Theory and Decision Library B – serie

Measuring productive efficiency for nonprofit organizations has posed a great challenge to applied researchers today. The problem has many facets and diverse implications for a number of disciplines such as economics, applied statistics, management science and information theory. This monograph discusses four major areas, which emphasize the applied economic and econometric as. pects of the production frontier analysis: A. Stochastic frontier theory, B. Data envelopment analysis, C. Clustering and estimation theory, D. Economic and managerial applications Besides containing an up-to-date survey of the mos. t recent developments in the field, the monograph presents several new results and theorems from my own research. These include but are not limited to the following: (1) interface with parametric theory, (2) minimax and robust concepts of production frontier, (3) game-theoretic extension of the Farrell and Johansen models, (4) optimal clustering techniques for data envelopment analysis and (5) the dynamic and stochastic generalizations of the efficiency frontier at the micro and macro levels. In my research work in this field I have received great support and inspiration from Professor Abraham Charnes of the University of Texas at Austin, who has basically founded the technique of data envelopment analysis, developed it and is still expanding it. My interactions with him have been most fruitful and productive. I am deeply grateful to him. Finally, I must record my deep appreciation to my wife and two children for their loving and enduring support. But for their support this work would not have been completed.

This book presents the content of a year's course in decision processes for third and fourth year students given at the University of Toronto. A principal theme of the book is the relationship between normative and descriptive decision theory. The distinction between the two approaches is not clear to everyone, yet it is of great importance. Normative decision theory addresses itself to the question of how people ought to make decisions in various types of situations, if they wish to be regarded (or to regard themselves) as 'rational'. Descriptive decision theory purports to describe how people actually make decisions in a variety of situations. Normative decision theory is much more formalized than descriptive theory. Especially in its advanced branches, normative theory makes use of mathematicallanguage, mode of discourse, and concepts. For this reason, the definitions of terms encountered in normative decision theory are precise, and its deductions are rigorous. Like the terms and assertions of other branches of mathematics, those of mathematically formalized decision theory need not refer to anything in the 'real', i. e. the observable, world. The terms and assertions can be interpreted in the context of models of real li fe situations, but the verisimilitude of the models is not important. They are meant to capture only the essentials of adecision situation, which in reallife may be obscured by complex details and ambiguities. It is these details and ambiguities, however, that may be crucial in determining the outcomes of the decisions.

Decision making is certainly a very crucial component of many human activities. It is, therefore, not surprising that models of decisions play a very important role not only in decision theory but also in areas such as operations Research, Management science, social Psychology etc . . The basic model of a decision in classical normative decision theory has very little in common with real decision making: It portrays a decision as a clear-cut act of choice, performed by one individual decision maker and in which states of nature, possible actions, results and preferences are well and crisply defined. The only compo­ nent in which uncertainty is permitted is the occurence of the different states of nature, for which probabilistic descriptions are allowed. These probabilities are generally assumed to be known numerically, i. e. as single probabili­ ties or as probability distribution functions. Extensions of this basic model can primarily be conceived in three directions: 1. Rather than a single decision maker there are several decision makers involved. This has lead to the areas of game theory, team theory and group decision theory. 2. The preference or utility function is not single valued but rather vector valued. This extension is considered in multiattribute utility theory and in multicritieria analysis. 3.

In this volume we present some of the papers delivered at FUR-IV - the Fourth International Conference on Founda­ tions and Applications of Utility, Risk and Decision Theory in Budapest, June 1988. The FUR Conferences have provided an appreciated forum every two years since 1982 within which scientists can report recent issues and prospective applications of decision theory, and exchange ideas about controversial questions of this field. Focal points of the presented papers are: expected utility versus alterna­ tive utility models, concepts of risk and uncertainty, developments of game theory, and investigations of real decision making behaviour under uncertainty and/or in risky situations. We hope that this sample of papers will appeal to a wide spectrum of readers who are interested in and fami­ liar with this interesting and exciting issues of decision theory. A wide range of theoretical and practical questions is considered in papers included in this volume, and many of them closely related to economics. In fact, there were two Nobel-Laureates in economics among the participants: I. Herbert A. Simon (1978) and Maurice Allais (1988), who won the prize just after the conference. His paper deals with problems of cardinal utility. After a concise overview of the history and theory of cardinal utility he gives an estimate of the invariant cardinal utility function for its whole domain of variation (i. e.

