Just need full UML digram now.
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@ -366,12 +366,13 @@ can be active at one time is termed a {\textbf{unitary~state}} failure mode set.
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Let the set of all possible components be $ \mathcal{C}$
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and let the set of all possible failure modes be $ \mathcal{F}$.
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The set of failure modes of a particular component are of interest
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here. What is required is to define a property for
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a set of failure modes where only one failure mode can be active at a time,
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or borrowing from the terms of statistics, the failure mode is an event, and it is mutually exclusive
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with the a specific set $F$.
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here.
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What is required is to define a property for
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a set of failure modes where only one failure mode can be active at a time;
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or borrowing from the terms of statistics, the failure mode being an event that is mutually exclusive
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with a set $F$.
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We can define a set of failure mode sets called $\mathcal{U}$ to represent this
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property.
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property for a set of failure modes..
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\begin{definition}
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We can define a set $\mathcal{U}$ which is a set of sets of failure modes, where
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@ -475,36 +476,36 @@ to dealing with double simultaneous failure modes.}.
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To generalise, we may need to consider $N$ simultaneous
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failure modes when analysing a functional group. This involves finding
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all combinations of failures modes of size $N$ and less.
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The Powerset concept from Set theory is useful to model this.
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%The Powerset concept from Set theory is useful to model this.
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The powerset, when applied to a set S is the set of all subsets of S, including the empty set
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\footnote{The empty set ( $\emptyset$ ) is a special case for FMMD analysis, it simply means there
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is no fault active in the functional~group under analysis.}
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and S itself.
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In order to consider combinations for the set S where the number of elements in each sub-set of S is $N$ or less, a concept of the `cardinality constrained powerset'
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In order to consider combinations for the set S where the number of elements in each subset of S is $N$ or less, a concept of the `cardinality constrained powerset'
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is proposed and described in the next section.
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%\pagebreak[1]
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\subsection{Cardinality Constrained Powerset }
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\label{ccp}
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A Cardinality Constrained powerset is one where sub-sets of a cardinality greater than a threshold
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A Cardinality Constrained powerset is one where subsets of a cardinality greater than a threshold
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are not included. This threshold is called the cardinality constraint.
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To indicate this, the cardinality constraint $cc$ is subscripted to the powerset symbol thus $\mathcal{P}_{cc}$.
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Consider the set $S = \{a,b,c\}$.
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The powerset of S:
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$$ \mathcal{P} S = \{ \emptyset, \{a,b,c\}, \{a,b\},\{b,c\},\{c,a\},\{a\},\{b\},\{c\} \} $$
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$$ \mathcal{P} S = \{ \emptyset, \{a,b,c\}, \{a,b\},\{b,c\},\{c,a\},\{a\},\{b\},\{c\} \} $$.
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$\mathcal{P}_{2} S $ means all non-empty subsets of S where the cardinality of the subsets is
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$\mathcal{P}_{\le 2} S $ means all non-empty subsets of S where the cardinality of the subsets is
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less than or equal to 2 or less.
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$$ \mathcal{P}_{2} S = \{ \{a,b\},\{b,c\},\{c,a\},\{a\},\{b\},\{c\} \} $$
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$$ \mathcal{P}_{\le 2} S = \{ \{a,b\},\{b,c\},\{c,a\},\{a\},\{b\},\{c\} \} $$.
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Note that $\mathcal{P}_{1} S $ (non-empty subsets where cardinality $\leq 1$) for this example is:
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$$ \mathcal{P}_{1} S = \{ \{a\},\{b\},\{c\} \} $$
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$$ \mathcal{P}_{1} S = \{ \{a\},\{b\},\{c\} \} $$.
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\paragraph{Calculating the number of elements in a cardinality constrained powerset}
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@ -515,7 +516,7 @@ with $n$ elements (size $n$) is the binomial coefficient~\cite{probstat} shown i
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\begin{equation}
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C^n_k = {n \choose k} = \frac{n!}{k!(n-k)!}
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\label{bico}
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\end{equation}
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\end{equation} .
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To find the number of elements in a cardinality constrained subset S with up to $cc$ elements
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in each combination sub-set,
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@ -531,7 +532,7 @@ from $1$ to $cc$ thus
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\begin{equation}
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|{\mathcal{P}_{cc}S}| = \sum^{cc}_{k=1} \frac{|{S}|!}{ k! ( |{S}| - k)!}
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\label{eqn:ccps}
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\end{equation}
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\end{equation} .
