## 2 Interval Type-2 Fuzzy Sets
### Interval Type-2 Triangular Fuzzy Sets  
An interval type-2 fuzzy set \(\widetilde{\rm A}\) is represented by a lower membership function (LMF) \(\widetilde{A}^{L}\) and an upper membership function (UMF) \(\widetilde{\rm A}^{\rm U}\), where \(\widetilde{\rm A}=\widetilde{\rm A}^{\rm L}\), \(\widetilde{\rm A}^{\rm U}\). In particular, when triangular pertinence functions are used, it is illustrated as in Figure 2, where \(\widetilde{A}^{L}=\left(a_{i1}^{L},a_{i2}^{L},a_{i3}^{L};\ H\left(\widetilde{A }^{L}\right)\right)\), \(\widetilde{A}^{U}=\left(a_{i1}^{U},a_{i2}^{U},a_{i3}^{U};\ H\left(\widetilde{A }^{U}\right)\right)\), \((a_{i1}^{L},a_{i2}^{L},a_{i3}^{L})\) and \((a_{i1}^{U},a_{i2}^{U},a_{i3}^{U})\) denote three key points of LMF and UMF, and \(\widetilde{\rm A}^{\rm L}\) and \(\widetilde{\rm A}^{\rm U}\), where \(a_{1}^{U}\leq a_{1}^{L}\), \(a_{3}^{U}\leq a_{3}^{L}\), \(0<\ H\left(\ \widetilde{\rm A}^{\rm L}\right)\leq{\rm H}\left(\ \widetilde{\rm A}^{\rm U}\right)=1\), denote the highest membership function values. Upper membership and lower membership functions are described according to Equations (2) and (3), respectively:  
\[\mu_{\widetilde{\rm A}}^{\rm U}=\left\{ \begin{array}{ll}\frac{x-a_{1}^{\rm U }}{a_{2}^{\rm U}-a_{1}^{\rm U}};\text{if}\ a_{1}^{\rm U}\ \leq\ x\ \leq\ a_{2}^{\rm U}\\ \frac{a_{3}^{\rm U}-x}{a_{3}^{\rm U}-a_{2}^{\rm U}};\text{if}\ a_{2}^{\rm U}\ \leq\ x\ \leq\ a_{3}^{\rm U}\end{array} \right. \tag{2}\]  
\[\mu_{\widetilde{\rm A}}^{\rm L}=\left\{ \begin{array}{ll}\frac{x-a_{1}^{\rm L }}{a_{2}^{\rm U}-a_{1}^{\rm U}};\text{if}\ a_{1}^{\rm L}\ \leq\ x\ \leq\ a_{2}^{\rm L}\\ \frac{a_{3}^{\rm U}-x}{a_{3}^{\rm U}-x_{2}^{\rm U}};\text{if}\ a_{2}^{\rm L}\ \leq\ x\ \leq\ a_{3}^{\rm L}\end{array} \right. \tag{3}\]  
The closer the shapes of \(\widetilde{\rm A}^{\rm L}\) and \(\widetilde{\rm A}^{\rm U}\) are, the less uncertain information contained in \(\widetilde{\rm A}\) obviously is. When \(\widetilde{\rm A}^{\rm L}\) coincides with \(\widetilde{\rm A}^{\rm U}\), the interval type-2 fuzzy set becomes a type-1 fuzzy set [12]. Figure 1: Interval type-2 triangular fuzzy number (3, 8, 12, 15, 18) in a three-dimensional space. Figure 2: Interval type-2 triangular fuzzy number and the membership functions. Adapted from [12]. In order to simplify the model, the central IT2 TFN value \(a_{2}^{U}=a_{2}^{L}\) is considered, as it defines the singleton of an interval type-2 triangular fuzzy number, but height \(H\Big{(}\widetilde{A}^{U}\Big{)}\) of the upper membership function coincides with height \(\mathrm{H}\Big{(}\widetilde{A}^{\mathrm{L}}\Big{)}\) of the lower membership function. Particularly where \(a_{2}^{U}=a_{2}^{L}\) and \(H\Big{(}\widetilde{A}^{U}\Big{)}=H\Big{(}\widetilde{A}^{L}\Big{)}\), the so-called perfect interval type-2 triangular fuzzy number (PI2 TFN) is obtained [13]. The arithmetic operations of addition, subtraction and multiplication, respectively, are used in \(\overline{\mathrm{X}}\)-R control charts for IT2 TFN, which are given by Equations (4)-(6) according to [14]:  
\[\widetilde{A}_{1}+\ \widetilde{A}_{2} = \begin{pmatrix}\Big{(}a_{11}^{\mathrm{U}}+a_{21}^{\mathrm{U}},a_{ 12}^{\mathrm{U}}+a_{22}^{\mathrm{U}},a_{13}^{\mathrm{U}}+a_{23}^{\mathrm{U}}; \min\Big{(}\mathrm{H}\left(\widetilde{A}_{1}^{\mathrm{U}}\right),\Big{(} \mathrm{H}\left(\widetilde{A}_{2}^{\mathrm{U}}\right)\Big{)},\Big{)}\\ \Big{(}a_{11}^{\mathrm{L}}+a_{21}^{\mathrm{L}},a_{12}^{\mathrm{L}}+a_{22}^{ \mathrm{L}},a_{13}^{\mathrm{L}}+a_{23}^{\mathrm{L}};\min\Big{(}\mathrm{H} \left(\widetilde{A}_{1}^{\mathrm{L}}\right),\Big{(}\mathrm{H}\left(\widetilde{ A}_{2}^{\mathrm{L}}\right)\Big{)}\Big{)}\end{pmatrix}  \tag{4}\] \[\widetilde{A}_{1}-\ \widetilde{A}_{2} =