This book has a fundamental relationship to the International Seminar on Fuzzy Set Theory held each September in Linz, Austria. First, this volume is an extended account of the eleventh Seminar of 1989. Second, and more importantly, it is the culmination of the tradition of the preceding ten Seminars. The purpose of the Linz Seminar, since its inception, was and is to foster the development of the mathematical aspects of fuzzy sets. In the earlier years, this was accomplished by bringing together for a week small grou ps of mathematicians in various fields in an intimate, focused environment which promoted much informal, critical discussion in addition to formal presentations. Beginning with the tenth Seminar, the intimate setting was retained, but each Seminar narrowed in theme; and participation was broadened to include both younger scholars within, and established mathematicians outside, the mathematical mainstream of fuzzy sets theory. Most of the material of this book was developed over the years in close association with the Seminar or influenced by what transpired at Linz. For much of the content, it played a crucial role in either stimulating this material or in providing feedback and the necessary screening of ideas. Thus we may fairly say that the book, and the eleventh Seminar to which it is directly related, are in many respects a culmination of the previous Seminars.

Fuzzy data such as marks, scores, verbal evaluations, imprecise observations, experts' opinions and grey tone pictures, are common. In "Fuzzy Data Analysis" the authors collect their results, aiming to provide the reader with ideas, approaches and methods for processing such data when looking for sub-structures in knowledge basis for an evaluation of functional relationship, for example in order to specify diagnostic of control systems. The modelling presented uses ideas from fuzzy set theory and the suggested methods solve problems usually tackled by data analysis if the data are real numbers. "Fuzzy Data Analysis" is self-contained and is addressed to mathematicians oriented towards applications and to practitioners in any field of application who have some background in mathematics and statistics.

This volume collects the best work representative of the theory of decision making under risk and uncertainty. Thirteen articles present extensions and alternatives to expected utility theory explaining behavior in and out of the laboratory, including activity in financial markets, crimes of passion, and the Ellsberg paradox. These contributions on the theory of individual behavior are complemented by five studies reporting the design, execution, and implications of experiments that test alternative theories. Finally, the implications of the interaction of individuals under risk and uncertainty in markets and negotiation are explored in ten papers. Throughout, the approach taken reflects the multidisciplinary nature of the research. The contributors include economists, psychologists, statisticians, and management scientists, each writing to make their work critically comprehensible to the wider audience.

"Fundamentals of Convex Analysis" offers an in-depth look at some of the fundamental themes covered within an area of mathematical analysis called convex analysis. In particular, it explores the topics of duality, separation, representation, and resolution. The work is intended for students of economics, management science, engineering, and mathematics who need exposure to the mathematical foundations of matrix games, optimization, and general equilibrium analysis. It is written at the advanced undergraduate to beginning graduate level and the only formal preparation required is some familiarity with set operations and with linear algebra and matrix theory. Fundamentals of Convex Analysis is self-contained in that a brief review of the essentials of these tool areas is provided in Chapter 1. Chapter exercises are also provided. Topics covered include: convex sets and their properties; separation and support theorems; theorems of the alternative; convex cones, dual homogeneous systems; basic solutions and complementary slackness: extreme points and directions; resolution and representation of polyhedra; simplicial topology; and fixed point theorems.A strength of this work is how these topics are developed in a fully integrated fashion.

Econometrics as an applied discipline attempts to use information in a most efficient manner, yet the information theory and entropy approach developed by Shannon and others has not played much of a role in applied econometrics. "Econometrics of Information and Efficiency" aims to bridge the gap. Broadly viewed, information theory analyzes the uncertainty of a given set of data and its probabilistic characteristics. Whereas the economic theory of information emphasizes the value of information to agents in a market, the entropy theory stresses the various aspects of imprecision of data and their interactions with the subjective decision processes. The tools of information theory, such as the maximum entropy principle, mutual information and the minimum discrepancy are useful in several areas of statistical inference, for instance Bayesian estimation, expected maximum likelihood principle, and the fuzzy statistical regression. This volume analyzes the applications of these tools of information theory to the most commonly used models in econometrics.The features of the text include: a critical survey of the uses of information theory in economics and econometrics; an integration of applied information theory and economic efficiency analysis; and the development of a new economic hypotheses relating information theory to economic growth models. New lines of research are emphasized.

