## 3 Interval Type-2 Fuzzy \(\overline{\mathbf{X}}\) and R Control Charts  
Regarding a type of quality characteristic able to be expressed as a measure, it is common to monitor both the mean value and variability. The \(\overline{\mathbf{X}}\) control chart is widely used to monitor mean and variability of a process which can be controlled by an R chart [20].  
Fuzzy control charts must be used in the following cases: when statistical data under consideration are uncertain or vague or when available data about the process are incomplete or include human subjectivity [21].  
Regarding the characteristics of the fuzzy set, [22] proposed \(\overline{\mathbf{X}}\) and R and \(\overline{\mathbf{X}}\) and S control chart models using triangular fuzzy numbers. The alpha-cut midrange transformation technique was used for data defuzzification, and representative fuzzy values were found using control charts.  
Subsequently, [14] proposed \(\overline{\mathbf{X}}\) and R control charts using interval type-2 trapezoidal fuzzy numbers (IT2 TraFN). Adapting equations to triangular fuzzy numbers, parameters \(\overline{X}\) and \(R\) to be plotted can be more clearly understood by (7) and (8):  
\[\overline{\mathbf{x}}_{\text{IT2 ITFN}}=\left(\frac{\sum_{i=1}^{n}\mathbf{x}_{ 1}^{\text{U}}}{n},\;\frac{\sum_{i=1}^{n}\mathbf{x}_{1}^{\text{L}}}{n},\; \frac{\sum_{i=1}^{n}\mathbf{x}_{2}^{\text{U}}}{n},\;\frac{\sum_{i=1}^{n} \mathbf{x}_{3}^{\text{L}}}{n},\;\frac{\sum_{i=1}^{n}\mathbf{x}_{3}^{\text{U} }}{n}\right) \tag{7}\]  
\[\text{R}_{\text{IT2 ITFN}}=\left(\text{max}_{1}^{\text{U}}-\text{min}_{3}^{ \text{U}},\;\text{max}_{1}^{\text{L}}-\text{min}_{3}^{\text{L}},\;\text{max} _{2}^{\text{U}}-\text{min}_{2}^{\text{U}},\;\text{max}_{3}^{\text{L}}-\text {min}_{1}^{\text{L}},\;\text{max}_{3}^{\text{U}}-\text{min}_{1}^{\text{U}}\right) \tag{8}\]  
where variable x is an interval type-2 triangular fuzzy number in the format \(\left(\mathbf{x}_{11}^{\text{U}},\;\mathbf{x}_{11}^{\text{L}},\;\mathbf{x}_{ 12}^{\text{L}}=\mathbf{x}_{2}^{\text{U}},\;\mathbf{x}_{32}^{\text{L}},\; \mathbf{x}_{33}^{\text{L}}\right)\) and n is the sample size.  
The average of sample means and sample mean amplitude for m samples can be calculated as in Equations (9) and (10):  
\[\overline{\mathbf{x}}_{\text{IT2 ITFN}}=\left(\frac{\sum_{i=1}^{n}\mathbf{x}_{ 1}^{\text{U}}}{m},\;\frac{\sum_{i=1}^{n}\mathbf{x}_{1}^{\text{L}}}{m},\;\frac {\sum_{i=1}^{n}\mathbf{x}_{2}^{\text{U}}}{m},\;\frac{\sum_{i=1}^{n}\mathbf{x}_ {3}^{\text{L}}}{m},\;\frac{\sum_{i=1}^{n}\mathbf{x}_{3}^{\text{U}}}{m}\right) \tag{9}\]  
\[\overline{\mathbf{R}}_{\text{IT2 ITFN}}=\left(\frac{\sum_{i=1}^{n}\mathbf{R}_ {1}^{\text{U}}}{m},\;\frac{\sum_{i=1}^{n}\mathbf{R}_{1}^{\text{L}}}{m},\;\frac {\sum_{i=1}^{n}\mathbf{R}_{2}^{\text{U}}}{m},\;\frac{\sum_{i=1}^{n}\mathbf{R}_ {3}^{\text{L}}}{m},\;\frac{\sum_{i=1}^{n}\mathbf{R}_{3}^{\text{U}}}{m}\right) \tag{10}\]  
Upper (UCL) and lower control limits (LCL), and the center line (CL) for the \(\overline{\mathbf{X}}\) control chart can be obtained from Equations (11)-(13):  
\begin{table}
\begin{tabular}{c c c} \hline
**Target** & **FOU Values (\(\theta\))** & **FOU Relationship** \\ \hline Faster response & \(0.00<\theta_{1}=\theta_{2}=\theta<0.50\) & They should be equal and small \\ Response with lower or no overshoot & \(0.50\leq\theta_{1}=\theta_{2}=\theta<1.00\) & They should be equal and larg \\ Smaller overshoot and faster response & \(0.30\leq\theta_{1}=\theta_{2}=\theta<0.90\) & They should be equal and medium sized \\ \hline \end{tabular}
\end{table}
Table 1: FOU values according to objectives of a control system. Adapted from [18].