## 4 The Modifications and Extensions of Classic Hellwig's Method
### Hellwig-Based Methods with Data Represented by Intuitionistic Fuzzy Numbers
The formula is as follows:  
\[E_{4}(A)=-\frac{1}{nln2}\sum_{j=1}^{n}[\mu_{A}\big{(}x_{j}\big{)}ln\mu_{A} \big{(}x_{j}\big{)}+\
u_{A}\big{(}x_{j}\big{)}Inv_{A}\big{(}x_{j}\big{)}-\big{(} 1-\pi_{A}\big{(}x_{j}\big{)}\big{)}ln\big{(}1-\pi_{A}\big{(}x_{j}\big{)}\big{)} -\pi_{A}\big{(}x_{j}\big{)}ln2], \tag{46}\]  
where \(\frac{1}{nln2}-\)is the constant that assures \(0\leq E_{4}(A)\leq 1\). Below, we present the weighted Hamming and Euclidean distances formulas between two intuitionistic fuzzy sets (IFSs), initially introduced by [PERSON] [172]. These measures have been employed in studies utilizing the Hellwig-based method for data represented by intuitionistic fuzzy sets. The weighted Euclidean distance between \(A,B\in IFS\) with membership functions \(\mu_{A}(x)\), \(\mu_{B}(x)\) and non-membership functions \(\
u_{A}(x)\), \(\
u_{B}(x))\) is determined using the following formulas [172]:  
\[e_{IFS}^{2}(A,B)=\sqrt{\frac{1}{2}{\sum_{j=1}^{n}w_{j}}\Big{[}\big{(}\mu_{A} \big{(}x_{j}\big{)}-\mu_{B}\big{(}x_{j}\big{)}\big{)}^{2}+\big{(}\
u_{A}\big{(} x_{j}\big{)}-\
u_{B}\big{(}x_{j}\big{)}\big{)}^{2}\Big{]}} \tag{47}\]  
\[e_{IFS}^{3}(A,B)=\sqrt{\frac{1}{2}{\sum_{j=1}^{n}w_{j}}\Big{[}\big{(}\mu_{A} \big{(}x_{j}\big{)}-\mu_{B}\big{(}x_{j}\big{)}\big{)}^{2}+\big{(}\
u_{A}\big{(} x_{j}\big{)}-\
u_{B}\big{(}x_{j}\big{)}\big{)}^{2}+\big{(}\pi_{A}\big{(}x_{j} \big{)}-\pi_{B}\big{(}x_{j}\big{)}\big{)}^{2}\Big{]}}, \tag{48}\]  
where \(\sum_{j=1}^{n}w_{j}=1\). The weighted Hamming distance between \(A,B\in IFS\) with membership functions \(\mu_{A}(x)\), \(\mu_{B}(x)\) and non-membership functions \(\
u_{A}(x)\), \(\
u_{B}(x)\) is determined using the following formulas [172]:  
\[h_{IFS}^{2}(A,B)=\frac{1}{2}{\sum_{j=1}^{n}w_{j}}\big{[}\big{|}\mu_{A}\big{(} x_{j}\big{)}-\mu_{B}\big{(}x_{j}\big{)}\big{|}+\big{|}\
u_{A}\big{(}x_{j} \big{)}-\
u_{B}\big{(}x_{j}\big{)}\big{|}\big{]} \tag{49}\]  
\[h_{IFS}^{3}(A,B)=\frac{1}{2}{\sum_{j=1}^{n}w_{j}}\big{[}\big{|}\mu_{A}\big{(} x_{j}\big{)}-\mu_{B}\big{(}x_{j}\big{)}\big{|}+\big{|}\
u_{A}\big{(}x_{j} \big{)}-\
u_{B}\big{(}x_{j}\big{)}\big{|}+\big{|}\pi_{A}\big{(}x_{j}\big{)}- \pi_{B}\big{(}x_{j}\big{)}\big{|}\big{]} \tag{50}\]  
It is noteworthy that when the weights in Formulas (47)-(50) are equal, the result is the normalized Euclidean or Hamming distance. The combination of membership degree, non-membership degree, and hesitancy degree provides a more accurate and comprehensive representation of the decision-making context. Specifically, the hesitancy degree indicates the extent of uncertainty or lack of information during the decision-making process. The motivation to expand the classical Hellwig method through the use of intuitionistic sets was driven by the need to analyze complex problems, particularly when the data are represented in the form of surveys [12; 173]. This approach incorporates ordinal scales while taking into account the lack of responses or the response \"I don't know\". The preliminary stage involves representing the data collected from the questionnaire survey in the form of intuitionistic fuzzy values. Respondents' answers are collected through a questionnaire survey, which evaluates a set of objects (alternatives) and a set of questions (criteria) using an ordinal scale. Initially, the opinions of respondents regarding the \(i\)-th alternative for criterion \(C_{j}\) are classified into three categories: \"positive opinion about the alternative\", \"negative opinion about the alternative\", and \"no opinion or no answer\". Let \(p_{ij}\) represent the total number of \"positive opinions\", \(n_{ij}\) is the total number of \"negative opinions\" and \(h_{ij}\) is the total number of responses categorized as \"no opinion or no answer\". The total number of respondents is denoted as \(N_{ij}\), where \(i\) corresponds to the alternative number, and \(j\) corresponds to the criterion number. It is worth noting \(p_{ij}+n_{ij}+h_{ij}=N_{ij}\).