This collection of essays is written in honour of Maurice Allais. It focuses on considering his economic theories. Chapters include a theory of spatial general equilibrium in a fuzzy economy and some possible uses of household assets in the national economy with reference to American households.

"Non-Additive Measure and Integral" takes a systematic approach to the subject. Much of the additive theory (convergence theorems, Lebesgue spaces, representation theorems) is generalized, at least for submodular measures which are characterized by having a subadditive integral. The theory is of interest for applications to economic decision theory (decisions under risk and uncertainty), to statistics (including belief functions, fuzzy measures) and to co-operative game theory, artificial intelligence, insurance, and so on. "Non-Additive Measure and Integral" collects the results of scattered and often isolated approaches to non-additive measures and their integrals which originate in pure mathematics, potential theory, statistics, game theory, economic decision theory and other fields of application. It unifies, simplifies and generalizes known results and supplements the theory with new results, thus providing a sound basis for applications and further research in this growing field. It also contains fundamental results of s-additive and finitely additive measure and integration theory, and sheds new light on additive theory."Non-Additive Measure and Integral" employs distribution functions and quantile functions as basis tools, thus remaining close to the familiar language of probability theory. In addition to serving as a reference text, the book can be used as a mathematics textbook for graduate courses or seminars, containing many exercises to support or supplement the text.

The limit distribution of the empirical generalized variance, covariance matrices, their eigenvalues and solutions of the system of linear algebraic equations with random coefficients, are an important function of observations in multidimensional statistical analysis. This book is devoted to their investigation. A general statistical analysis is developed in which observed random vectors may not have density and their components have an arbitrary dependence structure. The methods of this theory have important advantages over existing methods of statistical processing. The results have applications in nuclear and statistical physics, multivariate statistical analysis in the theory of the stability of solutions of stochastic differential equations, control theory of linear stochastic systems, linear stochastic programming, and the theory of experiment planning.

This text presents original contributions to the areas of individual choice, experimental economics, operations and analysis, multiple-criteria decision-making, market uncertainty, game theory and social choice. The papers, which were presented at the FUR VI conference, are arranged to appear in order of increasing complexity of the decision environment or social context in which they situate themselves. The first section, "Psychological Aspects of Risk-Bearing", considers choice at the purely individual level and, for the most part, free of any specific economic or social context. The second section examines individual choice within the classical expected utility approach, while the third section works from a perspective that includes non-expected utility preferences over lotteries. Section Four, "Multiple Criteria Decision-Making Under Uncertainty", considers the more specialized, but crucial, context of uncertain choice involving tradeoffs between competing criteria - a field which is becoming of increasing importance in applied decision analysis. The final two sections examine uncertain choice in social or group contexts.

This decade has witnessed increasing interest in fuzzy technology both from academia and industry. It is often said that fuzzy theory is easy and simple so that engineers can progress quickly to real applications. However, the lack of knowledge of design methodologies and the theoretical results of fuzzy theory have often caused problems for design engineers. The aim of this book is to provide a rigorous background for uncertainty calculi, with an emphasis on fuzziness. "Fundamentals of Uncertainty Calculi with Applications to Fuzzy Inference" is primarily about the type of knowledge expressed in a natural language, that is, in linguistic terms. The approach to modelling such knowledge is based upon the mathematical theory of uncertainty related to the fuzzy measures and integrals and their applications. The book consists of two parts: Chapters 2-6 comprise the theory, and applications are offered in Chapters 7-10. In the theory section the exposition is mathematical in nature and gives a complete background on uncertainty measures and integrals, especially in a fuzzy setting.Applications concern recent ones of fuzzy measures and integrals to problems such as pattern recognition, decision making and subjective multicriteria evaluations.

This volume is an introduction to stochastic processes and their statistics. Basic stochastic processes are developed from real world situations to the need for generating mathematical models, while at the same time students learn to apply theoretical models. The lessons cover basic stochastic processes such as Poisson processes, Markov chains, random walks, renewal theory, queuing theory, ARMA models, martingales, Brownian motion and diffusion processes. The statistical topics treated include the basic aspects of statistics of point processes, stationary processes and diffusion processes.

