## 3 Discrete Event Simulation Model
### Equipment Modeling  
In the DES model, the behavior pattern of the machines is described by an analytical model. A previous research developed by [PERSON] and [PERSON] [12] considered this analytical model as a single equipment model. The present work takes into account the modeling of maintenance, quality and production speed loss costs simultaneously, with the benefit related to the production of non-defective products. The decision variables (\(\mathbf{x}\)) are the critical age levels when PM activities are carried out; thus, CBM deterioration thresholds, along with instants when a PM activity is not enough and a substitution of the component is needed (the As Good As New (GAN) model). They are optimized considering cost and profit criteria, and they affect directly all the terms mentioned above (modeling of maintenance, quality, production speed loss costs, etc.). The aforementioned equipment model is built under the following assumptions: (1) An imperfect maintenance model determines the effect of the maintenance activities. In such cases, a Proportional Age Reduction (PAR) [13] is considered. (2) The deterioration and failure processes are independent from each other. (3) When the process is out-of-control, the system only delivers non-conforming components, with a rate constant (\(\alpha\)). (4) Preventive maintenance and process inspection are carried out concurrently. (5) Inspections are free of errors. (6) When the preventive maintenance is performed, the process is brought back to under control state. (7) Productive speed drops from its initial speed (\(V_{0}\)) to another speed value (\(V^{*}(\mathbf{x})\)), which is directly related to the CBM decision variables (\(\mathbf{x}\)). (8) All deterioration processes related to the three studied components are independent. (9) Since the process delivers only one type of product, setup times of reference changes are not calculated. Unlike the Condition Monitoring (CMT) of the component, done continuously, PM is only performed when a component has a specific critical age or deterioration level (\(w_{c}\)). Moreover, PAR model assumes that the maintenance reduces the age of the component proportionally by a \(\epsilon\) factor, between the moment after executing the previous PM activity (\(w_{m-1}^{+}\)) and the moment before the last one (\(w_{m}^{-}\)). The assumptions \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Group**} & \(\lambda_{1}\) & \(\gamma_{1}\) & \(\lambda_{2}\) & \multirow{2}{*}{\(\gamma_{2}\)} \\  & (\(\mathbf{10}^{-2}\mathbf{h}^{-1}\)) & & (\(\mathbf{10}^{-2}\mathbf{h}^{-1}\)) & \\ \hline S1 & 5 & 2 & & \\ S2 & 2 & 2.9 & & \\ S3 & 4 & 2 & 4 & 2 \\ S4 & 6.6 & 2 & & \\ S5 & 7.7 & 3 & & \\ S6 & 10 & 3 & 110 & 3 \\ \hline \end{tabular}
\end{table} 
Table 2: Weibull data of the studied subsystems. to obtain the expressions for the PAR model are the same as in [12]. The results of these assumptions are stated in Figure 2, where the age of a component \(w\) is compared with its chronological time. Figure 2 shows that the component is continuously monitored, i.e. no preventive action is carried out until it reaches a critical age \(w_{c}\). It can be observed that corrective maintenance is performed with minimal repair (the As Bad As Old (BAO) model is adopted), and PM is modeled considering a PAR model with effectiveness \(\varepsilon\). Moreover, the time between two PM activities, M, gets shorter and shorter, following this equation:  
\[M_{m}=w_{c}\cdot\varepsilon^{m-1}. \tag{1}\]  
This makes is necessary to fix a limit value on time between two PM activities, \(M_{min}\), from which the component will be upgraded with an intervention of \(\varepsilon=1\) (GAN). The aim of the problem is to optimize the limit values \(M_{min}\) and the values \(W_{c}\). Hence, a component will be installed for a \(L_{c}\) time period and then it will be upgraded (or substituted) into a GAN situation:  
\[L_{c}=\sum_{k=1}^{e}M_{k}, \tag{2}\]  
where \(e\) represents the amount of PMs executed before a component is replaced.