Book Reviews

Red Plenty: Industry! Progress! Abundance! – Inside the Fifties' Soviet Dream

Spufford, Francis. 2010. Red Plenty: Industry! Progress! Abundance! – Inside the Fifties' Soviet Dream. Faber and Faber, London. 434 pp, £9.99.

Why review a novel in Interfaces? OK, let me follow that up with a second question. The one fact we all know about linear programming is that it was invented by George Dantzig, right? Wrong. It was first invented by the Soviet mathematical economist Leonid Kantorovich in 1939. (Dantzig independently rediscovered it in 1947 and crucially developed the simplex solution method. Kantorovich and Tjalling Koopmans received the Nobel prize for this work in 1975.)

I have a very personal interest in finding out what Kantorovich was like, because I admire his work and very nearly met him. In 1986, as President of the British OR Society, I proposed that our Silver Medal (the Society's highest honour) should be given in a simultaneous ceremony to both Kantorovich and Dantzig. We did not hear from Kantorovich and therefore gave a medal to Dantzig alone. Later I heard from Kantorovich's son-in-law that our invitation arrived just 10 days before his death. I was pleased to be able to present the medal posthumously to his widow the following year.

What is the connection between these two starter questions? Simply that Kantorovich is the unlikely hero of this remarkable novel. Actually it is not quite a novel and not quite a work of history. Real people populate the pages—not only Leonid Kantorovich but also Nikita Khrushchev; Alexei Kosygin, Khrushchev's eventual joint successor as boss of the USSR, but who was the chairman of the massive Soviet planning and allocation system GOSPLAN at the time of the novel's action; Sergei Lebedev, the pioneer Soviet computer designer; and many others. This Soviet drama also includes characters loosely based on real actors, but with added embellishments, and some purely imaginary but archetypal ones—the middle-ranking economist worried about his career, the factory manager faking the statistics, the fixer who enables the creaking GOSPLAN system to function after a fashion by trading favours, and so on. All are brought vividly to life.

Spufford comes from a literary background, and his previous works (which have won awards such as the UK's “Young Writer of the Year”) have all been nonfiction. He originally set out to write a history of the particular period in the 1950s when Soviet production was booming and Khrushchev and others believed that the USSR could beat the United States at its own game, producing abundance for its citizens. However, he evidently became fascinated by the varied cast of characters and the unlikely dance of bureaucrats and optimisers. So, he has invented events—Kantorovich's ride back from the Plywood Trust on a Leningrad tram in 1938, with his head full of the problem formulation that he would develop into linear programming; the party at the elite establishment at Akademgorodok where Kantorovich shyly but skillfully demonstrates his passion for dancing. Spufford also includes the arguments for and against the attempt to optimise the entire Soviet economy expressed vividly, even lyrically, and often put into the mouth of Kantorovich himself. But, if you worry that this is playing fast and loose with the historical truth, there is an easy remedy. Turn to the notes at the back (yes, this is a novel with 54 pages of notes and 14 pages of bibliography, including Saul Gass on linear programming!) and you will find that Kantorovich did say just that, but maybe not at the precise conference where Spufford has placed it.

How does Kantorovich emerge from this composite portrayal of Soviet society at a crucial turning point? In effect, as a holy innocent. Recognised as a genius by all who dealt with him, he was utterly unassuming. He took extraordinary risks to try and get his ideas and those of his school accepted and implemented. When the Soviet apparatus did not welcome his ideas, what did he do? He petitioned Stalin himself. (He also wrote in like mode to every Soviet leader from Stalin to Andropov.) A more dangerous strategy is hard to imagine. Kantorovich's wife told me that for a time he had to stop all his work related to the Soviet economy because his life was in danger.

Why was linear programming regarded with such distrust by the Soviet establishment? In brief, because of shadow prices. Orthodox Soviet economists and politicians feared that linear programming was introducing the market into the Soviet system by the back door. Kantorovich's attempts to publish an extensive treatise on optimal planning were repeatedly thwarted, and his book, The Best Use of Economic Resources, did not see the light of day until 1959.

