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\bs
  \heading{Unit 7: Orthogonal Arrays and Response Surface Methodology}

\bc Source : Sections 8.1 - 8.2, Appendix 8A, 8C; Sections 10.1 -
10.2 \ec

\vspace{.1 in}

\bi
 \item In Tables~1 and 2, the design used does not belong to the
 $2^{k-p}$ series (Chapter~5) or the $3^{k-p}$ series (Chapter~6),
 because the latter would require run size as a power of 2 or 3.
 These designs belong to the class of orthogonal arrays.
 \item An {\bf orthogonal array}  array $OA(N,s_1^{m_1}\ldots s_{\gamma}^{m_{\gamma}}, t)$ of strength $t$
is an $N\times m$ matrix, $m=m_1+\ldots + m_{\gamma}$, in which
$m_i$ columns have $s_i (\geq 2)$ symbols or levels such that, for
any $t$ columns, all possible combinations of symbols appear
equally often in the matrix.
 \item For $OA$ of strength two, the index $t=2$ is dropped for
 simplicity.
 \item An $OA(12,2^{11})$ is used in Table~1 and an $OA(18,2^13^7)$ is
 used in Table~2.
\ei
 \es



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

\bs
  \heading{Example : $OA(12,2^{11})$}
\begin{table}[htbp]
%\label{tab:ex5desdat}
\caption{Design Matrix and Lifetime Data, Cast Fatigue Experiment}
\begin{center}
%\renewcommand{\baselinestretch}{1}
%\large\normalsize

\centering\scriptsize

\begin{tabular}{|c|ccccccccccc|r|} \hline
\multicolumn{1}{|c|}{} & \multicolumn{11}{c}{Factor} &
\multicolumn{1}{|c|}{Logged} \\
\multicolumn{1}{|c|}{Run} & \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}{8} & \multicolumn{1}{c}{9} &
\multicolumn{1}{c}{10} & \multicolumn{1}{c}{11} &
\multicolumn{1}{|c|}{Lifetime} \\ \hline
1 & + & + & $-$ & + & + & + & $-$ & $-$ & $-$ & + & $-$&6.058\\
2 & + & $-$ & + & + & + & $-$ & $-$ & $-$ & + & $-$ & +&4.733\\
3 & $-$ & + & + & + & $-$ & $-$ & $-$ & + & $-$ & + & +&4.625\\
4 & + & + & + & $-$ & $-$ & $-$ & + & $-$ & + & + & $-$&5.899\\
5 & + & + & $-$ & $-$ & $-$ & + & $-$ & + & + & $-$ & +&7.000\\
6 & + & $-$ & $-$ & $-$ & + & $-$ & + & + & $-$ & + & +&5.752\\
7 & $-$ & $-$ & $-$ & + & $-$ & + & + & $-$ & + & + & +&5.682\\
8 & $-$ & $-$ & + & $-$ & + & + & $-$ & + & + & + & $-$&6.607\\
9 & $-$ & + & $-$ & + & + & $-$ & + & + & + & $-$ & $-$&5.818\\
10 & + & $-$ & + & + & $-$ & + & + & + & $-$ & $-$ & $-$&5.917\\
11 & $-$ & + & + & $-$ & + & + & + & $-$ & $-$ & $-$ & +&5.863\\
12 & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ &
$-$&4.809\\ \hline
\end{tabular}

\scriptsize


\end{center}
%\renewcommand{\baselinestretch}{1.5}
%\large\normalsize
\label{tab:ex5desdat}
\end{table}
\es



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

\bs
  \heading{Example : $OA(18,2^13^7)$}


\begin{table}[htbp]
%\label{tab:ex6desdat}
\caption{Design Matrix and Response Data, Blood Glucose
Experiment} \label{tab:ex6desdat}
\begin{center}
%\renewcommand{\baselinestretch}{1}
%\large\normalsize


