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Discrete fuzzy measures : computational aspects
Beliakov G., James S., Wu J., Springer International Publishing, New York, NY, 2020. 245 pp.  Type: Book (978-3-030153-04-5)
Date Reviewed: Mar 9 2020

Decision-making is a problem-solving activity aimed at reckoning a choice by taking into account a possibly large number of constraints, preferences, beliefs, and costs, among other things. It is a complex discipline that requires sophisticated operators based on solid theory. This book offers a clean and rigorous view of an important class of such operators--discrete fuzzy measures--to help specialists use them correctly, with a closed eye to the computational aspects of their implementation.

One important operation in decision-making is aggregation (of preferences, satisfaction degrees, beliefs, and so on), which combines multiple values--coming from several sources--into a single value that can be used to support a decision. In large part, aggregation is obtained by some kind of averaging, like weighted average or other forms of means. In most cases, however, these simple aggregation functions disregard interactions among the sources that provided the values. Nevertheless, interactions are commonplace, so ignoring them may lead to biased aggregations and, consequently, suboptimal decisions.

Fuzzy integrals, namely Choquet and Sugeno integrals, come in hand for dealing with interactive sources. They are not new (the Choquet integral dates back to 1953), but their use is not widespread, mainly because of the complexity of the underlying theory. Nevertheless, fuzzy integrals are of utmost utility; therefore, books like this are precious in making the topic more accessible to students and professionals. A fuzzy integral aggregates values based on a fuzzy measure (also known as capacity), which is a function that gives a distinct weight to each subset of sources. (The idea of considering subsets of sources is key to represent interactions.) Fuzzy measures generalize the classical idea of (additive) measure and unleash a tremendous flexibility at the price of increased complexity, both in theoretical and computational terms.

This book delivers rigorous yet clear guidance in the realm of discrete fuzzy measures (that is, fuzzy measures defined on a discrete number of sources). The treatment has a strong mathematical style, with emphasis on definitions and the recall of main theorems. There are few examples to explain the most important passages. (More examples would have been appreciated, anyway.) The main idea is to let the reader understand the main concepts and the principal properties that arise from the theory. The comprehensive treatment of all such properties is left to the literature, which is abundantly referenced at the end of each chapter.

The book first introduces the concept of fuzzy measures, with a generous list of the most important types, including measures coming from other theories, such as belief, possibility, and probability, so that connections among theories can be easily made. Then the idea of interaction is investigated with the introduction of interaction indexes. The book’s focus is in computational aspects; therefore, an entire chapter is devoted to representations of fuzzy measures, which could eventually be implemented on a computer. (The authors do not offer implementations of the methods they describe; however, they point to a couple of libraries in the preface of the book.)

Once the concepts related to fuzzy measures have been made known and discussed, the definitions of fuzzy integrals are introduced. The main focus is on Choquet and Sugeno integrals, but paragraphs are devoted to lesser-known fuzzy integrals, which are not further investigated (but this is not a weakness: the book is on fuzzy measures and the intended readership could be more interested in tightening their knowledge on more consolidated ideas).

An entire (and quite long) chapter is devoted to ordered weighted averages (OWAs) because of their striking importance in decision-making. (OWAs result from the application of the Choquet integral to symmetric fuzzy measures, that is, fuzzy measures that only depend on the cardinality of the evaluated subsets.) Many variants of OWAs are introduced and studied, in some cases by reporting algorithms for efficient computation. As an added value, a common notation is used to describe concepts that come from disparate works and authors, thus favoring a global understanding of the concepts.

In the last part of the book, the problem of efficiency is tackled because an unconstrained formulation of fuzzy measures leads to an exponential complexity, both in time and space (because each single subset of sources must be evaluated). To tame complexity, k-order fuzzy measures are introduced (which does not evaluate subsets of cardinality greater than a parameter k) and main concepts and properties are revisited in terms of this constraint by showing its impact on the complexity of their evaluation. The final chapter takes complexity from a different viewpoint, as it is tackled by learning fuzzy measures from a data-fitting perspective. In essence, the learning problem is translated into linear programming problems with different constraints in accordance to the type of fuzzy measure and fuzzy integral. In this way, all efficient techniques for solving linear programming problems can be applied to learn fuzzy measures from data.

Overall, the book is recommended to graduate-level students, knowledge engineers, and researchers who want to be introduced to the domain of fuzzy measures, with a special focus on their computational aspects such as efficient representation and processing. The clean writing and rigorous setting make even the most complex topics accessible and provide a starting point for more comprehensive studies.

Reviewer:  Corrado Mencar Review #: CR146923
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