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\bs
  \heading{Unit 8: Robust Parameter Design}

\bc Source : Chapter~11 (sections 11.1 - 11.6, part of sections
11.7 - 11.8 and 11.9). \ec

\vspace{.1 in}

\bi
 \item Revisiting two previous experiments.
 \item Strategies for reducing variation.
 \item Types of noise factors.
 \item Variation reduction through robust parameter design.
 \item Cross array, location-dispersion modeling, response modeling.
 \item Single arrays vs cross arrays.
 \item Signal-to-noise ratios and limitations.
\ei
 \es

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% 2 -----------------------------------------------------------------

\bs
  \heading{Robust Parameter Design}

\vspace{.1 in}

\bi
 \item Statistical/engineering method for product/process improvement
(G. Taguchi).
 \item Two types of factors in a system (product/process):
  \bi \item control factors: once chosen, values remain fixed.
       \item noise factors: hard-to-control during normal process or usage. \ei
  \item \textbf{Robust Parameter design} (\textbf{RPD} or \textbf{PD}): choose control factor settings to make response less sensitive
  (i.e.more robust) to noise variation; exploiting
  control-by-noise interactions.
 \ei

\es

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% 3 -----------------------------------------------------------------

\bs
  \heading{A Robust Design Perspective of Layer-growth and Leaf Spring Experiments}

\bi
 \item The original AT \& T layer growth experiment had 8 control factors, 2
 noise factors (location and facet). Goal was to achieve $uniform$
 thickness around 14.5 $\mu$m over the noise factors. See Tables 1
 and 2.
 \item The original leaf spring experiment had 4 control factors,
 1 noise factor (quench oil temperature). The quench oil
 temperature is not controllable; with efforts it can be set in two
 ranges of values 130-150, 150-170. Goal is to achieve \emph{uniform}
 free height around 8 inches over the range of quench oil
 temperature. See Tables 3 and 4.
 \item Must understand the role of \emph{noise factors} in achieveing
 \emph{robustness}.
 \ei

\es

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% 4 -----------------------------------------------------------------

\bs
  \heading{Layer Growth Experiment: Factors and Levels}

\begin{table}
\caption{Factors and Levels, Layer Growth Experiment}
$ $\\
\centering \scriptsize
\begin{tabular}{|ll|ll|} \hline
\multicolumn{2}{|c}{} &
\multicolumn{2}{|c|}{Level} \\
\multicolumn{2}{|c}{Control Factor} & \multicolumn{1}{|c}{$-$} &
\multicolumn{1}{c|}{+} \\ \hline
$A.$& susceptor-rotation method& continuous & oscillating\\
$B.$&code of wafers & 668G4& 678D4\\
$C.$&deposition temperature($^{\circ}$C)& 1210& 1220\\
$D.$&deposition time& short& long\\
$E.$&arsenic flow rate(\%)& 55& 59\\
$F.$&hydrochloric acid etch temperature($^{\circ}$C)&1180 & 1215 \\
$G.$&hydrochloric acid flow rate(\%) & 10& 14\\
$H.$&nozzle position& 2 & 6\\ \hline \multicolumn{2}{|c}{} &
\multicolumn{2}{|c|}{Level} \\
\multicolumn{2}{|c}{Noise Factor} & \multicolumn{1}{|c}{$-$} &
\multicolumn{1}{c|}{+} \\ \hline
$L.$&location& bottom & top\\
$M.$&facet& 1 $\;\;$ 2& 3 $\;\;$ 4 \\ \hline
\end{tabular}
\label{tab:ch8layerfactors}
\end{table}


\es

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% 5 -----------------------------------------------------------------

