# Analysis and Application of Selected Control Charts Suitable for Smart Manufacturing Processes
## 3 Methodological Framework
### Selection of Control Chart Performance Indicators  
Before simulations, attention had to be paid to the selection of the most precise control chart performance indicators that are most stable for different probability distributions and so the most convenient for appraising and comparing the performance of various NPCCs [52; 53; 54]. Indicators for statistically stable processes such as average run length ARL(0), median run length MRL(0), 5% quantile \(x_{5}\) and indicators for statistically unstable processes such as ARL(\(\delta\)), MRL(\(\delta\)) and 95% quantile \(x_{95}\) were analyzed. For a statistically stable process, the values of the performance indicators must be as high as possible [12]. Calculations of these indicators are based on RL(0), which represents the quantity of points, plotted in the control chart, that are located inside the control limits between two points lying outside of these limits. For a statistically unstable process, the values of the performance indicators must be as small as possible. To compute these indicators, it is necessary to set RL(\(\delta\)) (\(\delta\) is a standardized value of the shift that is supposed to be detected by the control chart as soon as possible), which is a quantity of points plotted in the control chart starting from the point when the shift in the process occurred until the shift is announced by the point outside of the control limit.  
To assess the quality of these control chart performance characteristics, their values were sorted in descending order. The course of the resulting pointed lines expresses the rate of stability of the analyzed performance indicators for various simulated deviations from the data prerequisites represented by 7 probability distributions. The more tardily the line goes down, the more stable the performance indicator can be considered. Based on this conclusion, it can be said that for statistically stable processes, the 5% quantile \(x_{5}\) is the most stable indicator for various deviations from the data prerequisites. Regarding ARL(0) and MRL(0), the latter is the more stable indicator. Accordingly, the following performance evaluation of analyzed NPCCs for statistically stable processes was made using only \(x_{5}\) and MRL(0) performance indicators [51]. As an example of this analysis, there are plots of performance indicators in relation to various violations of data assumptions represented by seven different probability distributions (see Table 1) for the SSCC control that can be found in Appendix A. Seven points on every curve in Figures 11-16 (Appendix A) correspond to the seven probability distributions, and three points in a vertical direction represent three analyzed performance indicators. Such plots were constructed for all the rest of the analyzed NPCCs.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Unmet Data Assumption** & **Distribution/Data Properties** & **Distribution Parameters** \\ \hline None & Normal distribution & N (0,1) \\ \hline Data normality & Pearson distribution/skewed distribution & \(\chi_{3}^{2}\) \\ \hline  & Uniform distribution/less kurtosis & \(R\left(-\sqrt{3};\sqrt{3}\right)\) \\ \hline  & Student’s distribution/greater kurtosis & \(\mathrm{t}_{3}\) \\ \hline Constant variance & Mixed distribution/different variances & 50\% N(0,1) + 50\% N(0,4) \\  & MIX 2 \\ \hline Constant mean & Mixed distribution/different means & 50\% N(0,1) + 50\% N(2,1) \\  & MIX 1 \\ \hline Data independence & Autocorrelated data & AR (1): \(x_{i}-0.5x_{i}+a_{i}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of various simulated violations of data prerequisites and their connection to the utilized probability distributions. Source: own research.  
Based on a similar analysis of statistically unstable processes, only \(x_{95}\) and \(\text{MRL}(\delta)\) were utilized for evaluation of NPCC performance [51].