## 3 Interval Valued Type-2 Fuzzy TOPSIS Method (IVT2F-TOPSIS)  
Interval-Valued Type-2 Fuzzy Sets (IVT2 FSs) aim to manage uncertainties. Since more flexibility could be provided with these, much more detailed and more healthy information transfer is gained [25]. The difference between IVT2 FSs and Type-2 Fuzzy Sets appears in the secondary membership functions. Primary Membership Function (Primary Uncertainty) is defined as \(\mu_{\widetilde{A}}\left(x\right)=\left[\mu^{L}(x),\ \mu^{U}(x)\right],\ \forall x\in X\), where \(\mu^{L}(x)\) and \(\mu^{U}(x)\) are defined as the lower and upper bounds of the primary membership interval and capture the imprecision in the membership degree of an element \(x\). In the IVT2 FS, the secondary membership functions (secondary uncertainty levels) are defined as \(\mu_{\widetilde{A}}\left(x,u\right)\in\left[0,1\right]\), \(u=\left[\mu^{L}(x),\ \mu^{U}(x)\right]\), where \(\mu_{\widetilde{A}}\left(x,u\right)\) represents the degree of confidence or likelihood that the primary membership degree \(u\) and defines the distribution of confidence over the primary membership interval [26]. IVT2 FSs simplify the uncertainty representation. Definition of an interval type-2 fuzzy set:  
\[ \begin{array}{ll}\text{[PERSON], and [PERSON] (2010) defines the IVTZFS as follows }\overset{\approx}{A_{i}}=\left(\overset{U}{A_{i}},\overset{\sim}{A_{i}} \right)=\\ \left(a_{1I}^{U},a_{12}^{U},a_{13}^{U},a_{14}^{U};H_{1}\left(\overset{\sim}{A_ {i}}\right),H_{2}\left(\overset{\sim}{A_{i}}\right)\right)\text{, }\left(a_{1I}^{L},a_{12}^{L},a_{13}^{L},a_{14}^{L};H_{1}\left(\overset{\sim}{A _{i}}\right),H_{2}\left(\overset{\sim}{A_{i}}\right)\right)\text{ in which}\\ \bullet\overset{\sim}{A_{i}}\overset{U}{\text{ and }}\overset{\sim}{A_{i}} \text{ represent the type-1 fuzzy sets,}\\ \bullet\quad a_{ip}^{U}\text{ and }a_{ip}^{L}\text{ denote the reference points of }\overset{\approx}{A_{i}}\text{, }(p=1,\ldots,4)\\ \bullet\quad H_{j}\left(\overset{\sim}{A_{i}}\right)\text{ indicates the upper trapezoidal membership function value of }a_{i(j+1)^{\prime}}^{U}\text{ }(j=1,\ 2)\text{,}\\ \bullet\quad H_{j}\left(\overset{\sim}{A_{i}}\right)\text{ indicates the upper trapezoidal membership function value of }a_{i(j+1)^{\prime}}^{L}\text{ }(j=1,\ 2)\text{,}\\ \bullet\quad H_{1}\left(\overset{\sim}{A_{i}}\right)\text{, }H_{2}\left(\overset{\sim}{A_{i}}\right)\text{, }H_{1}\left(\overset{\sim}{A_{i}}\right)\text{, and }H_{2}\left(\overset{\sim}{A_{i}} \right)\in[\ 0,\ 1]\text{, }(i=1,\ldots,\ n)\text{ [25].}\\ \end{array} \]  
The preliminaries of the arithmetic operations between IVTZFSs are defined in [25]. In this research, the defined arithmetic operations of IVTZFSs the applied for fuzzy multiple-attribute group decision-making. For this purpose, the TOPSIS method is enhanced by IVTZFSs to capture fuzzy environment conditions. The TOPSIS method attempts to select alternatives that are closest to the positive ideal solution while being farthest from the negative ideal solution [27]. In Table 1, linguistic variables and the corresponding Interval Type-2 Fuzzy Sets are shown [28]. In this research, it is assumed that there are \(n\) alternatives, \(A=\{A_{1},\ A_{2},\ \ldots,\ A_{n}\}\), and \(m\) criteria, \(C=\{C_{1},\ C_{2},\ldots,\ C_{m}\}\). The set \(C\) of criteria is divided into two main groups: benefit attributes \((C^{+})\) and cost attributes \((C^{-})\). In addition, \(k\) experts, \(E=\{E_{1},\ E_{2},\ \ldots,\ E_{k}\}\), evaluate alternatives under defined criteria in a fuzzy environment using Table 1[28]. The proposed Interval Type-2 Fuzzy TOPSIS (IT2F-TOPSIS) method steps are illustrated in Figure 1 and details of each step are given below as follows:  
Step 1: Obtain the decision matrix \(D_{p}\) of the \(p^{th}\) expert using linguistic terms as shown in Equation (1). \[D_{p}=\left(\overset{\sim}{D_{ij}^{p}}\right)_{m\times n}=