I cannot recommend this book too highly—as an introduction to one of the key controversies that shaped our discipline, as an enthralling description of one of our great pioneers, and as a stonking good read. For an account of Kantorovich's nonfictional life, see Gass and Rosenhead (2011).

Jonathan Rosenhead

London School of Economics, London, United Kingdom, [email protected]

Sports Data Mining

Schumaker, Robert P., Osama K. Solieman, Hsinchun Chen. 2010. Sports Data Mining. Springer Science+Business Media, New York. 174 pp. $119.00.

Sports Data Mining spans a variety of approaches and techniques for the rigorous analysis of sports data. Data mining is the process of extracting hidden patterns from data and, as the title suggests, there is interest in applying these techniques to sports.

The authors acknowledge that this is a young discipline; however, they have written a comprehensive introductory book on the subject. It focuses on team sports such as baseball, soccer, cricket, hockey, and American football, and uses greyhound racing as an interesting case study near the end of the book. Potential readers need to be aware that this book does not address endurance sports such as cycling, running, triathlon, and cross-country skiing. Nonetheless, some of the techniques discussed could be applied to these sports. The book also includes little discussion on the physiological aspects of any sport. It is more concerned with cataloging discrete events to track outcomes and behaviours. It is well-structured and incremental in nature and provides a coherent narrative without being too technical. Each chapter ends with a handful of relevant discussion questions.

The book has essentially three themes. The first is motivational in nature. In Chapter 1, the story begins appropriately with the data and compares different approaches to analysis, ranging from gut instinct to the use of mathematics. It introduces tool support through sabermetrics and mentions other applications such as financial markets. The second chapter is more concrete in nature and covers the methodology of data mining, starting with its historical roots in statistics, artificial intelligence, and machine learning. The third chapter is a bit dry; it deals with the sources of data but provides the necessary motivation for understanding metrics and the early tool support.

The second theme is that of techniques. Chapter 4, which delves into the underlying mathematical methods required to analyse baseball, basketball, and American football, is the most theoretical of the book. It introduces various statistical methods and metrics to show how players contribute to a win. Chapter 5 exemplifies this approach through a discussion of tool support for analysis and visual game capture. It is motivated by user-centric data mining, talent scouting, and sports-fraud detection. The subsequent chapter expands on this theme by looking at predictive modelling through statistical simulations and machine learning.

The third theme is based around the latest tool support available. It begins with multimedia and video analysis in Chapter 7 and covers the problem of searching the data to ensure that it is applicable through user or motion-analysis annotations. Chapter 8 consolidates this theme by discussing visualisation tools and repositories of sport data, and Chapter 9 looks at open source data mining tools. Chapters 10 and 11 use the example of greyhound racing to show how various machine learning techniques can be applied to a given scenario and lead to different predications. These are excellent demonstrations of the material illustrated by the second theme.

Overall, this is an excellent introduction to the field. It is reasonably broad and comprehensive without being too detailed. It would make an excellent undergraduate textbook in sports science. Although a mathematician or computer scientist might be left wanting, the bibliography is extensive and up-to-date. It is also accessible to those who do not have a sophisticated mathematical background but have an interest in game analysis and desire a formal presentation. My only criticism is that the chapter on visualisation is not very adventurous. It discusses state-of-the-art tool support; however, it does not allude to what could be provided with immersive visual environments and online data support. With ever more digital content available online and in clouds, coupled with access to high-performance grid clusters, the scope for analysing data will depend on techniques for presenting the content, both analytical and visual. I would have liked this book to include some discussion of this emerging issue. Nonetheless, it includes plenty of fascinating material, expertly presented to entertain anyone with a passing interest in sports analysis.

Steve McKeever

Department of Computing Science, Oxford University, Oxford, United Kingdom, [email protected]

Foundations of Optimization

Güler, Osman. 2010. Foundations of Optimization. Springer Science+Business Media, New York. 457 pp. $74.95.