\bs
  \heading{Layer Growth Experiment: Thickness Data}

\begin{table}
\caption{Cross Array and Thickness Data, Layer Growth Experiment}
$ $\\
\tiny \centering
\begin{tabular}{|rrrrrrrr|rrrr|rrrr|} \hline
\multicolumn{8}{|c|}{} &
\multicolumn{8}{c|}{Noise Factor} \\
\multicolumn{8}{|c|}{Control Factor} &
\multicolumn{4}{c}{L-Bottom} &
\multicolumn{4}{c|}{L-Top} \\
\multicolumn{1}{|c}{$A$} & \multicolumn{1}{c}{$B$} &
\multicolumn{1}{c}{$C$} & \multicolumn{1}{c}{$D$} &
\multicolumn{1}{c}{$E$} & \multicolumn{1}{c}{$F$} &
\multicolumn{1}{c}{$G$} & \multicolumn{1}{c|}{$H$} &
\multicolumn{1}{c}{$M$-1} & \multicolumn{1}{c}{$M$-2} &
\multicolumn{1}{c}{$M$-3} & \multicolumn{1}{c}{$M$-4}&
\multicolumn{1}{c}{$M$-1} & \multicolumn{1}{c}{$M$-2} &
\multicolumn{1}{c}{$M$-3} & \multicolumn{1}{c|}{$M$-4} \\ \hline
$-$&$-$&$-$&$+$&$-$&$-$&$-$&$-$&14.2908&14.1924&14.2714&14.1876&15.3182&15.4279&15.2657&15.4056\\
$-$&$-$&$-$&$+$&$+$&$+$&$+$&$+$&14.8030&14.7193&14.6960&14.7635&14.9306&14.8954&14.9210&15.1349\\
$-$&$-$&$+$&$-$&$-$&$-$&$+$&$+$&13.8793&13.9213&13.8532&14.0849&14.0121&13.9386&14.2118&14.0789\\
$-$&$-$&$+$&$-$&$+$&$+$&$-$&$-$&13.4054&13.4788&13.5878&13.5167&14.2444&14.2573&14.3951&14.3724\\
$-$&$+$&$-$&$-$&$-$&$+$&$-$&$+$&14.1736&14.0306&14.1398&14.0796&14.1492&14.1654&14.1487&14.2765\\
$-$&$+$&$-$&$-$&$+$&$-$&$+$&$-$&13.2539&13.3338&13.1920&13.4430&14.2204&14.3028&14.2689&14.4104\\
$-$&$+$&$+$&$+$&$-$&$+$&$+$&$-$&14.0623&14.0888&14.1766&14.0528&15.2969&15.5209&15.4200&15.2077\\
$-$&$+$&$+$&$+$&$+$&$-$&$-$&$+$&14.3068&14.4055&14.6780&14.5811&15.0100&15.0618&15.5724&15.4668\\
$+$&$-$&$-$&$-$&$-$&$+$&$+$&$-$&13.7259&13.2934&12.6502&13.2666&14.9039&14.7952&14.1886&14.6254\\
$+$&$-$&$-$&$-$&$+$&$-$&$-$&$+$&13.8953&14.5597&14.4492&13.7064&13.7546&14.3229&14.2224&13.8209\\
$+$&$-$&$+$&$+$&$-$&$+$&$-$&$+$&14.2201&14.3974&15.2757&15.0363&14.1936&14.4295&15.5537&15.2200\\
$+$&$-$&$+$&$+$&$+$&$-$&$+$&$-$&13.5228&13.5828&14.2822&13.8449&14.5640&14.4670&15.2293&15.1099\\
$+$&$+$&$-$&$+$&$-$&$-$&$+$&$+$&14.5335&14.2492&14.6701&15.2799&14.7437&14.1827&14.9695&15.5484\\
$+$&$+$&$-$&$+$&$+$&$+$&$-$&$-$&14.5676&14.0310&13.7099&14.6375&15.8717&15.2239&14.9700&16.0001\\
$+$&$+$&$+$&$-$&$-$&$-$&$-$&$-$&12.9012&12.7071&13.1484&13.8940&14.2537&13.8368&14.1332&15.1681\\
$+$&$+$&$+$&$-$&$+$&$+$&$+$&$+$&13.9532&14.0830&14.1119&13.5963&13.8136&14.0745&14.4313&13.6862\\
%$-$&$-$&$-$&$-$&$-$&$-$&$-$&$-$&14.812&14.774&14.772&14.860\\
%$-$&$-$&$-$&$-$&$+$&$+$&$+$&$+$&14.886&14.810&14.868&14.958\\
%$-$&$-$&$+$&$+$&$-$&$-$&$+$&$+$&13.996&13.988&14.044&14.108\\
%$-$&$-$&$+$&$+$&$+$&$+$&$-$&$-$&13.860&13.876&13.932&13.896\\
%$-$&$+$&$-$&$+$&$-$&$+$&$-$&$+$&14.182&14.172&14.126&14.154\\
%$-$&$+$&$-$&$+$&$+$&$-$&$+$&$-$&13.768&13.778&13.870&13.932\\
%$-$&$+$&$+$&$-$&$-$&$+$&$+$&$-$&14.722&14.736&14.774&14.682\\
%$-$&$+$&$+$&$-$&$+$&$-$&$-$&$+$&14.758&14.784&15.054&14.938\\
%$+$&$-$&$-$&$+$&$-$&$+$&$+$&$-$&14.324&14.092&13.536&13.964\\
%$+$&$-$&$-$&$+$&$+$&$-$&$-$&$+$&13.970&14.448&14.326&13.738\\
%$+$&$-$&$+$&$-$&$-$&$+$&$-$&$+$&14.184&14.402&15.544&15.036\\
%$+$&$-$&$+$&$-$&$+$&$-$&$+$&$-$&13.918&14.044&14.926&14.504\\
%$+$&$+$&$-$&$-$&$-$&$-$&$+$&$+$&14.648&14.350&14.682&15.384\\
%$+$&$+$&$-$&$-$&$+$&$+$&$-$&$-$&15.272&14.656&14.258&15.198\\
%$+$&$+$&$+$&$+$&$-$&$-$&$-$&$-$&13.614&13.202&13.704&14.432\\
%$+$&$+$&$+$&$+$&$+$&$+$&$+$&$+$&13.866&14.130&14.256&13.640\\
\hline
\end{tabular}
\label{tab:ch8layerdesdat}
\end{table}

