fuzzy Inference is primarily about the type of knowledge expressed in a natural language, that is, in linguistic terms. The approach to modeling such knowledge is based upon the mathematical theory of uncertainty related to fuzzy measures and integrals and their applications. The book consists of two parts: Chapters 2 - 6 comprise the theory, and applications are offered in Chapters 7 - 10. In the theory section the exposition is mathematical in nature and gives a complete background on uncertainty measures and integrals, especially in a fuzzy setting. Applications concern recent ones of fuzzy measures and integrals to problems such as pattern recognition, decision making and subjective multicriteria evaluations.

Fuzzy Measures and Integrals

Chapter 5 was concerned with the theory of fuzzy sets as a mathematical model to describe vague concepts, and as an extension of ordinary set theory. In a similar spirit, this chapter is about an extension of additive measures, in particular probability measures, to a more general class of non-additive set functions.

A review of methods for capacity identification in Choquet integral based multi-attribute utility theory: Applications of the Kappalab R package

The application of multi-attribute utility theory whose aggregation process is based on the Choquet integral requires the prior identification of a capacity. The main approaches to capacity identification proposed in the literature are reviewed and their advantages and inconveniences are discussed. All the reviewed methods have been implemented within the Kappalab R package. Their application is illustrated on a detailed example.

Classification by fuzzy integral: performance and tests

Abstract This paper presents an attempt to characterize the performance of methods of classification based on fuzzy integral. After an introductory explanation about the approach, a lower bound of the minimal number of training samples is found, and it is shown that a minimum squared error criterion leads to the best approximate for the optimal Bayes classifier. Some tests on simulated and real data are provided.

An Axiomatic Approach to the Concept of Interaction Among Players in Cooperative Games

Abstract. An axiomatization of the interaction between the players of any coalition is given. It is based on three axioms: linearity, dummy and symmetry. These interaction indices extend the Banzhaf and Shapley values when using in addition two equivalent recursive axioms. Lastly, we give an expression of the Banzhaf and Shapley interaction indices in terms of pseudo-Boolean functions.

Aggregation functions: Means

This two-part state-of-the-art overview on aggregation theory summarizes the essential information concerning aggregation issues. An overview of aggregation properties is given, including the basic classification on aggregation functions. In this first part, the stress is put on means, i.e., averaging aggregation functions, both with fixed arity (n-ary means) and with multiple arities (extended means).

Fusion: General concepts and characteristics

The problem of combining pieces of information issued from several sources can be encountered in various fields of application. This paper aims at presenting the different aspects of information fusion in different domains, such as databases, regulations, preferences, sensor fusion, etc., at a quite general level. We first present different types of information encountered in fusion problems, and different aims of the fusion process. Then we focus on representation issues which are relevant when discussing fusion problems. An important issue is then addressed, the handling of conflicting information. We briefly review different domains where fusion is involved, and describe how the fusion problems are stated in each domain. Since the term fusion can have different, more or less broad, meanings, we specify later some terminology with respect to related problems, that might be included in a broad meaning of fusion. Finally we briefly discuss the difficult aspects of validation and evaluation. © 2001 John Wiley & Sons, Inc.