## 3 Interval Valued Type-2 Fuzzy TOPSIS Method (IVT2F-TOPSIS)
\begin{array}{ c}C_{1}\\ C_{2}\\ \vdots\\ C_{m}\end{array}\begin{array}{cccc}D_{1}&D_{2}&\ldots&D_{m}\\ \overset{\approx}{\approx}&\overset{\approx}{\approx}\\ D_{11}^{p}&D_{12}^{p}&\ldots&D_{1n}^{p}\\ \overset{\approx}{\approx}&\overset{\approx}{\approx}&\overset{\approx}{ \approx}\\ D_{11}^{p}&D_{12}^{p}&\ldots&D_{1 nn}^{p}\end{array}  \tag{1}\] \begin{table}
\begin{tabular}{c c} \hline
**Linguistic Terms** & **Interval Type-2 Fuzzy Sets** \\ \hline Very low (VL) & ((0.0, 0.0, 0.0, 0.1; 1.0, 1.0), (0.0, 0.0, 0.0, 0.05; 0.9, 0.9)) \\ Low (L) & ((0.0, 0.1, 0.1, 0.3; 1.0, 1.0), (0.05, 0.1, 0.1, 0.2; 0.9, 0.9)) \\ Medium Low (ML) & ((0.1, 0.3, 0.3, 0.5; 1.0, 1.0), (0.2, 0.3, 0.3, 0.4; 0.9, 0.9)) \\ Medium (M) & ((0.3, 0.5, 0.5, 0.7; 1.0, 1.0), (0.4, 0.5, 0.5, 0.6; 0.9, 0.9)) \\ Medium High (MH) & ((0.5, 0.7, 0.7, 0.9; 1.0, 1.0), (0.6, 0.7, 0.7, 0.8; 0.9, 0.9)) \\ High (H) & ((0.7, 0.9, 0.9, 1.0; 1.0, 1.0), (0.8, 0.9, 0.9, 0.95; 0.9, 0.9)) \\ Very High (VH) & ((0.9, 1.0, 1.0; 1.0, 1.0), (0.95, 1.0, 1.0, 1.0; 0.9, 0.9)) \\ \hline \end{tabular}
\end{table} 
Table 1: Linguistic terms and the corresponding Interval Type-2 Fuzzy Sets. Figure 1: Flowchart of the proposed Interval Type-2 Fuzzy TOPSIS. Step 2: Combine each expert's assessments by calculating the average decision matrix \(\overline{D}\) as shown in Equation (2). \[\overline{D}=(\widetilde{D_{ij}^{\prime}})_{m\times n}=\frac{\widetilde{D_{ij}^{ \prime}}\bigoplus\widetilde{D_{ij}^{\prime}}\bigoplus\ldots\bigoplus\widetilde {D_{ij}^{\prime}}}{k} \tag{2}\]  
\(1\leq i\leq m\), \(1\leq j\leq n\), and \(1\leq p\leq k\). It is important to consider the specifics of arithmetic operations in type-2 fuzzy sets, including summation, subtraction, and multiplication, which are defined in [25]. Step 3: Determine the weights of the importance matrix \(Wp\) for the attributes of the \(p^{th}\) expert, as shown in Equation (3). Then, obtain the average weighting matrix \(\overline{W}\) as shown in Equation (4). \[W_{p}=(\widetilde{w_{i}^{p}})_{1\times n}= \begin{bmatrix}C_{1}&C_{2}&\ldots&C_{ m}\\ \left[\widetilde{w_{1}^{p}}\right]_{1}&\widetilde{w_{2}^{p}}&\ldots&\widetilde {w_{m}^{p}}\end{bmatrix}  \tag{3}\]  
\[\overline{W}=(\widetilde{w_{i}})_{1\times m} \tag{4}\]  
where \(\widetilde{w_{i}}=(\frac{\widetilde{w_{i}^{p}}\bigoplus\widetilde{w_{i}^{p}} \bigoplus\ldots\bigoplus\widetilde{w_{i}^{p}}}{k})\), \(\widetilde{w_{i}}\) is an IVT2 FS, \(1\leq i\leq m\), \(1\leq p\leq k\), and \(k\) represents the number of experts. Step 4: Calculate the weighted decision matrix based on each expert evaluation \(\overline{WD}_{w}\) as shown in Equation (5). \[\overline{WD}_{w}=(\widetilde{z})_{m\times n}= \begin{smallmatrix}C_{1}&a_{1}&a _{2}&\ldots&a_{n}\\ \begin{smallmatrix}\approx&\approx&\approx&\approx&\approx\\ 21&21&\ldots&21m\\ C_{2}&z_{21}^{n}&z_{22}&\ldots&z_{2n}\\ \vdots&\vdots&\vdots&\vdots\\ \approx&\approx&\approx&\approx\\ z_{m1}&z_{m2}&\ldots&z_{mn}\end{smallmatrix}\end{smallmatrix}  \tag{5}\]  
where \(1\leq i\leq m\), \(1\leq j\leq n\). Step 5: Determine the ranking value Rank \((z_{ij}^{n})\) to define the ranking of the weighted decision matrix \(\overline{WD}_{w}^{*}\) as shown in Equation (6). \[ \begin{array}{rl}Rank(\widetilde{z_{i}^{n}})=&M_{1}(\widetilde{z_{i}^{U}})+ M_{1}(\widetilde{z_{i}^{U}})+M_{2}(\widetilde{z_{i}^{U}})+M_{3}(\widetilde{z_{i}^{ U}})+M_{3}(\widetilde{z_{i}^{U}})+M_{3}(\widetilde{z_{i}^{U}})\\ &-\frac{1}{4}(S_{1}(\widetilde{z_{i}^{U}})+S_{1}(\widetilde{z_{i}^{U}})+S_{2} (\widetilde{z_{i}^{U}})+S_{2}(\widetilde{z_{i}^{U}})+S_{3}(\widetilde{z_{i}^{ U}})+S_{3}(\widetilde{z_{i}^{U}})\\ &+S_{4}(\widetilde{z_{i}^{U}})+S_{4}(\widetilde{z_{i}^{U}}))+H_{1}(\widetilde{ z_{i}^{U}})+H_{1}(\widetilde{z_{i}^{U}})+H_{2}(\widetilde{z_{i}^{U}})+H_{2}(\widetilde{z_{i}^{ U}})\end{array}  \tag{6}\]  
where \(M_{p}(\widetilde{z_{i}^{j}})\) shows the average of \(a_{ip}^{j}\) and \(a_{i(p+1)}^{j}\), \(S_{p}(\widetilde{z_{i}^{j}})\) shows the standard deviation of \(a_{ip}^{j}\) and \(a_{i(p+1)}^{j}\) which are calculated as shown in Equations (7)-(9), respectively.