\es

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\bs
  \heading{Leaf Spring Experiment}

\begin{table}
\caption{Factors and Levels, Leaf Spring Experiment}
\begin{center}
%\renewcommand{\baselinestretch}{1}
%\large\normalsize
\tiny
\begin{tabular}{|ll|ll|} \hline
\multicolumn{2}{|c}{} &
\multicolumn{2}{|c|}{Level} \\
\multicolumn{2}{|c}{Control Factor} & \multicolumn{1}{|c}{$-$} &
\multicolumn{1}{c|}{$+$} \\ \hline
$B$.& high heat temperature ($^{\circ}$F)& 1840& 1880\\
$C$.& heating time (seconds)& 23 & 25\\
$D$.& transfer time (seconds)& 10 & 12\\
$E$.& hold down time (seconds)& 2 & 3\\ \hline
\multicolumn{2}{|c}{} &
\multicolumn{2}{|c|}{Level} \\
\multicolumn{2}{|c}{Noise Factor} & \multicolumn{1}{|c}{$-$} &
\multicolumn{1}{c|}{$+$} \\ \hline $Q$.& quench oil temperature
($^{\circ}$F)& 130-150& 150-170\\ \hline
\end{tabular}
\end{center}
%\renewcommand{\baselinestretch}{1.5}
\large\normalsize \label{tab:robustspringfac}
\end{table}

\begin{table}
\caption{Cross Array and Height Data, Leaf Spring Experiment}
\begin{center}
%\renewcommand{\baselinestretch}{1}
%\large\normalsize
\tiny
\begin{tabular}{|rrrr|rrr|rrr|} \hline
\multicolumn{4}{|c|}{Control Factor} &
\multicolumn{6}{c|}{Noise Factor} \\
\multicolumn{1}{|c}{$B$} & \multicolumn{1}{c}{$C$} &
\multicolumn{1}{c}{$D$} & \multicolumn{1}{c|}{$E$} &
\multicolumn{3}{c}{$Q^{-}$}& \multicolumn{3}{c|}{$Q^{+}$} \\
\hline
$-$&$+$&$+$&$-$&7.78&7.78&7.81 &7.50&7.25&7.12\\
+  &$+$&$+$&+&8.15&8.18&7.88 &7.88&7.88&7.44\\
$-$&$-$&$+$&+&7.50&7.56&7.50 &7.50&7.56&7.50\\
+  &$-$&$+$&$-$&7.59&7.56&7.75 &7.63&7.75&7.56\\
$-$&$+$&$-$&+&7.94&8.00&7.88 &7.32&7.44&7.44\\
+  &$+$&$-$&$-$&7.69&8.09&8.06 &7.56&7.69&7.62\\
$-$&$-$&$-$&$-$&7.56&7.62&7.44 &7.18&7.18&7.25\\
+  &$-$&$-$&+&7.56&7.81&7.69 &7.81&7.50&7.59\\ \hline
\end{tabular}
\end{center}
\label{tab:robustspringdesdat}
\end{table}

\es

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% 7 -----------------------------------------------------------------

\bs
  \heading{Strategies for Variation Reduction}

\bi

 \item \emph{\textbf{Sampling inspection}}: passive, sometimes last resort.
 \item \emph{\textbf{Control charting and process monitoring}}: can remove special
 causes. If the process is stable, it can be \emph{followed} by using a
 \emph{designed experiment}.
 \item \emph{\textbf{Blocking, covariate adjustment}}: passive measures but useful
 in reducing variability, not for removing root causes.
 \item \emph{\textbf{Reducing variation in noise factors}}: effective as it may
 reduce variation in the response but can be expensive. Better
 approach is to change control factor settings (\emph{cheaper} and \emph{easier}
 to do) by exploiting control-by-noise interactions, i.e., use
 robust parameter design!

\ei \es

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% 8 -----------------------------------------------------------------

\bs
  \heading{Types of Noise Factors}

\begin{enumerate}

 \item Variation in process parameters.
 \item Variation in product parameters.
 \item Environmental variation.
 \item Load Factors.
 \item Upstream variation.
 \item Downstream or user conditions.
 \item Unit-to-unit and spatial variation.
 \item Variation over time.
 \item Degradation.

\end{enumerate}

\bi
 \item \textbf{Traditional design uses 7 and 8}.
\ei \es

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% 9 -----------------------------------------------------------------

\bs
  \heading{Variation Reduction Through RPD}

\bi

 \item Suppose $y = f(x,z)$, $x$ control factors and $z$ noise
 factors. If $x$ and $z$ interact in their effects on $y$, then
 the $var(y)$ can be reduced either by reducing $var(z)$ (i.e.
 method 4 on p.7) or by changing the $x$ values (i.e., RPD).
 \item An example:
\begin{eqnarray*}
y & =  &\mu + \alpha x_1 + \beta z + \gamma x_2 z + \epsilon, \\
&= & \mu + \alpha x_1 + (\beta  + \gamma x_2)  z + \epsilon.
\end{eqnarray*}
By choosing an appropriate value of $x$ to reduce the coefficient
$\beta + \gamma x_2$, the impact of $z$ on $y$ can be reduced.
Since $\beta$ and $\gamma$ are unknown, this can be achieved by
using the control-by-noise interaction plots or other methods to
be presented later.

