This book contains new and useful materials concerning fuzzy fractional differential and integral operators and their relationship. As the title of the book suggests, the fuzzy subject matter is one of the most important tools discussed. Therefore, it begins by providing a brief but important and new description of fuzzy sets and the computational calculus they require. Fuzzy fractals and fractional operators have a broad range of applications in the engineering, medical and economic sciences. Although these operators have been addressed briefly in previous papers, this book represents the first comprehensive collection of all relevant explanations. Most of the real problems in the biological and engineering sciences involve dynamic models, which are defined by fuzzy fractional operators in the form of fuzzy fractional initial value problems. Another important goal of this book is to solve these systems and analyze their solutions both theoretically and numerically. Given the content covered, the book will benefit all researchers and students in the mathematical and computer sciences, but also the engineering sciences.

This book identifies the important uncertainties to use in real-world problem modeling. Having information about several types of ambiguities, vagueness, and uncertainties is vital in modeling problems that involve linguistic variables, parameters, and word computing. Today, since most of our real-world problems are related to decision-making at the right time, we need to apply intelligent decision science. Clearly, in order to have an appropriate and flexible mathematical model, every intelligent system requires real data on our environment. Presenting problems that can be represented using mathematical models to create a system of linear equations, this book discusses the latest insights into uncertain information.

Numerical analysis forms a cornerstone of numeric computing and optimization, in particular recently, interval numerical computations play an important role in these topics. The interest of researchers in computations involving uncertain data, namely interval data opens new avenues in coping with real-world problems and deliver innovative and efficient solutions. This book provides the basic theoretical foundations of numerical methods, discusses key technique classes, explains improvements and improvements, and provides insights into recent developments and challenges. The theoretical parts of numerical methods, including the concept of interval approximation theory, are introduced and explained in detail. In general, the key features of the book include an up-to-date and focused treatise on error analysis in calculations, in particular the comprehensive and systematic treatment of error propagation mechanisms, considerations on the quality of data involved in numerical calculations, and a thorough discussion of interval approximation theory. Moreover, this book focuses on approximation theory and its development from the perspective of linear algebra, and new and regular representations of numerical integration and their solutions are enhanced by error analysis as well. The book is unique in the sense that its content and organization will cater to several audiences, in particular graduate students, researchers, and practitioners.

This book suggests that the numerical analysis subjects' matter are the important tools of the book topic, because numerical errors and methods have important roles in solving integral equations. Therefore, all needed topics including a brief description of interpolation are explained in the book. The integral equations have many applications in the engineering, medical, and economic sciences, so the present book contains new and useful materials about interval computations including interval interpolations that are going to be used in interval integral equations. The concepts of integral equations are going to be discussed in two directions, analytical concepts, and numerical solutions which both are necessary for these kinds of dynamic systems. The differences between this book with the others are a full discussion of error topics and also using interval interpolations concepts to obtain interval integral equations. All researchers and students in the field of mathematical, computer, and also engineering sciences can benefit the subjects of the book.

This book delivers a concise and carefully structured introduction to probability and random variables. It aims to build a linkage between the theoretical conceptual topics and the practical applications, especially in the undergraduate engineering area. The book motivates the student to gain full understanding of the fundamentals of probability theory and help acquire working problem-solving skills and apply the theory to engineering applications. Each chapter includes solved examples at varying levels (both introductory and advanced) in addition to problems that demonstrate the relevance of the probability and random variables in engineering. As authors, we focused on to find out the optimum ways in order to introduce the topics in probability and random variables area.

As the title of the book suggests, the topics of this book are organized into two parts. The first part points out the fuzzy differential equations and the second one is related to the fuzzy integral equations. The book contains nine chapters that six chapters are about fuzzy differential equations and three of them are about fuzzy integral equations. In each part, the chapters' authors are going to discuss the topics theoretically and numerically. All researchers and students in the field of mathematical, computer, and also engineering sciences can benefit from the subjects of the book.

This book presents the theory and application of the models presented in this regard and establishes a meaningful relationship between data envelopment analysis and multi-attribute decision making. The issue of "choice" using the aggregation of voters' votes is one of the most important group decision-making issues that are always considered by decision makers in electoral systems. Voting is a method of group decision making in a democratic society that expresses the will of the majority. Voting is perhaps the simplest way to gather the opinions of experts, and this ease of application has made it a multi-attribute decision-making method in group decisions. Preferential voting is a type of voting that may refer to electoral systems or groups of the electoral system. In preferential voting, voters vote for multiple candidates, and how the candidates are arranged on the ballot is important. Researchers have made many efforts to provide models of voter aggregation, and one of the best results of these efforts is the aggregation of votes based on the policy of data envelopment analysis. Thus, in group decisions, the opinions of experts are obtained in a simple structure and consolidated in an interactive and logical structure, and the results can be a powerful tool for decision support. This book provides a complete set of voting models based on data envelopment analysis and expressing its various applications in industry and society. However, most decision-making methods do not use the opinions of experts or reduce the motivation of experts to participate in complex interactions and time, while voting methods do not have this shortcoming. This book is suitable for graduate students in the fields of industrial management, business management, industrial engineering, applied mathematics, and economics. It can also be a good source for researchers in decision science, decision support systems, data envelopment analysis, supply chain management, healthcare management, and others. The methods presented in this book can not only offer a comprehensive framework for solving the problems of these areas but also can inspire researchers to pursue new innovative hybrid methods.

