# Real-Time Maintenance Optimization with Industrial Internet of Things
## 3 Models of Conventional and Real-Time Optimization
### Real-Time Maintenance Strategy Optimization in IIoT-Supported Production Systems  
If the IIoT framework outlined in Section 3.2 is available, it is possible to transform the mathematical model presented in Section 3.1 into a real-time optimization model, which can be used to determine optimal maintenance parameters in real time. Since real-time optimization is fundamentally based on the conventional model presented in Section 3.1, only those elements that show significant differences from the conventional model will be presented. The general structure (topology) of the production system can be described in the same way, as shown in Table 1. The input parameters are the same: layout configuration, degradation rate parameters, initial condition of the system component, and the impact of maintenance on the entity's state. Regarding the input parameters, the most important difference is that the real-time state information of all entities in the manufacturing system is known during operation from the optimization and user interface layer. The initial reliabilities of the three main types of nodes (entities, serial, and parallel connected entities) can be calculated using the same recursive algorithm, based on Equation (3). Based on the real-time state information, real-time decision-making can be performed to find the optimal maintenance performance for the next maintenance period. Based on this real-time decision, the reliability of the nodes and the reliability of the system, depending on the length of the maintenance period, assigned to maintenance period \(p\), can be calculated. The following equation defines this iterative and recursive formula in the  
Figure 1: IIoT-supported framework for real-time maintenance strategy optimization. case of exponential degradation to calculate the state of a node in the iteration step \(p\) before maintenance:  
\[r_{NODE}^{pPREM}= \begin{bmatrix}r_{i}^{p-1}\Big{(}s_{i}^{p-1}\Big{)}+e^{-DRP_{i} \cdot\tau_{p}^{*}},&\text{if the NODE is machine }i\\ \prod_{i=1}^{n}r_{i}^{p-1}\Big{(}child_{i}^{p-1}\Big{)}+e^{-DRP_{i}\cdot\tau_{p }^{*}},&\text{if the NODE is a serial connection }(SER)\\ 1-\prod_{i=1}^{n}\Big{(}1-r_{i}^{p-1}\Big{(}s_{i}^{p-1}\Big{)}+e^{-DRP_{i} \cdot\tau_{p}^{*}}\Big{)},&\text{if the NODE is a parallel connection }(PAR)\\ \end{bmatrix} . \tag{8}\]  
where \(\tau_{p}^{*}\) is the length of the maintenance period for period \(p\). We can use Equation (9) as an iterative recursive formula to compute the reliability of the nodes and the reliability of the system for each maintenance period. The following equation defines this iterative and recursive formula:  
\[r_{NODE}^{pPOSTM}= \begin{bmatrix}r_{i}^{p-1}\Big{(}s_{i}^{p-1}\Big{)}+e^{-DRP_{ i}\cdot\tau_{p}^{*}}+\delta_{i},&\text{if the NODE is machine }i\\ \prod_{i=1}^{n}r_{i}^{p-1}\Big{(}child_{i}^{p-1}\Big{)}+e^{-DRP_{i}\cdot\tau_{p }^{*}}+\delta_{i},&\text{if the NODE is a serial connection }(SER)\\ 1-\prod_{i=1}^{n}\Big{(}1-r_{i}^{p-1}\Big{(}s_{i}^{p-1}\Big{)}+e^{-DRP_{i} \cdot\tau_{p}^{*}}+\delta_{i}\Big{)},&\text{if the NODE is a parallel connection }(PAR)\\ \end{bmatrix} . \tag{9}\]  
We have chosen the described recursive-iterative algorithm because it makes it possible to simulate the impact of real-time decisions during the optimization process. This enables the development of an optimal maintenance strategy in real time, ensuring that the reliability of both system components and the entire system is maintained at an optimal level, while also taking cost optimization into account. The recursive nature of the model reflects the fact that the reliability of subsystems and the entire system can be determined recursively based on the reliability of individual components. In contrast, the iterative nature of the model expresses that the system state is determined iteratively as a function of time. In the case of this real-time maintenance strategy optimization model, the decision variables are the lengths of maintenance cycles for each period (\(\tau_{p}^{*}\)).