\section*{Metrics}

\section{Defining the concept of `comparison~complexity' in FMEA}

%
% DOMAIN == INPUTS
% RANGE == OUTPUTS
%

When performing FMEA, we have a system under investigation, which will be
comprised of a collection of components which have associated failure modes.
The object of FMEA is to determine cause and effect: 
from the failure modes (the causes, {\fms} of {\bcs}) to the effects (or symptoms of failure) at the top level.
%
To perform FMEA rigorously
we could stipulate that every failure mode must be checked for effects
against all the components in the system.
We could term this `rigorous~FMEA'~(RFMEA).
The number of checks we have to make to achieve this, gives an indication of the complexity of the task.
%
We could term this `comparison~complexity', as  the number of 
paths between failure modes and components necessary to achieve RFMEA for a given system/functional~group.


% (except its self of course, that component is already considered to be in a failed state!).
%
Obviously, for a small number of components and failure modes, we have a smaller number
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} $G$,  by 
$ | G | $,
(an indexing and sub-scripting notation to identify particular {\fgs}
within an FMMD hierarchy is given in section~\ref{sec:indexsub}).

The function $fm$ has a component as its domain and the components failure modes as its range (see equation~\ref{eqn:fm}).
We can represent the number of potential failure modes of 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 
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, $CC$, with its domain as the system 
or {\fg}, $\FG$, and
its range as the number of checks to perform to satisfy a rigorous FMEA inspection.

Where $\mathcal{\FG}$ represents the set of all {\fgs}, and $ \mathbb{N} $ any natural integer, $CC$ is defined by,
\begin{equation}
%$$
 CC:\mathcal{\FG} \rightarrow \mathbb{N}, 
%$$
\end{equation}

and, where n is the number of components in the system/{\fg},   $|fm(c_i)|$ is the number of failure modes
in component ${c_i}$, is given by

\begin{equation}
\label{eqn:CC}
%$$
 %%% when it was called reasoning distance -- 19NOV2011 -- RD(fg) = \sum_{n=1}^{|fg|} |fm(c_n)|.(|fg|-1) 
 CC(\FG) = (n-1) \sum_{1 \le i \le n} fm(c_i).
%$$
\end{equation}

This can be simplified if we can determine the total number of failure modes in the system $K$, (i.e. $ K = \sum_{n=1}^{|G|} {|fm(c_n)|}$);
equation~\ref{eqn:CC} becomes 

%$$
\begin{equation}
\label{eqn:rd2}
 CC(\FG) = K.(|\FG|-1).
\end{equation}
%$$
%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}

An FMMD Hierarchy will have reducing numbers of functional groups 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 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 

\begin{equation}
%$$
g(H, i) \rightarrow \forall {\FG}^{\xi} \;where\; ({\xi} = {i}) \wedge ({\FG}^{\xi} \in H) .
%$$
\end{equation}

Where $L$ represents the number of levels in the FMMD hierarchy,
$|g(\xi)|$ represents the number of functional groups on the level
and $H$ represents an FMMD hierarchy,
we overload the comparison complexity thus:
%$$
\begin{equation}
     \label{eqn:gf}
      CC(H) = \sum_{\xi=0}^{L} \sum_{j=1}^{|g(H,\xi)|} CC({\FG}_{j}^{\xi}).
%$$
\end{equation}


\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.
$$CC(potdiv) = \sum_{n=1}^{2} |2|.(|1|) = 4 $$

Even considering  a $example$ system with just 81 components (with these components
having 3 failure modes each) we would have an $CC$ of 

$$CC(example) = \sum_{n=1}^{81} |3|.(|80|) = 19440 .$$

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 by-hand/human activities. It can be seen that it is practically impossible to achieve 
RFMEA for anything but trivial systems.
%
% Next statement needs alot of justification
%
It is the authors belief that FMMD reduces the comparison complexity enough to make
rigorous checking feasible. 


\pagebreak[4]
%\subsection{Using the concept of Complexity Comparison to compare RFMEA with FMMD}

\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}



\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
use the general formula for comparing the number of checks to make for
RFMEA and FMMD.
%
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 compare the number of checks to make from an FMMD hierarchy to RFMEA applied to
all components in a system.
 
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 a (fixed parameter for $|{\FG}|$ and $|fm(c_i)|$) FMMD hierarchy
with equation~\ref{eqn:anscen}.
 
\begin{equation}
  \label{eqn:anscen}
 \sum_{n=0}^{L} {k}^{n}.k.f.(k-1)   
\end{equation}
 
The thinking behind equation~\ref{eqn:anscen}, is that for each level of analysis -- counting down from the top --
there are ${k}^{n}$ {\fgs} within each level; we need to apply RFMEA to each {\fg} on the level.
The number of checks to make for RFMEA is number of components $k$ multiplied by the number of failure modes $f$
checked against the remaining components in the {\fg} $(k-1)$.

