## 4 The Modifications and Extensions of Classic Hellwig's Method
### Hellwig-Based Methods with Data Represented by Fuzzy Numbers
\begin{table}
\begin{tabular}{c c c} \hline
**L** & **LT** & **FTN** \\ \hline
1 & Very poor & \((0,0,1)\) \\
2 & Poor & \((0,1,3)\) \\
3 & Medium poor & \((1,3,5)\) \\
4 & Fair & \((3,5,7)\) \\
5 & Medium good & \((5,7,9)\) \\
6 & Good & \((7,9,10)\) \\
7 & Very good & \((9,10,10)\) \\ \hline \end{tabular} Source: see [6; 163].  
\end{table} 
Table 6: Linguistic terms for the option ratings. \begin{table}
\begin{tabular}{c c c} \hline
**L** & **LT** & **FTN** \\ \hline
1 & Absolutely low importance & \((0.0,0.1,0.2)\) \\
2 & Very low importance & \((0.1,0.2,0.3)\) \\
3 & Low importance & \((0.2,0.3,0.4)\) \\ \hline \end{tabular}
\end{table} 
Table 7: Linguistic terms for the weight ratings. Step 3. Construction of the fuzzy normalized decision matrix  
The normalized fuzzy decision matrix is given by  
\[\widehat{ID}=\big{[}\hat{z}_{ij}\big{]}, \tag{33}\]  
where \(\hat{z}_{ij}\) is expressed as  
\[\widehat{z}_{ij}=\left(\frac{a_{ij}}{c_{j}^{max}},\frac{b_{ij}}{c_{j}^{max}}, \frac{c_{ij}}{c_{j}^{max}}\right)\text{ if }j\text{ is }a\text{ benefit criterion} \tag{34}\]  
\[\widehat{z}_{ij}=\left(\frac{a_{j}^{min}}{c_{ij}},\frac{a_{j}^{min}}{b_{ij}}, \frac{a_{j}^{min}}{a_{ij}}\right)\text{ if }j\text{ is }a\text{ cost criterion}. \tag{35}\]  
Here, \(a_{j}^{min}=\underset{i}{\underset{i}{\underset{j}{\underset{c}{\sum}}}}a_{ij}\), \(c_{j}^{max}=\underset{i}{\underset{c}{\sum}}c_{ij}\), and values \(a_{ij},b_{ij},c_{ij}\) represent the descriptors of the fuzzy option \(\widehat{x}_{ij}=\big{(}a_{ij},b_{ij},c_{ij}\big{)}\). Step 4. Construction of the weighted fuzzy normalized decision matrix  
The weighted normalized fuzzy decision matrix is expressed as  
\[\widehat{V}=\left[\widehat{r}_{ij}\right], \tag{36}\]  
where  
\[\widehat{r}_{ij}=w_{j}\otimes\widehat{z}_{ij}\text{ for }i=1,\ldots,m;j=1, \ldots,n. \tag{37}\]  
Step 5. Identification of the fuzzy ideal  
The fuzzy positive ideal (\(\  
The following formula is used to calculate the aggregation of the fuzzy [PERSON]'s measure:  
\[FH_{i}=1-\frac{d_{i}^{+}}{d_{o}} \tag{40}\]  
where \(d_{o}\) is determined as in Formula (6). Step 8. Ranking of alternatives according to the decreasing values of fuzzy [PERSON]'s measure  
All alternatives \(A_{i}\) (\(i=1,\ldots,\ m\)) are ranked in descending order based on \(FH(A_{i})\). Below, we will discuss an overview of approaches and applications of the Hellwig-based method in a fuzzy environment. [PERSON] and [PERSON] [131] introduced modifications to the classical Hellwig method by incorporating fuzzy numbers into the synthetic measure calculations, allowing for the aggregation of both quantitative and qualitative criteria. Additionally, it integrates weights for criteria and features derived from the Analytic Hierarchy Process (AHP). This approach was demonstrated through an evaluation of socio-economic development levels in districts of the Wielkopolska Poland Voivodsehip, utilizing survey and statistical data from all 31 county administrations in the region. Subsequently, [PERSON] [6] advanced the classical [PERSON]'s method by developing a fuzzy version. In this approach, values from the ordinal scale are converted into triangular fuzzy numbers, defined by three parameters: \(a\) for the pessimistic evaluation, \(b\) for the most probable evaluation, and \(c\) for the optimistic evaluation. After this conversion, the fuzzy numbers are normalized using Formulas (34) and (35). Next, a weighted normalized data matrix is established, incorporating the influence of a system of weights. The triangular fuzzy numbers are then converted into real numbers, typically using a simple arithmetic mean (see Formula (31)). The procedure continues with the application of the remaining steps of the classical [PERSON] method, as outlined in Formulas (5) and (6). Finally, the objects are ranked based on the non-increasing values of the fuzzy Hellwig's measure.