\ei \es

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% 10 -----------------------------------------------------------------

\bs
  \heading{Exploitation of Nonlinearity}

\bi
\small
 \item Nonlinearity between $y$ and ${\bf x}$ can be exploited for
 robustness if ${\bf x_0}$, nominal values of ${\bf x}$, are control factors
 and deviations of ${\bf x}$ around ${\bf x_0}$ are viewed as noise factors
 (called \emph{internal noise}). Expand $y = f({\bf x})$ around ${\bf x_0}$,
 \begin{eqnarray}
y \approx f({\bf x_0}) + \sum_i {\left.\frac{\partial f}{\partial
x_i}\right|}_{x_{i0}}(x_i - x_{i0}).\nonumber \\
\end{eqnarray}
This leads to
\begin{eqnarray*} \sigma^2
\approx \sum_i {\left({\left.\frac{\partial f}{\partial
x_i}\right|}_{x_{i0}}\right)}^2\sigma^2_i,
\end{eqnarray*}
where $\sigma^2 = var(y)$, $\sigma_i^2 = var(x_i)$, each component
$x_i$ has mean $x_{i0}$ and variance $\sigma_i^2$.

\item From (1), it can be seen that $\sigma^2$ can be reduced by
choosing $x_{i0}$ with a smaller slope ${\left.\frac{\partial
f}{\partial x_i}\right|}_{x_{i0}}$. This is demonstrated in Figure
1. Moving the nominal value $a$ to $b$ can reduce $var(y)$ because
the slope at $b$ is more flat. This is a \textbf{parameter design}
step. On the other hand, reducing the variation of $x$ around $a$
can also reduce $var(y)$. This is a \textbf{tolerance design}
step.


\ei \es

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% 11 -----------------------------------------------------------------

\bs
  \heading{Exploitation of Nonlinearity to Reduce Variation}

\begin{figure}
\centerline{\psfig{figure=picture/robustnonlinear.ps,width=2.75in,height=4.5in,angle=-90}}
%\vspace{3.5in}
\caption{Exploiting the Nonlinearity of $f(x)$ to Reduce
Variation} \label{fig:vartrans}
\end{figure}


\es

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% 12 -----------------------------------------------------------------

\bs
  \heading{Cross Array and Location-Dispersion Modeling}

\vspace{.2 in}
 \bi
 \item Cross array = control array $\times$ noise array, \\
 control array = array for control factors, \\
 noise array = array for noise factors.\\

 \item Location-dispersion modeling
  \bi
   \item compute $\bar{y_i}, s_i^2$ based on the noise
settings for the $i^{th}$ control setting,
   \item analyze $\bar{y_i}$ (location), and $\ln s_i^2$
(dispersion), identify significant location and dispersion
effects.
 \ei

\ei \es

%---------------------------------------------------------------------
% 13 -----------------------------------------------------------------

\bs
  \heading{Two-step Procedures for Parameter Design Optimization}
\small
\bi
 \item {\bf Two-Step Procedure for Nominal-the-Best Problem}
\begin{equation}
\begin{array}{l}
{(i)}\; select\; the\; levels\; of\; the\; dispersion\; factors\;
to\; minimize\;
dispersion, \\
{(ii)}\; select\; the\; level\; of\; the\; adjustment\; factor\;
to\; bring\; the\; location\; on \; target\;.
\end{array}
\label{eqn:NTBTwoStepProc}
\end{equation}

\item {\bf Two-Step Procedure for Larger-the-Better and
Smaller-the-Better Problems}
\begin{equation}
\begin{array}{l}
{(i)}\; select\; the\; levels\; of\; the\;
location\; factors\; to\; maximize\; (or\; minimize)\\ the\; location, \\
{(ii)}\; select\; the\; levels\; of\; the\; dispersion\;
factors\; that\; are\; not\; location\\ factors\;
 to\; minimize\; dispersion.
\end{array}
\label{eqn:LTBSTBTwoStepProc}
\end{equation}

Note that the two steps in (3) are in reverse order from those in
(2). \\Reason: It is usually harder to increase or decrease the
response $y$ in the latter problem, so this step should be the
first to perform. \ei \es

%---------------------------------------------------------------------
% 14 -----------------------------------------------------------------

\bs
  \heading{Analysis of Layer Growth Experiment}
\bi
 \item From the $\bar{y_i}$ and
$\ln s^2_i$ columns of Table 5, compute the factorial effects for
location and dispersion respectively. (These numbers are not given
in the book.) From the half-normal plots of these effects (Figure
2), $D$ is significant for location and $H$, $A$ for dispersion.
\begin{eqnarray*}
\hat{y}&=& 14.352 +0.402 x_D,  \\
\hat{z}&=& -1.822 + 0.619 x_A -0.982 x_H. \\
\end{eqnarray*}

\item Two-step procedure:\\
(i) choose $A$ at the ``$-$" level (continuous rotation) and $H$ at the ``$+$" level (nozzle position 6).\\
(ii) By solving
\begin{eqnarray*}
\hat{y}=14.352+0.402x_D =14.5,
\end{eqnarray*}
choose $x_D$= 0.368.