\centering\tiny


\begin{tabular}{|c|cccccccc|r|} \hline
\multicolumn{1}{|c|}{} & \multicolumn{8}{c|}{Factor} &
\multicolumn{1}{|c|}{Mean} \\
\multicolumn{1}{|c|}{Run} & \multicolumn{1}{c}{$A$} &
\multicolumn{1}{c}{$G$} & \multicolumn{1}{c}{$B$} &
\multicolumn{1}{c}{$C$} & \multicolumn{1}{c}{$D$} &
\multicolumn{1}{c}{$E$} & \multicolumn{1}{c}{$F$} &
\multicolumn{1}{c}{$H$} & \multicolumn{1}{|c|}{Reading} \\ \hline
1  & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 97.94\\
2  & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 83.40\\
3  & 0 & 0 & 2 & 2 & 2 & 2 & 2 & 2 & 95.88\\
4  & 0 & 1 & 0 & 0 & 1 & 1 & 2 & 2 & 88.86\\
5  & 0 & 1 & 1 & 1 & 2 & 2 & 0 & 0 & 106.58\\
6  & 0 & 1 & 2 & 2 & 0 & 0 & 1 & 1 & 89.57\\
7  & 0 & 2 & 0 & 1 & 0 & 2 & 1 & 2 & 91.98\\
8  & 0 & 2 & 1 & 2 & 1 & 0 & 2 & 0 & 98.41\\
9  & 0 & 2 & 2 & 0 & 2 & 1 & 0 & 1 & 87.56\\
10 & 1 & 0 & 0 & 2 & 2 & 1 & 1 & 0 & 88.11\\
11 & 1 & 0 & 1 & 0 & 0 & 2 & 2 & 1 & 83.81\\
12 & 1 & 0 & 2 & 1 & 1 & 0 & 0 & 2 & 98.27\\
13 & 1 & 1 & 0 & 1 & 2 & 0 & 2 & 1 & 115.52\\
14 & 1 & 1 & 1 & 2 & 0 & 1 & 0 & 2 & 94.89\\
15 & 1 & 1 & 2 & 0 & 1 & 2 & 1 & 0 & 94.70\\
16 & 1 & 2 & 0 & 2 & 1 & 2 & 0 & 1 & 121.62\\
17 & 1 & 2 & 1 & 0 & 2 & 0 & 1 & 2 & 93.86\\
18 & 1 & 2 & 2 & 1 & 0 & 1 & 2 & 0 & 96.10\\ \hline
\end{tabular}

\tiny


\end{center}
%\renewcommand{\baselinestretch}{1.5}
%\large\normalsize
\end{table}



 \es


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


\begin{slide}
\heading{Why Using Orthogonal Array}

\begin{itemize}

\item \textbf{Run size economy.} Suppose 8-11 factors at two
levels are to be studied. Using an $OA(12,2^{11})$ will save 4
runs over a 16-run $2^{k-p}$ design. Similarly, suppose 5-7
factors at three levels are to be studied. Using an $OA(18,3^{7})$
will save 9 runs over a 27-run $3^{k-p}$ design.

\item \textbf{Flexibility.} Many $OA$'s exist for flexible
combinations of factor levels. See the collection on next page.


\item Analysis strategy for experiments based on $OA$ can be found
in Chapter 9 of WH.

\end{itemize}

\end{slide}




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


\begin{slide}
\heading{Useful Orthogonal Arrays}
\begin{itemize}

\item Collection in Appendix 8A and 8C of WH:
\begin{center}
\begin{tabular}{ccc}
 *$OA(12,2^{11})$ & $OA(12,3^{1}2^4)$  & *$OA(18,2^{1}3^{7})$\\
 *$OA(18,6^{1}2^{6})$  & $OA(20,2^{19})$  & $OA(24,3^{1}2^{16})$\\
 $OA(24,6^{1}2^{14})$  & *$OA(36,2^{11}3^{12})$ & $OA(36,3^{7}6^{3})$\\
  $OA(36,2^{8}6^{3})$  & $OA(48,2^{11}4^{12})$ &  $OA(50,2^{1}5^{11})$\\
  &  $OA(54,2^{1}3^{25})$ & \\
\end{tabular}
\end{center}
* especially useful


\item Learn to choose and use the design tables in the collection.


