## 4 The Modifications and Extensions of Classic Hellwig's Method
### Hellwig-Based Methods with Data Represented by Intuitionistic Fuzzy Numbers
Following this, the aggregated opinions are expressed as intuitionistic fuzzy values \((\mu_{ij},\
u_{ij})\), represented by the equations:  
\[0\leq\mu_{A}(x)+\
u_{A}(x)\leq 1. \tag{42}\]0  
where \(i\) indicates the alternative number, and\(j\) indicates the criterion number. Now, we will review the intuitionistic fuzzy variants of the Hellwig-based methods, focusing on the steps of the algorithm (see Figure 1). Let \(A=\{A_{1},A_{2},\ldots,A_{m}\},i=1,2,\ldots,m\) represent the set of alternatives and \(C=\{C_{1},C_{2},\ldots,C_{n}\},j=1,2,\ldots,n\) represent the set of criteria. The representation of the \(i\)th alternative with respect to the \(j\)th criterion is given by an intuitionistic fuzzy value \(\big{(}\mu_{ij},\
u_{ij}\big{)}\), where \(\pi_{ij}=1-\mu_{ij}-\
u_{ij}\). As a result, alternative \(A_{i}\) is described by the vector: \(A_{i}=\big{[}(\mu_{i1},\
u_{1}),\ldots,(\mu_{in},\
u_{in})\big{]}\), where \(i=1,2,\ldots,m\). The general framework and intuitionistic fuzzy Hellwig-based method consist of the following steps [170; 173]:  
Step 1. Definition of an intuitionistic fuzzy decision matrix  
The intuitionistic fuzzy decision matrix (\(IFD\)) is defined as follows:  
\[IFD= \begin{bmatrix}(\mu_{11},\
u_{11})&(\mu_{12},\
u_{12})&\cdots&(\mu_{1n},\
u_ {1n})\\ (\mu_{21},\
u_{21})&(\mu_{22},\
u_{22})&\cdots&(\mu_{2n},\
u_{2n})\\ \vdots&\vdots&\ddots&\vdots\\ (\mu_{m1},\
u_{m1})&(\mu_{m2},\
u_{m2})&\cdots&(\mu_{mn},\
u_{mn})\end{bmatrix}  \tag{52}\]  
where \(\big{(}\mu_{ij},\
u_{ij}\big{)}\) is the intuitionistic fuzzy evaluation of the \(i\)th alternative in terms of the \(j\)th criterion and \(\pi_{ij}=1-\mu_{ij}-\
u_{ij}\) (\(i=1,2,\ldots,m\); \(j=1,2,\ldots,n\)). Step 2. Determination of the system of weights  
Table 8 describes the weighting methods that have been be utilized in the Hellwig-based method in intuitionistic fuzzy environments [170; 173; 12]. Step 3. Identification of the intuitionistic fuzzy ideal  
The determination of the intuitionistic fuzzy ideal can be approached in two ways, as outlined by [170; 173; 12; 15]. The proposals of intuitionistic fuzzy ideals are presented in Table 9.  
Step 4. Calculation of the distance of the \(i\)th alternative from the intuitionistic fuzzy ideal  
Following the selection of the distance measure, the computation of distances between objects and the intuitionistic fuzzy ideal, chosen in step 3, is conducted using one of the Formulas (47)-(50). The expression for the distance measure from the pattern object is given [174; 12; 15] as  
\[d_{i}^{+}(A_{i},IFI)=d(I_{IFS},A_{i}) \tag{53}\]  
where \(I_{IFS}\in\big{\{}I_{IFS}^{1},I_{IFS}^{2}\big{\}}\), \(d\in\big{\{}\varepsilon_{IFS}^{1},\varepsilon_{IFS}^{2},I_{IFS}^{2},I_{IFS}^{3} \big{\}}\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Weights** & **Formula** \\ \hline Equal weights & \(w_{j}=\frac{1}{\pi},j=1,2,\ldots,n\). \\ Intuitionistic fuzzy & \(w_{j}=\frac{1-S_{j}}{\sum_{i=1}^{n}(1-S_{j})}\), \(S_{j}=\frac{\sum_{i=1}^{n}E_{ij}}{\sum_{i=1}^{n}E_{ij}}\), \(E_{ij}\) is an entropy value of IF value \((\mu_{ij},\
u_{ij})\) from the matrix \(IFD\) (\(i=1,2,\ldots,m\); \(j=1,2,\ldots,n\)). Entropy value \(E_{ij}\) can be calculated using one of the Formulas (43)-(46). \\ \hline \hline \end{tabular} Source: own study.  
\end{table} 
Table 8: Weighting methods used in the Hellwig-based method in intuitionistic fuzzy environments.