## 3 Selection of Methods for Methodology
### Preliminaries of Intuitionistic Fuzzy Sets  
Intuitionistic fuzzy sets (IFSs) extend the concept of classical fuzzy sets by incorporating additional information to deal with uncertainty and hesitation. The main idea behind intuitionistic fuzzy sets is to represent the hesitancy and ambiguity that are often associated with decision-making processes. First, we discuss the preliminaries of the intuitionistic fuzzy set (IFS) and the algebraic operations (discussed in [54]) between the intuitionistic fuzzy numbers (IFNs) that are relevant to the proposed PROMETHE method, namely, PROMETHEE-IFS. **Definition 1**. : _If the domain of problem-related objects is denoted by \(X\), \(x\in X\) is a single object. In this research, \(X\) is a set of criteria modeled under the intuitionistic fuzzy environment and \(x\) is a value of a single criterion._ \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Experts**} & \multicolumn{4}{c}{**Pairwise Evaluation of Criteria Relative Importance**} \\ \cline{2-5}  & \(\mathbf{C_{1\leftrightarrow 2}}\) & \(\mathbf{C_{2\leftrightarrow 3}}\) & \(\mathbf{C_{3\leftrightarrow 4}}\) & \(\mathbf{C_{4\leftrightarrow 5}}\) \\ \hline
1 & 0.40 & 0.10 & 0.35 & 0.85 \\
2 & 0.10 & 0.15 & 0.30 & 0.25 \\
3 & 0.15 & 0.00 & 0.70 & 0.65 \\
4 & 0.25 & 0.33 & 0.79 & 0.74 \\
5 & 0.33 & 0.20 & 0.28 & 0.45 \\
6 & 0.52 & 0.48 & 0.69 & 0.63 \\
7 & 0.35 & 0.60 & 0.20 & 0.50 \\
8 & 0.20 & 0.39 & 0.30 & 0.62 \\
9 & 0.25 & 0.77 & 0.58 & 0.55 \\
10 & 0.30 & 0.65 & 0.65 & 0.75 \\ Average & 0.285 & 0.367 & 0.484 & 0.599 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Evaluation of the relative importance by criteria pairs. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Criteria**} & \multicolumn{2}{c}{**Average Values of**} & **Coefficients of Comparative** & **Resalculated (Intermediate)** & **Final Criteria Weights,** \\  & **Comparative Importance** & **Importance Criteria,** & **Criteria Weights,** & \(w_{j}\) \\  & **Criteria \(s_{j\leftrightarrow j}\)+1** & \(k_{j}\) & \(q_{j}\) & \(w_{j}\) \\ \hline \(C_{1}\) & - & 1 & 1 & 0.336585 \\ \(C_{2}\) & 0.285 & 1.285 & 0.778210 & 0.261934 \\ \(C_{3}\) & 0.367 & 1.367 & 0.569283 & 0.191612 \\ \(C_{4}\) & 0.484 & 1.484 & 0.383614 & 0.129119 \\ \(C_{5}\) & 0.599 & 1.599 & 0.239909 & 0.080750 \\ \(\Sigma\) & & & 2.971016 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Criteria weighting by SWARA method. In fuzzy sets, the membership function defines the degree of membership of each element in the set. This function maps each element to a value in the interval [0, 1]. The intuitionistic fuzzy set \(IFS\) consists of two membership functions (fuzzy components):  
\[T_{ifs}F_{ifs}:X\rightarrow[0,1] \tag{9}\]  
Here, \(T_{ifs}\) is the truth membership function that represents the degree to which an element \(x\) belongs to the set and \(F_{ifs}\) is the falsity membership function (or non-membership function) that represents the degree to which an element \(x\) does not belong to the set. Then, the intuitionistic fuzzy set [14] is defined as follows:  
\[IFS=\left\{\left\langle T_{ifs}(x),F_{ifs}(x)\right\rangle:x\in X\right\} \tag{10}\]  
The two membership functions also satisfy the following condition:  
\[0\leq T_{ifs}(x)+F_{ifs}(x)\leq 1\ \ \forall x\in X \tag{11}\]  
The IFS theory also introduces a third dimension known as hesitation function \(\pi\), which is also called the degree of non-determinacy (or uncertainty) [14]. This function, of the element \(x\in X\) to the intuitionistic fuzzy set \(IFS\) represents the degree of hesitation or uncertainty associated with the element \(x\)'s membership status. This allows for a more realistic representation of uncertainty that may arise during the assessment of segmentation methods. \[\pi_{ifs}(x)=1-T_{ifs}(x)-F_{ifs}(x) \tag{12}\]  
Here, \(0\leq\pi_{ifs}(x)\leq 1\), \(\forall x\in X\). A high value of \(\pi_{ifs}(x)\) indicates a higher degree of hesitation or uncertainty regarding whether the element \(x\) belongs to the set or not.