 \ei \es

%---------------------------------------------------------------------
% 15 -----------------------------------------------------------------

\bs
  \heading{Layer Growth Experiment: Analysis Results}

\begin{table}
\caption{Means, Log Variances and SN Ratios, Layer Growth
Experiment}
$ $\\
\centering\tiny
\begin{tabular}{|rrrrrrrr|rrrr|} \hline
\multicolumn{8}{|c|}{Control Factor} &
\multicolumn{4}{c|}{} \\
\multicolumn{1}{|c}{$A$} & \multicolumn{1}{c} {$B$} &
\multicolumn{1}{c} {$C$} & \multicolumn{1}{c} {$D$} &
\multicolumn{1}{c} {$E$} & \multicolumn{1}{c} {$F$} &
\multicolumn{1}{c} {$G$} & \multicolumn{1}{c|}{$H$} &
\multicolumn{1}{c}{$\bar{y}_i$} & \multicolumn{1}{c}{$\ln s^2_i$}
& \multicolumn{1}{c}{$\ln \bar{y}_i^2$} &
\multicolumn{1}{c|}{$\hat{\eta}_i$} \\ \hline
$-$&$-$&$-$&$+$&$-$&$-$&$-$&$-$& 14.79 & -1.018 & 5.389 &
6.41\\
$-$&$-$&$-$&$+$&$+$&$+$&$+$&$+$& 14.86 & -3.879 & 5.397 &
9.28\\
$-$&$-$&$+$&$-$&$-$&$-$&$+$&$+$& 14.00 & -4.205 & 5.278 &
9.48\\
$-$&$-$&$+$&$-$&$+$&$+$&$-$&$-$& 13.91 & -1.623 & 5.265 &
6.89\\
$-$&$+$&$-$&$-$&$-$&$+$&$-$&$+$& 14.15 & -5.306 & 5.299 &
10.60\\
$-$&$+$&$-$&$-$&$+$&$-$&$+$&$-$& 13.80 & -1.236 & 5.250 &
6.49\\
$-$&$+$&$+$&$+$&$-$&$+$&$+$&$-$& 14.73 & -0.760 & 5.380 &
6.14\\
$-$&$+$&$+$&$+$&$+$&$-$&$-$&$+$& 14.89 & -1.503 & 5.401 &
6.90\\
$+$&$-$&$-$&$-$&$-$&$+$&$+$&$-$& 13.93 & -0.383 & 5.268 &
5.65\\
$+$&$-$&$-$&$-$&$+$&$-$&$-$&$+$& 14.09 & -2.180 & 5.291&
7.47\\
$+$&$-$&$+$&$+$&$-$&$+$&$-$&$+$& 14.79 & -1.238 & 5.388 &
6.63\\
$+$&$-$&$+$&$+$&$+$&$-$&$+$&$-$& 14.33 & -0.868 & 5.324 &
6.19\\
$+$&$+$&$-$&$+$&$-$&$-$&$+$&$+$& 14.77 & -1.483 & 5.386 &
6.87\\
$+$&$+$&$-$&$+$&$+$&$+$&$-$&$-$& 14.88 & -0.418 & 5.400 &
5.82\\
$+$&$+$&$+$&$-$&$-$&$-$&$-$&$-$& 13.76 & -0.418 & 5.243 &
5.66\\
$+$&$+$&$+$&$-$&$+$&$+$&$+$&$+$& 13.97 & -2.636 & 5.274& 7.91\\
\hline
\end{tabular}
\label{tab:ch8layermeanvar}
\end{table}

\es

%---------------------------------------------------------------------
% 16 -----------------------------------------------------------------

\bs
  \heading{Layer Growth Experiment: Plots}

\begin{figure}
\centerline{\psfig{figure=picture/layer8hnp.ps,width=3.5in,height=2.8in}}
%\vspace{3.5in}
\caption{\scriptsize Half-Normal Plots of Location and Dispersion
Effects, Layer Growth Experiment} \label{fig:layer8hnp}
\end{figure}

\es

%---------------------------------------------------------------------
% 17 -----------------------------------------------------------------

\bs
  \heading{Analysis of Leaf Spring Experiment}
\bi
 \item Based on the half-normal plots in Figure 3, $B$,
$C$ and $E$ are significant for location, $C$ is significant for
dispersion:
\begin{eqnarray*}
\hat{y}&=& 7.6360 +0.1106 x_B +0.0881 x_C +0.0519 x_E, \\
\hat{z}&=& -3.6886 + 1.0901 x_C.
\end{eqnarray*}
 \item Two-step procedure:\\
(i) choose $C$ at $-$.\\
(ii) With $x_C = -1,\;\;\; \hat{y} = 7.5479 + 0.1106 x_B + 0.0519
x_E$.\\

To achieve $\hat{y} = 8.0$, $x_B$ and $x_E$ must be chosen beyond
$+1$, i.e., $x_B = x_E = 2.78$. This is too drastic, and not
validated by current data. An alternative is to select $x_B = x_E
= x_C = +1$ (not to follow the two-step procedure), then
$\hat{y}$=7.89 is closer to 8. (Note that $\hat{y} = 7.71$ with
$B_+C_-E_+$.) Reason for the breakdown of the 2-step procedure:
its second step cannot achieve the target 8.0.

