## 3 Interval Type-2 Fuzzy \(\overline{\mathbf{X}}\) and R Control Charts
### Type Reduction and Defuzzification Method  
The transformation process from an IT2 TFN into a crisp value is performed through a couple of steps according to Figure 5.  
Firstly, type reduction occurs by transforming a type-2 into type-1 fuzzy set by means of a type-reducing method. Subsequently, a defuzzification method is applied to obtain a crisp number. The defuzzification method for interval type-2 triangular fuzzy numbers proposed by [26] is used herein to generate a crisp value in order to facilitate the plotting of sample observations, maintain the traditional control chart format and decide whether the monitored process is in or out of statistical control. Let \(\widetilde{\text{A}}=(a_{1}^{U},\,a_{1}^{L},\,a_{2}^{U}\!=\!a_{2}^{L},\,a_{3}^ {L},\,a_{3}^{U})\) be an IT2 TFN. The defuzzification method can be applied using Equation (17):  
\[\text{D}_{\text{TrIT}}=\frac{\frac{\left(\frac{a_{3}^{U}-a_{1}^{U}}{a_{1}} \right)+\left(\frac{a_{2}^{U}-a_{1}^{U}}{a_{1}}\right)}{3}+a_{1}^{U}+\text{H} \Big{(}\,\widetilde{\text{A}}^{\text{L}}\Big{)}\left[\frac{\left(\frac{a_{3}^ {U}-a_{1}^{L}}{a_{1}}\right)+\left(\frac{a_{2}^{L}-a_{1}^{L}}{a_{1}}\right)}{ 3}\right]}{2} \tag{17}\]  
In order to compare \(\,\overline{\text{X}}\) and R interval type-2 fuzzy control charts (IT2 TFN) with traditional \(\,\overline{\text{X}}\) and R control charts, there must be type reduction followed by defuzzification. As calculated values are similar to crisp values, the interval type-2 fuzzy control charts for triangular fuzzy numbers generated by these methods are similar to traditional control charts. Therefore, the presented reduction and defuzzification methods can evaluate the process as being \"in control\" and \"out of control\" in the same way as the traditional (statistical) method. The defuzzification method proposed by [26] allows finding the plotted parameters which can be defuzzied by Equations (18) and (19):  
\[\overline{\text{x}}\,\text{D}_{\text{TrIT}}=\frac{\left(\frac{\,\overline{ \text{x}}_{3}^{U}-\overline{\text{x}}_{1}^{U}}{3}\right)+\left(\,\overline{ \text{x}}_{2}^{U}-\overline{\text{x}}_{1}^{U}\right)}{3}+\,\overline{\text{x }}_{1}^{U}+\text{H}\Big{(}\,\widetilde{\text{A}}^{\text{L}}\Big{)}\bigg{[} \frac{\left(\,\overline{\text{x}}_{3}^{U}-\overline{\text{x}}_{1}^{L}\right)+ \left(\,\overline{\text{x}}_{2}^{L}-\overline{\text{x}}_{1}^{L}\right)}{3} \bigg{]}}{2} \tag{18}\]  
\[\text{RD}_{\text{TrIT}}=\frac{\frac{\left(\,\overline{\text{R}}_{3}^{U}- \overline{\text{R}}_{1}^{U}\right)+\left(\,\overline{\text{R}}_{2}^{U}- \overline{\text{R}}_{1}^{U}\right)}{3}+\,\overline{\text{R}}_{1}^{U}+\text{H }\Big{(}\,\widetilde{\text{A}}^{\text{L}}\Big{)}\bigg{[}\frac{\left(\, \overline{\text{R}}_{3}^{U}-\overline{\text{R}}_{1}^{L}\right)+\left(\, \overline{\text{R}}_{2}^{L}-\overline{\text{R}}_{1}^{L}\right)}{3}\bigg{]}}{2} \tag{19}\]  
Performing the defuzzification of control limits of IT2 TFN \(\,\overline{\text{X}}\) results in Equations (20)-(22):  
\[\text{UCL}_{\text{IT2 TFN}}=\frac{\,\overline{\text{x}}_{1}^{U}+\overline{ \text{x}}_{2}^{U}+\overline{\text{x}}_{3}^{U}+\text{A}_{2}\Big{(}\,\overline{ \text{R}}_{1}^{U}+\,\overline{\text{R}}_{2}^{U}+\,\overline{\text{R}}_{3}^{U} \Big{)}}{6}+\,\text{H}\Big{(}\,\widetilde{\text{A}}^{\text{L}}\Big{)}\Bigg{[} \frac{\,\overline{\text{x}}_{1}^{L}+\overline{\text{x}}_{2}^{L}+\overline{ \text{x}}_{3}^{L}+\text{A}_{2}\Big{(}\,\overline{\text{R}}_{1}^{L}+\, \overline{\text{R}}_{2}^{L}+\,\overline{\text{R}}_{3}^{L}\Big{)}}{6}\Bigg{]}} \tag{20}\]  
Figure 5: Defuzzification method. Adapted from [25].