Foundations of Optimization builds the theory of continuous optimization from the ground up. It covers the most important aspects of the theory of optimization of smooth functions in finite-dimensional spaces. The book tries to be as self-contained as possible and is quite successful at meeting this goal. Starting with a proof of Taylor's theorem, it proceeds to the classical optimality conditions for nonlinear programs, the necessary notions from convex analysis, and duality theory and semi-infinite programming. The author carefully proves all such results with an exposition that flows naturally from one topic to the next. As such, the book is oriented toward graduate-level students (or possibly advanced undergraduates) who are interested in the theory of optimization. It will also appeal to researchers who are searching for a concise treatment of the topic, and possibly to practitioners who are interested in the background of the methods they apply or an overview of the subject area. However, the book's focus is not on applications or algorithms; of 14 chapters, only the last is devoted to algorithms (the steepest descent method, Newton's method, and conjugate gradient methods); even in that chapter, the text is devoid of practical hints or recommendations. Although a potential reader must be aware of this, it is not a disadvantage.

As mentioned above, the book starts with a thorough treatment of optimality conditions for unconstrained optimization and implicit function theorems, and proceeds to variational principles, especially Ekeland's variational principle, which provide a natural strategy for proofing various theorems of alternatives. It then uses these principles in the usual fashion to provide optimality conditions in constrained optimization, paving the way to (Lagrangian) duality theory. In between discussion of these topics, it develops the necessary theory of convex analysis, properties of convex sets and functions, convex polyhedral sets and cones, and separation results for convex sets. Each chapter closes with a list of problems; some are numerical examples illustrating the theory covered; others provide interesting theoretical results and special cases. Unfortunately, the book includes no solution hints. Throughout it, the theory developed is accompanied by various applications of the theory (e.g., the theorems of Banach, Grave, Lyusternik, and Courant-Fischer), thus showing the wealth and breadth of an optimization-based approach.

When writing a book on a lively research topic such as optimization, deciding what to include and not include is difficult. Naturally, the material in the final chapters appears to have a certain air of arbitrariness about it—and the same can be said about the material not included. For example, the book does not discuss polyhedral combinatorics or discrete optimization problems, although Chapters 7 and 8 give the underlying theory. Likewise, Chapter 12 covers some basic results from semi-infinite programming, an important but specialized subject. It would have been interesting to also include, for example, some results from the rich theory of multiobjective programming. Although conjugate, the book discusses gradient methods in some depth; however, it does not cover algorithms for constrained optimization. Moreover, several examples discuss topics (e.g., derivatives of matrix-valued functions) that are fine for experts and connoisseurs of optimization but might be taxing for a student who has only recently begun to learn the subject matter. Overall, these are trifling matters. Finally, the author treads a fine line between the treatment of finite-dimensional and infinite-dimensional spaces. Some results from the finite-dimensional case are easily generalized to infinite-dimensional spaces; others are not. The reader can easily get bogged down in technical details. The author appears to have chosen to include more general results provided that the exposition did not suffer. This worked well.

I recommend this book to the audience mentioned above: graduate students, researchers and practitioners interested in the topic, and possibly advanced undergraduate students.

Joerg Fliege

School of Mathematics, The University of Southampton, Southampton, United Kingdom, [email protected]

Stochastic Processes with Applications to Reliability Theory

Nakagawa, Toshio. 2011. Stochastic Processes with Applications to Reliability Theory. Springer, New York. 251 pp. $129.00.

As Nakagawa explains, the goal of Stochastic Processes with Applications to Reliability Theory is to offer an easy-to-read primer on stochastic processes for those interested in learning reliability theory. The preface claims that “the reader could learn both stochastic process and reliability theory from this book at the same time” (p. vi). The result is a reasonably thorough, nonmeasure, theoretic, yet still rigorous, treatment of stochastic processes in which all the examples and applications are reliability related. The author describes the target audience as (presumably graduate) students majoring in reliability and reliability engineers and managers in the field. The text does not offer new material, but rather a novel repackaging of parts of well-known graduate-level textbooks on stochastic processes and the author's own monographs on reliability and maintainability.

