## 7 Results and Discussion
### Maintenance Scheduling with Imperfect RUL Prognostics (PdMI) on the Aircraft Engine Use Case  
The availability of monitoring data from sensors offers a significant advantage in enhancing maintenance actions. By utilizing these data, it becomes possible to develop RUL prognostic models that accurately estimate the current degradation state of assets based on sensor measurements. The number of cycles an aircraft engine can perform until failure depends on a lot of variables and working conditions; thus, its value may exhibit high variability. Consequently, implementing an effective PM strategy is challenging since the optimal time between interventions is not a trivial problem. Furthermore, for this specific use case, there is a major safety concern since aircraft engines must not fail during a flight. For this reason, a PdM policy for such components is highly advantageous, since the forecasting of the RUL of components has the potential to have a considerable impact in reducing the costs related to maintenance, while ensuring the safety and integrity of the engines. The k-LSTM model was integrated into the maintenance scheduling framework to leverage the failure probability estimated by the GFT for calculating a dynamic safety margin in the RUL prediction. For testing and evaluation, the training set of FD001 was split, with 80 engines used for training the k-LSTM-GFT model and the remaining 20 engines for testing. The results of these test instances are presented in Figure 9, displaying the RUL prediction outcomes. Additionally, Figure 9 illustrates the root-mean-squared-error (RMSE) as an evaluation metric plotted against the number of flights until failure. The RMSE of the model is lower when the actual RUL decreases, which means that the model is more accurate near the failure point. This occurs because the degradation patterns are more pronounced near failure when the model's accuracy is more critical. Figure 8: Costs per working cycle of different PM maintenance policies for the testing engines. The costs obtained with the proposed PdMI policy are presented in Figure 10 for both cases, A and B. With this policy, the useful life of components is exploited as close to the maximum as possible and no failure is incurred (see Figure 11). The perfect maintenance policy (PdMP) can be used as a bound that may be reached in terms of costs. The costs with the proposed approach are only 3% and 5% higher than the PdMP for Cases A and B, respectively. Note that the PdMP policy is not achievable since there is no perfect RUL prognostic model. This demonstrates the effectiveness of the PdMI approach in optimizing maintenance scheduling since the obtained costs are very close to the best that may be attainable in an idealistic situation. The main advantage of using the k-LSTM-GFT model in the proposed scheduling policy, compared to other RUL prognostic models, is the fact that, in addition to being very accurate in RUL prediction, the failure probability estimated by the GFT allows the calculation of a dynamic safety margin to mitigate RUL prediction errors. When the estimated failure probability is higher, typically near the point of failure, the safety margin to account for RUL uncertainties is also bigger, to ensure that the RUL is not overestimated when the engine is near failure. On the other hand, when the estimated failure probability is lower, the safety margin is also lower, allowing the exploitation of the engines' useful life. Table 3 shows an example of a testing engine. If using the estimated RUL by the model \(RUL_{e}\), at the maintenance slot available, when performing 190 flights, no immediate action would be taken since the \(RUL_{e}\) falls between \(m=10\) and \(2m=20\), so a replacement would be scheduled for the next maintenance slot. However, the prediction in this case overestimates the actual RUL, and a failure would occur before the next maintenance slot if the engine was not replaced at the current time slot.