The 1980s and 1990s have been a period of major developments in the modelling of decision-making under risk and uncertainty. Extensions of the theory of expected utility and alternative theories of "non-expected utility" have been devised to explain many puzzles and paradoxes of individual and collective choice behaviour. This volume presents some recent work on the modelling of risk and uncertainty, with applications to problems in environmental policy, public health, economics and finance. Eighteen papers by economists, management scientists, and statisticians aims to shed new light on phenomena such as the Allais and St Petersburg paradoxes, the equity premium puzzle, the demand for insurance, the valuation of public health and safety, and environmental goods. This work should be of interest to economists, management scientists, risk and policy analysts, and others who study risky decision-making in economic and environmental contexts.

Semiorder is probably one of the most frequently ordered structures in science. It naturally appears in fields like psychometrics, economics, decision sciences, linguistics and archaeology. It explicitly takes into account the inevitable imprecisions of scientific instruments by allowing the replacement of precise numbers by intervals. The purpose of this book is to dissect this structure and to study its fundamental properties. The main subjects treated are the numerical representations of semiorders, the generalizations of the concept to valued relations, the aggregation of semiorders, and their basic role in a general theoretical framework for multicriteria decision-aid methods. This volume is intended for students and researchers in the fields of decision analysis, management science, operations research, discrete mathematics, classification, social choice theory, and order theory, as well as for practitioners in the design of decision tools.

Data fusion or information fusion are names which have been primarily assigned to military-oriented problems. In military applications, typical data fusion problems are: multisensor, multitarget detection, object identification, tracking, threat assessment, mission assessment and mission planning, among many others. However, it is clear that the basic concepts underlying such fusion procedures can often be used in nonmilitary applications as well. The purpose of this text is twofold: firstly, to point out present gaps in the way data fusion problems are conceptually treated, and secondly, to address this issue by exhibiting mathematical tools which treat a combination of evidence in the presence of uncertainty in a more systematic and comprehensive way. These techniques are based essentially on two ideas relating to probability theory: the newly developed fields of random set theory and conditional and relational event algebra. The book is intended to be both an update on research progress on data fusion and an introduction to potentially powerful new techniques: fuzzy logic, random set theory, and conditional and relational event algebra.The text may be useful as a reference book for researchers and practitioners in data fusion or expert systems theory, or for graduate students as text for a research seminar or graduate level course.

This volume is intended to be used as a textbook for a special topic course in computer science. It addresses contemporary research topics of interest such as intelligent control, genetic algorithms, neural networks, optimization techniques, expert systems, fractals, and computer vision. The work incorporates many new research ideas, and focuses on the role of continuous mathematics.

Presents a new branch of mathematical statistics aimed at constructing unimprovable methods of multivariate analysis, multi-parametric estimation, and discriminant and regression analysis. In contrast to the traditional consistent Fisher method of statistics, the essentially multivariate technique is based on the decision function approach by A. Wald. Developing this new method for high dimensions, comparable in magnitude with sample size, provides stable approximately unimprovable procedures in some wide classes, depending on an arbitrary function. A fact is established that, for high-dimensional problems, under some weak restrictions on the variable dependence, the standard quality functions of regularized multivariate procedures prove to be independent of distributions. This opens the possibility to construct unimprovable procedures free from distributions.

As the world enters the new millennium, mankind faces a series ofnew prob­ lems, many of them created by man himself. These include overpopulation, air and water pollution, global warming, accumulation of greenhouse gases, darnage to the ozone layer and loss of biodiversity. Perhaps these problems were around even earlier in an incipient stage, but they have now assumed global proportions and are uppermost in the minds of all. A natural con­ sequence is enhanced interest in sciences connected with these problems. Ecology is a field that is immensely useful in understanding many of them. In the seventies, nature conservation became a concern of wide sections of society, well beyond the small group of expert ecologists. Species extinc­ tion and depletion of biological resources were seen as major threats to human welfare. It was therefore natural for scientists from different disci­ plines to seek reasons behind these developments. We were no exceptions and when opportunity to interact with ecologists as statistreal consultants came, we found ourselves reading more and more of ecology and evolution­ ary biology. Several years ago we proposed starting of an elective one semester course on statistical ecology for graduate students of statistics of Pune University.