The integrated approach is unique compared to existing texts on stochastic processes, which typically draw on only a limited number of examples from reliability applications and existing texts on reliability, which usually assume some prior knowledge of stochastic processes and review these topics as needed. The target audience of this text, however, might be somewhat narrow given that most courses in stochastic processes are taught to students with interests in a broad variety of application areas, and most courses in reliability theory are taught to students who have already taken a course in stochastic processes. Although the concept of combining these two topics in one text will appeal to some, its execution might limit its usefulness. Overall, the text is not particularly well-written. Although the mathematics are clear, the prose tends at times to be choppy and difficult to read, distracting the reader from the main ideas. For example, Subsection 1.2 describes “stochastic processes as examples of reliability systems with maintenance” (p. 2), whereas most people in the field would describe the latter as an example of the former.

Unlike most texts on stochastic processes, Stochastic Processes with Applications to Reliability Theory does not begin with background material on probability and random variables. Such a chapter would have been a welcome addition and avoided some unnatural placement of key concepts. For example, fundamental definitions of failure-rate functions and cumulative hazard functions appear midway through Chapter 2, and Poisson processes are in a subsection on nonhomogeneous Poisson processes. Similarly, general definitions for generic stochastic processes are located within Subsection 2.2, Poisson Process. Instead, the first chapter presents a high-level overview of reliability models and stochastic processes, the latter of which previews the subsequent chapters. The terse overview of reliability models seems incomplete (e.g., the subsection on maintenance does not mention condition-based maintenance), whereas the overview of stochastic processes seems overly detailed and too technical for a reader who is not familiar with the topic.

Past the introduction, however, the flow of the text greatly improves as Nakagawa presents chapters on classic concepts including Poisson processes, renewal theory, Markov processes, Brownian motion, and Levy processes. The text makes good use of insightful examples. However, the downside to including many examples is that key results are sometimes buried within an example. For example, memorylessness of the exponential distribution is presented entirely within Example 2.3.

Each chapter includes a useful reference list to key books and journal publications on the chapter's specific topic. Most chapters also include a reasonable number of exercises (with the exception of Chapters 4 and 5 on Markov chains and semi-Markov and Markov renewal processes, respectively, which offer only 11 problems). The problems are almost entirely theoretical, although some do require computational work. Answers to selected problems are provided, as are references that contain the derivations of these answers.

The majority of the chapters provide a reasonably thorough treatment of their respective topics. The exception is the continuous-time Markov chain material in Chapter 4; this material is limited to birth-death processes and does not cover the important concept of uniformization (although embedded chains are discussed in the following chapter on semi-Markov processes). Chapter 6 focuses solely on shock models and presents interesting material not typically included in standard texts on stochastic processes. The final chapter, Redundant Systems, acts as a capstone to the text. This chapter takes a very broad definition of redundant systems and begins by considering an exponentially failing component that is replaced instantaneously upon failure. The chapter then proceeds to reiterate some material presented in the previous chapters by gradually adding complexity to the system of interest.

In summary, although this combination of stochastic processes and reliability theory is successful, it could be improved by more polishing, some reorganization, and some additions. The obvious topical omission is queueing models, which are only mentioned in passing. It seems difficult to imagine a course on stochastic processes that does not expose students to even the most basic queueing models. As a faculty member, I think the text is too narrowly focused to use in a first-year graduate stochastic processes course. However, it could be useful to individuals seeking a graduate degree with a focus on reliability and maintainability.

Lisa M. Maillart

Department of Industrial Engineering, Swanson School of Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, [email protected]

Data Mining and Knowledge Discovery via Logic-Based Methods

Triantaphyllou, Evangelos. 2010. Data Mining and Knowledge Discovery via Logic-Based Methods. Springer Science+Business Media, New York. 348 pp. $124.00.

In recent years, many monographs and textbooks with titles that include the words “data mining” and “knowledge discovery” have been published. It would seem that this sector of the book market should be saturated. Even a potential reader who is superficially knowledgeable about these subjects can predict the contents of such books. However, the title of the book being reviewed also has two words that are unusual in this context—“logic-based.” Indeed, the contents of Data Mining and Knowledge Discovery via Logic-Based Methods differ from what we would expect based on the first part of the title.

