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\title{What are Interval Computations, and How Are They Related
to Quality in Manufacturing?}
\author{Andrew Bernat$^1$, Vladik Kreinovich$^1$, 
Thomas J. McLean$^2$, and\\ Gennady N.
Solopchenko$^{3,4}$}

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\markright{APIC'95, El Paso,
Extended Abstracts,
A Supplement to the international journal of {\rm Reliable
Computing}\ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ }

\maketitle

\auffil{The authors are with the $^1$ Department of Computer Science
and $^2$ Department of Manufacturing and Industrial Engineering,
University of Texas at El Paso, El Paso, TX 79968, with $^3$St. Petersburg 
Technical University, Russia, and with the 
$^4$Academy of Metrological Sciences. This work was supported in part by
NSF Grant No. CDA-9015006 and NASA Research Grant No. 9-757.}

\section{One of the Major Aspects of Quality in Manufacturing}
\centerline{\it One of the major aspects of 
quality}

\centerline{\it of the manufacturing product}

\centerline{\it  is that 
all relevant numerical characteristics}

\centerline{\it of this
product}

\centerline{\it should lie in the
prescribed limits,}
\smallskip

\noindent
be it the fuel efficiency of an engine, the size of
a machine part, etc. So, to ensure the quality, we must measure the
values of these characteristics, and check that the characteristics
fit into the prescribed limits. 

Since measurements cannot be absolutely accurate, the resulting
estimate $\tilde y$ may differ from the actual value $y$ of the
desired characteristic. To be sure that the actual value $y$ 
satisfies our requirements (i.e., that it lies in
the required tolerance interval $[y^-,y^+]$), it is thus not
sufficient to know
that the estimate $\tilde y$ belongs to this interval. We must
also know the {\it accuracy} of this estimate.

It is important to distinguish between the following two cases:
\begin{itemize}
\item
Some of these characteristics (e.g., size) can be measured {\it directly},
i.e., by directly applying a
measuring instrument. In these cases, we can read the accuracy from the
documentation that has been supplied with this measuring device
(and is thus guaranteed by the manufacturer of this device; if the
manufacturer does not guarantee anything, then it is not a measuring
device at all). 

\item For other characteristics 
(e.g., for fuel efficiency) there is no sensor that
would immediately produce the desired result. Such a 
characteristic $y$ can be measured {\it indirectly.} Namely, we measure some
other characteristics $x_1,...,x_n$ 
that are directly measurable, and then use a
computer to transform the results $\tilde x_1,...\tilde x_n$ 
of these direct measurements into an
estimate $\tilde y$ of the desired characteristic. 
\smallskip

\item[]In this case,
from the documentation, we can only extract the
accuracies of the direct measurements $x_i$. We need to transform
these accuracies into the accuracy of $y$. 
\end{itemize}

So, we arrive at the following {\it problem}:

\section{Problem}

\centerline{\it Estimate the accuracy}
\centerline{\it of indirect measurements.}
\smallskip

For simple transformations, the solution to this 
problem is well known in engineering.
However, in many cases, the transformations from $x_i$ to $y$ involve
complicated computations (e.g., filtering, image processing, etc), and
thus the problem of estimating the accuracy of $y$ becomes
non-trivial. Here, we also have to consider two cases:
\begin{itemize}

\item
If we {\it know the probability} distribution of errors, then this problem
can be solved by using existing statistical techniques. 

\item However, in
many cases, the {\it only information} about the errors that the
manufacturer guarantees is the {\it upper bound for the error}. In these
cases, new methods are needed. 
\end{itemize}

Such methods have been developed 
in applied mathematics. Since our ultimate goal is to check 
whether a characteristic fits into a given tolerance
interval, one of the names for this field is {\it Interval Computations}.

\section{Summarizing: What Is Interval Computations (From the
Viewpoint of Manufacturing Applications)}

\centerline{\it Interval Computations is a methodology}

\centerline{\it that
estimates
accuracy of indirect measurements,}
\centerline{\it 
and thus helps to check the
quality.}
\newpage

\section{What Manufacturing Quality Problems Can Be Solved By Interval
Computations Methodology}

The need for interval computation methods appears if the following
conditions are satisfied: 
\begin{itemize}

\item We are checking the value of the characteristic $y$ that
cannot be measured directly. Therefore, we have to use an indirect
measurement, in which we first measure some other other characteristics
$x_1,...,x_n$, and then apply an algorithm $f$ to estimate $y$.