This book contains new and useful materials concerning fuzzy fractional differential and integral operators and their relationship. As the title of the book suggests, the fuzzy subject matter is one of the most important tools discussed. Therefore, it begins by providing a brief but important and new description of fuzzy sets and the computational calculus they require. Fuzzy fractals and fractional operators have a broad range of applications in the engineering, medical and economic sciences. Although these operators have been addressed briefly in previous papers, this book represents the first comprehensive collection of all relevant explanations. Most of the real problems in the biological and engineering sciences involve dynamic models, which are defined by fuzzy fractional operators in the form of fuzzy fractional initial value problems. Another important goal of this book is to solve these systems and analyze their solutions both theoretically and numerically. Given the content covered, the book will benefit all researchers and students in the mathematical and computer sciences, but also the engineering sciences.

This book contains the topics of artificial intelligence and deep learning that do have much application in real-life problems. The concept of uncertainty has long been used in applied science, especially decision making and a logical decision must be made in the field of uncertainty or in the real-life environment that is formed and combined with vague concepts and data. The chapters of this book are connected to the new concepts and aspects of decision making with uncertainty. Besides, other chapters are involved with the concept of data mining and decision making under uncertain computations.

This book identifies the important uncertainties to use in real-world problem modeling. Having information about several types of ambiguities, vagueness, and uncertainties is vital in modeling problems that involve linguistic variables, parameters, and word computing. Today, since most of our real-world problems are related to decision-making at the right time, we need to apply intelligent decision science. Clearly, in order to have an appropriate and flexible mathematical model, every intelligent system requires real data on our environment. Presenting problems that can be represented using mathematical models to create a system of linear equations, this book discusses the latest insights into uncertain information.

Soft Numerical Computing in Uncertain Dynamic Systems is intended for system specialists interested in dynamic systems that operate at different time scales. The book discusses several types of errors and their propagation, covering numerical methods-including convergence and consistence properties and characteristics-and proving of related theorems within the setting of soft computing. Several types of uncertainty representation like interval, fuzzy, type 2 fuzzy, granular, and combined uncertain sets are discussed in detail. The book can be used by engineering students in control and finite element fields, as well as all engineering, applied mathematics, economics, and computer science students. One of the important topics in applied science is dynamic systems and their applications. The authors develop these models and deliver solutions with the aid of numerical methods. Since they are inherently uncertain, soft computations are of high relevance here. This is the reason behind investigating soft numerical computing in dynamic systems. If these systems are involved with complex-uncertain data, they will be more practical and important. Real-life problems work with this type of data and most of them cannot be solved exactly and easily-sometimes they are impossible to solve. Clearly, all the numerical methods need to consider error of approximation. Other important applied topics involving uncertain dynamic systems include image processing and pattern recognition, which can benefit from uncertain dynamic systems as well. In fact, the main objective is to determine the coefficients of a matrix that acts as the frame in the image. One of the effective methods exhibiting high accuracy is to use finite differences to fill the cells of the matrix.

This book delivers a concise and carefully structured introduction to probability and random variables. It aims to build a linkage between the theoretical conceptual topics and the practical applications, especially in the undergraduate engineering area. The book motivates the student to gain full understanding of the fundamentals of probability theory and help acquire working problem-solving skills and apply the theory to engineering applications. Each chapter includes solved examples at varying levels (both introductory and advanced) in addition to problems that demonstrate the relevance of the probability and random variables in engineering. As authors, we focused on to find out the optimum ways in order to introduce the topics in probability and random variables area.

As the title of the book suggests, the topics of this book are organized into two parts. The first part points out the fuzzy differential equations and the second one is related to the fuzzy integral equations. The book contains nine chapters that six chapters are about fuzzy differential equations and three of them are about fuzzy integral equations. In each part, the chapters' authors are going to discuss the topics theoretically and numerically. All researchers and students in the field of mathematical, computer, and also engineering sciences can benefit from the subjects of the book.

This book suggests that the numerical analysis subjects' matter are the important tools of the book topic, because numerical errors and methods have important roles in solving integral equations. Therefore, all needed topics including a brief description of interpolation are explained in the book. The integral equations have many applications in the engineering, medical, and economic sciences, so the present book contains new and useful materials about interval computations including interval interpolations that are going to be used in interval integral equations. The concepts of integral equations are going to be discussed in two directions, analytical concepts, and numerical solutions which both are necessary for these kinds of dynamic systems. The differences between this book with the others are a full discussion of error topics and also using interval interpolations concepts to obtain interval integral equations. All researchers and students in the field of mathematical, computer, and also engineering sciences can benefit the subjects of the book.