Convexity of sets in linear spaces, and concavity and convexity of functions, lie at the root of beautiful theoretical results that are at the same time extremely useful in the analysis and solution of optimization problems, including problems of either single objective or multiple objectives. Not all of these results rely necessarily on convexity and concavity; some of the results can guarantee that each local optimum is also a global optimum, giving these methods broader application to a wider class of problems. Hence, the focus of the first part of the book is concerned with several types of generalized convex sets and generalized concave functions. In addition to their applicability to nonconvex optimization, these convex sets and generalized concave functions are used in the book's second part, where decision-making and optimization problems under uncertainty are investigated. Uncertainty in the problem data often cannot be avoided when dealing with practical problems. Errors occur in real-world data for a host of reasons. However, over the last thirty years, the fuzzy set approach has proved to be useful in these situations. It is this approach to optimization under uncertainty that is extensively used and studied in the second part of this book. Typically, the membership functions of fuzzy sets involved in such problems are neither concave nor convex. They are, however, often quasiconcave or concave in some generalized sense. This opens possibilities for application of results on generalized concavity to fuzzy optimization. Despite this obvious relation, applying the interface of these two areas has been limited to date. It is hoped that the combination of ideas and results from the field of generalized concavity on the one hand and fuzzy optimization on the other hand outlined and discussed in Generalized Concavity in Fuzzy Optimization and Decision Analysis will be of interest to both communities. Our aim is to broaden the classes of problems that the combination of these two areas can satisfactorily address and solve.

Linear programming attracted the interest of mathematicians during and after World War II when the first computers were constructed and methods for solving large linear programming problems were sought in connection with specific practical problems for example, providing logistical support for the U.S. Armed Forces or modeling national economies. Early attempts to apply linear programming methods to solve practical problems failed to satisfy expectations. There were various reasons for the failure. One of them, which is the central topic of this book, was the inexactness of the data used to create the models. This phenomenon, inherent in most practical problems, has been dealt with in several ways. At first, linear programming models used average values of inherently vague coefficients, but the optimal solutions of these models were not always optimal for the original problem itself. Later researchers developed the stochastic linear programming approach, but this too has its limitations. Recently, interest has been given to linear programming problems with data given as intervals, convex sets and/or fuzzy sets. literature has not presented a unified theory. Linear Optimization Problems with Inexact Data attempts to present a comprehensive treatment of linear optimization with inexact data, summarizing existing results and presenting new ones within a unifying framework.

The papers presented at the Symposium focused mainly on two fields of interest. First, there were papers dealing with the theoretical background of fuzzy logic and with applications of fuzzy reasoning to the problems of artificial intelligence, robotics and expert systems. Second, quite a large number of papers were devoted to fuzzy approaches to modelling of decision-making situations under uncertainty and vagueness and their applications to the evaluation of alternatives, system control and optimization.Apart from that, there were also some interesting contributions from other areas, like fuzzy classifications and the use of fuzzy approaches in quantum physics.This volume contains the most valuable and interesting papers presented at the Symposium and will be of use to all those researchers interested in fuzzy set theory and its applications.

The theory of fuzzy sets has become known in Czechoslovakia in the early seventies. Since then, it was applied in various areas of science, engineering and economics where indeterminate concepts had to be handled. There has been a number of national semi- nars and conferences devoted to this topic. However, the International Symposium on Fuzzy Approach to Reasoning and Decision-Making, held in 1990, was the first really representative international meeting of this kind organized in Czechoslovakia. The symposium took place in the House of Scientists of the Czechoslovak Academy of Sciences in Bechyne from June 25 till 29, 1990. Its main organizer was Mining In- stitute of the Czechoslovak Academy of Sciences in Ostrava in cooperation and support of several other institutions and organizations. A crucial role in preparing of the Sym- posium was played by the working group for Fuzzy Sets and Systems which is active in the frame of the Society of Czechoslovak Mathematicians and Physicists. The organizing and program committee was headed by Dr. Vilem Novak from the Mining Institute in Ostrava. Its members (in alphabetical order) were Dr. Martin Cerny (Prague), Prof. Bla- hoslav Harman (Liptovsky Mikulas), Ema Hyklova (Prague), Prof. Zdenek Karpfsek (Brno), Jan Laub (Prague), Dr. Milan MareS - vice-chairman (Prague), Prof. Radko Mesiar (Bratislava), Dr. Jifi Nekola - vice-chairman (Prague), Daria Novakova (Os- trava), Dr. Jaroslav Ramfk (Ostrava), Prof. Dr. Beloslav Riecan (Bratislava), Dr. Jana TalaSova (Pi'erov) and Dr. Milos Vitek (Pardubice).

