From 097da091c8303804b24e4d866f59eabe129543cc Mon Sep 17 00:00:00 2001 From: Robin Clark Date: Sun, 20 Jan 2013 17:47:14 +0000 Subject: [PATCH] CH7 (although in CH6 directory) --- submission_thesis/CH6_Evaluation/Makefile | 2 +- .../CH6_Evaluation/components_81_euler.dia | Bin 0 -> 4752 bytes submission_thesis/CH6_Evaluation/copy.tex | 32 +++++++++++------- 3 files changed, 21 insertions(+), 13 deletions(-) create mode 100644 submission_thesis/CH6_Evaluation/components_81_euler.dia diff --git a/submission_thesis/CH6_Evaluation/Makefile b/submission_thesis/CH6_Evaluation/Makefile index 5743eec..a1e839a 100644 --- a/submission_thesis/CH6_Evaluation/Makefile +++ b/submission_thesis/CH6_Evaluation/Makefile @@ -3,7 +3,7 @@ # # Place all .dia files here as .png targets # -DIA = +DIA = components_81_euler.png doc: $(DIA) diff --git a/submission_thesis/CH6_Evaluation/components_81_euler.dia b/submission_thesis/CH6_Evaluation/components_81_euler.dia new file mode 100644 index 0000000000000000000000000000000000000000..e8b7d988fc3044260909a5182bfdf54e0958272f GIT binary patch literal 4752 zcmX|DcOctc)HY+U+M8;LS#8V`E2y?1h}~B0^;$v2ZYiROy?0QtYn9k)k1D0S)!K?0 zQF|6&+V}gu-yip%-@WHP=eg%O&*6`uB>MA_km8?>cs+SQKX)N+G;HE>bivfGH!9(v z6q9M*3XDlq#AvZ=ecyBF2qrq5>Z#)eb(6C{(@$M7(^^=899fh8Bt5*-d9~;6y^na` z`pm<_GVi`E3}zzl(P+Ary_Jr=K08tUw6*c>o#k(@_m)?*oz=(@q{yzkyZavO4s2oP z{j(T;Nb1nlkeYU+w88-E=}*8ee&MzjJE`}Z?akXcwcT} zJNxp_Rh`fKd+Ixm7%rs-ujagv?Hy-a6!Y1-FNNC5<^%meh83e}X0vmF`~NOU#nUt? zO3m)zJh#68`>Uz-(!F{9C7plsoV&?Lc3jIsuo;+t{dm-i^zrtX(_ORQjq|^nR8LQ? z8)s+vGzwl`Sh~G5ZvMCRXeyuPB*rvoT{M5%^rGqAHDf^Q`J^fh_;TgkH@Mk6cDYnt)dmcu($0T%-&^(&8;lIZl~*)8dKnyuyQ=B1n7gvd3_i>HrM!VZtc&4q z--10y{^XwQJhH17jvC!RaFGXJ{bKljICMywr-IN+^Elp$N4^X=%yYeS-5OAzKX~;| zXIrzyPPN`A*j2~n%Ap`2l+rHvX#X((ctn0n6zs*hl z+8RsvW&PArCE_dbo7K@Hha)0e(1-Ab-A|dZ1%0^n&}z8~iQ%hp7W5>=cZgvgj}X^RQ-Jpxg(!c8Fr5G z$CFzgWK|T~P77D3IKT1X`o5LZ<)MDS)Be#n(IS~9OeJd(O{2HETnhxu@t)0PzJTdC z$A1RQlW*dQt8}6%JZ-iV_YqJKj`};mBDJYwv51I#xT1I3X__OlsKb@ zdje2_jGA4QuyD@@HO5BDuI?=dw3WHN@v~XA;KFiU8Lxl9CZsu(n|s2t(X&b0^F3JX zIj5OwX6K0-a1{vBDGY>3i!#$jVbu=R&0$G%0k|(l_emdPcboDwQl)ihG-=!^nSrZA z1-ULZ4>JyoW^ia-J@35GKKR-4n|_FLDw-9%Nz6g92fs%r$7eQKGyG|{v+moH*Cg7} z3NO>)+lDx+%(}%@Cm0Q35o@rIny=ieW8zNF*mU^8Y}nD+WuGh?q1$nOUDF3zwFpELA9m8m-9+VK`sf!B(ulfCYs5J0syDe)xT!bU#mGs>S zN>YXl&U^kFOaNtft_HM3-o1gJq7vG=rP!2e&TJJo<1r% zw!4)zHqqOMn~|5rQE+x#wNRLYZN$21{mcfNOkK!nH7NrPm#uhuEIZvYQB=1anTt^< zO*nTkS_vd=#Dtb87>EyCUJKd$$Dh_R0x3e{0JIHpX6{pD+z=Rv6%ulK0t z2IxzUlAihEHqZm%=sDp=6BLp~fvzmzl!BLndroT7v`z%bqIoGFMD&KOmrtiiDCMNX zV7kRZ;!$`_Pr6oHs|e#6MdK*j#{*?F`7u=Zk031LZ+$gTjgqEnGv~FRsg9+-DwX55 z>NL#oQuzV^?_k2TggkJ+S7)-7mUhf;HO7yBy;!7n^bFL0|te z9pwQ5Ca^%0y|p0nT|>cU?qMIXHq~m$<#tW{a*C4eXbJufa*S5XrT`WDJY!_!>hcUj z@qr&+9bjj0RMWgM|EFfIUa#rm)s2~so(t*47d|38vtvF#qBdx}M*T

