\appendix 
\section{Actual Screening Designs}
%%%\vspace*{-0.2in}
\label{exampleSc}

\iffalse
Moreover, this fractional factorial design have two desirable
properties: it is orthogonal and it is balanced. An experimental
design is orthogonal if the effects of any factor sum to zero across
the effects of the other factors. And, an experimental design is
balanced if any factor setting combinations have the same number of
observations. These properties led reliable assessment of factorial
effects by ensuring that for every pair of factors all their setting
combinations appear in the same number of runs.

%% Cemal -- explain orthogonality and balance are important

One-factor-at-a-time approach investigates the effect of each factor
in isolation of other factors. This approach selects a potentially
important factor, holds all the remaining factors fixed, and analyzes
the observations to understand the effect of the free factor by
varying it. It then moves onto the next factor and repeats the same
steps.
  
One-factor-at-a-time approach has the following disadvantages when
compared to screening designs.

In this approach, in what order the factors should be examined and
what values the remaining factors should have is at question. Even in
situations where we have answers to these questions,
one-factor-at-a-time approach has the following disadvantages when
compared to screening designs:

\begin{itemize} 

\item It does not conduct a systematic search over the factor settings
  and/or the combinations of factor settings. In what order the
  factors should be examined and what values the remaining factors
  should have is not clear. This can lead to unbalanced and
  none-orthogonal observations, which, in turn, results in unreliable
  effect computations. Therefore conclusions from one-factor-at-a-time
  experiments are not general. Moreover, this approach can led optimal
  settings  of factors escape detection.
 
\item It requires more runs for the same precision in effect
  estimation. For example, in our simple example given above, the main
  effect of an option is computed by using 8 observations (see the
  ME(.) formula). In order the one-factor-at-a-time approach has the
  same precision, it should make 8 observations for each options,
  which corresponds to a total of 32 observations. 
  
\end{itemize}
\fi  

The screening designs used in Section~\ref{step1} were calculated
using the SAS statistical package.  (\url{www.sas.com}).

\noindent
{\bf $Scr_{32}$} is a 
$2^{14-9}_{IV}$ with design generators
$F=ABC$, $G=ABD$, $H=ACD$, $I=BCD$, $J=ABE$, $K=ACE$, $L=BCE$,
$M=ADE$, $N=BDE$. 

\noindent
{\bf $Scr_{64}$} is a
$2^{14-8}_{IV}$ with design generators
$G=ABC$, $H=ABD$, $I=ABE$, $J=ACDE$, $K=ABF$, $L=ACDF$,
$M=ACEF$, $N=ADEF$. 

\noindent
{\bf $Scr_{128}$} is a
$2^{14-7}_{IV}$ with design generators
$H=ABC$, $I=ABDE$, $J=ABDF$, $K=ACEF$, $L=ACDG$,
$M=ABEFG$, $N=BCDEFG$.
 
The screening designs used in Section~\ref{extensions} were calculated
using the SAS statistical package.  (\url{www.sas.com}).


\noindent
It is a $2_{VI}^{14-3}$ (2048-run) design with design generators
$L = ABCDEFGHIJK$, $M = EFGHIJK$, and $N = CDGHIJK$.
