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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 usual usual "implementation" "implementation" ofreal numbers as
floating point numbers on exist- iing ng computers computers has
the well-known disadvantage that most of the real numbers are not
exactly representable in floating point. Also the four basic
arithmetic operations can usually not be performed exactly. For
numerical algorithms there are frequently error bounds for the
computed approximation available. Traditionally a bound for the
infinity norm is estima- ted using ttheoretical heoretical
ccoonncceeppttss llike ike the the condition condition number
number of of a a matrix matrix for for example. example. Therefore
Therefore the error bounds are not really available in practice
since their com- putation requires more or less the exact solution
of the original problem. During the last years research in
different areas has been intensified in or- der to overcome these
problems. As a result applications to different concrete problems
were obtained. The LEDA-library (K. Mehlhorn et al.) offers a
collection of data types for combinatorical problems. In a series
of applications, where floating point arith- metic fails, reliable
results are delivered. Interesting examples can be found in
classical geometric problems. At the Imperial College in London was
introduced a simple principle for "exact arithmetic with real
numbers" (A. Edalat et al.), which uses certain nonlinear
transformations. Among others a library for the effective
computation of the elementary functions already has been
implemented.
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.
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