This book considers a theoretical problem—the dichotomy of a set of Boolean vectors. Its aim is to infer a Boolean function with the values “1” for the items of one subset and the values “0” for the values of the other subset by using examples of both subsets as the training data. In addition to the theoretical and algorithmic components in the solution of the dichotomy problem, it includes a discussion of the reducibility of different data mining and knowledge discovery problems to a problem that addresses the dichotomy of a set of Boolean vectors. It also presents several examples of applications of the proposed approach to related problems in the real world.

The book has two parts: Algorithmic Issues and Applications Issues. Algorithmic Issues consists of eight chapters. The first chapter provides a general introduction to the problems of data mining and knowledge discovery; the other chapters in this part discuss the inference of Boolean functions from positive and negative examples. Because the majority of applied (i.e., classification) problems are related to continuous data, to apply the logic-based methods to such problems, the data should be converted into Boolean vectors. This conversion is briefly discussed in the second chapter, the main focus of which is the inference of a Boolean function from positive and negative examples (i.e., from the examples that belong to two subsets that are supposed to separate them using the inferred function). Although the construction of a conjunctive normal form (CNF) with the value “0” for all training examples of the negative subset is a trivial task, such a construction is insufficient because it results in formulae that are too long. It also lacks the generalization property necessary for the supposed classification of items not belonging to the training set. This chapter also describes sophisticated methods to construct short expressions (e.g., “one clause at the time” and a version of the branch-and-bound method). The other chapters of Part I address various improvements of the initial version of the branch-and-bound algorithm and the development of various heuristic methods. The performance of the methods considered is assessed using experimental testing.

The second part of the book addresses the general problems of applying data mining, with a focus on applications of the methods that the author and his collaborators developed. This part has nine chapters and is approximately half the book's volume. Two chapters discuss the general problems of data mining: reliability issues and fuzzy logic application to transform fuzzy (especially medical) data into numerical (Boolean) attributes. Two chapters deal with the issues of applications of the monotone Boolean functions. Four chapters describe particular applied problems. The data related to these problems are used in the experimental comparison of the methods described in the book with those based on other approaches. The final chapter contains conclusions.

From the point of view of applications, reliability is one of the most important characteristics of data mining methods. The reliability problem consists not only of the methods. Potential reliability depends a great deal on the representativeness of the sample of items available for training (the training data). The author suggests analyzing the reliability of data mining methods taking into account the ratio S/N (index of potential reliability) and the ratio S/P (i.e., index of actual reliability), where N denotes the number of potential items (i.e., the size of the state space), P denotes the population size (i.e., number of the items of interest), and S denotes the size of training data. A method can be reliable (with respect to the index of actual reliability) only in the case of the narrow vicinity hypothesis: “All real possible cases are in a narrow vicinity of an assessable small training sample” (p. 179). The author claims that “The main advantage of the methods which we used is that they allow one to identify and evaluate the reliability of CAD methods” (p.184) (note that CAD is an acronym for “computer-aided diagnostic”).

The author claims that the class of general Boolean functions that Part I considers is excessively complex with respect to many applications. The subclass of monotone Boolean functions is inherently adequate to many applications. The monotonicity is a formalization of the property illustrated by the following example: “with all the other factors constant, a student with a high grade point average (GPA) is more likely to be accepted into a particular college than a student with a low GPA” (p. 191). The subclass of monotone Boolean functions is advantageous because of the relative simplicity of algorithms for their inference from examples; two chapters analyze those functions to facilitate the development of the practically applicable inference algorithms.

