## 4 Numerical Example
### Maintenance and Quality Problem  
Example 1:  
To resolve the maintenance and quality problem, we developed the following numerical optimization procedure based on the obtained optimal production plan and following different parameters arbitrarily chosen:  
* The average duration of each preventive maintenance action, at each out-of-control state, is 1.5 days = \(\Delta t/20\). * In our study, we have used the Weibull distribution for the evolution of the production system failure rate in its wear-out life, where its failure rate increases quickly with both time and use, and which requires preventive maintenance actions unlike the other phases (early life, useful life) which are not needed. So, the degradation law of the production system characterized by a Weibull distribution with scale and shape parameters are respectively \(\beta_{1}\)= 100 and \(\alpha_{1}\)= 2. * \(C_{CM}\) = 3000 mu, \(C_{PM}\) = 500 mu, \(c_{r}\) = 70 \(\mu\)/unit lost, \(c_{F}\) = 120 \(\mu\)/false alarm, \(c_{i}\) = 15 \(\mu\)/product, \(c_{r}\) = 70 \(\mu\)/defective item. * \(\mu_{0}\) = 5, \(\sigma_{0}\) = 1.5, \(\mu_{pc}\)= 1, \(\delta_{1}\)= 0.4. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(Q_{2}\)*(1) & \(Q_{2}\)*(2) & \(Q_{2}\)*(3) & \(Q_{2}\)*(4) & \(Q_{2}\)*(5) & \(Q_{2}\)*(6) \\ \hline
181 & 112 & 130 & 162 & 155 & 222 \\ \hline \(Q_{2}\)*(7) & \(Q_{2}\)*(8) & \(Q_{2}\)*(9) & \(Q_{2}\)*(10) & \(Q_{2}\)*(11) & \(Q_{2}\)*(12) \\ \hline
205 & 225 & 182 & 100 & 70 & 110 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Optimal delivery plan for warehouse 2 (\(\tau\)* = 1). Figure 7: Production rates as a function of standard deviation, \(\sigma d_{i}\)  
Using a numerical procedure, the optimum values of \(m,n,\ h,\ kp\), and \(N\) are presented in the Table 14 and Figure 8 below. \[UCL=\mu+4.6\times\frac{\sigma}{\sqrt{12}}\]  
The best strategy is to take one sample every 60 h. The optimal number of samples taken is 144. The model, therefore, reacts so as to minimize the average total cost by increasing the interval and frequency of sampling. Furthermore, the ideal number of standard deviations between the control chart centerline and limits is around 4.6. The total cost average corresponding to the optimum policy is \(45,000\ \mu\). The average number of samples taken to have a false alarm is equal to \(AOP_{2}=120\) h. The average number of samples taken to detect when the control limits are exceeded is equal to \(AOP_{1}=84\ h=7\cdot\Delta t\). Thus, the control card can be considered efficient because it allows to give a false alarm after a relatively high number of samples and quickly detects exceeding limits. From Table 7, the total best number of preventive maintenance activities without duration equals 3. Concerning the optimal number of preventive maintenance activities between the out-of-control point, maintenance cycle 1 (\(N_{2}=2\)) and for maintenance cycle 2 (\(N_{2}=1\)), as shown in Table 15.  
The first maintenance cycle is characterized by the production period before the out-of-control point, where the \(AOP_{1}=84\ h=7.\Delta t\), and we applied two preventive maintenance actions without duration according to the average number of failures. The second cycle is characterized by the production periods from \(AOP_{1}+tp=7.\Delta t+t/20\) to \(H.\Delta t\), where we applied one preventive maintenance action. Example 2 (increased average of demand):  
Using the economic plan of production obtained in Table 8, the optimal delivery plan for warehouses (1 and 2) at \(\tau=1\) in Tables 9 and 10, and the exact costs of maintenance and quality, the optimal values of \(n,\ h,\ m,\ kp\), and \(N\) are presented in Table 16 and Figure 9 below. \begin{table}
\begin{tabular}{c c c} \hline
**Cycle 1 of Maintenance** & **Cycle 2 of Maintenance** \\
[0,\(AOP_{1}\)]=[0,\(7.\Delta t\)] & [\(AOP_{1}\), \(H\)]=[\(7.\Delta t\)+t/20, \(12.\Delta t\)] \\ \hline \(N_{2}=2\) & \(N_{2}=1\) \\ \hline \end{tabular}
\end{table} 
Table 15: Optimal number of preventive maintenance activities (\(N=3\)).