# Classification of Systems and Maintenance Models
## 4 Classification of Maintenance Models
### Optimal Inspection Schedule and Replacement Thresholds  
Various criteria can be used to frame the challenge of determining the optimal inspection schedule and replacement thresholds throughout a finite period of the system operation. Some examples of optimization criteria are maximum availability, minimum possible average system operation costs over the interval (0, \(T\)), provision of the necessary level of OPFFO with the lowest possible average maintenance costs over the interval (0, \(T\)), and the maximum level of OPFFO while limiting the average maintenance costs over the interval (0, \(T\)).  
The criterion of the maximum achieved availability is formulated as follows:  
\[A_{a}^{r}\Big{(}t_{1}^{opt},\ldots,t_{N}^{opt},RT_{1}^{opt},\ldots,RT_{N}^{ opt}\Big{)}=\max\limits_{RT_{1},\ldots,RT_{N}\cap t_{1},\ldots,t_{N}}A_{a}^{r}(t_{1}, \ldots,t_{N},RT_{1},\ldots,RT_{N}) \tag{120}\]  
where \(t_{1}^{opt},\ldots,t_{N}^{opt}\) are the optimal inspection instants over the interval (0, \(T\)) and \(RT_{1}^{opt},\ldots,RT_{N}^{opt}\) are the optimal replacement thresholds at instants \(t_{1}^{opt},\ldots,t_{N}^{opt}\)  
We present the criterion for the minimum AMCUT over the interval (0, \(T\)) as follows:  
\[V_{l}^{r}\Big{(}t_{1}^{opt},\ldots,t_{N}^{opt},RT_{1}^{opt},\ldots,RT_{N}^{ opt}\Big{)}=\min\limits_{RT_{1},\ldots,RT_{N}\cap t_{1},\ldots,t_{N}}V_{l}^{r}(t_{1}, \ldots,t_{N},RT_{1},\ldots,RT_{N}) \tag{121}\]where \(l=1\) for the continuous mode and \(l=2\) for the intermittent mode of operation.  
Since the two indicators evaluate the effectiveness of safety-critical systems, we can formulate two optimization criteria for the subgroups of systems \(I_{1111}\), \(I_{1112}\), \(I_{2111}\), and \(I_{2112}\). For example, the average maintenance cost of \(C_{l}^{\tau}\)\((l=1,\ 2)\) can be reduced by specifying the lowest permissible OPFFO \(P_{0}^{*}\) for subgroups \(I_{1111}\) and \(I_{1112}\). In this case, the optimization criterion has the following form:  
\[\left\{\begin{array}{l}C_{l}^{\tau}\Big{(}t_{1}^{opt},\ldots,t_{N}^{opt},RT_ {1}^{opt},\ldots,RT_{N}^{opt}\Big{)}=\min_{RT_{1},\ldots,RT_{N}\cap t_{1}, \ldots,t_{N}}C_{l}^{\tau}\{t_{1},\ldots,RT_{1},\ldots,RT_{N}\}\\ P_{0}\Big{(}t_{k}^{opt},t_{k+1}^{opt}\Big{)}\geq P_{0}^{*},\ k=0,\ldots,N\\ \end{array}\right. \tag{122}\]  
If the maximum allowed average maintenance cost \(C_{l}^{\tau*}\) is specified, the optimization criterion is as follows:  
\[\left\{\begin{array}{l}t_{1}^{opt},\ldots,t_{N}^{opt}\cap RT_{1}^{opt}, \ldots,RT_{N}^{opt}\Rightarrow\max_{RT_{1},\ldots,RT_{N}\cap t_{1},\ldots,t_{ N}}P_{0}(t_{k},t_{k+1})(k=0,\ldots,N)\\ C_{l}^{\tau}\Big{(}t_{1}^{opt},\ldots,t_{N}^{opt},RT_{1}^{opt},\ldots,RT_{N}^ {opt}\Big{)}\leq C_{l}^{\tau*}\end{array}\right. \tag{123}\]  
We can simplify criteria (120)-(123) for the case of a periodic inspection schedule and determine an optimal threshold \(RT^{opt}\) for all checking moments instead of the optimal replacement threshold for each inspection instant. For example, the optimization criterion (120) in this case can be expressed as:  
\[A_{a}^{\tau}\big{(}\tau_{opt},RT^{opt}\big{)}=\max_{\tau,RT}\{A_{a}^{\tau}(\tau,RT)\} \tag{124}\]  
where \(\tau\) is the periodicity of the CBM inspections and \(\tau^{opt}\) is the optimal periodicity of inspections.