Changed the notation FG now G, RD now CC

and entered a general formula for finding the
Comparison Complexity of an FMMD hierarchy.

Sent it ( ref section 9.1) to Andrew to see
idf its a clear description.

opamp_circuits_C_GARRETT/opamps.tex

@@ -19,7 +19,7 @@
 \newcommand{\bcs}{\em base~components}
 \newcommand{\irl}{in~real~life}
 \newcommand{\abslevel}{\ensuremath{\psi}}
-
+\newcommand{\FG}{\ensuremath{G}}
 
 %\usepackage{glossary}
 %opening
@@ -880,20 +880,20 @@ of components.
 %We thus define $FG$ as a set of chosen components defining 
 %a {\fg}; all functional groups 
 We can state that
-$FG$ is a member of the power set of all components, $ FG \in \mathcal{P} \mathcal{C}. $
+{\FG} is a member of the power set of all components, $ \FG \in \mathcal{P} \mathcal{C}. $
 
-We can overload the $fm$ function for a functional group $FG$
+We can overload the $fm$ function for a functional group {\FG}
-where it will return all the failure modes of the components in $FG$
+where it will return all the failure modes of the components in {\FG}
 
 
 given by
 
 $$ fm (FG) = F. $$
 
-Generally, where $\mathcal{FG}$ is the set of all functional groups,
+Generally, where $\mathcal{{\FG}}$ is the set of all functional groups,
 
 \begin{equation}
-fm : \mathcal{FG} \rightarrow \mathcal{P}\mathcal{F}.
+fm : \mathcal{{\FG}} \rightarrow \mathcal{P}\mathcal{F}.
 \end{equation}
 
 
@@ -914,7 +914,7 @@ We can further define the abstraction level of a {\fg}.
 We can say that it is the maximum abstraction level of any of its
 components. Thus a functional group containing only base components
 would have an abstraction level zero and could be represented with a superscript of zero thus
-`$FG^0$'. The functional group set may also be indexed.
+`${\FG}^0$'. % The functional group set may also be indexed.
 
 We can apply symptom abstraction to a {\fg} to find
 its symptoms. 
@@ -930,7 +930,12 @@ The symptom abstraction process must always raise the abstraction level
 for the newly created {\dc}.
 Using $\abslevel$ to symbolise the fault abstraction level, we can now state:
 
-$$ \bowtie(FG^{\abslevel}) \rightarrow c^{{\abslevel}+N} | N \ge 1. $$ 
+$$ \bowtie({\FG}^{\abslevel}) \rightarrow c^{{\abslevel}+N} | N \ge 1. $$ 
+
+\paragraph{Functional Groups may be indexed}
+We will typically have more than one {\fg} on each level of FMMD hierarchy ( expect the top level where there will only be one)
+we could index the {\fgs} with a sub-script, and can then uniquely identify them using their level and their index.
+For example ${\FG}^{3}_{2}$ would be the second  {\fg} at the third level of abstraction in an FMMD hierarchy.
 
 \paragraph{The symptom abstraction process in outline.} 
 The $\bowtie$ function processes each component in the {\fg} and 
@@ -978,6 +983,8 @@ Idea stage on this section
 \end{itemize}
 
 \clearpage
+Two areas that cannot be automated. Choosing {\fgs} and the analysis/symptom collection process itself.
+
 \section{Side Effects: A Problem for FMMD analysis}
 A problem with modularising according to functionality is that we can have component failures that would 
 intuitively be associated with one {\fg} that may cause unintended side effects in other
@@ -1060,7 +1067,7 @@ Our logic circuit may be able to cope with $LOW\_VOLTAGE$ and $NOISE\_LF$, but r
 But in order to process these failure modes it must be at a higher stage in the FMMD hierarchy.
 