\end{itemize}

\end{slide}

 %---------------------------------------------------------------------
%  6-------------------------------------------------------------------
\bs
  \heading{$OA(18,2^13^7)$ and $OA(18,6^13^6)$}
 {\tiny
 \begin{table}[h]
 \begin{center}
\vspace*{-20pt} \caption{$OA(18, 2^{1}3^{7})$ (columns 1--8) and
$OA(18, 6^{1}3^{6})$ (columns $1'$ and 3--8)} \tabcolsep=12pt
\begin{tabular}{@{}c|ccccccccc@{}} \hline
 Run & $1'$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
1 &0 &0 &0 &0 &0 &0 &0 &0 &0 \\
2 &0 &0 &0 &1 &1 &1 &1 &1 &1 \\
3 &0 &0 &0 &2 &2 &2 &2 &2 &2 \\
4 &1 &0 &1 &0 &0 &1 &1 &2 &2 \\
5 &1 &0 &1 &1 &1 &2 &2 &0 &0 \\
6 &1 &0 &1 &2 &2 &0 &0 &1 &1 \\
7 &2 &0 &2 &0 &1 &0 &2 &1 &2 \\
8 &2 &0 &2 &1 &2 &1 &0 &2 &0 \\
9 &2 &0 &2 &2 &0 &2 &1 &0 &1 \\
10 &3 &1 &0 &0 &2 &2 &1 &1 &0 \\
11 &3 &1 &0 &1 &0 &0 &2 &2 &1 \\
12 &3 &1 &0 &2 &1 &1 &0 &0 &2 \\
13 &4 &1 &1 &0 &1 &2 &0 &2 &1 \\
14 &4 &1 &1 &1 &2 &0 &1 &0 &2 \\
15 &4 &1 &1 &2 &0 &1 &2 &1 &0 \\
16 &5 &1 &2 &0 &2 &1 &2 &0 &1 \\
17 &5 &1 &2 &1 &0 &2 &0 &1 &2 \\
18 &5 &1 &2 &2 &1 &0 &1 &2 &0 \\ \hline
\end{tabular}
\end{center}
\end{table}
} \es
 %-------------------------------------------------------------------------
%  7-----------------------------------------------------------------------
\bs
   \heading{$OA(36,2^{11}3^{12})$}
{\tiny
\begin{table}[h]
\begin{center}
\caption{$OA(36, 2^{11}3^{12})$}{%
\tabcolsep=3pt\begin{tabular}{@{}c|ccccccccccccccccccccccc@{}}
\hline
 Run & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 \\
\hline
1 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 \\
2 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 \\
3 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 \\
4 &0 &0 &0 &0 &0 &1 &1 &1 &1 &1 &1 &0 &0 &0 &0 &1 &1 &1 &1 &2 &2 &2 &2 \\
5 &0 &0 &0 &0 &0 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &2 &2 &2 &2 &0 &0 &0 &0 \\
6 &0 &0 &0 &0 &0 &1 &1 &1 &1 &1 &1 &2 &2 &2 &2 &0 &0 &0 &0 &1 &1 &1 &1 \\
7 &0 &0 &1 &1 &1 &0 &0 &0 &1 &1 &1 &0 &0 &1 &2 &0 &1 &2 &2 &0 &1 &1 &2 \\
8 &0 &0 &1 &1 &1 &0 &0 &0 &1 &1 &1 &1 &1 &2 &0 &1 &2 &0 &0 &1 &2 &2 &0 \\
9 &0 &0 &1 &1 &1 &0 &0 &0 &1 &1 &1 &2 &2 &0 &1 &2 &0 &1 &1 &2 &0 &0 &1 \\
10 &0 &1 &0 &1 &1 &0 &1 &1 &0 &0 &1 &0 &0 &2 &1 &0 &2 &1 &2 &1 &0 &2 &1 \\
11 &0 &1 &0 &1 &1 &0 &1 &1 &0 &0 &1 &1 &1 &0 &2 &1 &0 &2 &0 &2 &1 &0 &2 \\
12 &0 &1 &0 &1 &1 &0 &1 &1 &0 &0 &1 &2 &2 &1 &0 &2 &1 &0 &1 &0 &2 &1 &0 \\
13 &0 &1 &1 &0 &1 &1 &0 &1 &0 &1 &0 &0 &1 &2 &0 &2 &1 &0 &2 &2 &1 &0 &1 \\
14 &0 &1 &1 &0 &1 &1 &0 &1 &0 &1 &0 &1 &2 &0 &1 &0 &2 &1 &0 &0 &2 &1 &2 \\
15 &0 &1 &1 &0 &1 &1 &0 &1 &0 &1 &0 &2 &0 &1 &2 &1 &0 &2 &1 &1 &0 &2 &0 \\
16 &0 &1 &1 &1 &0 &1 &1 &0 &1 &0 &0 &0 &1 &2 &1 &0 &0 &2 &1 &2 &2 &1 &0 \\
17 &0 &1 &1 &1 &0 &1 &1 &0 &1 &0 &0 &1 &2 &0 &2 &1 &1 &0 &2 &0 &0 &2 &1 \\
18 &0 &1 &1 &1 &0 &1 &1 &0 &1 &0 &0 &2 &0 &1 &0 &2 &2 &1 &0 &1 &1 &0 &2 \\
\hline
\end{tabular}}
\end{center}
\end{table}
}
\es


