## 5 Results and Discussion
### Run Length Percentile Analysis for \(\overline{\mathrm{X}}\)-R Control Charts  
Table 4 presents the theoretical values of RL percentiles found for the traditional control chart. It should be noted that the 50 th percentile for the in-control situation, i.e., considering \(\delta\) = 0.0 and \(\lambda\) = 1.0, indicates that there is a false alarm within the first 257 samples, at least half the time. The initial 5 th percentile for \(\delta\) = 0.0 and \(\lambda\) = 1.0 allowed evidencing early false alarms. There is a 5% probability that there will be a first false alarm in less than 19 measurements. Table 5 presents the simulated values of RL percentiles found for IT2 TFN control charts. It should be noted that MDRL (50 th percentile) considering a scenario which is in control, i.e., \(\delta\) = 0.0 and \(\lambda\) = 1.0, indicates that there is a false alarm within the first 260 samples at least half the time. The initial 5 th percentile for  
and R control charts, as can be seen in Table 6, all the samples are in control, which indicates that the process is stable. Applying the proposed fuzzification method, considering the lower and upper bounds \(L_{1}=L_{2}=0.05\), according to experts, FOU equals 0.30 and considering \(H(\widetilde{A}^{U})=H(\widetilde{A}^{L})=1\). The crisp numbers obtained for the traditional \(\overline{\mathrm{X}}\) and R control charts was used with \(x_{2}^{U2}=x_{2}^{L}\) (center value) and is used as a basis for obtaining the other interval type\(-\)2 fuzzy values. After the fuzzification process, as can be seen in Figure 4, we calculate the sample means \(\overline{\mathrm{x}}_{\mathrm{IT2 TFN}}=\left( \begin{array}{cccc}\overline{ \mathrm{x}}_{11}^{U},&\overline{\mathrm{x}}_{11}^{U},&\overline{\mathrm{x}}_{ 20}^{U},&\overline{\mathrm{x}}_{13}^{U},&\overline{\mathrm{x}}_{13}^{U} \end{array} \right)\) and sample ranges \(R_{IT2 TFN}=\left( \begin{array}{cccc}R_{11}^{U},&R_{11}^{L},&R_{12}^{L},&R_{ 13}^{L},&R_{13}^{U}\end{array} \right)\), where \(i\) is the number of the measurements obtained (Table 7). In order to compare IT2 TFN \(\overline{\mathrm{X}}\) and R control charts with the traditional control charts, it is important that the fuzzified values of \(\overline{\mathrm{X}}_{\mathrm{IT2 TFN}}\) and \(\mathrm{R}_{\mathrm{IT2 TFN}}\) are defuzzied, according to Equations (18) and (19). The defuzzied \(\overline{\mathrm{X}}\mathrm{D}_{\mathrm{TIT}}\) and \(\mathrm{RD}_{\mathrm{TIT}}\) values are plotted in the control charts. The fuzzified and defuzzied data can be seen in Table 8.  
Since the values of \(\overline{\mathrm{X}}=\) (0.033, 0.041, 0.059, 0.077, 0.084) and \(\overline{\mathrm{R}}=\) (2.617, 2.632, 2.666, 2.700, 2.715), and considering the variables A\({}_{2}=0.577\), D\({}_{3}=0.000\) and D\({}_{4}=2.114\), it is possible to calculate the defuzzied IT2 TFN \(\overline{\mathrm{X}}\) and R control limits. Based on the data presented in Table 8, one can calculate the control limits for the IT2 TFN \(\overline{\mathrm{X}}\) control chart using Equations (20)-(22). Similarly, to calculate the control limits of the IT2 TFN \(R\) control chart, we use Equations (23)-(25). The IT2 TFN \(\overline{\mathrm{X}}\) and IT2 TFN \(R\) control charts are illustrated in Figures 12 and 13, respectively. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Sample**} & \multicolumn{4}{c}{**Measures (Crisp Values)**} & \multicolumn{4}{c}{**Variables**} & \multirow{2}{*}{**Decision**} \\ \cline{2-2} \cline{4-9}  & \(\mathbf{x_{1}}\) & \(\mathbf{x_{2}}\) & \(\mathbf{x_{3}}\) & \(\mathbf{x_{4}}\) & \(\mathbf{x_{5}}\) & \(\overline{\mathbf{x}}\) & \(\mathbf{R}\) \\ \hline  
0.551\) & \(-0.362\) & \(0.309\) & \(0.449\) & \(0.027\) & \(1.000\) & In control \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Example for traditional \(\overline{\mathrm{X}}\) and \(R\) control charts (normal distribution, \(\mu=0.0\) and \(\sigma=1.0\).