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Consider the FMEA type methodologies
where we look at all the failure modes in a system, and then 
see how they can affect all other components within it,
to determine its system level symptom or failure mode.
We need to look at a large number of failure scenarios
to do this completely (all failure modes against all components).
This is represented in equation~\ref{eqn:fmea_state_exp},
where $N$ is the total number of components in the system, and 
$cfm$ is the number of failure modes per component.

\begin{equation}
  \label{eqn:fmea_state_exp}
  N.(N-1).cfm % \\
  %(N^2 - N).cfm 
\end{equation}


The FMMD methodology breaks the analysis down into small stages,
by making the analyst choose functional groups, and then when analysed the groups
are treated as components to be used for a higher stage.
This is designed to address the state explosion (where $O$ is order
of complexity) $O=N^2$ inherent in equation~\ref{eqn:fmea_state_exp}.

\clearpage

We can view the functional groups in FMMD as forming a hierarchy.
If for the sake of example we consider each functional group to
be three components, figure~\ref{fig:three_tree} shows
how the levels work and converge to a top or system level.

\begin{figure}
 \centering
 \includegraphics[width=300pt]{./three_tree.png}
 % three_tree.png: 780x226 pixel, 72dpi, 27.52x7.97 cm, bb=0 0 780 226
 \caption{Functional Group Tree example}
 \label{fig:three_tree}
\end{figure}

\clearpage
We can represent the number of failure scenarios to check in an FMMD hierarchy
with equation~\ref{eqn:anscen}.

\begin{equation}
  \label{eqn:anscen}
  \sum_{n=0}^{L} {fgn}^{n}.fgn.cfm.(fgn-1) 
\end{equation}

Where $fgn$ is the number of components in each functional group,
and $cfm$ is the number of failure modes per component
and L is the number of levels, the number of
analysis scenarios to consider is show in equation~\ref{eqn:anscen}.


So for a very simple analysis with three components forming a functional group where
each component has three failure modes, we have only one level (zero'th).
So to check every failure modes against the other components in the functional group
requires 18 checks.

\begin{equation}
  \label{eqn:anscen2}
  \sum_{n=0}^{0} {3}^{0}.3.3.(3-1) = 18 
\end{equation}
\clearpage



In other words, we have three components in our functional group, 
and nine failure modes to consider.
So taking each failure mode and looking at how that could affect the functional group,
we must compare each failure mode against the two other components (the `$fgn-1$' term).

For the one `zero' level FMMD case we are doing the same thing as FMEA type analysis
(but on a very simple small sub-system).
We are looking at how each failure~mode can effect the system/top level.
We can use equation~\ref{eqn:fmea_state_exp} to represent 
the number of checks to rigorously perform FMEA, where $N$ is the total
number of components in the system, and $cfm$ is the number of failures per component.



Where $N=3$ and $cfm=3$ we can see that  the number of checks for this simple functional 
group is the same for equation~\ref{eqn:fmea_state_exp}
and equation~\ref{eqn:anscen}.
\clearpage

\section{Example}

To see the effects of reducing `state~explosion' we need to look at a larger system.
Let us take a system with 3 levels and apply these formulae.
Having three levels (in addition to the top zero'th level)
will require 81 base level components.

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

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

Thus for FMMD we needed to examine 720 failure mode scenarios, and for traditional FMEA
type analysis methods 19440.
% In practical example followed through, no more than 9 components have ever been required for a functional
% group and the largest known number of failure modes has been 6.
% If we take these numbers and double them (18 components per functional group
% and 12 failure modes per component) and apply the formulas for a 4 level analysis
% (i.e. 

\clearpage
Note that for all possible double simultaneous failures the equation~\ref{eqn:fmea_state_exp} becomes
equation~\ref{eqn:fmea_state_exp2} essentially making the order $N^3$.
The FMMD case (equation~\ref{eqn:anscen2}), is cubic within the functional groups only, 
not all the components in the system.


\begin{equation}
  \label{eqn:fmea_state_exp2}
  N.(N-1).(N-2).cfm % \\
  %(N^2 - N).cfm 
\end{equation}

\begin{equation}
  \label{eqn:anscen2}
  \sum_{n=0}^{L} {fgn}^{n}.fgn.cfm.(fgn-1).(fgn-2)
\end{equation}


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