Added XFMEA FMMD comparison graph

mybib.bib

@@ -987,11 +987,18 @@ ISSN={1530-2059},}
 
 @MISC{gnuplot,
   author =       "Various Open~source~Project",
-  title =        "",
+  title =        "Gnuplot: graph plotting utility",
   howpublished = "Available from http://www.gnuplot.info/",
   year =         "2005"
 }
-
+@book{Janert:2009:GAU:1631269,
+ author = {Janert, Philipp K.},
+ title = {Gnuplot in Action: Understanding Data with Graphs},
+ year = {2009},
+ isbn = {1933988398, 9781933988399},
+ publisher = {Manning Publications Co.},
+ address = {Greenwich, CT, USA},
+} 
 @MISC{eulerviz,
   author =       "Peter~Rodgers, John~Howse, Andrew~Fish",
   title =        "Visualization of Euler Diagrams",

submission_thesis/CH7_Evaluation/Makefile

@@ -6,7 +6,7 @@
 DIA = components_81_euler.png
 
 
-doc: $(DIA)
+doc: $(DIA)  xfmea_fmmd_comp.png
 
 #
 #bib:	
@@ -14,6 +14,9 @@ doc: $(DIA)
 #	bibtex HR230003_combined_o2_co_sensor
 #
 
+xfmea_fmmd_comp.png:
+	gnuplot <  xfmea_comp.gpt
+
 
 %.png:%.dia
 	dia -t png $<

submission_thesis/CH7_Evaluation/copy.tex

@@ -14,7 +14,14 @@ and then formulae are presented for calculating the
 complexity of applying FMEA to a group of components.
 %
 These formulae are then used for a hypothetical example, which is analysed by both FMEA and FMMD.
-After analysing hypothetical examples, the FMMD examples from chapter~\ref{sec:chap5} are
+%
+%After analysing hypothetical examples, the 
+The hypothetical example gives a general formula, which shows that the reasoning distance
+goes from a polynomial to a logarithmic order comparing XFMEA with FMMD.
+%
+%This means that for 
+%
+The reasoning distances obtained from the FMMD examples (see chapter~\ref{sec:chap5}) are
 compared against {\XFMEA}.
 %
 Following on from the formal definitions, `unitary state failure modes' are defined. In short these
@@ -209,7 +216,7 @@ $i$ for identification and a superscript for  the $\alpha$~level (see section~\r
 For example  the first {\fg} in a hierarchy containing base components only
 i.e. at the zero'th level of an FMMD hierarchy where $\alpha=0$, would have the superscript 0 and a subscript of 1: $FG^{0}_{1}$.
 %
-The {\fg} representing the potential divider in section~\ref{sec:pd}
+The {\fg} representing the potential divider in section~\ref{subsec:potdiv}
 has an $\alpha$ level of 0 (as it contains base components). The {\fg}
 with the potential divider and the operational amplifier has an $\alpha$ level of 1.
 %$$
@@ -303,8 +310,8 @@ process are by-hand/human activities. It can be seen that it is practically impo
 %
 % Next statement needs alot of justification
 %
-It is the author's belief that FMMD reduces the comparison complexity enough to make
-exhaustive checking (within {\fgs}) entirely feasible. 
+%It is the author's belief that FMMD reduces the comparison complexity enough to make
+%exhaustive checking (within {\fgs}) entirely feasible. 
 
 
 %\pagebreak[4]
@@ -337,8 +344,9 @@ use the general formula for comparing the number of checks to make for
 %
 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 {\XFMEA} applied to
-all components in a system.
+to compare the number of checks to make from an FMMD hierarchy to {\XFMEA}.
+%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 
@@ -437,6 +445,53 @@ $$
 For FMMD (where within {\fgs} the analysis \textbf{is exhaustive}) we only require
 720 reasoning paths.
 
