Aggregation, the authors tell us, is the process of combining several numerical values to get a number that is, in some sense, representative of them all. Taking the arithmetic mean is a trivial example. We are all aware, of course, of the pressures on anyone who handles complex numerical data to “reduce it to a number,” so that one can get a linear ranking of the data points. Decision making under multiple criteria, multiobjective optimization, pattern classification, construction of rule-based systems, and fuzzy inference are just a few of the areas in which aggregation is used, often extensively, and often without proper analysis.

Obviously, some methods of aggregation are going to be better than others, and the authors set out to construct a comprehensive and rigorous mathematical theory to classify and understand possible aggregation functions. For simplicity, they assume that all of the numerical values come from the same interval (open, closed, half-closed) I on the real line R. Let n be a positive integer. An aggregation function A is a function on n variables that is nondecreasing in each variable and that fulfills the boundary conditions that inf A(x) = inf (I), and sup A(x) = sup (I).

Note that if I = [0 , 1], the initial conditions just say that A(0, ... , 0) = 0 and A(1, ... ,1) = 1. The arithmetic mean, min, and max are simple examples of aggregation functions. Triangular norms and copulas are more interesting, and more sophisticated, examples of such functions. Another extremely interesting class of such functions comes from integrals defined with respect to nonadditive measures, such as Choquet and Sugeno integrals. An extended aggregation function is then defined to be a function F, whose restriction for each n is an aggregation function.

The mathematical theory of aggregation functions, which the authors construct in great detail, shows that this abstract definition is both natural and interesting. While quite abstract, it builds on concrete examples from statistics, mathematical economics, game theory, and other applications. The book is very impressive in its scope. The authors dedicate over 250 pages to methods for the construction of aggregation functions, their properties, and their classification. They then spend another 60 pages on aggregation functions with various scale types, including ordinal and bipolar scales, and an additional 50 pages on the behavioral analysis and identification of aggregation functions. Aggregation with infinitely many arguments is dealt with briefly in an appendix. Throughout the entire work, many examples and applications of aggregation are given. The presentation is abstract, formal, and mathematically rigorous, providing a firm foundation for specific applications.

This book should be taken very seriously by anyone whose work involves aggregation.