## 2 Materials and Methods
### Opportunistic Grouping Strategy for the Preventive Maintenance Plan
#### 2.2.1 Maintenance Grouping Strategy Framework Formulation
A set of values then establishes an arbitrary instance for the activities and executions \(\left\{(i,\ j)\right\}_{(i\in I,\ j\in I)}\), from which is possible to obtain each feasible grouping pair \(\left\{(i,\ j,\ i^{\prime},\ j^{\prime}):\ i,i^{\prime}\in I;j\in J_{i},\ j^{ \prime}\in J_{j^{\prime}}\right\}\) for a given instance. The sample for each scheme can be approached through an exhaustive searching process, iterating over the Cartesian product \(R=P^{C}\), where \(P=\left\{p_{i}:\ i\in I\right\};C=\sum_{i\in I}|J_{i}|\), obtaining in this way all possible grouping combinations. Each scheme allows the construction of the set \(N_{i}\), defined as:  
\[N_{i}=\left\{\left[t_{i,1}^{f}+r_{i,1}+T_{i}-e_{i}T_{i},\ t_{i,1}^{f}+r_{i,1}+ T_{i}+e_{i}T_{i}\right],\ldots,\ \left[t_{i,|I|}^{f}+r_{i,|J|}+T_{i}-e_{i}T_{i},\ t_{i,|I|}^{f}+r_{i,|J|}+T_{i}+ e_{i}T_{i}\right]\right\},\ \forall i\in I \tag{6}\]  
In turn, the set \(N_{p}\) contains all the information of the feasible grouping pairs for each arbitrary iteration instance \(\left\{r_{i,\ j}\right\}_{i\in I,\ j\in I_{i}}\). This is formally defined by the following expression:  
\[N_{p}=\left\{\left(i,j,i^{\prime},f^{\prime}\right):i>i^{\prime}\wedge N_{i} \cap N_{i^{\prime}}\
eq\varnothing\right\},\ \forall j\in J_{i},\forall j^{\prime}\in J_{i^{\prime}};\ i,i^{\prime}\in I \tag{7}\]  
Once all the possible grouping pairs \((i,\ j,\ i^{\prime},j^{\prime})\) have been determined for each instance, the tuples are incorporated and stored in the definitive set of grouping schemes \(N\). According to [PERSON] et al. [46], the set \(N\) defines the solution space (feasible grouping pairs) for the clustering binary decision variables \(z_{i,\ j,\ i^{\prime},\ j^{\prime}}\). Therefore, the presented framework seeks to address two fundamental planning decisions: the grouping schemes (i.e., the optimal set \(\left\{z_{i,\ j^{\prime},\ i^{\prime},\ j^{\prime}}\right\}_{(i,j,i^{\prime}, j^{\prime})\in N}\)), and the execution time scheduling of the activities (i.e., the optimal set \(\left\{\left(t_{i,j}^{s},t_{i,j}^{f}\right)\right\}_{j\in J_{i},\ i\in I}\)), considering the use of time-windowtolerances. In this regard, a conservative approach is taken into account for the use of the aforementioned tolerance, to maintain as much as possible the tentative PM execution plan. This approach is applied considering that the advancement or delay of the executions implies incurring in suboptimal economic performance (advancing), or in a greater occurrence of failures (delaying) [46]. Considering the assumption that the system unavailability corresponds exclusively to the occurrence of PM activities (i.e., system downtime is the exclusive result of repair times associated with scheduled activities), the objective function corresponds to the minimization of system unavailability, or equivalently, the time out of service. The framework will be modeled through the MILP paradigm for optimization, considering the following assumptions:  
**General assumptions for the integrated framework:**  
* The system contemplates a multi-component series arrangement, where each preventive maintenance activity involves a system shutdown;
* The impact of stochastic and structural interactions on the system can be neglected;
* The system considers a uniform continuous (24 h) operational regime;
* The system considers a two-state operational condition (i.e., the system is necessarily under repair or under operation);
* The execution of grouped PM activities always involves a simultaneous/parallel execution;
* The execution of the PM activity returns the related equipment to its initial operating conditions (perfect maintenance).