## 4 The Modifications and Extensions of Classic Hellwig's Method
### The Hellwig-Based Methods with Data Represented by Real Numbers or Linguistic Evaluation
Construction of the normalized decision matrix  
The normalized decision matrix has the form  
\[\overline{D}=[\overline{x}_{ij}], \tag{14}\]  
where \(\overline{x}_{ij}\) is the normalized value of \(x_{ij}(i=1,2,\ldots,m;\ j=1,2,\ldots,n)\). Data normalization is a crucial step in [PERSON]'s method, as it standardizes the data related to the criteria onto a uniform scale. Various approaches for data normalization are discussed in the literature [62; 63; 127]. However, in Hellwig-based methods, max-min normalization, vector normalization, and sum normalization have sometimes been utilized instead of Formula (4) [128; 129]. When individual ideals are considered (Formula (10)), two scenarios must be accounted for [9]:  
For benefit criteria,  
* if the ideal is reached, i.e., \(x_{j}^{-}<x_{ij}^{+}\leq x_{ij}\) then \(\frac{x_{ij}-x_{j}^{-}}{x_{ij}^{+}-x_{j}^{-}}\geq 1\);
* if the ideal is not reached, i.e., \(x_{j}^{-}<x_{ij}<x_{ij}^{+}\) then \(\frac{x_{ij}-x_{j}^{-}}{x_{ij}^{+}-x_{j}^{-}}<1\). For cost criteria,
* if the ideal is reached, i.e., \(x_{ij}\leq x_{ij}^{+}<x_{j}^{-}\) then \(\frac{x_{ij}-x_{j}^{-}}{x_{ij}^{+}-x_{j}^{-}}\geq 1\);
* if the ideal is not reached, i.e., \(x_{ij}^{+}\leq x_{ij}<x_{j}^{-}\) then \(\frac{x_{ij}-x_{j}^{-}}{x_{ij}^{+}-x_{j}^{-}}<1\). As a result, the adopted max-min normalization (see Table 3, last method) is applied. In the Hellwig method, based on [PERSON]'s median, the Weber standardization is applied [6; 130; 131]. A summary of the normalization approaches used in [PERSON]'s modified procedures is presented in Table 3.  
Let us observe that in the case of linguistic evaluation, this step can be omitted. Step 5. Construction of the weighted normalized decision matrix  
The weighted normalized matrix has the form  
\begin{table}
\begin{tabular}{c c} \hline
**Normalization Procedure** & **Formula** \\ \hline Standardization & \(\overline{x}_{ij}=\frac{x_{ij}-\overline{x}_{j}}{\overline{x}_{j}},\overline{x }_{j}=\frac{1}{m}\sum_{i=1}^{m}x_{ij}\), \(S_{j}=\sqrt{\frac{1}{m}\sum_{i=1}^{m}\left(x_{ij}-\overline{x}_{j}\right)^{ 2}},i=1,2,\ldots,m;\  
\[\widetilde{D}=\left[\widetilde{x}_{ij}\right], \tag{15}\]  
where  
\[\widetilde{x}_{ij}=w_{j}\overline{x}_{ij} \tag{16}\]  
for \(i=1,2,\ldots,m;j=1,2,\ldots,n\). Step 6. Calculation of the distance of the \(i\)th alternative from the ideal point  
In the classic [PERSON]'s method, distances are computed using the Euclidean norm, with the implicit assumption that the criteria are independent. However, in practical scenarios, achieving independence among criteria is seldom the case. To address this limitation, an extension of [PERSON]'s method has been suggested, involving the incorporation of the Mahalanobis distance [132; 133]. The Mahalanobis distance considers the covariance structure of the data, offering a more accurate representation of distances when criteria exhibit correlations, particularly in situations involving asymmetrical relationships between them. The distance measures used in Hellwig-based methods for data represented by real numbers [11; 133; 134] are presented in Table 4.  
Practically, the selection of a distance measure hinges on the characteristics of the data and the details of the multi-criteria analysis. The Euclidean distance is chosen when variables are assumed to be independent, but if the dataset exhibits correlations or asymmetrical distributions, the Mahalanobis distance becomes a more suitable option. In the dynamic approach to [PERSON]'s measure of development for calculating the distance, the values of the common pattern were taken in the form of the minimum and maximum from the entire period under study [124]. The ordinal nature of the data required the use of a specialized metric to measure the distance between objects. Object evaluation is based on the numerical relationships of \"equal to\", \"greater than\", and \"less than\". In the linguistic [PERSON]'s method the distance metric for ordinal data proposed by [PERSON] [135; 136] can be applied.