# Production and Maintenance Planning for a Deteriorating System with Operation-Dependent Defectives
## 4 Optimization Method Description
### Production Policy  
We present the obtained production policy \(u^{*}(t)\) in Figure 5a. In such policy the production rate applied in the operational mode is defined in function of the system state that is determined in this case by the stock level and the age of the machine, \((x,a\ )\). At examining this figure, we can realize that it exists a specific threshold for each production rate. To better interpret the production policy, we present its trace in Figure 5b. From the analysis of this figure, we have the following observations:  
1. The computational domain is divided in four zones, where the production rate is set to \(0,u_{1},u_{2}\), or \(u_{max}\), respectively. Such zones are delimited by the production thresholds \(Z_{1},\ Z_{2}\), and \(Z_{3}\). 2. The production rate is decremented gradually (i.e., \(u_{max}>u_{2}>u_{1}>0\)) as the stock level increases and surpasses the production thresholds \(Z_{3},\ Z_{2}\), and \(Z_{1}\) with \(Z_{1}>Z_{2}>Z_{3}\). We note that the minimum production rate \(u_{1}\) is recommended for high levels of inventory, when the stock level is between the production threshold \(Z_{1}\) and \(Z_{2}\). Since given the existence of inventory, the system can manage to satisfy product demand operating at a reduced pace and mainly because it avoids further deterioration. Moreover, we note that as the stock level decreases, the production rate increases, thus the machine operates at its second production rate \(u_{2}\) when the inventory level is between the thresholds \(Z_{2}\) and \(Z_{3}\), with the aim to replace inventory faster. Additionally, we note that the use of the maximum production rate \(u_{max}\) is limited, because it deteriorates more rapidly than the machine, and so its use is only recommended in scenarios where the stock level is almost depleted and there is a need to rapidly replace the inventory level and avoid further shortages. 3. From the obtained results we observe that the production thresholds \(Z_{1},\ Z_{2}\), and \(Z_{3}\) increases as the machine ages, this reflects the impact of the deterioration process on the production control rule. For instance at age \(a\ =0\), the production threshold \(Z_{1}\) has a value of \(Z_{1}=4\), and as the machine deteriorates, at age \(a\ =100\), such threshold increases to \(Z_{1}=7\). The same pattern is observed for thresholds \(Z_{2}\) and \(Z_{3}\). The increment is explained by the fact that as the machine deteriorates, it produces more defectives and also it is subject to experience more frequent failures. Thus, more inventory is needed as protection to mitigate shortages. Our results differ from the classical results of [PERSON] et al. [23] with only three production rules. In our case, the production policy defines the following control rules:  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(u_{max}\) & \(d\) & \(\rho\) & \(h_{x}\) & \(h_{a}\) & \(u_{1}\) & \(u_{2}\) & \(\lambda_{1}\) & \(b_{1}\) & \(  
1. Decrease the production rate to zero, if the current stock level is above the first production threshold \(Z_{1}\). 2. Set the production rate to its first productivity \(u_{1}\), when the current stock level is under the first production threshold \(Z_{1}\). 3. Increase the production rate to its second productivity \(u_{2}\), when the stock level is below the second production threshold \(Z_{2}\). 4. Increase the production rate to its maximum rate \(u_{max}\), when the current stock level is under the third production threshold \(Z_{3}\). In a practical sense, the results of Figure 5 imply a multi-hedging policy form, where the production rate is defined as follows:  
\[u(1,x,a)^{*}=\left\{ \begin{array}{ll}u_{max}&if\ x(t)<Z_{3}(\cdot)\\ u_{2}&if\ x(t)<Z_{2}(\cdot)\\ u_{1}&if\ x(t)<Z_{1}(\cdot)\\ 0&if\ x(t)>Z_{1}(\cdot)\end{array} \right. \tag{25}\]  
where \(Z_{1}(\cdot)>Z_{2}(\cdot)>Z_{3}(\cdot)\) and \(0<u_{1}<u_{2}<u_{max}\).