## 4 The Modifications and Extensions of Classic Hellwig's Method
### The Hellwig-Based Methods with Data Represented by Real Numbers or Linguistic Evaluation
The first modification is the positional [PERSON] method, which uses positional measures to normalize input variables. This version, based on [PERSON]'s spatial median [6; 130; 138], generalizes the classic median concept to multiple dimensions by minimizing the sum of Euclidean distances to the data points, thereby positioning itself centrally and remaining robust against outliers. The Weber normalization is defined by the formula (see Table 3)  
\[\overline{x}_{ij}=\frac{x_{ij}-\theta_{j}}{1.4826 MAD_{j}} \tag{20}\] \begin{table}
\begin{tabular}{c c} \hline
**Normalization Factor** & **Formula** \\ \hline Classical factor & \(N=d_{0}=\overline{d}+2S\), \(\overline{d}=\frac{1}{m}\sum_{i=1}^{m}d_{i}^{+}\), \(S=\sqrt{\frac{1}{m}\sum_{i=1}^{m}\left(d_{i}^{+}-\overline{d}\right)^{2}}\), \(i=1,2,\ldots,m\) \\ Max value of measure & \(N=max(d_{i})\) \\ Distance between ideal and anti-ideal & \(N=\sqrt{\sum_{j=1}^{n}\left(\widetilde{x}_{j}^{-}-\widetilde{x}_{j}^{+} \right)^{2}}\), where \(\widetilde{x}_{j}^{-},\widetilde{x}_{j}^{+}-\)normalized weighted values of \\  & anti-ideal and ideal for \(j\)th criterion \\ \hline \end{tabular} Source: own study.  
\end{table} 
Table 5: Normalization factors used in Hellwig-based methods for data represented by real numbers. where \(\theta_{j}\) is the value of a particular component of the multidimensional median vector, known as the Weber median, and \(MAD_{j}\) is the median absolute deviation of the \(j\)th criterion, calculated as \(MAD_{j}=\underset{i=1,\dots,m}{\text{med}}\left|x_{ij}-\theta_{j}\right|\), and \(med\) is the median. The constant 1.4826 is an empirically derived coefficient, and \(x_{ij}\) is the value of the \(i\)th alternative for the \(j\)th criterion \((i=1,2,\dots,m;\ j=1,2,\dots,n)\). The positional Hellwig measure of normalized variables (Formula (20)) is calculated as follows:  
\[HP(A_{i})=1-\frac{d_{i}}{med(D)+2.5 mad(D)} \tag{21}\]  
where \(d_{i}\) represents the individual values of the distance vector \(D\), \(D=[d_{1},\dots,d_{n}]\) is the distance vector, \(med(D)\) is the median of the distance vector \(D\), \(mad(D)\) is the median absolute deviation of the distance vector \(D\), and 2.5 is a constant value (immune threshold value). The distance vector \(d_{i}\) is calculated using the Manhattan distance formula  
\[d_{i}=\sum_{j=1}^{n}\left|\overline{x}_{ij}-\overline{x}_{j}^{+}\right|, \tag{22}\]  
where \(\overline{x}_{j}^{+}\) is the \(j\)th coordinate of the ideal point constituted of the maximum values of the normalized features \((i=1,2,\dots,m;\ j=1,2,\dots,n)\). [PERSON] [139] modified the classical development measure by introducing the Taxonomic Spatial Measure of Development (TSMD), which enables simultaneous analysis across three dimensions: section, time, and space. This modification involves adding a spatial weight matrix to the classic TMD [4]. [PERSON] [140] suggested a modification spatial development measure considering the potential strength of interactions among regions, rather than modifying the standardization formula, while [PERSON] et al. [141] included the locational component in the analysis. In a different vein, [PERSON] and [PERSON] [142] introduced the Multi-Criteria Method Integrating Distances to Ideal and Anti-Ideal Points (MIDIA), which serves as an extension of [PERSON]'s approach. MIDIA enhances the original method by incorporating a weighted system that accounts for both balance and asymmetry in evaluating alternatives based on their distances from ideal and anti-ideal points. This approach allows for a more nuanced ranking of alternatives, focusing on their relative distances from these reference points, rather than relying solely on positional normalization. To understand the developments and applications of [PERSON]'s method for real data, it is essential to review several notable and representative contributions in the field. In a series of five papers, the [PERSON] method with max-min normalization was utilized.