James J. Buckley Bücher

1. 1 Introduction This book is written in four major divisions. The first part is the introductory chapters consisting of Chapters 1 and 2. In part two, Chapters 3-11, we develop fuzzy estimation. For example, in Chapter 3 we construct a fuzzy estimator for the mean of a normal distribution assuming the variance is known. More details on fuzzy estimation are in Chapter 3 and then after Chapter 3, Chapters 4-11 can be read independently. Part three, Chapters 12- 20, are on fuzzy hypothesis testing. For example, in Chapter 12 we consider the test Ho : /1 = /10 verses HI : /1 f=- /10 where /1 is the mean of a normal distribution with known variance, but we use a fuzzy number (from Chapter 3) estimator of /1 in the test statistic. More details on fuzzy hypothesis testing are in Chapter 12 and then after Chapter 12 Chapters 13-20 may be read independently. Part four, Chapters 21-27, are on fuzzy regression and fuzzy prediction. We start with fuzzy correlation in Chapter 21. Simple linear regression is the topic in Chapters 22-24 and Chapters 25-27 concentrate on multiple linear regression. Part two (fuzzy estimation) is used in Chapters 22 and 25; and part 3 (fuzzy hypothesis testing) is employed in Chapters 24 and 27. Fuzzy prediction is contained in Chapters 23 and 26. A most important part of our models in fuzzy statistics is that we always start with a random sample producing crisp (non-fuzzy) data.

This book is the companion text to Simulating Fuzzy Systems which investigated discrete fuzzy systems through crisp discrete simulation. The current book studies continuous fuzzy dynamical systems using crisp continuous simulation. We start with a crisp continuous dynamical system whose evolution depends on a system of ordinary differential equations (ODEs). The system of ODEs contains parameters many of which have uncertain values. Usually point estimators for these uncertain parameters are used, but the resulting system will not display any uncertainty associated with these estimators. Instead we employ fuzzy number estimators, constructed from expert opinion or from data, for the uncertain parameters. Fuzzy number estimators produces a system of fuzzy ODEs to solve whose solution will be fuzzy trajectories for the variables. We use crisp continuous simulation to estimate the trajectories of the support and core of these fuzzy numbers in a variety of twenty applications of fuzzy dynamical systems. The applications range from Bungee jumping to the AIDS epidemic to dynamical models in economics.

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This book serves as an undergraduate introduction to the theory of fuzzy sets, aimed at establishing a foundation for a future curriculum in fuzzy systems theory across various fields such as computer science, mathematics, engineering, and economics. It is designed for beginners, focusing on the basics rather than research. The content parallels a pre-calculus course, covering fuzzy algebra, functions, trigonometry, geometry, and solving fuzzy equations in chapters 3 through 11. After this foundational course, students can pursue specialized topics in fuzzy systems theory, including fuzzy clustering, pattern recognition, database management, image processing, robotics, intelligent agents, soft computing, and fuzzy decision-making, among others. While the book does not delve deeply into these advanced topics, it introduces several new concepts, such as mixed fuzzy logic, methods for solving fuzzy equations, fuzzy inequalities, inverse fuzzy functions, fuzzy plane geometry, fuzzy trigonometry, and fuzzy optimization using genetic algorithms. This approach ensures that students are well-prepared for more advanced studies in the field.

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This book aims to introduce Monte Carlo methods for finding approximate solutions to fuzzy optimization problems, an area lacking established algorithms compared to crisp optimization. It addresses key topics such as the comparison of fuzzy numbers and the evaluation of max/min values for fuzzy objective functions. The structure of the book is divided into four parts: Part I serves as an introduction, covering Chapters 1-5; Part II, spanning Chapters 6-16, applies the Monte Carlo method to fuzzy optimization problems; Part III, consisting of Chapters 17-27, discusses unresolved fuzzy optimization challenges where the Monte Carlo method has yet to be applied; and Part IV offers a summary, conclusions, and suggestions for future research. In Part I, readers will become acquainted with fuzzy sets, with Chapter 2 providing the essential concepts needed for the book. For those new to fuzzy sets and fuzzy logic, a preliminary introduction is suggested. Additionally, Chapter 2 includes three significant topics related to fuzzy sets: Section 2.5 discusses past approaches to determining max/min values for fuzzy sets representing objective functions in fuzzy optimization. This foundational knowledge is crucial for understanding the subsequent applications and discussions throughout the book.

This book combines material from our previous books FP (Fuzzy Probabilities: New Approach and Applications, Physica-Verlag, 2003) and FS (Fuzzy Statistics, Springer, 2004), plus has about one third new results. From FP we have material on basic fuzzy probability, discrete (fuzzy Poisson, binomial) and continuous (uniform, normal, exponential) fuzzy random variables. From FS we included chapters on fuzzy estimation and fuzzy hypothesis testing related to means, variances, proportions, correlation and regression. New material includes fuzzy estimators for arrival and service rates, and the uniform distribution, with applications in fuzzy queuing theory. Also, new to this book, is three chapters on fuzzy maximum entropy (imprecise side conditions) estimators producing fuzzy distributions and crisp discrete/continuous distributions. Other new results are: (1) two chapters on fuzzy ANOVA (one-way and two-way); (2) random fuzzy numbers with applications to fuzzy Monte Carlo studies; and (3) a fuzzy nonparametric estimator for the median.

Simulating Fuzzy Systems demonstrates how many systems naturally become fuzzy systems and shows how regular (crisp) simulation can be used to estimate the alpha-cuts of the fuzzy numbers used to analyze the behavior of the fuzzy system. This monograph presents a concise introduction to fuzzy sets, fuzzy logic, fuzzy estimation, fuzzy probabilities, fuzzy systems theory, and fuzzy computation. It also presents a wide selection of simulation applications ranging from emergency rooms to machine shops to project scheduling, showing the varieties of fuzzy systems.

1.1 Introduction This book is written in five major divisions. The first part is the introduc tory chapters consisting of Chapters 1-3. In part two, Chapters 4-10, we use fuzzy probabilities to model a fuzzy queuing system . We switch to employ ing fuzzy arrival rates and fuzzy service rates to model the fuzzy queuing system in part three in Chapters 11 and 12. Optimization models comprise part four in Chapters 13-17. The final part has a brief summary and sug gestions for future research in Chapter 18, and a summary of our numerical methods for calculating fuzzy probabilities, values of objective functions in fuzzy optimization, etc., is in Chapter 19. First we need to be familiar with fuzzy sets. All you need to know about fuzzy sets for this book comprises Chapter 2. Two other items relating to fuzzy sets, needed in Chapters 13-17, are also in Chapter 2: (1) how we plan to handle the maximum/minimum of a fuzzy set; and (2) how we will rank a finite collection of fuzzy numbers from smallest to largest.