\documentclass{beamer}
\title[Failure Mode Effects Analysis]{Failure Mode Effects Analysis\\A critical view}
\usetheme{Warsaw}
\usepackage[latin1]{inputenc}
\author{Robin Clark -- Energy Technology Control Ltd}
\institute{Brighton University}
\setbeamertemplate{footline}[page number]
\begin{document}

\section{F.M.E.A.}

\begin{frame}
\begin{itemize}
  \pause \item Failure
   \pause \item Mode
   \pause \item Effects
  \pause \item Analysis
\end{itemize}

\end{frame}

% % \begin{itemize}
%  \item Failure 
%  \item Mode 
%  \item Effects 
%  \item Analysis
% \end{itemize}


\subsection{FMEA basic concept}

\begin{frame}
\begin{itemize}
  \pause \item \textbf{F - Failures of given component} Consider a component in a system
   \pause \item \textbf{M - Failure Mode} Look at one of the ways in which it can fail (i.e. determine a component `failure~mode')
   \pause \item \textbf{E - Effects} Determine the effects this failure mode will cause to the system we are examining
  \pause \item \textbf{A - Analysis} Analyse how much impact this symptom will have on the environment/people/the system its-self
\end{itemize}
\end{frame}


\begin{frame}

Example: Let us consider a system, in this case a milli-volt reader, consisting
of instrumentation amplifiers connected to a micro-processor
that reports its readings via RS-232.
Let us perform an FMEA and consider how one of its resistors failing could affect
it.
For the sake of example let us choose a resistor in an OP-AMP
reading the milli-volt source and that if it were to go open, we would have a gain 
of 1 from the amplifier.

\begin{itemize}
  \pause \item \textbf{F - Failures of given component} The resistor could fail by going OPEN or SHORT (EN298 definition).
   \pause \item \textbf{M - Failure Mode} Consider the component failure mode OPEN
   \pause \item \textbf{E - Effects} This will disconnect the feedback loop in the amplifier causing a LOW READING
  \pause \item \textbf{A - Analysis} The reading will be out of normal range, and we will have an erroneous milli-volt reading
\end{itemize}
\end{frame}



\begin{frame}
Note here that we have had to look at the failure~mode
in relation to the entire circuit.
We have used intuition to determine the probable
effect of this failure mode.
We have not examined this failure mode
against every other component in the system.
Perhaps we should.... this would be a more rigorous and complete
approach in looking for system failures.

\end{frame}


\begin{frame}
Consider the analysis
where we look at all the failure modes in a system, and then 
see how they can affect all other components within it.

 
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_single}
  N.(N-1).cfm % \\
  %(N^2 - N).cfm 
\end{equation}

This would mean an order of $N^2$ number of checks to perform
to perform `rigorous~FMEA'. Even small systems have typically
100 components, and they typically have 3 or more failure modes each.
$100*99*3=29,700$.
\end{frame}




\begin{frame}
 
For looking at potential double failure scenarios (two components
failing within a given time frame) and the order becomes
$N^3$.

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

$100*99*98*3=2,910,600$.

The European Gas burner standard (EN298:2003), demands the checking of 
double failure scenarios (for burner lock-out scenarios).

\end{frame}

\section{Production FMEA : 1940's to present}

\begin{frame}
 

Describe process, the probability times the cost
\end{frame}




\subsection{Production FMEA : Example Ford Pinto : 1975}
\begin{frame}
 
\end{frame}
 


\subsection{FMEA and complexity of each failure scenario analysis}
\begin{frame}
 
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}.
\end{frame}


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}


\end{document}