Convexity of sets in linear spaces, and concavity and convexity of functions, lie at the root of beautiful theoretical results that are at the same time extremely useful in the analysis and solution of optimization problems, including problems of either single objective or multiple objectives. Not all of these results rely necessarily on convexity and concavity; some of the results can guarantee that each local optimum is also a global optimum, giving these methods broader application to a wider class of problems. Hence, the focus of the first part of the book is concerned with several types of generalized convex sets and generalized concave functions. In addition to their applicability to nonconvex optimization, these convex sets and generalized concave functions are used in the book's second part, where decision-making and optimization problems under uncertainty are investigated. Uncertainty in the problem data often cannot be avoided when dealing with practical problems. Errors occur in real-world data for a host of reasons. However, over the last thirty years, the fuzzy set approach has proved to be useful in these situations. It is this approach to optimization under uncertainty that is extensively used and studied in the second part of this book. Typically, the membership functions of fuzzy sets involved in such problems are neither concave nor convex. They are, however, often quasiconcave or concave in some generalized sense. This opens possibilities for application of results on generalized concavity to fuzzy optimization. Despite this obvious relation, applying the interface of these two areas has been limited to date. It is hoped that the combination of ideas and results from the field of generalized concavity on the one hand and fuzzy optimization on the other hand outlined and discussed in Generalized Concavity in Fuzzy Optimization and Decision Analysis will be of interest to both communities. Our aim is to broaden the classes of problems that the combination of these two areas can satisfactorily address and solve.

Linear programming attracted the interest of mathematicians during and after World War II when the first computers were constructed and methods for solving large linear programming problems were sought in connection with specific practical problems for example, providing logistical support for the U.S. Armed Forces or modeling national economies. Early attempts to apply linear programming methods to solve practical problems failed to satisfy expectations. There were various reasons for the failure. One of them, which is the central topic of this book, was the inexactness of the data used to create the models. This phenomenon, inherent in most pratical problems, has been dealt with in several ways. At first, linear programming models used "average" values of inherently vague coefficients, but the optimal solutions of these models were not always optimal for the original problem itself. Later researchers developed the stochastic linear programming approach, but this too has its limitations. Recently, interest has been given to linear programming problems with data given as intervals, convex sets and/or fuzzy sets. The individual results of these studies have been promising, but the literature has not presented a unified theory. Linear Optimization Problems with Inexact Data attempts to present a comprehensive treatment of linear optimization with inexact data, summarizing existing results and presenting new ones within a unifying framework."

The book is a benefit for graduate and postgraduate students in the areas of operations research, decision theory, optimization theory, linear algebra, interval analysis and fuzzy sets. The book will also be useful for the researchers in the respective areas. The first part of the book deals with decision making problems and procedures that have been established to combine opinions about alternatives related to different points of view. Procedures based on pairwise comparisons are thoroughly investigated. In the second part we investigate optimization problems where objective functions and constraints are characterized by extremal operators such as maximum, minimum or various triangular norms (t-norms). Matrices in max-min algebra are useful in applications such as automata theory, design of switching circuits, logic of binary relations, medical diagnosis, Markov chains, social choice, models of organizations, information systems, political systemsand clustering. The input data in real problems are usually not exact and can be characterized by interval values."