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CH7 (although in CH6 directory)

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Robin Clark 2013-01-20 17:47:14 +00:00
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submission_thesis/CH6_Evaluation/Makefile
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@ -3,7 +3,7 @@
#
# Place all .dia files here as .png targets
#
DIA =
DIA = components_81_euler.png
doc: $(DIA)

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submission_thesis/CH6_Evaluation/components_81_euler.dia Normal file
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submission_thesis/CH6_Evaluation/copy.tex
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@ -123,11 +123,11 @@ The function $fm$ has a component as its domain and the components failure modes
We can represent the number of potential failure modes of a component $c$, to be $ | fm(c) | .$
\paragraph{Indexing components with the group $G$.}
If we index all the components in the system under investigation $ c_1, c_2 \ldots c_{|\FG|} $ we can express
If we index all the components in the system under investigation $ c_1, c_2 \ldots c_{|G|} $ we can express
the number of checks required to rigorously examine every
failure mode against all the other components in the system.
We can define this as a function, Comparison Complexity, can be represented by a function $CC$, with its domain as $G$, and
Comparison Complexity can be represented by a function $CC$, with its domain as $G$, and
its range as the number of checks---or reasoning stages---to perform to satisfy a rigorous FMEA inspection.
Where $\mathcal{G}$ represents the set of all {\fgs}, and $ \mathbb{N} $ any natural integer, $CC$ is defined by,
@ -174,10 +174,10 @@ to identify the hierarchy level.
% \end{equation}
\subsection{A general formula for counting Comparison Complexity in an FMMD hierarchy}
An FMMD Hierarchy will have reducing numbers of functional groups as we progress up the hierarchy.
An FMMD Hierarchy will have reducing numbers of {\fgs} as we progress up the hierarchy.
In order to calculate its comparison~complexity we need to apply equation~\ref{eqn:CC} to
all {\fgs} on each level.
We can define an FMMD hierarchy as a set of {\fgs}, $H$.
We define a helper function $g$ with a domain of the level $i$ in an FMMD hierarchy $H$, and a co-domain of a set of {\fgs} (specifically all the {\fgs} on the given level),
defined by
@ -188,7 +188,7 @@ g(H, i) \rightarrow \forall {\FG}^{\xi} \;where\; ({\xi} = {i}) \wedge ({\FG}^{\
\end{equation}
Where $L$ represents the number of levels in the FMMD hierarchy,
$|g(\xi)|$ represents the number of functional groups on the level
$|g(\xi)|$ represents the number of {\fgs} on the level
and $H$ represents an FMMD hierarchy,
we overload the comparison complexity thus:
%$$
@ -202,10 +202,11 @@ we overload the comparison complexity thus:
\pagebreak[4]
\subsection{Complexity Comparison Examples}
The potential divider discussed in section~\ref{potdivfmmd} has four failure modes and two components and therefore has $CC$ of 4.
The potential divider discussed in section~\ref{subsec:potdiv} has four failure modes and two components and therefore has $CC$ of 4.
$$CC(potdiv) = \sum_{n=1}^{2} |2|.(|1|) = 4 $$
Even considering a $example$ system with just 81 components (with these components
%Even considering a $example$
A system, $example$, with just 81 components (with these components
having 3 failure modes each) we would have an $CC$ of
$$CC(example) = \sum_{n=1}^{81} |3|.(|80|) = 19440 .$$
@ -228,16 +229,23 @@ rigorous checking feasible.
\pagebreak[4]
%\subsection{Using the concept of Complexity Comparison to compare RFMEA with FMMD}
\begin{figure}
% \begin{figure}
% \centering
% \includegraphics[width=400pt,keepaspectratio=true]{CH5_Examples/three_tree.png}
% % three_tree.png: 851x385 pixel, 72dpi, 30.02x13.58 cm, bb=0 0 851 385
% \caption{FMMD Hierarchy with number of components in {\fg} fixed to 3 $(|G| = 3)$ } % \wedge (|fm(c)| = 3)$}
% \label{fig:three_tree}
% \end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=400pt,keepaspectratio=true]{CH5_Examples/three_tree.png}
% three_tree.png: 851x385 pixel, 72dpi, 30.02x13.58 cm, bb=0 0 851 385
\caption{FMMD Hierarchy with number of components in {\fg} fixed to 3 $(|G| = 3)$ } % \wedge (|fm(c)| = 3)$}
\includegraphics[width=400pt]{./CH6_Evaluation/components_81_euler.png}
% components_81_euler.png: 3056x2532 pixel, 72dpi, 107.81x89.32 cm, bb=0 0 3056 2532
\caption{FMMD Hierarchy with number of compnents in each $FG$ fixed to three ($|FG|=3$)}
\label{fig:three_tree}
\end{figure}
\subsection{Comparing FMMD and RFMEA comparison complexity}
Because components have variable numbers of failure modes,