## 3 Detailed Operators  
### Subsystem Division Based on Failure Decoupling  
This process corresponds to Stage II in Figure 1 and is described in Section 2. Reliability-based and cost-oriented product optimization deserves enough attention. In practical engineering, the product reliability optimization design only needs to be performed to the level of its subsystems because it will call for too much time and cost to apportion the product reliability to its modules and components. Hence, the reasonable subsystem division based on the function module classification is an effective guarantee for the reasonable reliability optimization. As the tight connections between product failures and reliability, the subsystem division is carried out on the module failure decoupling based on reachability matrix. Most of the product failures can be related to certain modules, but whether the product failure is caused by the corresponding module or the connection between the corresponding module and other modules is difficult to judge. Therefore, the failure decoupling is necessary to conduct subsystem division which is used to put the function modules whose failures are related closely together to form the subsystems. We perform the classification of function modules according to the composition and working principle of products, and then conduct the data collection of module failures based on the observation of sample products. The causal relationship between modules failures is analyzed and the module failure-related mathematical model considering causal failure analysis is established. \(F\) is the set of product module failures which consist of the collected data about module failures. \(a\) and \(b\) are the subsets of \(F\). If the module failures in the subset \(b\) are caused by the module failures in the subset \(a\) and all the failures in subset \(b\) compose the set \(A(a)\), \(a\) is the root failure sets of \(A(a)\) and \(A(a)\) is the relevant zone of \(a\). Failure decoupling is the process to find the root failures according to the collected failure data whose description is:  
\[\xi=f[F;\forall x\in F,A(x);CF] \tag{1}\]  
where \(CF\) is the set of product module failure in practice, and \(\xi\) is root failure sets with the calculation of function \(f\). The failure correlation diagram consists of a set of points \(V=\{v_{i}\}\) (\(i\) = 1, 2,..., \(M\)) and a set of directed edges \(E=\{e_{ij}\}\) (\(j\) = 1, 2,..., \(N\)). The points represent the elements of the set \(F\). There is a directed edge from \(a\) to \(b\) if \(\forall a,b\in F,b\in A(a)\). Therefore, if the failure \(j\) is caused by the failure \(i\), there is a directed edge from \(i\) to \(j\) and there is a connection between failures \(i\) and \(j\). These directed edges and points form the failure correlation graph. Product failure correlation graph is drawn and translated into an adjacency matrix \(A=\left[a_{ij}\right]_{N\times M^{\prime}}\) and:  
\[a_{ij}=\left\{ \begin{array}{c}1,\text{ if there is a directed edge form the failure }i\text{ to the failure }j\text{ }(i\
eq j)\text{ or }i=j\\ 0,\text{ otherwise}\end{array} \right. \tag{2}\]  
Reachability matrix is defined to describe the extent from each connection point to another one in a directed graph via a path, which can be obtained using the time characteristics and laws of Boolean algebra. For a system consisting of \(n\) connection points which can be represented as \(p_{i}\) (\(i\) = 1, 2,..., \(n\)), its reachability matrix can be denoted as \(R=\left[a_{ij}\right]_{n\times m}\), and  
\[a_{ij}=\left\{ \begin{array}{c}1,\text{ if }p_{i}\text{ can reach }p_{j}\text{ via a path }(i\
eq j)\text{ or }i=j\\ 0,\text{ otherwise}\end{array} \right. \tag{3}\]  
The reachability matrix has a significant feature which can be called as transfer characteristic. It can be described as that if the point \(p_{i}\) can reach the point \(p_{k}\) and the \(p_{k}\) can reach the point \(p_{j}\), the point \(p_{i}\) can reach the point \(p_{j}\).