The mining of several data sets, which are related to real-world problems, is used to illustrate the properties of the methods considered. The first applied problem considered is the mining of association rules, which aims to find the correlation between the items in a customer's shopping cart. To apply the developed algorithms, the association rules are modeled by the clauses of disjunctive normal form (DNF) expressions. A version of the logic-based method that the author describes is compared with two other algorithms using different approaches. The second problem, related to the data mining of text documents, is formulated as a dichotomy in which the methods presented in the book are applicable without any preprocessing. The experimental comparison of efficiency of the developed methods with a standard vector space method was performed on a small set of text documents. The two last data sets relate to medical diagnosis of breast cancer statistics and to localization of muscle fatigue. Many data mining experts analyze the former set of data, which Wingo et al. (1995) describe. Waly et al. (1997) had used the latter set of data earlier in a research project. To apply the logic-based methods to the mining of the considered medical data, preprocessing was needed to transform the original data into binary attributes. The experimental results in these four chapters demonstrate advantages of the logic-based methods, as the book describes, over their competitors. The author is critical about the application of the known data mining methods in CAD: “Strictly speaking, all CAD/data mining methods are still very unreliable in spite of the apparent, and possibly fortuitous, high accuracy of cancer diagnosis reported in the literature” (p. 183). His assessment of the proposed combination of the developed logic-based methods with fuzzy logic is different: “The proposed fuzzy logic approach has the potential to open a new and very exciting direction for effective and early breast cancer diagnosis and, in general, in data mining and knowledge discovery research and applications” (p. 308). The results of testing with the real-world data corroborate such an optimistic assessment; however, the corroboration would be much stronger if references (even preliminary) were given to the real application of those methods in clinical practice or to other real applications.

Every method and algorithm has some favorable application areas and some limitations. In this book, the limitations of the methods discussed are not explicitly formulated. Some are obvious; for example, logic-based methods can hardly substitute for visualization methods in knowledge discovery problems similar to those that Zilinskas et al. (2006) consider. A clear definition of the limitations of the described methods would be helpful to the reader and would not spoil the optimistic prediction by the author: “Another reason to support the argument that logic-based methods have a bright future comes from the huge literature currently available on such methods for other applications” (p. 10).

From the preface, we see that the author expects a very broad audience for this book: “This book can be used as a textbook for senior undergraduate or graduate courses in data mining in engineering, computer science, and business schools; it can also provide a panoramic and systematic exposure of related methods and problems to researchers. Finally, it can become a valuable guide for practitioners who wish to take a more effective and critical approach to the solution of real-life data mining and knowledge discovery problems” (p. xiii). Such a pretension can be considered as a marketing cliché. Nevertheless, it seems reasonable to include this book in a library because it complements the other recent and popular books on data mining, including Berry (2004), Cios et al. (2010), Han et al. (2011), and Witten et al. (2011), which do not discuss the same methods addressed in this book.

Antanas Zilinskas

Department of Mathematics and Informatics, Vilnius University, Vilnius, Lithuania, [email protected]

Random-Like Multiple Objective Decision Making

Xu, Jiuping, Liming Yao. 2011. Random-Like Multiple Objective Decision Making. Springer Science+Business Media, New York. 456 pp. $139.00.

Consider the problem of deciding what proportion of one's budget to allocate to each of a number of prospective investments to maximise profit under some allocation constraints. If returns are fixed and known, the problem is a textbook linear programming problem. If some returns are uncertain, a next step might be to define probability distributions by describing their possible outcomes, assess parameters for these distributions, and then use any number of stochastic programming methods to frame and solve the problem.

What if the parameters of the probability distributions are uncertain? This is the essence of the problem addressed by Jiuping Xu and Liming Yao's book, published as a volume of the Springer Lecture Notes in Economics and Mathematical Systems. Their solution is to provide a framework in which second-order uncertainty about the parameters of probability distributions can be described using one of three uncertainty languages: further probability distributions, fuzzy sets, or rough sets. A variable described by a probability distribution whose parameters are uncertain is known as a two-fold uncertainty variable—more specifically as a random-random (Ra-Ra), random-fuzzy (Ra-Fu), or random-rough (Ra-Ro) variable. This book thoroughly covers these well-established developments. The authors' contribution is to comprehensively review the application of two-fold uncertainty variables to multiobjective linear and nonlinear programming. The result is a reference work that is broad in scope and captures much of the flavour of this relatively new area of research.

