## 2 Imperfect Preventive Maintenance
The algorithms determine the optimal times for maintenance activities on system components and the work compares the performance of different methods to find the global optimum. A case study is presented using data from an Algerian power distributor with a mathematical model for reliability constraints. The algorithms generate different solutions that are evaluated based on performance criteria. The results indicate that reliability constraints are crucial for obtaining viable and efficient solutions, and the study highlights that evolutionary algorithms are an effective tool for solving maintenance problems in complex systems. A mathematical model to optimize product availability with imperfect preventive maintenance was developed by [PERSON] et al. [13]. The model maximizes availability by optimizing product replacement time and ideal preventive maintenance intervals. The researchers compared numerical and evolutionary algorithms to find the global optimum and reported that the evolutionary algorithm was faster but slightly less precise than the numerical algorithm. The study demonstrates the applicability of the evolutionary algorithm for this problem. Future work includes constructing a model that considers warranty and availability costs, predictive risk of failure, and equipment utilization rate using operational research algorithms and different distributions, such as normal and Weibull distributions. [PERSON] et al. [14] proposed a mathematical optimization model to schedule a flow-shop system in the disposable appliance manufacturing industry while considering a pre-ventive maintenance policy (PMP). Real data from Pilka Plast Haraz Polymer Company in Iran were used to develop the model, which aims to minimize production and maintenance costs and considers uncertainties associated with the processes. The model uses operation times and system availability to optimize the processing sequence. Evolutionary algorithms were also used to solve the proposed model, indicating that such methods are effective in obtaining the optimal global solution that minimizes the total maintenance costs. The authors suggest future work on rigorous problem-solving techniques, introducing the development of new products, and implementing quality control (QC) in the mathematical model. [PERSON] et al. [15] proposed a statistical model to estimate system availability and optimize the times for preventive maintenance to minimize maintenance costs and maximize system availability. The proposed approach was applied in a power generation plant and evaluated using multi-objective optimization algorithms, such as particle swarm optimization (PSO) and the genetic algorithm (GA). The results showed that these algorithms can significantly reduce maintenance costs compared to existing preventive maintenance policies. Future work could involve using real data to determine the distribution and parameters, such as component failure detection rates, resource utilization rates, and factors related to component quality, that can be considered. In the research conducted by [PERSON] and [PERSON] [16], the authors propose a maintenance scheduling model based on metaheuristic algorithms, considering reliability constraints, to minimize the total number of maintenance activities throughout the lifetime of a nuclear power plant. The proposed model was applied to find the best maintenance plans, the global optimum, for components that constitute a water injection function. Comparing the results obtained with data from an existing method, it was possible to verify that the proposed model produced a superior maintenance schedule, in which the total number of activities was minimized, and the system's unreliability was restricted within a predefined upper limit. In future work, the researchers mention that they intend to consider the non-linearity of component failure rates and the difference in cost of each type of maintenance to optimize the schedule under more realistic conditions.