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@ -584,14 +585,14 @@ $$ \mathcal{P}_{2}(fm(FG)) = \{
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\}
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$$
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And % by inspection
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$$
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|
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\{
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\{R_o T_o\}, \{R_o T_s\}, \{R_o T_h\}, \{R_s T_o\}, \{R_s T_s\}, \{R_s T_h\}, \{R_o \}, \{R_s \}, \{T_o \}, \{T_s \}, \{T_h \}
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\}
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| = 11
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$$
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And whose cardinality is 11. % by inspection
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%$$
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%|
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%\{
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% \{R_o T_o\}, \{R_o T_s\}, \{R_o T_h\}, \{R_s T_o\}, \{R_s T_s\}, \{R_s T_h\}, \{R_o \}, \{R_s \}, \{T_o \}, \{T_s \}, \{T_h \}
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%\}
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%| = 11
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%$$
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\pagebreak[1]
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@ -600,29 +601,36 @@ cardinality calculation}
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The cardinality constrained powerset in equation \ref{eqn:ccps}, can be modified for % corrected for
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unitary state failure modes.
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This is written as a general formula in equation \ref{eqn:correctedccps}.
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%This is written as a general formula in equation \ref{eqn:correctedccps}.
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%\indent{
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To define terms :
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\begin{itemize}
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\item Let $C$ be a set of components (indexed by $j \in J$)
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%To define terms :
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%\begin{itemize}
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%\item
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Let $C$ be a set of components (indexed by $j \in J$)
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that are members of the functional group $FG$
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i.e. $ \forall j \in J | C_j \in FG $
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\item Let $|fm({C}_{j})|$
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i.e. $ \forall j \in J | C_j \in FG $.
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%\item
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Let $|fm({C}_{j})|$
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indicate the number of mutually exclusive fault modes of component $C_j$.
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\item Let $fm(FG)$ be the collection of all failure modes
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%\item
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Let $fm(FG)$ be the collection of all failure modes
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from all the components in the functional group.
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\item Let $SU$ be the set of failure modes from the {\fg} where all $FG$ is such that
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%\item
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Let $SU$ be the set of failure modes from the {\fg} where all $FG$ is such that
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components $C_j$ are in
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`unitary state' i.e. $(SU = fm(FG)) \wedge (\forall j \in J | fm(C_j) \in \mathcal{U}) $
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\end{itemize}
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`unitary state' i.e. $(SU = fm(FG)) \wedge (\forall j \in J | fm(C_j) \in \mathcal{U}) $, then
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%\end{itemize}
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%}
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\begin{equation}
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|{\mathcal{P}_{cc}SU}| = {\sum^{cc}_{k=1} \frac{|{SU}|!}{k!(|{SU}| - k)!}}
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- {\sum_{j \in J} {|FM({C_{j})}| \choose 2}}
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\label{eqn:correctedccps}
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\end{equation}
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\end{equation} .
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Expanding the combination in equation \ref{eqn:correctedccps}
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@ -631,7 +639,7 @@ Expanding the combination in equation \ref{eqn:correctedccps}
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|{\mathcal{P}_{cc}SU}| = {\sum^{cc}_{k=1} \frac{|{SU}|!}{k!(|{SU}| - k)!}}
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- {{\sum_{j \in J} \frac{|FM({C_j})|!}{2!(|FM({C_j})| - 2)!}} }
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\label{eqn:correctedccps2}
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\end{equation}
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\end{equation} .
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\paragraph{Use of Equation \ref{eqn:correctedccps2} }
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Equation \ref{eqn:correctedccps2} is useful for an automated tool that
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@ -639,11 +647,12 @@ would verify that a single or double simultaneous failures model has complete fa
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By knowing how many test cases should be covered, and checking the cardinality
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associated with the test cases, complete coverage would be verified.
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\paragraph{N Venn disallowed combinations}
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The general case of equation \ref{eqn:correctedccps2}, involves not just dis-allowing pairs
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of failure modes within components, but also ensuring that combinations across components
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do not involve any pairs of failure modes within the same component.
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A recursive algorithm and proof is described in appendix \ref{chap:vennccps}.
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%\paragraph{Multiple simultaneous failure modes disallowed combinations}
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%The general case of equation \ref{eqn:correctedccps2}, involves not just dis-allowing pairs
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%of failure modes within components, but also ensuring that combinations across components
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%do not involve any pairs of failure modes within the same component.
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%%%%- NOT SURE ABOUT THAT !!!!!
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%%%- A recursive algorithm and proof is described in appendix \ref{chap:vennccps}.
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%%\paragraph{Practicality}
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%%Functional Group may consist, typically of four or five components, which typically
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@ -701,7 +710,9 @@ Thus the statistical sample space $\Omega$ for a component or derived~component
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$$ \Omega(C) = \{OK, failure\_mode_{1},failure\_mode_{2},failure\_mode_{3}, \ldots ,failure\_mode_{N}\} $$
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The failure mode set $F$ for a given component or derived~component $C$
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is therefore
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$$ F = \Omega(C) \backslash \{OK\} $$
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$ fm(C) = \Omega(C) \backslash \{OK\} $
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(or expressed as
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$ \Omega(C) = fm(C) \cup \{OK\} $).
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The $OK$ statistical case is the largest in probability, and is therefore
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of interest when analysing systems from a statistical perspective.
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