 %-------------------------------------------------------------------------
% 8------------------------------------------------------------------------
\bs
   \heading{$OA(36,2^{11}3^{12})$}
{\tiny

\setcounter{table}{3}
\begin{table}[h]
\begin{center}
\caption{Continued}{%
\tabcolsep=3pt\begin{tabular}{@{}c|ccccccccccccccccccccccc@{}}
\hline
 Run & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 \\
\hline
19 &1 &0 &1 &1 &0 &0 &1 &1 &0 &1 &0 &0 &1 &0 &2 &2 &2 &0 &1 &1 &0 &1 &2 \\
20 &1 &0 &1 &1 &0 &0 &1 &1 &0 &1 &0 &1 &2 &1 &0 &0 &0 &1 &2 &2 &1 &2 &0 \\
21 &1 &0 &1 &1 &0 &0 &1 &1 &0 &1 &0 &2 &0 &2 &1 &1 &1 &2 &0 &0 &2 &0 &1 \\
22 &1 &0 &1 &0 &1 &1 &1 &0 &0 &0 &1 &0 &1 &1 &2 &2 &0 &1 &0 &0 &2 &2 &1 \\
23 &1 &0 &1 &0 &1 &1 &1 &0 &0 &0 &1 &1 &2 &2 &0 &0 &1 &2 &1 &1 &0 &0 &2 \\
24 &1 &0 &1 &0 &1 &1 &1 &0 &0 &0 &1 &2 &0 &0 &1 &1 &2 &0 &2 &2 &1 &1 &0 \\
25 &1 &0 &0 &1 &1 &1 &0 &1 &1 &0 &0 &0 &2 &1 &0 &1 &2 &2 &0 &2 &0 &1 &1 \\
26 &1 &0 &0 &1 &1 &1 &0 &1 &1 &0 &0 &1 &0 &2 &1 &2 &0 &0 &1 &0 &1 &2 &2 \\
27 &1 &0 &0 &1 &1 &1 &0 &1 &1 &0 &0 &2 &1 &0 &2 &0 &1 &1 &2 &1 &2 &0 &0 \\
28 &1 &1 &1 &0 &0 &0 &0 &1 &1 &0 &1 &0 &2 &1 &1 &1 &0 &0 &2 &1 &2 &0 &2 \\
29 &1 &1 &1 &0 &0 &0 &0 &1 &1 &0 &1 &1 &0 &2 &2 &2 &1 &1 &0 &2 &0 &1 &0 \\
30 &1 &1 &1 &0 &0 &0 &0 &1 &1 &0 &1 &2 &1 &0 &0 &0 &2 &2 &1 &0 &1 &2 &1 \\
31 &1 &1 &0 &1 &0 &1 &0 &0 &0 &1 &1 &0 &2 &2 &2 &1 &2 &1 &1 &0 &1 &0 &0 \\
32 &1 &1 &0 &1 &0 &1 &0 &0 &0 &1 &1 &1 &0 &0 &0 &2 &0 &2 &2 &1 &2 &1 &1 \\
33 &1 &1 &0 &1 &0 &1 &0 &0 &0 &1 &1 &2 &1 &1 &1 &0 &1 &0 &0 &2 &0 &2 &2 \\
34 &1 &1 &0 &0 &1 &0 &1 &0 &1 &1 &0 &0 &2 &0 &1 &2 &1 &2 &0 &1 &1 &2 &0 \\
35 &1 &1 &0 &0 &1 &0 &1 &0 &1 &1 &0 &1 &0 &1 &2 &0 &2 &0 &1 &2 &2 &0 &1 \\
36 &1 &1 &0 &0 &1 &0 &1 &0 &1 &1 &0 &2 &1 &2 &0 &1 &0 &1 &2 &0 &0 &1 &2 \\
\hline
\end{tabular}}
\end{center}
\end{table}
}
\es






 %---------------------------------------------------------------------
%  9 -----------------------------------------------------------------
\bs
  \heading{Poorman's Response Surface Methodology}