If, for the sake of example, we fix the number of components in a {\fg} to three and 
the number of failure modes per component to three, an FMMD hierarchy
would look like figure~\ref{fig:three_tree}.

\subsection{RFMEA FMMD Comparison Example}

Using the diagram in figure~\ref{fig:three_tree}, we have three levels of analysis.
Starting at the top, we have a {\fg} with three derived components, each of which has
three failure modes.
Thus the number of checks to make in the top level is $3^0.3.2.3=18$.
On the level below that, we have three {\fgs} each with a 
an identical number of checks, $3^1.3.2.3=56$.%{\fg}
On the level below that we have nine {\fgs}, $3^2.3.2.3=168$.
Adding these together gives $242$ checks to make to perform FMMD (i.e. RFMEA {\em{within the}}
{\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:CC},
$CC(G) = \sum_{n=1}^{|G|} |fm(c_n)|.(|G|-1)$, where $|G|$ is 27, $fm(c_n)$ is 3
and $(|G|-1)$ is 26.
This gives:
$CC(G) = \sum_{n=1}^{27} |3|.(|27|-1) = 2106$.

In order to get general equations with which to compare RFMEA with FMMD,
we can re-write equation~\ref{eqn: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:CC} as:
 

\begin{equation}
  \label{eqn:fmea_state_exp21}
  \sum_{n=1}^{k^{L+1}}.(k^{L+1}-1).f  \; , % \\
  %(N^2 - N).f 
\end{equation}

or

\begin{equation}
  \label{eqn:fmea_state_exp22}
  k^{L+1}.(k^{L+1}-1).f  \;. % \\
  %(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 $|G|$ and $|fm(c)|$)
the two approaches, for the work required to perform rigorous checking.


For instance, having four levels 
of FMMD analysis, with these fixed numbers,
%(in addition to the top zeroth level)
will require 81 base level components.

$$
%\begin{equation}
  \label{eqn:fmea_state_exp22}
  3^4.(3^4-1).3 = 81.(81-1).3 = 19440 % \\
  %(N^2 - N).f 
%\end{equation}
$$

$$
%\begin{equation}
 % \label{eqn:anscen}
  \sum_{n=0}^{3} {3}^{n}.3.3.(2) = 720
%\end{equation}
$$

% \subsection{Exponential squared to Exponential}
% 
% can I say that ?

\section{Problems in choosing membership of functional groups}

\subsection{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
{\fgs}.
For instance were we to have a component that on failing $SHORT$ could bring down 
a voltage supply rail, this could have drastic consequences for other 
functional groups in the system we are examining.

\pagebreak[3]
\subsubsection{Example de-coupling capacitors in logic circuits}

A good example of this, are de-coupling capacitors, often used 
over the power supply pins of all chips in a digital logic circuit.
Were any of these capacitors to fail $SHORT$, they could bring down 
the supply voltage to the other logic chips.


To a power-supply, shorted capacitors on the supply rails 
are a potential source of the symptom, $SUPPLY\_SHORT$.
In a logic chip/digital circuit {\fg} open capacitors are a potential 
source of symptoms caused by the failure mode $INTERFERENCE$.
So we have a `symptom' of the power-supply, and a `failure~mode' of
 the logic chip to consider.

A possible solution to this is to include the de-coupling capacitors
in the power-supply {\fg}.
% decision, could they be included in both places ????
% I think so 


Because the capacitor has two potential failure modes (EN298),
this raises another issue for FMMD. A de-coupling capacitor going $OPEN$ might not be considered relevant to
a power-supply module (but there might be additional noise on its output rails).
But in {\fg} terms the power supply, now has a new symptom that of  $INTERFERENCE$.

Some logic chips are more susceptible to  $INTERFERENCE$ than others.
A logic chip with de-coupling capacitor failing, may operate correctly 
but interfere with other chips in the circuit.

There is no reason why the de-coupling capacitors could not be included {\em in the {\fg} they would intuitively be associated with as well}.

This allows for the general principle of a component failure affecting more than one {\fg} in a circuit.
This allows functional groups to share components where necessary.
This does not break the modularity of the FMMD technique, because, as {\irl},
one component failure may affect more than one sub-system.
It does uncover a weakness in the FMMD methodology though.
It could be very easy to miss the side effect and include
the component causing the side effect into the wrong {\fg}, or only one germane {\fg}.

\section{Critiques}



\section{Evaluation}
