The book provides a thorough treatment of set functions, games and capacities as well as integrals with respect to capacities and games, in a mathematical rigorous presentation and in view of application to decision making. After a short chapter introducing some required basic knowledge (linear programming, polyhedra, ordered sets) and notation, the first part of the book consists of three long chapters developing the mathematical aspects. This part is not related to a particular application field and, by its neutral mathematical style, is useful to the widest audience. It gathers many results and notions which are scattered in the literature of various domains (game theory, decision, combinatorial optimization and operations research). The second part consists of three chapters, applying the previous notions in decision making and modelling: decision under uncertainty, decision with multiple criteria, possibility theory and Dempster-Shafer theory.

The book provides a thorough treatment of set functions, games and capacities as well as integrals with respect to capacities and games, in a mathematical rigorous presentation and in view of application to decision making. After a short chapter introducing some required basic knowledge (linear programming, polyhedra, ordered sets) and notation, the first part of the book consists of three long chapters developing the mathematical aspects. This part is not related to a particular application field and, by its neutral mathematical style, is useful to the widest audience. It gathers many results and notions which are scattered in the literature of various domains (game theory, decision, combinatorial optimization and operations research). The second part consists of three chapters, applying the previous notions in decision making and modelling: decision under uncertainty, decision with multiple criteria, possibility theory and Dempster-Shafer theory.

With the vision that machines can be rendered smarter, we have witnessed for more than a decade tremendous engineering efforts to implement intelligent sys tems. These attempts involve emulating human reasoning, and researchers have tried to model such reasoning from various points of view. But we know precious little about human reasoning processes, learning mechanisms and the like, and in particular about reasoning with limited, imprecise knowledge. In a sense, intelligent systems are machines which use the most general form of human knowledge together with human reasoning capability to reach decisions. Thus the general problem of reasoning with knowledge is the core of design methodology. The attempt to use human knowledge in its most natural sense, that is, through linguistic descriptions, is novel and controversial. The novelty lies in the recognition of a new type of un certainty, namely fuzziness in natural language, and the controversality lies in the mathematical modeling process. As R. Bellman [7] once said, decision making under uncertainty is one of the attributes of human intelligence. When uncertainty is understood as the impossi bility to predict occurrences of events, the context is familiar to statisticians. As such, efforts to use probability theory as an essential tool for building intelligent systems have been pursued (Pearl [203], Neapolitan [182)). The methodology seems alright if the uncertain knowledge in a given problem can be modeled as probability measures.

With the vision that machines can be rendered smarter, we have witnessed for more than a decade tremendous engineering efforts to implement intelligent sys tems. These attempts involve emulating human reasoning, and researchers have tried to model such reasoning from various points of view. But we know precious little about human reasoning processes, learning mechanisms and the like, and in particular about reasoning with limited, imprecise knowledge. In a sense, intelligent systems are machines which use the most general form of human knowledge together with human reasoning capability to reach decisions. Thus the general problem of reasoning with knowledge is the core of design methodology. The attempt to use human knowledge in its most natural sense, that is, through linguistic descriptions, is novel and controversial. The novelty lies in the recognition of a new type of un certainty, namely fuzziness in natural language, and the controversality lies in the mathematical modeling process. As R. Bellman [7] once said, decision making under uncertainty is one of the attributes of human intelligence. When uncertainty is understood as the impossi bility to predict occurrences of events, the context is familiar to statisticians. As such, efforts to use probability theory as an essential tool for building intelligent systems have been pursued (Pearl [203], Neapolitan [182)). The methodology seems alright if the uncertain knowledge in a given problem can be modeled as probability measures.

Aggregation is the process of combining several numerical values into a single representative value, and an aggregation function performs this operation. These functions arise wherever aggregating information is important: applied and pure mathematics (probability, statistics, decision theory, functional equations), operations research, computer science, and many applied fields (economics and finance, pattern recognition and image processing, data fusion, etc.). This is a comprehensive, rigorous and self-contained exposition of aggregation functions. Classes of aggregation functions covered include triangular norms and conorms, copulas, means and averages, and those based on nonadditive integrals. The properties of each method, as well as their interpretation and analysis, are studied in depth, together with construction methods and practical identification methods. Special attention is given to the nature of scales on which values to be aggregated are defined (ordinal, interval, ratio, bipolar). It is an ideal introduction for graduate students and a unique resource for researchers.