## 6 The Mathematical Approach for Computing the Dependability of the Solar Tracking System
### Maintenance-Oriented Metric System for Reducing the Mean Time to Repair
Additionally, the sum of all weights must be equal to 1, as presented in Equation (19):  
\[w_{1}+w_{2}+w_{3}=1 \tag{19}\]  
Therefore, the final formula for the maintainability metric can be rewritten as shown in Equation (20):  
\[MI_{Overall}=\frac{w_{1}\cdot MF+w_{2}\cdot AS+w_{3}\cdot PME}{w_{1}+w_{2}+w_{ 3}} \tag{20}\]  
In essence, the _MI_ is a useful tool for assessing, optimizing, and sustaining complex systems such as solar tracking systems. It provides stakeholders with actionable insights for improving system performance, dependability, and durability through proper maintenance methods. Moreover, higher modularity, accessibility, and predictive maintenance facilities may significantly improve system uptime by reducing mean time to repair (MTTR). _MTTR_ is a key maintenance metric that indicates the average time taken to diagnose and rectify faulty equipment. It essentially measures an organization's efficiency in handling and resolving unplanned equipment breakdowns. The _MTTR_ encompasses the following steps: (a) alerting the technicians about the issue; (b) diagnosing the root cause of the problem; (c) undertaking necessary repairs; (d) giving the equipment time to cool down, if necessary; (e) reassembling, aligning, and calibrating the equipment; (f) finally, setting up, testing, and restarting the equipment for regular production. Notably, _MTTR_ does not factor in the waiting period for replacement parts. The _MTTR_ formula can be expressed as in Equation (21):  
\[MTTR=\frac{T_{MT}}{T_{NF}}, \tag{21}\]  
where \(T_{MT}\) represents the total unplanned maintenance time spent on an asset, and \(T_{NF}\) designates the total number of failures that the asset experienced over a specific period. _MTTR_ is most commonly represented in hours and assumes that tasks are performed sequentially by appropriately trained personnel. By combining the _MTTR_ standard metrics with the maintenance-oriented metric system, we can estimate an improved \(MTTR\) index, which is mathematically expressed as in Equation (22):  
\[MTTR_{I}=MTTR_{B}-\sum_{i=1}^{n}TR, \tag{22}\]  
where \(MTTR_{I}\) designates the enhanced MTTR variable, \(MTTR_{B}\) is the baseline \(MTTR\), \(n\) is the number of maintenance-oriented metrics considered, and \(TR\) represents the hour reduction associated with each metric \(i\). To develop a mathematical model that shows the exact time reduction resulting from each metric based on the percentage of \(MF\), \(AS\), and \(PME\), we need to establish relationships between these metrics and the corresponding hour reductions. Thus, we correlate a mathematical function with the \(MF\), \(AS\), and \(PME\) metrics to measure the hour decrease based on their respective percentage values. Regarding the modularity metric, let us consider a function that represents the hour reduction resulting from the MI percentage. To compute this function, we need to establish a relationship between the percentage of MI and the corresponding hour reduction. Given that the \(MF\) represents the ease of replacing faulty components, we can assume a linear relationship where a higher \(MF\) percentage leads to a greater hour reduction. For example, if \(MF=100\%\) corresponds to a 2 h reduction, then \(50\%\)\(MF\) might correspond to a 1 h reduction. Therefore, we can define the modularity function according to Equation (23):  
\[f_{MF}(MF)=k_{1}\times\frac{MF}{100}, \tag{23}\]  
where \(f_{MF}\) denotes the hour reduction function, \(MF\) is the value of modularity expressed in percentage, and \(k_{1}\) s is a constant representing the hour reduction per percentage of \(MF\). Regarding the accessibility metric, similar to the approach for the \(MF\), we can assume a linear relationship between the \(AS\) percentage and the hour reduction. A higher \(AS\) percentage indicates better accessibility, leading to a greater reduction in repair time. For example, if \(AS=100\%\) corresponds to a 3 h reduction, then \(50\%\)\(AS\) might correspond to a 1.5 h reduction.