\bi
 \item Consider an experiment to study three quantitative factors
 with up to 5 levels.
 \begin{table}[htbp]
\caption{Factors and Levels, Ranitidine Experiment}
\begin{center}
\begin{tabular}{|l|l|} \hline
\multicolumn{1}{|c|}{Factor} & \multicolumn{1}{c|}{Levels} \\
\hline
$A$. pH & 2, 3.42, 5.5, 7.58, 9\\
$B$. voltage (kV)& 9.9, 14, 20, 26, 30.1\\
$C$. $\alpha$-CD (mM)& 0, 2, 5, 8, 10\\
\hline
\end{tabular}
\end{center}
\label{tab:RanitidineFactor}
\end{table}
 \item The design matrix and the data are given on the next page.
 The design differs from $2^{k-p}$ design in two respects :
\bi \item 6 replicates at the center,

\item 6 runs along the three axes. \ei
 It belongs to the class of {\it central composite designs}.




\ei \es



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


\bs \heading{Ranitidine Experiment}

\begin{table}[htbp]
\caption{Design Matrix and Response Data} \centering


\centering\tiny



\begin{tabular}{|r|rrr|rr|} \hline
\multicolumn{1}{|c|}{} & \multicolumn{3}{c|}{Factor} &
\multicolumn{2}{c|}{} \\
\multicolumn{1}{|c|}{Run} & \multicolumn{1}{c}{$A$} &
\multicolumn{1}{c}{$B$} & \multicolumn{1}{c|}{$C$} &
\multicolumn{1}{c}{CEF} & \multicolumn{1}{c|}{$\ln$ CEF} \\ \hline
 1&$-$1 & $-$1 & $-$1 & 17.293 & 2.850\\
 2& 1 & $-$1 & $-$1 & 45.488 & 3.817\\
 3&$-$1 &  1 & $-$1 & 10.311 & 2.333\\
 4& 1 &  1 & $-$1 & 11757.084 & 9.372\\
 5&$-$1 & $-$1 &  1 & 16.942 & 2.830\\
 6& 1 & $-$1 &  1 & 25.400 & 3.235\\
 7&$-$1 &  1 &  1 & 31697.199 & 10.364\\
 8& 1 &  1 &  1 & 12039.201 & 9.396\\
 9&0 & 0 & $-$1.67 & 7.474 & 2.011\\
10&0 & 0 &  1.67 & 6.312 & 1.842\\
11&0 & $-$1.68 & 0 & 11.145 & 2.411\\
12&0 &  1.68 & 0 & 6.664 & 1.897\\
13&$-$1.68 & 0 & 0 & 16548.749 & 9.714\\
14& 1.68 & 0 & 0 & 26351.811 & 10.179\\
15&0 & 0 & 0 & 9.854 & 2.288\\
16&0 & 0 & 0 & 9.606 & 2.262\\
17&0 & 0 & 0 & 8.863 & 2.182\\
18&0 & 0 & 0 & 8.783 & 2.173\\
19&0 & 0 & 0 & 8.013 & 2.081\\
20&0 & 0 & 0 & 8.059 & 2.087\\
\hline
\end{tabular}
\label{tab:RanitidineDesign} \tiny

\end{table}


\es



%---------------------------------------------------------------------
%  11 -----------------------------------------------------------------
\bs
 \heading{Central Composite Designs}
\bi
 \item General definition in Section~10.7 and designs in
 Table~10A.1. (Not required for the course).

 \item A simple CCD is shown graphically below. It has three parts

 (1) $cube$ ( or corner) points,
 (2) $axial$ (or star) points,
 (3) $center$ points.