\ei \es

%---------------------------------------------------------------------
% 18 -----------------------------------------------------------------

\bs
  \heading{Leaf Spring Experiment: Analysis Results}

\begin{table}
\caption{Means and Log Variances, Leaf Spring Experiment}
\begin{center}
%\renewcommand{\baselinestretch}{1}
%\large\normalsize
\small
\begin{tabular}{|rrrr|rr|} \hline
\multicolumn{4}{|c|}{Control Factor} &
\multicolumn{2}{c|}{} \\
\multicolumn{1}{|c}{$B$} & \multicolumn{1}{c}{$C$} &
\multicolumn{1}{c}{$D$} & \multicolumn{1}{c|}{$E$} &
\multicolumn{1}{c}{$\bar{y_i}$} & \multicolumn{1}{c|}{$\ln s^2_i$}
\\ \hline
$-$&$+$&$+$&$-$&  7.540 & -2.4075\\
+  &$+$&$+$&+&  7.902 & -2.6488\\
$-$&$-$&$+$&+&  7.520& -6.9486\\
+  &$-$&$+$&$-$&  7.640 & -4.8384\\
$-$&$+$&$-$&+&  7.670& -2.3987\\
+  &$+$&$-$&$-$&  7.785& -2.9392\\
$-$&$-$&$-$&$-$&  7.372& -3.2697\\
+  &$-$&$-$&+&  7.660& -4.0582 \\ \hline
\end{tabular}
\end{center}
\label{tab:leafmeanvar}
\end{table}
\es

%---------------------------------------------------------------------
% 19 -----------------------------------------------------------------

\bs
  \heading{Leaf Spring Experiment: Plots}

\begin{figure}
\centerline{\psfig{figure=picture/leaf8hnp.ps,width=3.5in,height=2.8in}}
%\vspace{3.5in}
\caption{\scriptsize Half-Normal Plots of Location and Dispersion
Effects, Leaf Spring Experiment} \label{fig:leaf8hnp}
\end{figure}

\es

%---------------------------------------------------------------------
% 20 -----------------------------------------------------------------

\bs
  \heading{Response Modeling and Control-by-Noise Interaction
  Plots}
\bi
 \item Response Model: model $y_{ij}$ directly in terms of control,
noise effects and control-by-noise interactions.
 \bi
  \item half normal plot of various effects.
  \item regression model fitting, obtaining $\hat{y}$.
 \ei
 \item Make control-by-noise interaction plots for significant effects in $\hat{y}$,
choose \textbf{robust} control settings at which y has a flatter
relationship with noise.
 \item Compute $Var(\hat{y})$ with respect to variation in the noise
 factors. Call $Var(\hat{y})$ the \textbf{transmitted variance model}. Use it to
 identify control factor settings with small transmitted variance.
\ei \es

%---------------------------------------------------------------------
% 21 -----------------------------------------------------------------

\bs
  \heading{Half-normal Plot, Layer Growth Experiment}
\bi
 \item Define
\begin{eqnarray*}
M_l = (M_1+M_2) - (M_3+M_4), \\
M_q = (M_1+M_4) - (M_2+M_3), \\
M_c = (M_1+M_3) - (M_2+M_4),
\end{eqnarray*}
 \item From Figure 4, select $D$, $L$, $HL$ as the most
 significant effects.
 \item How to deal with the next cluster of effects in Figure 4?
 Use \textbf{step-down multiple comparisons}.
 \item After removing the top three points in Figure 4, make a
 half-normal plot (Figure 5) on the remaining points. The cluster
 of next four effects ($M_l, H, CM_l, AHM_q$) appear to be
 significant.

\ei

\es

%---------------------------------------------------------------------
% 22 -----------------------------------------------------------------

\bs
  \heading{Half-normal Plot of Factorial Effects}

\begin{figure}
\centerline{\psfig{figure=picture/rmlayer8hnp.ps,width=3.5in,height=2.8in}}
%\vspace{3.5in}
\caption{\scriptsize Half-Normal Plot of Response Model Effects,
Layer Growth Experiment} \label{fig:rmlayer8hnp}
\end{figure}

\es

% 22-2nd -----------------------------------------------------------------

\bs
  \heading{Second Half-normal Plot of Factorial Effects}

\begin{figure}
\centerline{\psfig{figure=picture/half-normal2.ps,width=3in,height=2.5in}}
%\vspace{3.5in}
\caption{\scriptsize Second Half-Normal Plot of Response Model
Effects, Layer Growth Experiment} \label{fig:rmlayer8hnp}
\end{figure}

\es

%---------------------------------------------------------------------
% 23 -----------------------------------------------------------------

\bs
  \heading{Control-by-noise Interaction Plots}

\begin{figure}
\centerline{\psfig{figure=picture/newlayer.hlcm.ps,width=3.5in,height=2.8
in}}
%\vspace{3.5in}
\caption{\scriptsize $H \times L$ and $C \times M$ Interaction
Plots, Layer Growth Experiment} \label{fig:rmlayer8hnp}
\end{figure}

\es

%---------------------------------------------------------------------
% 24 -----------------------------------------------------------------

