## 4 Discussion  
Although the results obtained from the computational implementation are consistent, it is necessary to discuss the limitations and computational complexity issues that the framework includes. In the first term, the values associated with the deviation indicators from the tentative maintenance plan cannot be underestimated. Although the optimal maintenance plan establishes a considerable reduction in the unavailability of the system as the tolerance factor increases, the number of activities maintaining their tentative execution date does not exceed 20% of the total executions. In this sense, it is necessary to consider that the deviation from the tentative execution date of the PM activities implies a risk: the advance of activities on its part implies a poor economic performance, while its delay may increase the probability of failure occurrence in certain assets [52]. This may bring a considerable impact on the degradation of the assets, and therefore, on the probability of critical failures of the components if a previous estimation of the tolerance factor is undertaken. Hence the need to estimate, based on advanced optimization techniques and expert recommendations, the use of tolerance based on a controlled risk zone, which does not involve malfunctions that affect the economic performance of the plant, as well as the potential failures that may put the health of the personnel and the environment at risk. Figure 8: Optimal grouping schemes for the case study under a tolerance factor \(e=5\%\), considering a 52-week planning horizon. (**a**) First semester (weeks 1–26); (**b**) Second semester (weeks 27–52). On the other hand, it is necessary to discuss the resolution times reported in the computational implementation. Although the different instances have reported a reasonable computing time, what is observed is a consistent increase in the number of decision variables, as well as in the number of constraints and the computing times reported (See Figure 9). This must be taken into account when considering an increase in the number of PM activities to plan and schedule, which may eventually cause an increase in computing times or a suboptimal grouping scheme. In this sense, it is necessary to establish that the size of the MILP model previously presented, and therefore its complexity does not depend exclusively on the tolerance factor. Considering that the proposed MILP model is implemented through the use of a commercial solver, what is proposed as a measure of complexity is a theoretical expression for calculating the maximum number of variables and constraints, which directly depend on the number of executions, and indirectly, on the number and periodicity of activities, as well as the tolerance of execution of each one of them. In order to quantify the problem complexity, consider the following expression for the determination of the number of variables and constraints involved in the framework. Let the quantities \(\hat{j}\), \(\hat{n}\), \(\hat{k}\), and \(\hat{v}\) be defined as:  
\[\hat{j}=\sum_{i\in I}\lvert J_{i}\rvert\]  
\[\hat{n}=\left( \begin{array}{c}\hat{j}\\ 2\end{array} \right)=\frac{\hat{j}!}{2!(\hat{j}-2)! }\]  
\[\hat{k}=V_{\hat{j}2}=\frac{\hat{j}!}{(\hat{j}-2)!}=2\hat{n}\]  
\[\hat{v}=\left( \begin{array}{c}\hat{j}\\ 3\end{array} \right)=\frac{\hat{j}!}{3!(\hat{j}-3)! }\]  
Then it is possible to calculate, considering the set of values \(\left\{\hat{j},\ \hat{n},\ \hat{k},\ \hat{v}\right\}\), the maximum theoretical number of variables and constraints, considering the limit case when \(\ e\ \rightarrow\ \infty\) (i.e., there are no grouping feasibility constraints by the use of tolerance time windows). Table 6 shows the experimental values and the maximum theoretical value for the evaluated instance, considering that \(\hat{j}=43\). As can be seen, the number of variables and constraints increases progressively and approaches to maximum theoretical values when the tolerance factor increases.