# Towards Optimizing Multi-Level Selective Maintenance via Machine Learning Predictive Models
## 3 Problem Formulation
### Problem Description  
The problem studied in this work concerns the multi-series parallel system with \(n\) independent components (\(i=\{1,\ 2,\ \ldots,n\}\)), which is required to perform successive missions alternating with intermission breaks devoted to carrying out selective maintenance actions. The overall aim consists of selecting the best maintenance strategy (also called level) \(k_{i}\) (\(k_{i}\in\{1,\ 2,\ \ldots,K_{i}\}\)) to perform on component \(i\) (\(i=1,\ \ldots,n\)) according to the prediction model, while minimizing the total maintenance cost and maintaining a minimum system reliability. Each component \(i\) has a set of maintenance strategy \(k_{i}\), ranging from doing nothing (\(k_{i}=1\)) to perfect repair (AGAN with \(k_{i}=K_{i}\)), in which each \(k_{i}\) is associated with a defined cost. Obviously, the level of component replacement maintenance (\(k_{i}=K_{i}\)) is the most expensive of the other levels.  
Furthermore, the system state components can degrade due to a number of factors, either functional or environmental, or both, which can lead to a complete system failure and result in its unavailability. As the multi-component system is required to achieve successive missions under a reliability threshold, the actual degradation coefficients of components should be predicted at each break according to the real conditions. These predictions will be used in order to select the best maintenance strategy \(k_{i}\) that ensures a desired reliability level and successfully completes the next mission. Figure 1 illustrates the proposed selective maintenance approach. Actually, the component age provided by the manufacturer does not accurately count the real operational and environmental conditions such as activity frequency, operating time, wear, etc. The real age of a component can be accurately established by predicting its degradation factor according to the real conditions. So, firstly, the proposed approach aims to determine the components real age \(\gamma_{i}^{*}\) by computing the age reduction factor \(b_{i}\) using a k-means clustering algorithm, so that the collected data during the operating time can be represented as a matrix form \(X_{i}=[x_{1},\)\(x_{2},\)\(\ldots,\)\(x_{c},\)\(\ldots,\)\(x_{n}]\), \(X_{i}\in R_{s}\times n\), where \(x_{c}=[x_{c}^{1},\)\(x_{c}^{2},\)\(\ldots,\)\(x_{c}^{s}]\) is an \(s\)-dimensional vector of the component \(c\) features. Then, the system data collected during previous missions will be used to train the regression model. The model is evaluated with a new dataset in order to be used during the break in building the real system behavior and predicting the actual age components. The ultimate goal is to determine, at each break, the appropriate maintenance strategy \(k_{i}\) for each component based on their actual predicted age. This aims to establish an economical maintenance plan that prevents breakdowns and ensures the system's reliability.  
In addition, the following assumptions are considered to solve the proposed problem:  
* Maintenance actions are performed only during the intermission breaks;
* The duration of missions and breaks are assumed known and fixed;
* A minimum threshold of system reliability should be ensured throughout the operating period.