\bs
  \heading{$A \times H \times M$ Plot}

\begin{figure}
\centerline{\psfig{figure=picture/layerintplot.ahm.ps,width=3.5in,height=2.8
in}}
%\vspace{3.5in}
\caption{\scriptsize $A \times H \times M$ Interaction Plot, Layer
Growth Experiment} \label{fig:rmlayer8hnp}
\end{figure}

\es

%---------------------------------------------------------------------
% 24 -----------------------------------------------------------------

\bs
  \heading{Response Modeling, Layer Growth Experiment}

\bi

 \item The following model is obtained:
 \begin{eqnarray}
\hat{y}&= &14.352+ 0.402 x_D +0.087  x_H  +0.330 x_L - 0.090 x_{M_l} \nonumber \\
&& - 0.239 x_H x_L  - 0.083 x_C x_{M_l} -0.082 x_A x_H x_{M_q}.
\end{eqnarray}
\item Recommendations:\\
 $H$: $-$ (position 2) to $+$ (position 6)\\
 $A$: $+$ (oscillating) to $-$ (continuous)\\
 $C$: $+$ (1210) to $-$ (1220)\\
resulting in 37\% reduction of thickness standard variation.

\ei

\es

%---------------------------------------------------------------------
% 25 -----------------------------------------------------------------

\bs
  \heading{Predicted Variance Model}

\bi
 \item Assume $L$,
$M_l$ and $M_q$ are random variables, taking $-1$ and $+1$ with
equal probabilities. This leads to

\begin{equation}
\begin{array}{l}
x^2_L = x^2_{M_l} = x^2_{M_q} = x^2_A = x^2_C=x^2_H=1, \\
E(x_L)=E(x_{M_l})=E(x_{M_q})=0, \;\;  \\
Cov(x_L, x_{M_l}) =Cov(x_L, x_{M_q})=Cov(x_{M_l}, x_{M_q})=0.
\end{array}
\label{eqn:layerrelations}
\end{equation}

\item From (4) and (5), we have
\begin{eqnarray*}
Var( \hat{y} ) & = & (.330-.239 x_H)^2 Var(x_L)
+ (-.090-.083 x_C)^2 Var(x_{M_l})\nonumber \\
& &+ (.082 x_A x_H)^2 Var(x_{M_q})\nonumber \\
& = & {\rm constant} + {( .330 - .239 x_H )}^2
+ {( -.090 - .083 x_C )}^2 \nonumber \\
& = & {\rm constant} - {2(.330)(.239) x_H }
+ {2(.090)(.083)x_C } \nonumber \\
& = & {\rm constant} - {.158 x_H } + {.015 x_C }.
\label{eqn:layervarianceyhat10}
\end{eqnarray*}

\item Choose $H+$ and $C-$. But factor $A$ is not present here.
(Why? See explanation on p. 532).

\ei \es


%---------------------------------------------------------------------
% 26 -----------------------------------------------------------------

\bs
  \heading{Estimation Capacity for Cross Arrays}

\bi
 \item Example 1. Control array is a $2^{3-1}_{III}$ design with ${\bf I=ABC}$
 and the noise array is a $2^{3-1}_{III}$ design with ${\bf I=abc}$. The resulting cross array is
a 16-run $2^{6-2}_{III}$ design with ${\bf I=ABC=abc=ABCabc}$.
Easy to show that all 9 control-by-noise interactions are clear,
(but not the 6 main effects). This is indeed a general result
stated next.

\textbf{Theorem:}
%\noindent {\bf Theorem {\thechapter}.1.}
\rm Suppose a $2^{k-p}$ design $d_C$ is chosen for the control
array, a $2^{m-q}$ design $d_N$ is chosen for the noise array, and
a cross array, denoted by $d_C \otimes d_N ,$ is constructed from
$d_C$ and
$d_N.$\\
(i)  If $\alpha_1, \ldots, \alpha_A$ are the estimable factorial
effects (among the control factors) in $d_C$ and $\beta_1, \ldots,
\beta_B$ are the estimable factorial effects (among the noise
factors) in $d_N ,$ then $\alpha_i, \beta_j, \alpha_i \beta_j$ for
$i = 1, \ldots, A, \ j = 1, \ldots , B$ are estimable in
$d_C \otimes d_N.$\\
(ii)  All the $km$ control-by-noise interactions (i.e., two-factor
interactions between a control factor main effect and a noise
factor main effect) are clear in $d_C \otimes d_N.$

\ei \es

%---------------------------------------------------------------------
% 27 -----------------------------------------------------------------

\bs
  \heading{Cross Arrays or Single Arrays?}

\vspace{0.3in}

 \bi \item Three control factors $A$, $B$, $C$ two noise
factors $a$, $b$: $2^3 \times 2^2$ design, allowing all main
effects and two-factor interactions to be clearly estimated.\\

\item Use a single array with 16 runs for all five factors: a
resolution V $2^{5-1}$ design with ${\bf I}=ABCab$ or ${\bf
I}=-ABCab$, all main effects and two-factor interactions are
clear. (See Table 7)\\

\item Single arrays can have smaller runs, but cross arrays are
easier to use and interpret.