Aggregation of individual opinions into a social decision is a problem widely observed in everyday life. For centuries people tried to invent the "best" aggregation rule. In 1951, young American scientist and future Nobel Prize winner Kenneth Arrow formulated the problem in an axiomatic way, in that he specified a set of axioms which every reasonable aggregation rule has to satisfy, and obtained that these axioms are inconsistent. This result, often called Arrow's Paradox or General Impossibility Theorem, had become a cornerstone of social choice theory. The main condition used by Arrow was his famous independence of irrelevant alternatives. This very condition pre-defines the "local" treatment of the alternatives (or pairs of alternatives, or sets of alternatives) in aggregation procedures. Remaining within the framework of the axiomatic approach and based on the consideration of local rules, this book investigates three formulations of the aggregation problem according to the form in which the individual opinions about the alternatives are defined, as well as to the form of desired social decision. In other words, the book looks at three aggregation models.What is common between them is that in all models some analogue of the independence of irrelevant alternatives condition is used, which is why these models are called Arrovian aggregation models. The first chapter presents a general description of the problem of axiomatic synthesis of local rules, and introduces problem formulations for various versions of formalization of individual opinions and collective decision. Chapter two formalizes precisely the notion of "rationality" of individual opinions and social decision, while chapter three deals with the aggregation model for the case of individual opinions and social decisions formalized as binary relations. Chapter four deals with Functional Aggregation Rules which transform into a social choice function individual opinions defined as choice functions. Chapter five considers another model, Social Choice Correspondences, when the individual opinions are formalized as binary relations, and the collective decision is looked for as a choice function.

This text mixes a selection of papers, presented at the Eighth Foundations and Applications of Utility and Risk Theory ("FUR VIII") conference in Mons, Belgium, together with a few solicited papers from well-known authors in the field. It addresses some of the questions that have recently emerged in the research on decision-making and risk theory. In particular, authors have modeled more as interactions between the individual and the environment or between different individuals have emerged rapidly. This text analyzes several cases of such an interaction and derives consequences for the future of decision theory and risk theory. In the last ten years, modelling beliefs has become a specific sub-field of decision making, particularly with respect to low probability events. Rational decision making has also been generalized in order to encompass, in new ways and in more general situations than it used to be fitted to, multiple dimensions in consequences. It also addresses the difficult question to incorporate several of these recent advances simultaneously into one single decision model. And it offers perspectives about the future trends of modelling such complex decision questions.The volume is organized in three main blocks: the first block is the more "traditional" one. It deals with new extensions of the existing theory, as is always demanded by scientists in the field; a second block handles specific elements in the development of interactions between individuals and their environment, as defined in the most general sense; the last block confronts real-world problems in both financial and non-financial markets and decisions, and tries to show what kind of contributions can be brought to them by the type of research reported.

This title presents a clear, systematic treatment of convergence theorems of set-valued random variables (random sets) and fuzzy set-valued random variables (random fuzzy sets). Topics such as strong laws of large numbers and central limit theorems, including results in connection with the theory of empirical processes, are covered. The author's own recent developments on martingale convergence theorems and their applications to data processing are also included. The mathematical foundations along with a clear explanation such as H"olmander's embedding theorem, notions of various convergence of sets and fuzzy sets, Aumann integrals, conditional expectations, selection theorems, measurability and integrability arguments for both set-valued and fuzzy set-valued random variables and newly obtained optimizations techniques based on invariant properties are also given.

Case-based reasoning (CBR) has received a great deal of attention in recent years and has established itself as a core methodology in the field of artificial intelligence. The key idea of CBR is to tackle new problems by referring to similar problems that have already been solved in the past. More precisely, CBR proceeds from individual experiences in the form of cases. The generalization beyond these experiences typically relies on a kind of regularity assumption demanding that 'similar problems have similar solutions'.Making use of different frameworks of approximate reasoning and reasoning under uncertainty, notably probabilistic and fuzzy set-based techniques, this book develops formal models of the above inference principle, which is fundamental to CBR. The case-based approximate reasoning methods thus obtained especially emphasize the heuristic nature of case-based inference and aspects of uncertainty in CBR. This way, the book contributes to a solid foundation of CBR which is grounded on formal concepts and techniques from the aforementioned fields. Besides, it establishes interesting relationships between CBR and approximate reasoning, which not only cast new light on existing methods but also enhance the development of novel approaches and hybrid systems.This books is suitable for researchers and practioners in the fields of artifical intelligence, knowledge engineering and knowledge-based systems.