+
+
+\subsubsection{Plotting XFMEA and FMMD reasoning distance}
+
+Using the gnuplot utility~\cite{gnuplot,Janert:2009:GAU:1631269} and implementing equation~\ref{eqn:fmea_state_exp22} for
+XFMEA and equation~\ref{eqn:anscen} for FMMD reasoning distances, we can (using a logarithmic axis for reasoning distance)
+compare them graphically. The gnuplot script used to
+produce figure~\ref{fig:xfmeafmmdcomp} may be found in section~\ref{sec:gnuplotxfmeafmmdcomp}.
+
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=400pt]{./CH7_Evaluation/xfmea_fmmd_comp.png}
+ % xfmea_fmmd_comp.png: 640x480 pixel, 72dpi, 22.58x16.93 cm, bb=0 0 640 480
+ \caption{XFMEA and FMMD reasoning distance comparison graph.}
+ \label{fig:xfmeafmmdcomp}
+\end{figure}
+
+Looking at the graph in figure~\ref{fig:xfmeafmmdcomp} we see that the reasoning distance
+for large numbers of components becomes extremely difficult to achieve
+for FMEA.
+%
+It can be seen that the reasoning distance has gone from a polynomial to a logarithmic order.
+%
+By applying FMMD we have effectively decimated the large group for analysis into
+a hierarchy of much smaller groups and applied FMEA {\em within} those.
+%
+In mathematical terms this means we have converted the polynomial order
+to logarithmic by being able to convert exponentiation
+to constants of integration.
+%
+This process can be viewed as similar to the order of processing
+that occurs in the decimation in time FFT~\cite{fftoriginal} when
+compared to the DFT algorithm.
+%
+%We have been able to successively take constants of integration
+%out of the equations in the process of de-composition, resulting
+%in a saving in the number of processing steps (here hand analysis FMEA stages).
+
+
+
+
+
+
+
+
+
+
 %\clearpage
 \subsection{Complexity Comparison applied to FMMD electronic circuits analysed in chapter~\ref{sec:chap5}.}
 
@@ -1365,6 +1420,6 @@ 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{Evaluation}
+%\section{Evaluation}
 
-TO DO
+%TO DO

submission_thesis/CH7_Evaluation/xfmea_comp.gpt

@@ -0,0 +1,48 @@
+##
+# GNUPLOT SCRIPT to plot XFMEA FMMD reasoning distance
+# comparisons.
+#
+#
+# alway define fp in declaration, as in 'C'
+# or gnuplot treats these as integers.
+#
+# number of failure modes per component
+fm = 3.0
+
+# number of components in each functional group
+k = 3.0
+
+# place the functional group size and failure mode per components
+# size into a string to use as the graph title
+#
+tt = sprintf("reasoning distance comparison for |fg| = %d and |fm| = %d", k, fm)
+set title tt
+
+a = 0.0
+b = 0.0
+
+# formula for reasoning distance in one level of FMMD
+# hierarchy (as given by ll)
+#
+fmmd(ll)=k**ll * k * fm * (k - 1)
+
+# set up iterative sum in gnuplot syntax
+# to iterate over FMMD levels
+#
+sum(a,b) = (a > b) ? 0 : fmmd(a) + sum(a+1, b)
+sig_fx(c) = sum(a,c)
+
+# reasoning distance for exhaustive case in FMEA
+# where ll is the hierarchy level
+xfmea(ll) = k**(ll+1) * ( k**(ll+1) -1 ) * fm
+
+
+set xrange [0:1000]
+set xlabel "Component count"
+set ylabel "reasoning distance"
+set logscale y
+
+set terminal png
+set output 'xfmea_fmmd_comp.png'
+plot sig_fx(x**(1/k)), xfmea(x**(1/k))
+#!sleep 20

submission_thesis/CH8_Conclusion/copy.tex

@@ -51,7 +51,7 @@ the FMMD process strictly enforced this throughout the hierarchy of a model.
 %
 Finally the FMMD process was described algorithmically using set theory in appendix~\ref{sec:algorithmfmmd}.%{app:alg}.
 
-In conclusion then a new method of failure analysis has been devised  which improves on established techniques in the following ways:
+In conclusion then, a new method of failure analysis has been devised  which improves on established techniques in the following ways:
 % \begin{itemize}
 %     \item Must be able to analyse hybrid software/hardware systems, 
 %     \item no state explosion (which has rendered exhaustive analysis impractical),  
@@ -75,7 +75,7 @@ Under the following assumptions and constraints:
 \begin{itemize}
  \item Failure modes are available for all {\bcs},
  \item Analysts are capable of finding suitable {\fgs} from electronic schematics,
- \item Software is hierarchical and its elements (functions) can be modelled using contract programming,
+ \item Software is hierarchical and its elements (functions) can be modelled using contract programming.
  %\item 
 \end{itemize}
 
@@ -128,7 +128,8 @@ into its hierarchical model.
 %By way of example, the Pt100 analysis %example 
 %from section~\{sec:pt100} has been used to demonstrate this.
 Because we can use an FMMD model to generate an FMEA report, with additional {\bc} failure mode statistics
-we can therefore used FMMD to produce an FMEDA report.
+we can %therefore 
+use FMMD to produce an FMEDA report.
 
