## 4 The Modifications and Extensions of Classic Hellwig's Method
### Hellwig-Based Methods with Data Represented by Intuitionistic Fuzzy Numbers  
The basic definitions of intuitionistic fuzzy sets, intuitionistic fuzzy entropy, and intuitionistic fuzzy distances are provided below. The intuitionistic fuzzy set (IFS) theory, introduced by [PERSON] [165], extends [PERSON]'s Fuzzy Set (FS) theory [162] to address issues related to uncertainty by including the notion of hesitation or indeterminacy. [PERSON] [165] defined intuitionistic fuzzy sets as follows. Let \(X\) be a universe of discourse of objects. An intuitionistic fuzzy set \(A\) in \(X\) is given by  
\[A=\{<x,\mu_{A}(x),\
u_{A}(x)>|x\in X\} \tag{41}\]  
where \(\mu_{A},\
u_{A}\): \(X\rightarrow[0,1]\) are functions with the condition for every \(x\in X\)  
\[0\leq\mu_{A}(x)+\
u_{A}(x)\leq 1. \tag{42}\]  
The values \(\mu_{A}(x)\) and \(\
u_{A}(x)\) represent degrees of membership and non-membership, respectively, of the element \(x\in X\) to the set \(A\). Additionally, \(\pi_{A}(x)=1-\mu_{A}(x)-\
u_{A}(x)\) denotes the intuitionistic fuzzy index (hesitation margin) of the element \(x\) in the set \(A\). When \(\pi_{A}(x)=0\) for \(x\in X\), the intuitionistic fuzzy set \(A\) simplifies to an ordinary fuzzy set. A higher value of \(\pi_{A}(x)\) signifies an increase in uncertainty. An intuitionistic fuzzy value (IFV) denoted as \(A=(\mu_{A},\
u_{A})\), refers to an intuitionistic fuzzy set where the universe \(X\) consists of only one element \(x\). The intuitionistic value (1, 0) is the largest, while (0, 1) is the smallest in the set of all intuitionistic fuzzy values (IFVs). The entropy method for determining weights has been adapted to represent data using intuitionistic fuzzy numbers. Many researchers have explored entropy measures within the context of intuitionistic fuzzy systems [166; 167; 168; 169]. [PERSON] and [PERSON] [169] provides an overview of various approaches and formulas for defining entropy measures for intuitionistic fuzzy sets, with a focus on their application in determining objective weights. In this context, we present four intuitionistic entropy measures that have been employed to establish criteria weights in the Hellwig-based method within an intuitionistic fuzzy environment in [PERSON] et al. [170]. The first proposition is by [PERSON] and [PERSON] [166], who defined the entropy measure as the degree of hesitancy in intuitionistic fuzzy sets (IFS):  
\[E_{1}(A)=\sum_{j=1}^{n}(1-\big{(}\mu_{A}\big{(}x_{j}\big{)}+\
u_{A}\big{(}x_{ j}\big{)}\big{)}=\sum_{j=1}^{n}\pi_{A}\big{(}x_{j}\big{)}. \tag{43}\]  
The next approach was introduced by [PERSON] and [PERSON] [167] who developed a non-probabilistic entropy measure based on a geometric interpretation of intuitionistic fuzzy sets and applied a ratio of distances between them in the following way:  
\[E_{2}(A)=\frac{a}{b}, \tag{44}\]  
where a is the distance (\(A\); \(A_{near}\)) from \(A\) to the nearer point \(A_{near}\) among points (1, 0, 0) and (0, 1, 0), and \(b\) is the distance (\(A\); \(A_{far}\)) from \(A\) to the farther point \(A_{far}\) among points (1, 0, 0) and (0, 1, 0). The third definition, proposed by [PERSON] and [PERSON] [171], is based on the concept of cardinality for intuitionistic fuzzy sets and utilizes an axiomatic framework. The formula is as follows:  
\[E_{3}(A)=\frac{\sum_{j=1}^{n}\big{(}min\Big{(}\mu_{A}\big{(}x_{j}\big{)},\
u_ {A}\big{(}x_{j}\big{)}\Big{)}+min\Big{(}1-\mu_{A}\big{(}x_{j}\big{)},1-\
u_{A} \big{(}x_{j}\big{)}\Big{)}\Big{)}}{\sum_{j=1}^{n}\big{(}max\Big{(}\mu_{A} \big{(}x_{j}\big{)},\
u_{A}\big{(}x_{j}\big{)}\Big{)}+max\Big{(}1-\mu_{A} \big{(}x_{j}\big{)},1-\
u_{A}\big{(}x_{j}\big{)}\Big{)}\Big{)}}. \tag{45}\]  
The last approach, also proposed by [PERSON] and [PERSON] [168], introduced intuitionistic entropy measures for IFSs based on the information theory.