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Where $\mathcal{G}$ represents the set of all {\fgs}, and $ \mathbb{N} $ any natural integer, $CC$ is defined by, @@ -174,10 +174,10 @@ to identify the hierarchy level. % \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. +An FMMD Hierarchy will have reducing numbers of {\fgs} 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 can define an FMMD hierarchy as a set of {\fgs}, $H$. 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 @@ -188,7 +188,7 @@ g(H, i) \rightarrow \forall {\FG}^{\xi} \;where\; ({\xi} = {i}) \wedge ({\FG}^{\ \end{equation} Where $L$ represents the number of levels in the FMMD hierarchy, -$|g(\xi)|$ represents the number of functional groups on the level +$|g(\xi)|$ represents the number of {\fgs} on the level and $H$ represents an FMMD hierarchy, we overload the comparison complexity thus: %$$ @@ -202,10 +202,11 @@ we overload the comparison complexity thus: \pagebreak[4] \subsection{Complexity Comparison Examples} -The potential divider discussed in section~\ref{potdivfmmd} has four failure modes and two components and therefore has $CC$ of 4. +The potential divider discussed in section~\ref{subsec:potdiv} has four failure modes and two components and therefore has $CC$ of 4. $$CC(potdiv) = \sum_{n=1}^{2} |2|.(|1|) = 4 $$ -Even considering a $example$ system with just 81 components (with these components +%Even considering a $example$ +A system, $example$, with just 81 components (with these components having 3 failure modes each) we would have an $CC$ of $$CC(example) = \sum_{n=1}^{81} |3|.(|80|) = 19440 .$$ @@ -228,16 +229,23 @@ rigorous checking feasible. \pagebreak[4] %\subsection{Using the concept of Complexity Comparison to compare RFMEA with FMMD} -\begin{figure} +% \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} + +\begin{figure}[h] \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)$} + \includegraphics[width=400pt]{./CH6_Evaluation/components_81_euler.png} + % components_81_euler.png: 3056x2532 pixel, 72dpi, 107.81x89.32 cm, bb=0 0 3056 2532 + \caption{FMMD Hierarchy with number of compnents in each $FG$ fixed to three ($|FG|=3$)} \label{fig:three_tree} \end{figure} - \subsection{Comparing FMMD and RFMEA comparison complexity} Because components have variable numbers of failure modes,