 
 \subsection{Pt100 Example: Single Failures and statistical data}. %Mean Time to Failure}
@@ -147,7 +148,7 @@ can give conservative reliability figures when applied to
 modern components}. 
 %
 Using the MIL-HDBK-217F\cite{mil1991}  specifications for resistor and thermistor
-failure statistics, we calculate the reliability of the Pt100 example ( see section~\ref{sec:pt100}).
+failure statistics, we calculate the reliability of the Pt100 example (see section~\ref{sec:Pt100}).
 
 
 \paragraph{Resistor FIT Calculations}
@@ -367,7 +368,7 @@ the FMMD model.
 A  system will be expected to perform in a given environment.
 %
 Environment in the context of this study
-means external influences under which the System could be expected to work. % under.
+means external influences under which the system could be expected to work. % under.
 %
 A typical data sheet for an electrical component will give 
 a working temperature range: %, for instance.
@@ -381,7 +382,7 @@ authorised human intervention takes place.
 A safety critical circuit may have a self test mode which could be operated externally:
 a micro-processor may have a SLEEP mode etc.
 %
-To make FMMD compatible with FTA perational states and environmental conditions should %can %must 
+To make FMMD compatible with FTA operational states and environmental conditions should %can %must 
 be factored into the UML model.
 %
 We may encounter a condition where we would want to inhibit some action of the system.
@@ -399,7 +400,7 @@ Environmental influences will affect specific components in specific ways\footno
 affected by environmental conditions, in this case temperature, is the opto-isolator~\cite{tlp181}
 which is typically affected at around {60 \oc}. Most electrical components are more robust to temperature variations.}.
 Environmental analysis is thus applicable to components.
-Environmental influences, such as over stress due to voltage
+Environmental influences, such as over-stress due to voltage
 can be eliminated by down-rating components as discussed in section~\ref{sec:determine_fms}.
 With given environmental constraints, we can therefore eliminate some failure modes from the model.
 
@@ -508,7 +509,7 @@ With the contracts in place for the software functions, we can then integrate th
 FMMD models both software and hardware;
 we can thus verify that all
 failure modes from the electronics module have been dealt
-by the controlling software. 
+with by the controlling software. 
 %
 If not they are an un-handled error condition relating to the software hardware interface.
 %

submission_thesis/appendixes/detailed_analysis.tex

@@ -828,3 +828,58 @@ FMMD analysis tables from chapter~\ref{sec:chap6}.
 }
 \clearpage/pr
 
+\subsection{Gnuplot script for hypothetical XFMEA FMMD reasoning distance comparision}
+\label{sec:gnuplotxfmeafmmdcomp}
+
+\begin{verbatim}
+ ##
+# GNUPLOT SCRIPT to plot XFMEA FMMD reasoning distance
+# comparisons.
+#
+#
+# alway define fp in declaration, as in 'C'
+# or gnuplot treats these as integers.
+#
+# number of failure modes per component
+fm = 3.0
+
+# number of components in each functional group
+k = 3.0
+
+# place the functional group size and failure mode per components
+# size into a string to use as the graph title
+#
+tt = sprintf("reasoning distance comparison for |fg| = %d and |fm| = %d", k, fm)
+set title tt
+
+a = 0.0
+b = 0.0
+
+# formula for reasoning distance in one level of FMMD
+# hierarchy (as given by ll)
+#
+fmmd(ll)=k**ll * k * fm * (k - 1)
+
+# set up iterative sum in gnuplot syntax
+# to iterate over FMMD levels
+#
+sum(a,b) = (a > b) ? 0 : fmmd(a) + sum(a+1, b)
+sig_fx(c) = sum(a,c)
+
+# reasoning distance for exhaustive case in FMEA
+# where ll is the hierarchy level
+xfmea(ll) = k**(ll+1) * ( k**(ll+1) -1 ) * fm
+
+
+set xrange [0:1000]
+set xlabel "Component count"
+set ylabel "reasoning distance"
+set logscale y
+
+set terminal png
+set output 'xfmea_fmmd_comp.png'
+plot sig_fx(x**(1/k)), xfmea(x**(1/k))
+#!sleep 20
+ 
+\end{verbatim}
+