The book proceeds systematically, with each of three main chapters considering a different two-fold uncertainty variable. In each chapter, three models are developed to treat the uncertainty: expected value, chance-constrained, and dependent-chance models, although the authors note that other approaches are possible. Aggregation over multiple objectives is handled using 12 models; standard approaches such as weighted additive, lexicographic, or max-min aggregation are included among more esoteric methods such as the surrogate worth trade-off and maximin point methods. Analytical results are presented for more tractable problems (e.g., where objective functions and constraints are linear and distributions are normal). Nonlinear problems are treated using a combination of Monte Carlo simulation and heuristic optimization. Each chapter introduces a different heuristic optimization approach—simulated annealing, particle swarm optimization, genetic algorithms, and tabu search—and different variants of each are presented for each multiobjective model. The constant changing of solution method can at times be overwhelming; however, the reward is a wide survey of the current state-of-the-art methods. Short examples are scattered throughout the book, and each chapter concludes with a more substantial case study.

The heart of the book is contained in the three chapters that cover multiobjective programming with two-fold uncertainty. Expected-value models are generally easily implemented because expected values can be calculated in a piecewise fashion, for example, the expected value of a Ra-Ra variable $\phi \sim N(\mu ,{\sigma }^2)$ with $\mu \sim U\left[1,3\right]$ is $E\left[E\left[\phi \right]\right]=E\left[\mu \right]=2$. For chance-constrained models, the user follows the usual chance-constrained approach by specifying a probability giving the desired percentile to be maximized; the problem is that this percentile is itself random. This requires that the user defines a second cut-off indicating a desire to achieve the random percentile. Formally, this can be quite easily stated: the key constraint for the Ra-Ra case becomes $\text{Pr}\{\omega |\text{\hspace{0.17em}Pr}(\phi \ge f)\ge \beta \}\ge \alpha$, where ω is the probability space of the Ra-Ra variable φ representing the objective function, f is the quantity to be maximized, and β and α are user-specified confidence limits. However, after some effort I am still unable to convey this in a way that I think is satisfactory for general decision aid; herein lies the biggest (and, I think, unresolved) challenge for the field. The authors refer to a “confidence level α at which it is desired that the stochastic constraints hold” (p. 145); this is true but too brief for the nonspecialist. My effort—“the level α such that there is at least a 100 α% chance that the probability of exceeding f is greater than β”—would probably confuse more than enlighten. At the minimum, this suggests a crucial role for post hoc sensitivity analysis.

Multiobjective Ra-Fu or Ra-Ro models (using fuzzy or rough sets, respectively) require that one define fuzzy or rough analogues of expectation and probability; however, once this has been done results follow in much the same way as for probability distributions. For example, the key constraint in the chance-constrained Ra-Ra model remains unchanged in the fuzzy and rough models except that one talks of possibility and approximation rather than probability. Xu and Yao clearly and systematically develop all these results. However, similarities in the mathematical results can be misleading. Practically, the assessment of probability distributions, fuzzy sets, and rough sets are quite different, as is the way in which decision makers are likely to interpret them. This leads to my one criticism, which I touch on above, of an otherwise impressive book: the lack of detailed discussion around practical implementation and softer issues (e.g., interpretation of parameters, ease-of-use, opportunities for learning). Methodologically, the book's approach is typical of the field, offering an impressive and comprehensive set of mathematical properties for each uncertainty variable and associated mathematical program. It certainly gives enough detail to be of use to the practitioner. Yet the book, and I think this would apply to much of this branch of uncertainty modelling, would have benefited from a more critical self-assessment of its contribution by debating concerns that researchers unconverted to the fuzzy or rough paradigms might have about ambiguous parameter interpretation and practical assessment difficulties. For many practically minded researchers, the key question left at the end of the book will be whether models explicitly including second-order “uncertainty about uncertainty” really result in better decisions, or whether the increased complexity outweighs any benefit and the alternative—resolution of uncertainty via discussion and a return to problem structuring phases—might be preferred. Although this book does not provide answers to these particular questions, it raises them for others to do so. For this, and for bringing together a wide range of related developments in uncertainty modelling, Xu and Yao have done the field a considerable service.

Ian N. Durbach

Department of Statistical Sciences, University of Cape Town, Cape Town, South Africa, [email protected]