Fuzzy Relational Mathematical Programming

This book summarizes years of research in the field of fuzzy relational programming, with a special emphasis on geometric models. It discusses the state-of-the-art in fuzzy relational geometric problems, together with key open issues that must be resolved to achieve a more efficient application of this method. Though chiefly based on research conducted by the authors, who were the first to introduce fuzzy geometric problems, it also covers important findings obtained in the field of linear and non-linear programming. Thanks to its balance of basic and advanced concepts, and its wealth of practical examples, the book offers a valuable guide for both newcomers and experienced researcher in the fields of soft computing and mathematical optimization.

Optimization is of central concern to a number of discip lines. Operations Research and Decision Theory are often consi dered to be identical with optimizationo But also in other areas such as engineering design, regional policy, logistics and many others, the search for optimal solutions is one of the prime goals. The methods and models which have been used over the last decades in these areas have primarily been "hard" or "crisp", i. e. the solutions were considered to be either fea sible or unfeasible, either above a certain aspiration level or below. This dichotomous structure of methods very often forced the modeller to approximate real problem situations of the more-or-less type by yes-or-no-type models, the solutions of which might turn out not to be the solutions to the real prob lems. This is particularly true if the problem under considera tion includes vaguely defined relationships, human evaluations, uncertainty due to inconsistent or incomplete evidence, if na tural language has to be modelled or if state variables can only be described approximately. Until recently, everything which was not known with cer tainty, i. e. which was not known to be either true or false or which was not known to either happen with certainty or to be impossible to occur, was modelled by means of probabilitieso This holds in particular for uncertainties concerning the oc currence of events.

Fuzzy Sets in Decision Analysis, Operations Research and Statistics includes chapters on fuzzy preference modeling, multiple criteria analysis, ranking and sorting methods, group decision-making and fuzzy game theory. It also presents optimization techniques such as fuzzy linear and non-linear programming, applications to graph problems and fuzzy combinatorial methods such as fuzzy dynamic programming. In addition, the book also accounts for advances in fuzzy data analysis, fuzzy statistics, and applications to reliability analysis. These topics are covered within four parts: Decision Making, Mathematical Programming, Statistics and Data Analysis, and Reliability, Maintenance and Replacement. The scope and content of the book has resulted from multiple interactions between the editor of the volume, the series editors, the series advisory board, and experts in each chapter area. Each chapter was written by a well-known researcher on the topic and reviewed by other experts in the area. These expert reviewers sometimes became co-authors because of the extent of their contribution to the chapter. As a result, twenty-five authors from twelve countries and four continents were involved in the creation of the 13 chapters, which enhances the international character of the project and gives an idea of how carefully the Handbook has been developed.

Since their inception, fuzzy sets and fuzzy logic became popular. The reason is that the very idea of fuzzy sets and fuzzy logic attacks an old tradition in science, namely bivalent (black-or-white, all-or-none) judg ment and reasoning and the thus resulting approach to formation of scientific theories and models of reality. The idea of fuzzy logic, briefly speaking, is just the opposite of this tradition: instead of full truth and falsity, our judgment and reasoning also involve intermediate truth values. Application of this idea to various fields has become known under the term fuzzy approach (or graded truth approach). Both prac tice (many successful engineering applications) and theory (interesting nontrivial contributions and broad interest of mathematicians, logicians, and engineers) have proven the usefulness of fuzzy approach. One of the most successful areas of fuzzy methods is the application of fuzzy relational modeling. Fuzzy relations represent formal means for modeling of rather nontrivial phenomena (reasoning, decision, control, knowledge extraction, systems analysis and design, etc. ) in the pres ence of a particular kind of indeterminacy called vagueness. Models and methods based on fuzzy relations are often described by logical formulas (or by natural language statements that can be translated into logical formulas). Therefore, in order to approach these models and methods in an appropriate formal way, it is desirable to have a general theory of fuzzy relational systems with basic connections to (formal) language which enables us to describe relationships in these systems.

This book examines fuzzy relational calculus theory with applications in various engineering subjects. The scope of the text covers unified and exact methods with algorithms for direct and inverse problem resolution in fuzzy relational calculus. Extensive engineering applications of fuzzy relation compositions and fuzzy linear systems (linear, relational and intuitionistic) are discussed. Some examples of such applications include solutions of equivalence, reduction and minimization problems in fuzzy machines, pattern recognition in fuzzy languages, optimization and inference engines in textile and chemical engineering, etc. A comprehensive overview of the authors' original work in fuzzy relational calculus is also provided in each chapter.The attached CD-Rom contains a toolbox with many functions for fuzzy calculations, together with an original algorithm for inverse problem resolution in MATLAB. This book is also suitable for use as a textbook in related courses at advanced undergraduate and graduate levels.

This book includes 70 selected papers from the Ninth International Conference on Fuzzy Information and Engineering (ICFIE) Satellite, which was held on December 26–30, 2018; and from the 9th International Conference on Fuzzy Information and Engineering (ICFIAE), which was held on February 13–15, 2019. The two conferences presented the latest research in the areas of fuzzy information and engineering, operational research and management, and their applications.

This book combines computational intelligence and mathematics to solve theoretical and real-world problems. The real challenges of engineering and other applied sciences, e.g. economics and management, the social sciences, etc., and even everyday life, are increasingly raising complex problems – both in the usual sense, but also in the mathematical and theoretical computer science sense, which is referred to as intractability. Finding exact solutions to the latest problems in mathematics is impossible, and it has been also shown that no further technical advance will ever make it possible to find general and exact solutions to such complex problems. Rather, the goal is to find solutions that are “good enough” or “acceptably accurate,” including models and corresponding algorithms, which is most often achieved by combining traditional mathematical techniques and computational intelligence tools, such as fuzzy systems, evolutionary and memetic algorithms, and artificial neural networks. Consequently, international funding programs, such as the European Commission’s current framework program for research and innovation (Horizon 2020), and the preliminary research team building COST Actions, are devoted to developing new instruments for tackling the challenges that we face in the current technological age. And it goes without saying that research topics concerning the interactions between computational intelligence and traditional mathematics play a key role in overcoming the obstacles associated with the intractability of complex problems. In this book, mathematicians, engineers, and other scientists highlight novel methodological results connecting these two main research areas, and focusing on solving real-life problems.