 \pagebreak[4]
-\section{Defining the concept of `reasoning distance' in FMEA}
+\section{Defining the concept of `comparison~complexity' in FMEA}
 
 %
 % DOMAIN == INPUTS
@@ -1088,73 +1095,99 @@ Obviously, for a small number of components and failure modes we have a smaller
 of checks to make than for a complicated larger system.
 %
 We can consider the system as a large {\fg} of components.
-We represent the number of components in the {\fg} by 
+We represent the number of components in the {\fg} $G$,  by 
-$ | fg | .$
+$ | G | .$
 
 The function $fm$ has a component as its domain and the components failure modes as its range.
 We can represent the number of failure modes in a component $c$, to be  $ | fm(c) | .$
 
-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, $RD$, with its domain as the system 
+We can define this as a function, Comparison Complexity, $CC$, with its domain as the system 
-or {\fg}, $fg$, and
+or {\fg}, $G$, and
 its range as the number of checks to perform to satisfy a rigorous FMEA inspection.
 
 \begin{equation}
-\label{eqn:rd}
+\label{eqn:CC}
 %$$
- RD(fg) = \sum_{n=1}^{|fg|} |fm(c_n)|.(|fg|-1) 
+ %%% when it was called reasoning distance -- 19NOV2011 -- RD(fg) = \sum_{n=1}^{|fg|} |fm(c_n)|.(|fg|-1) 
+ CC(G) = (n-1) \sum_{1 \le i \le n} fm(c_i)
 %$$
 \end{equation}
 
-This can be simplified if we can determine the total number of failure modes in the system $fT$, (i.e. $ fT = \sum_{n=1}^{|fg|} {|fm(c_n)|}$);
+This can be simplified if we can determine the total number of failure modes in the system $K$, (i.e. $ K = \sum_{n=1}^{|G|} {|fm(c_n)|}$);
-equation~\ref{eqn:rd} becomes $$ RD(fg) = fT.(|fg|-1).$$
+equation~\ref{eqn:CC} becomes $$ CC(G) = K.(|G|-1).$$
-Equation~\ref{eqn:rd} can also be expressed as
+%Equation~\ref{eqn:rd} can also be expressed as
+% 
+% \begin{equation}
+% \label{eqn:rd2}
+% %$$
+%  CC(G) = {|G|}.{|fm(c_n)|}.{(|fg|-1)} .
+% %$$
+% \end{equation}
+\subsection{A general formula for counting Comparison Complexity in an FMMD hierarchy}
 
-\begin{equation}
+An FMMD Hierarchy will have reducing numbers of functional groups as we progress up the hierarchy.
-\label{eqn:rd2}
+In order to calculate its comparison~complexity we need to apply equation~\ref{eqn:CC} to
-%$$
+all {\fgs} on each level.
- RD(fg) = {|fg|}.{|fm(c_n)|}.{(|fg|-1)} .
+
-%$$
+Where $L$ represents the number of levels in the FMMD hierarchy,
-\end{equation}
+$|\xi|$ represents the number of functional groups on the level
+and $H$ represents an FMMD hierarchy,
+we overload the comparison complexity equation thus:
+$$
+     \label{eqn:gf}
+      CC(H) = \sum_{\xi=0}^{L} \sum_{j=1}^{|\xi|} CC({G}_{j}^{\xi}).
+$$
 
 
 \pagebreak[4]
-\subsection{Reasoning Distance Examples}
+\subsection{Complexity Comparison Examples}
 
-The potential divider discussed in section~\ref{potdivfmmd} has a four failure modes and two components and therefore has an $RD$ of 4.
+The potential divider discussed in section~\ref{potdivfmmd} has a four failure modes and two components and therefore has an $CC$ of 4.
-$$RD(potdiv) = \sum_{n=1}^{2} |2|.(|1|) = 4 $$
+$$CC(potdiv) = \sum_{n=1}^{2} |2|.(|1|) = 4 $$
 
 Were we to consider a $fictitious$ system with 81 components, with these components
-having 3 failure modes each, we would have an $RD$ of 
+having 3 failure modes each, we would have an $CC$ of 
 
-$$RD(fictitious) = \sum_{n=1}^{81} |3|.(|80|) = 19440 .$$
+$$CC(fictitious) = \sum_{n=1}^{81} |3|.(|80|) = 19440 .$$
 
-This would be the polynomial ($O(N^2)$) result of applying FMEA rigorously (we could term this
+Ensuring all component failure modes are checked against all other components in a system
+-- applying FMEA rigorously -- could be termed
 Rigorous FMEA (RFMEA). 
+The computational order for RFMEA would be polynomial ($O(N^2.K)$) (where $K$ is the variable number of failure modes).
+
+This order may be acceptable in a computational environment: However, the choosing of {\fgs} and the analysis
+process are human activities. It can be seen that it is practically impossible to achieve 
+RFMEA for anything but trival systems. FMMD reduces the comparison complexity enough to make
+rigorous checking feasible. 
 