Aggregation of individual opinions into a social decision is a problem widely observed in everyday life. For centuries people tried to invent the 'best' aggregation rule. In 1951 young American scientist and future Nobel Prize winner Kenneth Arrow formulated the problem in an axiomatic way, i.e., he specified a set of axioms which every reasonable aggregation rule has to satisfy, and obtained that these axioms are inconsistent. This result, often called Arrow's Paradox or General Impossibility Theorem, had become a cornerstone of social choice theory. The main condition used by Arrow was his famous Independence of Irrelevant Alternatives. This very condition pre-defines the 'local' treatment of the alternatives (or pairs of alternatives, or sets of alternatives, etc.) in aggregation procedures. Remaining within the framework of the axiomatic approach and based on the consideration of local rules, Arrovian Aggregation Models investigates three formulations of the aggregation problem according to the form in which the individual opinions about the alternatives are defined, as well as to the form of desired social decision.In other words, we study three aggregation models. What is common between them is that in all models some analogue of the Independence of Irrelevant Alternatives condition is used, which is why we call these models Arrovian aggregation models. Chapter 1 presents a general description of the problem of axiomatic synthesis of local rules, and introduces problem formulations for various versions of formalization of individual opinions and collective decision. Chapter 2 formalizes precisely the notion of 'rationality' of individual opinions and social decision. Chapter 3 deals with the aggregation model for the case of individual opinions and social decisions formalized as binary relations. Chapter 4 deals with Functional Aggregation Rules which transform into a social choice function individual opinions defined as choice functions. Chapter 5 considers another model -- Social Choice Correspondences when the individual opinions are formalized as binary relations, and the collective decision is looked for as a choice function. Several new classes of rules are introduced and analyzed.

Beliefs, Interactions and Preferences in Decision Making mixes a selection of papers, presented at the Eighth Foundations and Applications of Utility and Risk Theory ('FUR VIII') conference in Mons, Belgium, together with a few solicited papers from well-known authors in the field. This book addresses some of the questions that have recently emerged in the research on decision-making and risk theory. In particular, authors have modeled more and more as interactions between the individual and the environment or between different individuals the emergence of beliefs as well as the specific type of information treatment traditionally called 'rationality'. This book analyzes several cases of such an interaction and derives consequences for the future of decision theory and risk theory. In the last ten years, modeling beliefs has become a specific sub-field of decision making, particularly with respect to low probability events. Rational decision making has also been generalized in order to encompass, in new ways and in more general situations than it used to be fitted to, multiple dimensions in consequences.This book deals with some of the most conspicuous of these advances. It also addresses the difficult question to incorporate several of these recent advances simultaneously into one single decision model. And it offers perspectives about the future trends of modeling such complex decision questions. The volume is organized in three main blocks: The first block is the more 'traditional' one. It deals with new extensions of the existing theory, as is always demanded by scientists in the field. A second block handles specific elements in the development of interactions between individuals and their environment, as defined in the most general sense. The last block confronts real-world problems in both financial and non-financial markets and decisions, and tries to show what kind of contributions can be brought to them by the type of research reported on here.

This book presents the main applied aspects of stochas­ tic optimization in economic models. Stochastic processes and control theory are used under optimization to illustrate the various economic implications of optimal decision rules. Unlike econometrics which deals with estimation, this book emphasizes the decision-theoretic basis of uncertainty specified by the stochastic point of view. Methods of ap­ plied stochastic control using stochastic processes have now reached an exciti~g phase, where several disciplines like systems engineering, operations research and natural reso- ces interact along with the conventional fields such as mathematical economics, finance and control systems. Our objective is to present a critical overview of this broad terrain from a multidisciplinary viewpoint. In this attempt we have at times stressed viewpoints other than the purely economic one. We believe that the economist would find it most profitable to learn from the other disciplines where stochastic optimization has been successfully applied. It is in this spirit that we have discussed in some detail the following major areas: A. Portfolio models in ·:finance, B. Differential games under uncertainty, c. Self-tuning regulators, D. Models of renewable resources under uncertainty, and ix x PREFACE E. Nonparametric methods of efficiency measurement. Stochastic processes are now increasingly used in economic models to understand the various adaptive behavior implicit in the formulation of expectation and its application in decision rules which are optimum in some sense.