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

\title{Quality Improvement Via the Optimization of Tolerance Intervals During
the Design Stage}

\author{S. Hadjihassan, E. Walter, and
L. Pronzato}

\maketitle
  
\auffil{The authors are with the Laboratoire des Signaux et
Syst\`emes, CNRS-ESE, Plateau de Moulon, 91192 Gif-sur-Yvette Cedex,
France (S. Hadjihassan, E. Walter), and 
with the Laboratoire I3S, CNRS URA-1376, Sophia Antipolis,
06560 Valbonne, France (L. Pronzato).}


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\def\RR{I\!\!R}
\def\bm#1{\mbox{\boldmath $#1$}}
\def\fb{{\bm f}}
\def\xb{{\bm x}}
\def\yb{{\bm y}}
\def\mtb{{\bm \theta}}
\def\meb{{\bm \eta}}
\def\mfb{{\bm \phi}}
\def\SF{{\cal F}}
\def\SS{{\cal S}}

In a world where competition for markets is ever more intense, ensuring a
satisfactory quality to products at the lowest possible cost is a vital
issue. This is the main reason why industry finds statistical quality
control increasingly appealing. Until recently, most of the effort has been
concentrated on the control of the quality of products at the end of the
production line. It is one of the merits of G. Taguchi's celebrated work (see
e.g.\ \cite{K85,T86}) to
have stressed that much money could be saved by taking quality requirements
into account right from the design stage. By combining elementary statistical
notions with a good engineering intuition, he showed how to make products
robust to fluctuations in the quality of their components. Good quality
products can then be obtained by combining low quality (i.e.\ cheap)
components. Some criticisms have been expressed about several aspects of
Taguchi's approach. They include the following:
\begin{itemize}
\item the criterion used is not clearly related to the problem to be solved,
\item too many experiments are required,
\item there is no explicit optimization, due to the absence of a mathematical
model of the process,
\item interactions between factors are difficult to handle.
\end{itemize}

Combining Taguchi's ideas with the well-established response surface
methodology, the model-based approach to quality improvement during the design
stage developed by I. Vuchkov and L. Boyadjieva (see e.g.\ \cite{VB88,VB89}) 
can be seen as a major
improvement to Taguchi's approach on these four counts. The resulting method
involves the construction of an explicit model $\meb(\fb,\mtb)$, linear in its
parameters $\mtb$, where $\meb(.,.)$ defines the performance characteristic(s)
of interest, $\fb$ is a vector of factors, and where the parameter vector $\mtb$
may be given {\em a priori} or estimated from experiments performed in
controlled conditions, i.e.\ with $\fb$ known exactly. In mass production, the
factors cannot be set precisely
to their desired levels but fluctuate randomly, e.g.\ due to the variability of
the product components. This is why $\fb$ is then assumed to be distributed
according to the normal distribution 
${\cal N}(\xb,\Sigma)$, with the covariance 
$\Sigma$ equal to some known (e.g.\ diagonal) matrix and
with the averages $\xb$ (the nominal value of the factors) 
to be optimized. The model
$\meb(.,\mtb)$ is then used to construct predictors of the mean and variance
of the performance characteristics. These predictions permit for instance to
find the feasible nominal value for the factors that minimizes an
approximation of the mean square error
$$
MSE(\xb) = E_{\fb} \| \yb^* - \meb(\fb(\xb),\mtb)\|^2_2 \,,
$$
where $\yb^*$ is the desired value of the performance characteristics.

The aim of this paper is to present a deterministic counterpart to this
model-based statistical approach. It will permit to take into account
fluctuations that are characterized in terms of tolerance intervals without
having to approximate them as gaussian random variables. It must be noted that
information on the fluctuations of the characteristics of components is most
often expressed in terms of tolerance intervals, so that an interval approach
to quality improvement seems highly desirable (for instance, tolerance
intervals on resistors are specified in terms of percentages of nominal
values). Moreover, whenever components are sorted according to their
tolerances from some initial population, the factors corresponding to the
components with the lowest quality (i.e.\ the largest tolerance intervals and
the smallest price) will be far from following a normal distribution, even if
the initial population was approximately gaussian.

The paper will be organized as follows. Attention will be restricted to the
scalar case, with $\eta(\fb,\mtb)$ linear in $\mtb$ and quadratic in $\fb$
(for the problem to have any meaning, it is essential that $\eta(.,.)$ be
nonlinear in the factors, so that nonlinearity can be put at work to
compensate for their fluctuations). In Section 2, the model
$\eta(.,\mtb)$ is assumed to be known {\em a priori}, and no uncertainty on
$\mtb$ is taken into account. This model may for instance have been
constructed according to the response surface methodology. The criterion to be
minimized is then
$$
j(\xb) = \max_{\fb \in \SF(\xb)} |y^* - \eta(\fb,\mtb)| \,,
$$
where $\SF(\xb)$ is the vector interval (i.e., the axis-aligned
parallelepiped, also known as a {\it box})
defining the tolerances in mass production around the nominal value $\xb$ of 
the factors. Section 3 considers the case where $\mtb$ is estimated from
measurements obtained in controlled conditions, where the $i$th observation
satisfies
$$
y_i = \eta(\xb_i,\mtb) + \epsilon_i\,, \;\; i=1,\ldots,N\,,
$$
with $\epsilon_i$ belonging to some known uncertainty interval and $\xb_i$
known without error. A parameter bounding methodology will be used to build the
set $\SS^N$ of all values of $\mtb$ that are consistent with these
experimental data. The criterion to be minimized then becomes
$$
j(\xb) = \max_{\mtb \in \SS^N,\fb \in \SF(\xb)} |y^* - \eta(\fb,\mtb)| \,.
$$
It takes into account, in a guaranteed way, the fluctuations of the quality of
the product components as well as the uncertainty on the model parameters.
Section 4 addresses the situation where measurements are performed with errors
on the factors, i.e.\
$$
y_i = \eta(\fb_i,\mtb) + \epsilon_i\,, \;\; i=1,\ldots,N\,,
$$
with $\epsilon_i$ as in Section 3 and $\fb_i = \xb_i + \mfb_i$, where $\mfb_i$
belongs to some vector uncertainty interval. It will be shown that when
$\eta(\fb,\mtb)$ is quadratic in $\fb$, the set $\SS^N$ can be included in a
finite union of convex polyhedra (of polytopes if trivial to check
identifiability conditions are met). This permits the replacement of the set
$\SS^N$ by an approximation guaranteed to contain it. A very simple example
will illustrate these notions in Section 5. Remaining open questions will be
offered for discussion.

\begin{thebibliography}{99}

\bibitem{K85}
R. N. Kackar, ``Off-line quality control, parameter design and the
Taguchi method'', {\em J. Quality Technology}, 1985, Vol. 17, No. 4, 
pp. 176--209 (with discussion).
\bibitem{T86}
G. Taguchi, {\em Introduction to Quality Engineering}, APO, Tokyo, 1986.
\bibitem{VB88}
I. N. Vuchkov and L. N. Boyadjieva, 
``The robustness against tolerances of
performance characteristics described by second order polynomials''. In:
Y. Dodge, V. V. Fedorov and H. P. Wynn (eds.),
{\bf Optimal design and analysis of experiments},
North Holland, Amsterdam, 1988, pp. 293--309.
\bibitem{VB89}
I. N. Vuchkov and L. N. Boyadjieva, ``A model-based approach to the
robustness against tolerances'', {\it 
Proceedings of 33th Annual EOQ Conference},
Vienna, 1989, pp. 585--592.

\end{thebibliography}

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