\ei \es

%---------------------------------------------------------------------
% 28 -----------------------------------------------------------------

 \bs
  \heading{32-run Cross Array and 16-run Single Arrays}
\begin{table}
\caption{32-Run Cross Array}
$ $\\
\centering \small
\begin{tabular*}{10cm}{|r@{--}l@{\extracolsep{\fill}}%
rrrrrrr|} \hline \multicolumn{2}{|c}{\mbox{}}&&& $\;\;\;a$ & + & +
& $-$ &
\multicolumn{1}{c|} {$-$} \\
\multicolumn{2}{|c}{\mbox{}}&&& $\;\;\;b$ & + & $-$ & + &
\multicolumn{1}{c|} {$-$} \\
\multicolumn{2}{|c}{Runs} & $A$ & $B$ & $C$ & \multicolumn{4}{c|}
{} \\ \hline
 1&4 &+ &+ &+  &$\bullet$ &$\circ$      &$\circ$      &$\bullet$ \\
 5&8 &+ &+  &$-$ &$\circ$      &$\bullet$ &$\bullet$ &$\circ$     \\
 9&12&+  &$-$&+  &$\circ$      &$\bullet$ &$\bullet$ &$\circ$     \\
 13&16&+  &$-$ &$-$ &$\bullet$ &$\circ$      &$\circ$      &$\bullet$ \\
 17&20 &$-$&+ &+  &$\circ$      &$\bullet$ &$\bullet$ &$\circ$     \\
 21&24 &$-$&+  &$-$ &$\bullet$ &$\circ$      &$\circ$      &$\bullet$ \\
 25&28 &$-$ &$-$&+  &$\bullet$ &$\circ$      &$\circ$      &$\bullet$ \\
 29&32 &$-$ &$-$ &$-$ &$\circ$      &$\bullet$ &$\bullet$ &$\circ$      \\  \hline
\end{tabular*}
\label{tab:productarray}
\end{table}

\begin{center}
$\bullet$ : ${\bf I}=ABCab$, $\circ$ : ${\bf I}=-ABCab$,
\end{center}
\es

%---------------------------------------------------------------------
% 29 -----------------------------------------------------------------

\bs
  \heading{Comparison of Cross Arrays and Single Arrays}
  \vspace{.5 in}
\bi

\item Example 1 (continued) An alternative is to choose a single
array $2^{6-2}_{IV}$ design with ${\bf I=ABCa}$ ${\bf
=ABbc=abcC}$. This is not advisable because no 2fi's are clear and
only main effects are clear. (Why? We need to have some clear
control-by-noise interactions for robust optimization.) A better
one is to use a $2^{6-2}_{III}$ design with ${\bf I=ABCa}$ ${\bf
=abc=ABCbc}$. It has 9 clear effects: $A, B, C, Ab, Ac, Bb, Bc,
Cb, Cc$ (3 control main effects and 6 control-by-noise
interactions).

\ei \es



% 30 -----------------------------------------------------------------

\bs
  \heading{Signal-to-Noise Ratio}

\bi
 \item Taguchi's SN ratio $\hat{\eta} = \ln \frac{\bar{y}^{2}}{s^2}$
 \item Two-step procedure:
 \begin{enumerate}
 \item Select control factor levels to maximize SN
ratio, \item Use an adjustment factor to move mean on target.
\end{enumerate}
\item \textbf{Limitations}
 \bi
  \item maximizing $\bar{y}^2$ not always desired.
  \item little justification outside linear circuitry.
  \item statistically justifiable only when $Var(y)$ is proportional
  to $E(y)^2$
\ei \item \textbf{Recommendation}: Use SN ratio sparingly. Better
to use the location-dispersion modeling or the response modeling.
The latter strategies can do whatever SN ratio analysis can
achieve.
 \ei \es

% 31 -----------------------------------------------------------------

\bs
  \heading{Half-normal Plot for S/N Ratio Analysis}
  \begin{figure}
\centerline{\psfig{figure=picture/layer8hnpSN.ps,width=2.5in,height=3.8in,angle=-90}}
\caption{Half-Normal Plots of Effects Based on SN Ratio, Layer
Growth Experiment} \label{fig:layer8hnpSNRatio}
\end{figure}
\es

% 32 -----------------------------------------------------------------

\bs
  \heading{S/N Ratio Analysis for Layer Growth Experiment}

\vspace{.4in}

 \bi \item Based on the $\hat{\eta}_i$ column in
Table 5, compute the factorial effects using SN ratio. From Figure
7, the conclusion is similar to location-dispersion analysis. Why?
Using
\begin{eqnarray*}
\hat{\eta}_i = \ln \bar{y_i}^2 - \ln s_i^2,
\end{eqnarray*}
and from Table 5, the variation among $\ln s_i^2$ is much larger
than the variation among $\ln \bar{y_i}^2$; thus maximizing SN
ratio is equivalent to minimizing $\ln s_i^2$ in this case. \ei
\es

% last ------------------------------------------------------------


 \bs
  \heading{Comments on Board}

%\begin{theorem}
%\end{theorem}
\vspace{3 in} \es


\end{document}
