## 3 Model Formulation
### Reliability Metrics  
In this section, the reliability metrics for benchmarking the framework are defined. The metrics included Conditional Reliability, Remaining Useful Life (RUL), Mean Time Between Interventions (MTBI), Mean Time Between Failures (MTBF), maintenance costs, and the warning-limit function for defining the maintenance policy.  
The conditional reliability serves as an effective tool for illustrating how each state or survival time impacts the operational reliability of equipment, as visualized in Equation (9).  
\[R(t,Z(t))=\exp\left(-\left(\frac{t}{\eta}\right)^{\beta}\exp\left(\sum_{i=1}^ {m}\gamma_{i}Z_{i}(t)\right)\right). \tag{9}\]  
However, this form can often be challenging to calculate. Therefore, the product-integral method, as outlined by Ref. [44], was employed to approximate the conditional reliability. To do so, it was necessary to estimate the transition rate matrix \(\Lambda\) and the transition probability matrix \(P(x)\) as shown in Equations (10) and (11). Here, \(n_{ij}\) represents transitions from state \(i\) to \(j\), and \(A_{i}\) represents the total time spent in state \(i\).  
\[\Lambda=\lambda_{ij}(t),\qquad\lambda_{ij}=\frac{n_{ij}}{A_{i}},\quad i\
eq j, \quad\lambda_{ii}=-\sum_{i\
eq j}\lambda_{ij}, \tag{10}\]  
\[P(x)=\exp(\Lambda x)=\sum_{i=0}^{\infty}\frac{(\Lambda x)^{i}}{i!}. \tag{11}\]  
Subsequently, following this method, the conditional reliability can be expressed as:  
\[R(t|x,i)=\sum_{i}L_{ij}(x,t). \tag{12}\]  
The is also a crucial metric for CBM, as it evaluates the remaining lifespan of an asset. The calculation method is shown in Equation (13) based on values obtained from reliability curves.  
\[RUL(t,Z(t))=\int_{t}^{\infty}R(t,Z(t))\,dt. \tag{13}\]  
Other metrics that were also used included the MTBF, which represents the average expected time between failures, while the MTBI is a metric similar to the MTBF but with broader applicability. It considers uncertainties in an asset's operating environment, autonomy, and the reliability of inherent components.  
With respect to the optimization of the CBM model, it was demonstrated that the expected average cost in the long run depends on the optimal threshold risk as proposed by the authors of Ref. [37]. However, their initial approach involved complex mathematical calculations and typically required an iterative process to obtain the optimal risk. Therefore, the expected cost can be approximated using the MTBI, as represented in Equation (14).  
\[\phi(T_{p},Z(T_{p}))=\frac{C_{p}\cdot R(T_{p},Z(T_{p}))+C_{c}\cdot(1-R(T_{p},Z (T_{p})))}{MTBI(T_{p},Z(T_{p}))}, \tag{14}\]  
where \(C_{p}\) represents the cost of preventive maintenance, \(C_{c}\) represents the cost of corrective maintenance, \(R(t,Z(t))\) represents the conditional reliability, \(MTBI(t,Z(t))\) represents the expected time between interventions, and \(T_{p}\) represents the instant of time when the optimal cost of \(\phi(t)\) is achieved.  
Once the optimal time and cost have been calculated, the optimal risk can be derived by replacing \(t\) in Equation (1) with \(T_{p}\). Thus, the expression for the optimal hazard is described in Equation (15).  
\[d^{*}=\frac{\beta}{\eta}\bigg{(}\frac{T_{p}^{*}}{\eta}\bigg{)}^{\beta-1}\exp \bigg{(}\sum_{i=1}^{n}\gamma_{i}Z_{i}(T_{p}^{*})\bigg{)}. \tag{15}\]  
Then, the warning-limit function can be defined in terms of \(d^{*}\) in Equation (16).  
\[\gamma Z(t)\geq\varphi(t)=\ln\bigg{(}\frac{d^{*}\eta}{\beta K^{\beta}}-(\beta -1)\ln(t)\bigg{)}. \tag{16}\]  
This function allows the definition of a threshold that represents the operational state of equipment based on historical conditional data. This enables the establishment of a predictive maintenance policy using real-time data. If the data surpass this threshold, intervention is necessary to avoid imminent failure. The number of thresholds depends on the number of operational states provided by the band estimation process, with the threshold representing immediate equipment intervention determined by the state with the lowest conditional reliability.