It is frequently observed that most decision-making problems involve several objectives, and the aim of the decision makers is to find the best decision by fulfilling the aspiration levels of all the objectives. Multi-objective decision making is especially suitable for the design and planning steps and allows a decision maker to achieve the optimal or aspired goals by considering the various interactions of the given constraints. Multi-Objective Stochastic Programming in Fuzzy Environments discusses optimization problems with fuzzy random variables following several types of probability distributions and different types of fuzzy numbers with different defuzzification processes in probabilistic situations. The content within this publication examines such topics as waste management, agricultural systems, and fuzzy set theory. It is designed for academicians, researchers, and students.

In this innovative approach to the practice of social science, Charles Ragin explores the use of fuzzy sets to bridge the divide between quantitative and qualitative methods. Paradoxically, the fuzzy set is a powerful tool because it replaces an unwieldy, "fuzzy" instrument--the variable, which establishes only the positions of cases relative to each other, with a precise one--degree of membership in a well-defined set.
Ragin argues that fuzzy sets allow a far richer dialogue between ideas and evidence in social research than previously possible. They let quantitative researchers abandon "homogenizing assumptions" about cases and causes, they extend diversity-oriented research strategies, and they provide a powerful connection between theory and data analysis. Most important, fuzzy sets can be carefully tailored to fit evolving theoretical concepts, sharpening quantitative tools with in-depth knowledge gained through qualitative, case-oriented inquiry. This book will revolutionize research methods not only in sociology, political science, and anthropology but in any field of inquiry dealing with complex patterns of causation.

An introduction to category theory as a rigorous, flexible, and coherent modeling language that can be used across the sciences. Category theory was invented in the 1940s to unify and synthesize different areas in mathematics, and it has proven remarkably successful in enabling powerful communication between disparate fields and subfields within mathematics. This book shows that category theory can be useful outside of mathematics as a rigorous, flexible, and coherent modeling language throughout the sciences. Information is inherently dynamic; the same ideas can be organized and reorganized in countless ways, and the ability to translate between such organizational structures is becoming increasingly important in the sciences. Category theory offers a unifying framework for information modeling that can facilitate the translation of knowledge between disciplines. Written in an engaging and straightforward style, and assuming little background in mathematics, the book is rigorous but accessible to non-mathematicians. Using databases as an entry to category theory, it begins with sets and functions, then introduces the reader to notions that are fundamental in mathematics: monoids, groups, orders, and graphs-categories in disguise. After explaining the "big three" concepts of category theory-categories, functors, and natural transformations-the book covers other topics, including limits, colimits, functor categories, sheaves, monads, and operads. The book explains category theory by examples and exercises rather than focusing on theorems and proofs. It includes more than 300 exercises, with solutions. Category Theory for the Sciences is intended to create a bridge between the vast array of mathematical concepts used by mathematicians and the models and frameworks of such scientific disciplines as computation, neuroscience, and physics.

Fuzzy set theory deals with sets or categories whose boundaries are blurry or, in other words, "fuzzy." This book presents an accessible introduction to fuzzy set theory, focusing on its applicability to the social sciences. Unlike most books on this topic, Fuzzy Set Theory: Applications in the Social Sciences provides a systematic, yet practical guide for researchers wishing to combine fuzzy set theory with standard statistical techniques and model-testing. Key Features: Addresses Basic Concepts: Fuzzy set theory is an analytic framework for handling concepts that are simultaneously categorical and dimensional. Starting with a rationale for fuzzy sets, this book introduces readers with an elementary knowledge of statistics to the necessary concepts and techniques of fuzzy set theory and fuzzy logic. Introduces Novel Ways of Analyses: Researchers are shown alternative methods to conventional models, especially for testing theories that are expressed in set-wise terms. Issues of operationalizing graded membership in a fuzzy set and the measurement of the properties of such sets are a few of the topics addressed. Illustrates Techniques and Applications: Real examples and data-sets from various disciplines in the social sciences are used to demonstrate the connections between fuzzy sets and other data analytic techniques, empirical applications of the technique, and the critiques of fuzzy set theory. Intended Audience: Ideal for researchers in the social sciences, education, and behavioral sciences; as well as graduate students in the applied social sciences

Information granules, as encountered in natural language, are implicit in nature. To make them fully operational so they can be effectively used to analyze and design intelligent systems, information granules need to be made explicit. An emerging discipline, granular computing focuses on formalizing information granules and unifying them to create a coherent methodological and developmental environment for intelligent system design and analysis. Granular Computing: Analysis and Design of Intelligent Systems presents the unified principles of granular computing along with its comprehensive algorithmic framework and design practices. Introduces the concepts of information granules, information granularity, and granular computing Presents the key formalisms of information granules Builds on the concepts of information granules with discussion of higher-order and higher-type information granules Discusses the operational concept of information granulation and degranulation by highlighting the essence of this tandem and its quantification in terms of the associated reconstruction error Examines the principle of justifiable granularity Stresses the need to look at information granularity as an important design asset that helps construct more realistic models of real-world systems or facilitate collaborative pursuits of system modeling Highlights the concepts, architectures, and design algorithms of granular models Explores application domains where granular computing and granular models play a visible role, including pattern recognition, time series, and decision making Written by an internationally renowned authority in the field, this innovative book introduces readers to granular computing as a new paradigm for the analysis and synthesis of intelligent systems. It is a valuable resource for those engaged in research and practical developments in computer, electrical, industrial, manufacturing, and biomedical engineering. Building from fundamentals, the book is also suitable for readers from nontechnical disciplines where information granules assume a visible position.

The formal description of non-precise data before their statistical analysis is, except for error models and interval arithmetic, a relatively young topic. Fuzziness is described in the theory of fuzzy sets but only a few papers on statistical inference for non-precise data exist. In many cases, for example when very small concentrations are being measured, it is necessary to describe the imprecision of data. Otherwise, the results of statistical analysis can be unrealistic and misleading. Fortunately, there is a straightforward technique for dealing with non-precise data. The technique - the generalized inference method - is explained in Statistical Methods for Non-Precise Data. Anyone who understands elementary statistical methods and simple stochastic models will be able to use this book to understand and work with non-precise data.
The book includes explanations of how to cope with non-precise data in different practical situations, and makes an excellent graduate level text book for students, as well as a general reference for scientists and practitioners.

Features

Ever since fuzzy logic was introduced by Lotfi Zadeh in the mid-sixties and genetic algorithms by John Holland in the early seventies, these two fields widely been subjects of academic research the world over. During the last few years, they have been experiencing extremely rapid growth in the industrial world, where they have been shown to be very effective in solving real-world problems. These two substantial fields, together with neurocomputing techniques, are recognized as major parts of soft computing: a set of computing technologies already riding the waves of the next century to produce the human-centered intelligent systems of tomorrow; the collection of papers presented in this book shows the way. The book also contains an extensive bibliography on fuzzy logic and genetic algorithms.