## 2 Imperfect Preventive Maintenance  
A repairable system can be defined as \"a system that, after failing to perform an activity, can be restored to proper functioning satisfactorily by some method\" [7]. As defined in [8], a repairable system can be fixed by replacing some of its components, rather than restoring the entire system after a failure. The research conducted by [5; 9] reports five categories of maintenance activities, which are:  
1. Perfect maintenance: the maintenance operation restores the system to a status \"as good as new\". The maintenance operation restores the system such that its operational performance and integrity are indistinguishable from when it was brand new. This maintenance effectively resets the reliability curve, bringing it back to the initial high reliability associated with a new system. All accumulated wear and tear or degradation is completely reversed. 2. Minimal maintenance: maintenance operation that restores the system to a state of \"as bad as old\". The maintenance operation addresses only immediate faults or damages, but the cumulative wear, age, and minor degradations persist, rendering the system's condition nearly unchanged post-maintenance. There's little to no improvement in the reliability after the maintenance. The curve remains near its current level, which is closer to the system's end-of-life reliability. 3. Imperfect maintenance: maintenance operation that restores the system to a state of \"worse than new, but better than old\". The maintenance operation rejuvenates the system, but not to its original state. While certain wear and degradations are mitigated, the system does not return to its brand-new condition. The reliability improves post-maintenance but does not reach the peak reliability of a new system. The curve is lifted but stays between its original high (new system) and its current degraded state. 4. More than perfect maintenance: the maintenance operation returns the system to a state of \"better than new\". The maintenance operation not only restores the system but also enhances its performance, durability, and other attributes, making it surpass its original specifications. Post-maintenance, the reliability exceeds that of the original system. The curve is elevated beyond the initial reliability, indicating enhanced performance and longevity due to maintenance upgrades or innovations. 5. Destructive maintenance: maintenance operation that restores the system to a state of \"worse than old\". Instead of rejuvenating or restoring the system, the maintenance operation inadvertently introduces flaws, damages, or reduces the system's overall integrity. The reliability deteriorates post-maintenance, moving the curve downwards, closer to complete system failure. This is a non-ideal outcome, often resulting from erroneous maintenance procedures or unforeseen complications. According to [9], the fourth and fifth categories have little presence in reality; almost all correspond to specific events in the life of the system. In the first category, the most commonly used model that precisely fits the described scenario is the perfect renewal process (PRP) as mentioned by [4; 10]. In the second category, a non-homogeneous Poisson process (NHPP) [4; 10] is used, among others. In the third category, several failure process models, including variants of generalized geometric processes [11], are used. Recent works on reliability and imperfect preventive maintenance models have introduced new approaches where artificial intelligence (AI) algorithms are used together with statistical models to assist in optimizing maintenance policies. Many of these works focus on finding possible solutions for specific problems identified in certain industrial sectors. In this context, it is observed that the use of evolutionary and metaheuristic algorithms is widespread. In [12], optimization techniques using evolutionary algorithms for complex system maintenance costs are presented. The aim is to minimize maintenance costs while maintaining system reliability.