# Real-Time Maintenance Optimization with Industrial Internet of Things
## 3 Models of Conventional and Real-Time Optimization
### Maintenance Strategy Optimization in Conventional Production Systems  
In a conventional production system, the input parameters of the maintenance strategy optimization are the following (see Table 1):  
* Layout configuration: The structure of the production system is represented in a tabular format that details the interconnections and dependencies between individual machines or system components. This format allows for the application of a recursive algorithm to calculate the overall system reliability. The reliability of the system is derived based on the reliability of its subsystems and individual elements (entities), taking into account both serial and parallel dependencies within the production line;
* Degradation rate parameter: This parameter characterizes the speed at which a specific system component (entity) deteriorates over time as a result of usage, environmental exposure, or operational stress. It is typically modeled as a deterministic or stochastic function, depending on the system's nature and the availability of historical data. In simpler models, a linear degradation rate may be assumed, while more advanced models can employ exponential, Weibull, or other statistical distributions to capture component-specific behavior more accurately. The degradation rate directly influences the frequency of required maintenance operations; components with higher degradation rates demand more frequent maintenance operations to avoid system failures and ensure consistent performance and reliability;
* Initial condition (state) of the system component: This refers to the baseline operational state of the component at the beginning of the time window of the analysis. It defines the starting point of the degradation process and is often quantified as a normalized performance score (e.g., 100% health or functionality) or as a failure probability close to zero. The initial condition is essential for modeling time-dependent changes in reliability and performance;
* The impact of maintenance on the entity's state: When a more expensive and complex maintenance operation is performed, the state (condition) of the equipment (entity) can improve significantly. This is because advanced maintenance techniques can address deeper issues or provide more thorough repairs, enhancing the overall performance and extending the lifespan of the entity. However, this approach may not always be ideal, as excessive maintenance could lead to over-improvement, meaning the equipment might reach a condition better than necessary for optimal functioning. In such cases, the resources and costs spent on maintenance may outweigh the actual benefits, making it an inefficient strategy for achieving the desired operational state. Based on the data presented in Table 1, the system reliability can be determined using a recursive algorithm. If we define the structure of the production system according to Table 1, the entire system forms a tree, where each internal node is one of the following:  
* the form _SER_ (_ID\({}_{1}\)_, _ID\({}_{2}\)_, _ID\({}_{3}\)_,...) (series connection);
* _PAR_ (_ID\({}_{1}\)_, _ID\({}_{3}\)_,...) (parallel connection). \begin{table}
\begin{tabular}{c c c c c} \hline
**ID** & **Type** & **Child 1** & **Child 2** & **Initial State** & **Degradation Rate Parameter** \\ \hline \(ID_{1}\) & entity & & \(s_{1}^{INI}\) & \(DRP_{1}\) \\... & or & & & \\ \(ID_{i}\) & serial & \(Child_{1,y}\in\{ID_{1},\ldots,ID_{i},\ldots,ID_{n}\}\) & & \\ \(\ldots\) & or & & & \\ \(ID_{n}\) & parallel & & \(s_{n}^{INI}\) & \(DRP_{n}\) \\ \hline \end{tabular}
\end{table} 
Table 1: General structure of parent–child relationships of resources in the case of 2 children. The reliability functions of the tree's individual leaves are denoted as \(r_{i}(s_{i}^{INI})\) representing the reliability of machine \(i\) as a function of its state. Let us consider a simple example to calculate the reliability of a basic system.