 \ei


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%\put(64,59.5){\circle{5}}
%%\put(109,59.5){\circle*{3}}
%%\put(109,59.5){\Large $\times$}
%\put(107.2,57.7){\line(1,1){3.6}}
%\put(107.2,61.3){\line(1,-1){3.6}}
%%\put(19,59.5){\circle*{3}}
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%%\put(64,14.5){\circle*{3}}
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%%\put(64,104.5){\circle*{3}}
%\put(62.2,102.7){\line(1,1){3.6}}
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%%\put(37,46){\circle*{3}}
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%%\put(91,73){\circle*{3}}
%\put(90.2,72.2){\line(1,1){3.6}} \put(90.2,75.8){\line(1,-1){3.6}}
%}
%\end{picture}
%}
%
%\caption{\small A Central Composite Design in Three Dimensions
%(cube point (dot), star point(cross), center point (circle)) }
%\label{fig:CCD3Dimensions}
%\end{center}
%\end{figure}
\begin{figure}
\begin{center}
%\vskip1in
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\put(79,59.5){\line(1,0){30}} \put(40,59.5){\line(-1,0){21}}
\put(64,40){\line(0,-1){25.5}} \put(64,74.5){\line(0,1){30}}
\put(55,55){\line(-2,-1){18}} \put(88,72){\line(2,1){4}}
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%\multiput(41.5,59.5)(5,0){5}{\line(1,0){3.5}}
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%\multiput(56.4,55.7)(5.4,2.7){2}{\line(2,1){3.6}}
\multiput(56.4,55.7)(5.4,2.7){1}{\line(2,1){3.6}}
\multiput(87.6,71.8)(-5.4,-2.7){4}{\line(-2,-1){3.6}}
%\multiput(64,41.5)(0,5){4}{\line(0,1){3.5}}
\multiput(64,41.5)(0,5){3}{\line(0,1){3.5}}
%\multiput(64,73)(0,-5){3}{\line(0,-1){3.5}}
\multiput(64,73)(0,-5){2}{\line(0,-1){3.5}}
\put(40,40){\circle*{4}} \put(70,40){\circle*{4}}
\put(40,70){\circle*{4}} \put(70,70){\circle*{4}}
\put(88,49){\circle*{4}} \put(88,79){\circle*{4}}
\put(58,49){\circle*{4}} \put(58,79){\circle*{4}}
\put(64,59.5){\circle{5}}
%\put(109,59.5){\circle*{3}}
%\put(109,59.5){\Large $\times$}
\put(107.2,57.7){\line(1,1){3.6}}
\put(107.2,61.3){\line(1,-1){3.6}}
%\put(19,59.5){\circle*{3}}
\put(17.2,57.7){\line(1,1){3.6}} \put(17.2,61.3){\line(1,-1){3.6}}
%\put(64,14.5){\circle*{3}}
\put(62.2,12.7){\line(1,1){3.6}} \put(62.2,16.3){\line(1,-1){3.6}}
%\put(64,104.5){\circle*{3}}
\put(62.2,102.7){\line(1,1){3.6}}
\put(62.2,106.3){\line(1,-1){3.6}}
%\put(37,46){\circle*{3}}
\put(35.2,44.2){\line(1,1){3.6}} \put(35.2,47.8){\line(1,-1){3.6}}
%\put(91,73){\circle*{3}}
\put(90.2,72.2){\line(1,1){3.6}} \put(90.2,75.8){\line(1,-1){3.6}}
}
\end{picture}
}\caption{A Central Composite Design in Three Dimensions [cube
point (dot), star point (cross), center point (circle)].}
\end{center}
\end{figure}
 \es



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


\bs
 \heading{Sequential Nature of RSM}
\bi
 \item[1.] \textbf{Screening Experiment :} When many variables are
 considered, some are likely to be inert. Use \emph{a $2^{k-p}$ design
 or an $OA$.} If the experimental region is far from the optimum,
 use the \textbf{first-order model}\begin{equation}
y= \beta_0 + \sum_{i=1}^{k} \beta_i x_i + \epsilon ,
\label{eqn:RSMFirstOrderModel}
\end{equation}
to fit the data.
 \item[2.] Based on the fitted model, find the steepest ascent
 direction and perform a search along  this direction (called \textbf{steepest
 ascent search}).

 Steps 1 and 2 may be repeated until reaching the optimum
 region (e.g. peak of the surface).
\ei


\es





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


\bs
 \heading{Sequential Nature of RSM (Contd.)}
\bi
 \item[3.] To capture the curvature effects, use a \textbf{second-order
 design} (like the central composite design). Fit a \textbf{second-order
 model} \begin{equation}
y= \beta_0 + \sum_{i=1}^{k} \beta_i x_i + \sum_{i<j}^{k}
\beta_{ij} x_i x_j + \sum_{i=1}^{k} \beta_{ii} x_i^2 + \epsilon
\label{eqn:RSMSecondOrderModel}
\end{equation}
to data. Use the fitted model (with insignificant terms dropped)
to do \emph{contour plots} and find the \emph{optimum} conditions.

A graphical illustration of these steps is given on next page. \ei
\es

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


\bs \heading{Sequential Exploration of Response Surface}


\begin{figure}[htbp]
\centerline{\psfig{figure=picture/fakeFOseq.ps,height=3.76in,width=2.76in,angle=-90}}
%\caption{Sequential Exploration of the  Response Surface}
\label{fig:SyntheticSeqRS}
\end{figure}

\es


\end{document}
