## 4 The Modifications and Extensions of Classic Hellwig's Method
### The Hellwig-Based Methods with Data Represented by Real Numbers or Linguistic Evaluation
Let \(OS=\{1,2,\ldots,k\}\) be a numerical representation of the ordinal scale, where the higher number means \"more preferable\", and let \(O=\{O_{1},\ldots O_{m}\}\) be a set of objects evaluated on the ordinal scale \(OS\) with respect to \(n\) criteria. Let \(O_{i}=\left[x_{i1},x_{2j},\ldots,x_{in}\right],O_{k}=\left[x_{k1},x_{k2}, \ldots,x_{in}\right]\) represent the \(i\)th (\(k\)th) object, where \(x_{ij}(x_{kj})\) are the evaluations of \(i\)th (\(k\)th) object with respect to \(j\)th criterion, and \(x_{ij},x_{kj}\in OS(j=1,\ldots n;\ i,k=1,2,\ldots,m)\). The GDM2 distance between objects \(O_{i}\) and \(O_{k}\) are characterized by \(n\) criteria and has the following form [135; 136]:  
\[\small GDM2_{ik}=\frac{1}{2}-\frac{\sum_{j=1}^{n}\mathrm{w}_{j}a_{ikj}b_{kij}+ \sum_{j=1}^{n}\sum_{l=1;\ j\
eq i,k}^{m}\mathrm{w}_{j}a_{ilj}^{2}b_{klj}}{2 \left[\left(\sum_{j=1}^{n}\mathrm{w}_{j}a_{ikj}^{2}+\sum_{j=1}^{n}\sum_{l=1; \ l\
eq i,k}^{m}\mathrm{w}_{j}a_{ilj}^{2}\right)\left(\sum_{j=1}^{n}\mathrm{w }_{j}b_{kij}^{2}+\sum_{j=1}^{n}\sum_{l=1;\ l\
eq i,k}^{m}\mathrm{w}_{j}b_{klj} ^{2}\right)\right]^{\frac{1}{2}}} \tag{17}\] \begin{table}
\begin{tabular}{c l} \hline \multicolumn{1}{c}{**Distance Measures**} & \multicolumn{1}{c}{**Formula**} \\ \hline Euclidean & \(d_{i}(A_{i},I)=\mathrm{E}\left(\widetilde{A_{i}},\widetilde{I}\right)=\sum_{j =1}^{n}\left(\widetilde{x}_{ij}-\widetilde{x}_{j}^{+}\right)^{2},i=1,\ldots,m\) \\  & \(\widetilde{x}_{ij},\widetilde{x}_{j}^{+}\) are weighted normalized values \(x_{ij}\) and \(x_{j}^{+}\), respectively. \\ Mahalanobis & \(dM_{i}(A_{i},I)=\mathrm{M}(\overline{A}_{i},\overline{I})=\sqrt{(\overline{A }_{i}-\overline{I})WC^{-1}W^{T}(\overline{A}_{i}-\overline{I})^{T}}\), \\  & \(C\) is the variance-covariance matrix of the data matrix \(\overline{D},W=\mathrm{diag}(\sqrt{w}_{1},\ldots,\sqrt{w}_{n})\) is the diagonal matrix, where \(w_{1},w_{2},\ldots,w_{n}\) are the weights assigned to the criteria. \\ \hline \multicolumn{1}{c}{Source: own study.} \\ \end{tabular}
\end{table} 
Table 4: Distance measures used in Hellwig-based methods for data represented by real numbers. where \(i,k,l=1,\ldots,m\) is the number of objects, \(j=1,\ldots,n\) is the number of criteria, w\({}_{j}\) is the weights of the \(j\)th criterion, and \(x_{ij}\Big{(}x_{kj},x_{lj}\Big{)}\) is the evaluation of the \(i\)th (\(k\)th, \(l\)th) object with respect \(j\)th criterion. For the ordinal scale, \(a_{ipj},b_{krj}\) are given as  
\[a_{ipj}\Big{(}b_{krj}\Big{)}=\left\{ \begin{array}{l l l}1&\text{if}&x_{ij}>x_{ pj}\Big{(}x_{kj}>x_{rj}\Big{)}\\ 0&\text{if}&x_{ij}=x_{pj}\Big{(}x_{kj}=x_{rj}\Big{)},\text{ for }p=k,l;\,r=i,l\\ -1&\text{if}&x_{ij}<x_{pj}\Big{(}x_{kj}<x_{rj}\Big{)}\end{array} \right. \tag{18}\]  
The distances \(GDM2_{i}^{+}\) between alternative \(A_{i}\) (\(i=1,2,\ldots,m\)) and the ideal \(I\) are later calculated using the GDM2 measure. Step 7. Calculation of [PERSON]'s measure for the \(i\)th alternative. The following formula is used to calculate the aggregation of [PERSON]'s measure:  
\[H(A_{i})=1-\frac{d_{i}^{+}}{N} \tag{19}\]  
where \(N\) is a normalization factor of the procedure. Instead of the classical approach (see Formula (6)), other normalization factors are used. [PERSON] [125] proposed the distance between the pattern (ideal point) and anti-pattern (anti-ideal point), while [PERSON] and [PERSON] [137] suggested using \(max(d_{i})\). The normalization factors used in Hellwig-based methods are presented in Table 5.  
For data represented on an ordinal scale, \(N=GDM2^{+-}\) where \(GDM2^{+-}\) is the generalized distance measure between the ideal and anti-ideal points calculated using Formula (17). Step 8. Ranking of alternatives according to the decreasing values of Hellwig-based algorithm  
Synthetic measure \(H\) typically range between 0 and 1. Higher values indicate that the object is closer to the pattern of development. Finally, it is worth mentioning other methods derived from [PERSON]'s approach, each offering a different way of evaluating alternatives.