 
 \pagebreak[4]
-\subsection{Using the concept of Reasoning Distance to compare RFMEA with FMMD}
+\subsection{Using the concept of Complexity Comparison to compare RFMEA with FMMD}
 
 \begin{figure}
  \centering
  \includegraphics[width=400pt,keepaspectratio=true]{./three_tree.png}
  % three_tree.png: 851x385 pixel, 72dpi, 30.02x13.58 cm, bb=0 0 851 385
- \caption{FMMD Hierarchy with $(|fg| = 3)$ } % \wedge (|fm(c)| = 3)$}
+ \caption{FMMD Hierarchy with number of components in {\fg} fixed to 3 $(|G| = 3)$ } % \wedge (|fm(c)| = 3)$}
  \label{fig:three_tree}
 \end{figure}
 
+
+
+\subsection{Comparing FMMD and RFMEA comparison complexity}
+
 Because components have variable numbers of failure modes,
 and {\fgs} have variable numbers of components it is difficult to
-come up with a general formula for comparing the number of checks to make for
+use the general formula for comparing the number of checks to make for
 RFMEA and FMMMD.
 If we were to create an example by fixing the number of components in a {\fg}
 and the number of failure modes per component, we can derive formulae
 to represent the number of checks to make.
  
-Consider $k$ to be the number of components in a {\fg} (i.e. $k=|fg|$), 
+Consider $k$ to be the number of components in a {\fg} (i.e. $k=|{\FG}|$), 
 $f$ is the number of failure modes per component (i.e. $f=|fm(c)|$), and 
 $L$ to be the number of levels in the hierarchy of an FMMD analysis.
 We can represent the number of failure scenarios to check in an FMMD
@@ -1187,19 +1220,19 @@ Adding these together gives $242$ checks to make to perform FMMD (i.e. RFMEA \te
 {\fgs}).
 
 If we were to take the system represented in  figure~\ref{fig:three_tree}, and 
-apply RFMEA on it as a whole system, we can use equation~\ref{eqn:rd},
+apply RFMEA on it as a whole system, we can use equation~\ref{eqn:CC},
-$ RD(fg) = \sum_{n=1}^{|fg|} |fm(c_n)|.(|fg|-1)$, where $|fg|$ is 27, $fm(c_n)$ is 3
+$CC(G) = \sum_{n=1}^{|G|} |fm(c_n)|.(|G|-1)$, where $|G|$ is 27, $fm(c_n)$ is 3
-and $(|fg|-1)$ is 26.
+and $(|G|-1)$ is 26.
 This gives:
-$RD(fg) = \sum_{n=1}^{27} |3|.(|27|-1) = 2106$.
+$CC(G) = \sum_{n=1}^{27} |3|.(|27|-1) = 2106$.
 
 In order to get general equations with which to compare RFMEA with FMMD
-we can re-write equation~\ref{eqn:rd} in terms of the number of levels 
+we can re-write equation~\ref{eqn:CC} in terms of the number of levels 
 in an FMMD hierarchy. 
 %
 The number of components in the system, is number of components
 in a {\fg} raised to the power of the level plus one.
-Thus we re-write equation~\ref{eqn:rd} as:
+Thus we re-write equation~\ref{eqn:CC} as:
  
 
 \begin{equation}
@@ -1216,7 +1249,7 @@ or
   %(N^2 - N).f 
 \end{equation}
 
-We can now use  equation~\ref{eqn:anscen} and \ref{eqn:fmea_state_exp22} to compare (for fixed sizes of $|fg|$ and $|fm(c)|$)
+We can now use  equation~\ref{eqn:anscen} and \ref{eqn:fmea_state_exp22} to compare (for fixed sizes of $|G|$ and $|fm(c)|$)
 the two approaches, for the work required to perform rigorous checking.