\item This algorithm $f$ is complicated.

\item Statistical characteristics of the errors of measuring
$x_i$ are unknown. 
\end{itemize}

\section{Interval Computations Are Useful Not Only For
Controlling the Quality of the Result, But Also for Checking the Quality
of the Manufacturing Process}

We have shown that Interval Computations are
useful for checking the quality of the final product. Analyzing the
final product is a necessary part of {\it quality testing}, but the
most important goal of quality control is to {\it improve quality}.
In many cases, we know the manufacturing process that can guarantee the
desired quality on condition that all parameters of this process
(e.g., temperature, pressure, etc) are
maintained within the prescribed limits. So, in order to guarantee the
quality of the manufacturing process, we must monitor the relevant
parameters and check that they are within the limits. 

This problem is similar to the above-described problem of checking the
quality of the final product, and thus, it also requires some complicated
algorithms. Such algorithms have been developed in interval
computations.

\section{A Brief History of Interval Computations}
\begin{itemize}
\item {\it It started here, in the US.}
\begin{itemize}
\item[]
The main ideas of Interval Computations appeared in the USA, in the Ph.D.
Dissertation of R. E. Moore that was defended at Stanford in 1962. 
The first application of interval computation 
was presented by R. E. Moore in 1959 \cite{Moore 1959}. 
The first monograph, also by R. E. Moore,
appeared in the USA in 1966 \cite{Moore 1966}. 
\end{itemize}
\item {\it Moved to Europe.}
\begin{itemize}
\item[]
Later, the center of interval computations moved to Europe, mainly to Germany.
One of the reasons was that in the US, manufacturers were, in
average, less
cost-conscious, and they were thus less
worried about inaccuracy of sensors: ``if a sensor is not good enough,
let's spend some more money and buy a better one''. 
The main users of this techniques were scientists, for whom this
solution did not work, because they were working at
the cutting edge of accuracy, and they were already using the best
possible sensors to measure their micro-quantities. 

\item[]As a result, Interval Computations is not widely 
known in the US, while in Germany, it is a part of the standard
qualifying exam for several areas of 
Numerical Mathematics. Germany was the place where
the first specialized journal appeared. Germany still hosts regular
conferences in interval computations.
\end{itemize}
\item{\it A recent outburst of activity.}
\begin{itemize}
\item[]Recently, there has been an outburst of activity in the USA and
internationally, related to Interval Computations:
\begin{itemize} 
\item[$\bullet$]
A new international journal {\it Interval Computation} has been launched in
1991 (starting from 1995, it is issued under the new title {\it
Reliable Computing}). 

\item[$\bullet$]
In 1993, a well-represented International Conference on Interval
Computations was held in Lafayette, LA.
\end{itemize}
\end{itemize}
\end{itemize}

\section{There Are Already Many Intersting Applications, But Interval
Computations Still Have a Large Unexploited Application Potential} 
In spite of many successful applications (some of them are presented
in this volume), there are still many areas where interval methods can be
applied. To our viewpoint, the main reason obstacle to future
applications in manufacturing is lack of communications:
\begin{itemize}

\item Engineers who are involved in real manufacturing 
rarely have a knowledge of this field. So, when they encounter the accuracy
problems that can be in principle solved by interval methods, then, instead
of using the existing techniques, they try to invent techniques of
their own.

\item People who are experts in interval computations are
not very knowledgeable in manufacturing problems. Therefore,
although they would love to apply their methods for solving real-life
problems, instead, they apply them to simulated ``toy'' problems. 
\end{itemize}

\section{The Goal of This Workshop}
The main goal of this workshop was to bring together people who are
interested in different applications of interval computations, and
thus, to promote cooperation between them.  

This volume contains applications in different fields. We hope that it
will be helpful to researchers. 

\begin{thebibliography}{99}
\bibitem{Moore 1959} R. E. Moore, {\bf Automatic error analysis in digital
computation}, Lockheed Missiles and Space Co. Technical Report
LMSD-48421, Palo Alto, CA, 1959.

\bibitem{Moore 1966} 
R. E. Moore, {\bf Interval analysis}, Prentice Hall, Englewood Cliffs,
NJ, 1966. 
\end{thebibliography}

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