## 2 Production, Maintenance, and Quality Problem
### Control Chart Parameters  
Unlike the usual economic production models, simplifying the manufacturing process for improvement involves non-influence of the production rate of the machine to its deterioration conditions and limits their costs analysis only based on inventory issues, and decisions can be made based on ordering and storage costs. In the actual manufacturing process, the production system often starts in the \"in-control\" state operation and may shift to an \"out-of-control\" state due to many factors. An 'x-bar' control chart with a mean and standard deviation tool is utilized to supervise the quality characteristics and notify when the production process shifts to an out-of-control situation. The Gaussian distribution is the most useful distribution in statistical quality control for an incurred manufacturing process. The \"statistical process control\" (SPC) is an element of quality assurance and is considered among the six-sigma method, which aims to satisfy the customer in terms of quality, quantity, and service at the lowest cost. The SPC is a method of monitoring a process to identify specific causes of random variation and signals the need to take corrective actions, when appropriate and based on control charts. The normal law is widely used in statistical control, especially in its reduced centered form. It is used to model the random variation of the process, and before using the control chart, it must verify the normality of the samples used during process control. The state of the machine can be assessed by measuring the quality indicator \(X_{t}\) of the manufactured product. The evolution of this indicator follows a normal law of mean, \(\mu_{0}\), and standard deviation, \(\sigma_{0}\). Every \(h\) units of time, a sample of \(n\) size is taken and checked to monitor the evolution of the indicator \(X_{t}\). For each sample, \(X_{s}\), the average value shown in the control chart is characterized as follows:  
\[\overline{X}_{s}=\frac{1}{n}\times\sum_{i=1}^{n}X_{i} \tag{1}\]  
Figure 3: The two scenarios of quality control and evolution of maintenance. Besides, for each sample, \(S\), we define the associated standard deviation given by:  
\[\sigma_{s}=\sqrt{\frac{\sum_{i=1}^{n}(X_{i}-\overline{X}_{s})}{n-1}} \tag{2}\]  
Then, we calculate the mean of the means (central value) and the average of the standard deviations of the indicator \(X_{s}\) for all m samples:  
\[\mu_{0}=\frac{1}{m}\times\sum_{s=1}^{m}\overline{X}_{s} \tag{3}\]  
\[\sigma_{0}=\frac{1}{m}\times\sum_{s=1}^{m}\sigma_{s} \tag{4}\]  
So, we define the control limits as follows:  
Upper control chart limits:  
\[UCL=\mu_{0}+\frac{k_{p}}{\sqrt{n}}\times\sigma_{0} \tag{5}\]  
Lower control limit:  
\[LCL=\mu_{0}-\frac{k_{p}}{\sqrt{n}}\times\sigma_{0} \tag{6}\]  
Following an inspection of a sample of \(n\) size, two scenarios are possible depending on whether the limits of control are exceeded or not. Scenario 1: When the process is under control. If the measurements' average of the quality indicator \(\overline{X}_{s}\) is located between the control limits \(LCL\leq\overline{X}_{s}\leq UCL\), the production process is considered to be under control, and a maintenance strategy characterized by PMMR maintenance actions is planned. So, the production process continues as normal. Scenario 2: When the process is out of control. If the measurements' average of the quality indicator is situated outside control limits (\((UCL<\overline{X}_{s})\) or \((\overline{X}_{s}<LCL)\)), the production process is considered in critical condition. In this case, the process is stopped and a perfect preventive maintenance action with a mean duration \(t_{p}\) is performed. The probability of each of these two scenarios depends on the preventive maintenance activities number corresponding to the number of assignable causes, \(n_{pc}\), and the optimal number of inspections, \(m\).