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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."
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."
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.
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