Our world has become a system of networks. It is not astonishing that analysis of such complex structures is necessary in fields such as communications and computer technology. One important aspect of such an analysis is considering what is a relative share or the criticality of a selected element in the overall structural performance. When we look at it from the reliability perspective, we might want to know the overall sensitivity of the structure to changes or faults in an element. This can be theoretically treated by the general notion of importance measures, which is the topic of this book by Kuo and Zhu. The former co-authored a well-received textbook on optimization problems related to reliability [1]; this new book can be seen as an extension of it.

The title of the book refers to reliability, risk, and optimization, but all should be especially connected to the theoretical treatment of reliability. Some suggestions on how to use the presented methods in practice are given here and there. The book consists of five parts and a total of 19 chapters, plus a theoretical appendix with a few proofs. Part 1 introduces the basics of the topic in two chapters, showing the significance of reliability studies and presenting main results on the modeling of structural reliability. The next six chapters, forming Part 2, focus on various aspects of the theory of importance measures, such as fundamental definitions and the most frequently used measures of various kinds, with their properties. The next part takes up optimization issues, with five chapters covering interesting aspects such as redundancy allocation, upgrading system performance, and component assignment in coherent and k-out-of-n systems, along with some heuristics basing the design on the importance measure usage. Part 4 presents more advanced material in two chapters. Various importance measures are briefly compared from the viewpoint of various relations, and a general theory of such measures is formulated. The last part presents four fields related to the potential use of importance measures. Here, the book shows a real engineering context, especially in the last chapter, which describes applications in nuclear plants. The other three, while they can be used by practitioners, are quite theoretical and general, including applications to network flows, mathematical programming (mainly with linear constraints), and general sensitivity analysis.

As far as I know, this is the first (recent) book that treats importance measure problems. The book is quite demanding, and is recommended to researchers who are well acquainted with probability and statistics, and doctoral students specializing in reliability theory. For such readers, the work will be profitable, as it elaborates on those difficult problems as simply as possible, without sacrificing the necessary formalism. It will definitely be very useful for those interested in studying various structures. If another edition is planned, I hope the authors will go one step further and focus more on practical applications in various fields.