## 2 Methodology
Using the Weibull distribution, the latter is represented by:  
\[\lambda_{i}=\frac{1}{\alpha\times\Gamma\times\left(1+\frac{1}{\beta}\right)} \tag{4}\]  
\[MTTR_{parallel}=\frac{\sum_{i=1}^{n}MTTR_{i}}{n} \tag{5}\]  
and finally,  
\[A_{parallel}=\frac{MTTF_{parallel}}{MTTF_{parallel}+MTTR_{parallel}} \tag{6}\]  
Cold standby is considered regarding the subsystems in the standby configuration, which means that at any time, only one piece of equipment operates until failure, at which point the following standby piece of equipment starts operation. It is also considered that maintenance is performed at the same time for every piece of equipment under this configuration. In this sense, the following analysis is used:  
\[MTTF_{sandby}=\frac{1}{\lambda_{1}}+\frac{1}{\lambda_{2}} \tag{7}\]  
\[MTTR_{sandby}=\min\{MTTR_{i}\} \tag{8}\]  
and finally,  
\[A_{standby}=\frac{MTTF_{standby}}{MTBF_{standby}+MTTR_{standby}} \tag{9}\]  
For partial redundancy, which is characterized by the ability of the subsystem to meet 100% of the workload using only a fraction of the items embedded in the subsystem, the expression used is an extension of the reliability model:  
\[A_{partial}=\sum_{j=r}^{n}\left( \begin{array}{c}n\\ j\end{array} \right)A^{j}(1-A)^{n-j} \tag{10}\]  
where  
\(n\): total number of elements. \(r\): minimum number of elements to meet the required load. Regarding the load-sharing configuration [23], which is characterized by the ability of the subsystem to meet more than 100% of the workload because of the capability of the joint items of the subsystem that sums up more than the required capacity. To evaluate the loss, a ratio of capacities is used to determine the equivalent subsystem availability:  
\[A_{load\ sharing}=\sum_{i=1}^{n}\left(A_{i}\times\frac{Q_{i}}{Q_{T}}\right) \tag{11}\]  
where  
\(Q_{i}\): capacity of element \(i\). \(Q_{T}\): total capacity of the subsystem. Stage 2: Downstream impact determination starts the analysis from the highest level (system) to the lower levels, where single units can be found. During this phase, the P-OEI of each identified element and level of the system is determined by breaking down the global index (i.e., \(POEI_{system}\)), assigning relative weight to each element according to its availability calculated in phase 1 as follows:  
\[POEI_{system}=1 \tag{12}\]  
where  
\(POEI_{system}\): global plant operational effectiveness impact index. Weight assigning for each level is performed by applying the following mathematical expression:  
\[\sum_{j=1}^{n}POEI_{i,j}=POEI_{i-1},\ \ \forall\ i\in\{1,\ \ldots,\ r\}\ \wedge\ \forall\ j\in\{1,\ \ldots,\ n\} \tag{13}\]  
\[\frac{POEI_{i,j}}{POEI_{i,j+1}}=\frac{1-A_{i,j}}{1-A_{i,j+1}},\ \ \forall\ i\in\{1,\ \ldots,\ r\}\ \wedge\ \forall\ j\in\{1,\ \ldots,\ n\} \tag{14}\]  
where  
\(POEI_{i,j}\): P-OEI of element \(j\) (from 1 to \(n\)) within level \(i\) (from 1 to \(r\)) of the decomposition. In this regard, it is possible to observe that P-OEI simply assigns a relative weight to the contribution coming from each level and unit to the ineffectiveness of the system by quantifying the potential production loss. Accordingly, the sum of all the relative weights starting from level 1 (which is different from the system level) to the lowest level (\(r\)) is equal to 100% (i.e., the P-OEI of the system). \[\sum_{i=1}^{r}\sum_{j=1}^{n}POEI_{i,j}=1=POEI_{system} \tag{15}\]  
Finally, once the \(POEI_{i,j}\) is known for each and every element of the system, the potential contribution to the production loss can be broken down into two main aspects, these are frequency (i.e., unavailability index of the element) and consequence (through the impact of said unavailability). This article refers to the latter as the expected operational impact (\(EOI_{i,j}\)), which is the quantification of the effect of an unscheduled stoppage of element \(j\) from level \(i\) on the system.