## 3 Classic Hellwig's Method and Its Applications
### The Taxonomic Measure of Development  
The concept of the Taxonomic Measure of Development, introduced by Professor [PERSON], was published in 1968 in the Polish journal, Statistical Review [4]. It is noteworthy that this idea was initially presented at the UNESCO conference in Warsaw and discussed in English in an unpublished UNESCO working paper [64]. [PERSON]'s work has motivated many Polish researchers to investigate linear ordering methods, resulting in adaptations of his approach and the creation of new measures, as outlined in [65; 66; 67; 68]. Below, we will present the classic TMD algorithm, hereafter referred to as the [PERSON] method. Let \(O=\{O_{1},\dots,\ O_{m}\}\), where \(i=1,\ 2,\dots,\ m\) is the set of objects under assessment, and \(C=\{C_{1},\dots,\ C_{n}\}\), where \(j=1,\ 2,\dots,\ n\) is the set of diagnostic variables (indicators) comprising a complex phenomenon. Additionally, consider \(P\) and \(N\) as the sets of stimulating (positive) and destimulating (negative) indicators, respectively, influencing the complex phenomenon. The original [PERSON]'s method involves the following steps [4]: Step 1. Definition of a decision matrix  
\[D= \begin{bmatrix}x_{11}&x_{12}&\cdots&x_{1n}\\ x_{21}&x_{22}&\cdots&x_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ x_{m1}&x_{m2}&\cdots&x_{mn}\end{bmatrix} , \tag{1}\]  
where \(x_{ij}\) is the assessment of \(i\)th object concerning the \(j\)th diagnostic variable (indicator) \((i=1,2,\dots,m;j=1,2,\dots,n)\). Step 2. Identification of the pattern of development, \(I=\left[x_{1}^{+},x_{2}^{+},\dots,x_{n}^{+}\right]\), following the principle  
\[x_{j}^{+}= \begin{cases}\underset{i}{\max}x_{ij}&\text{if}\quad x_{ij}\in P,\\ \underset{i}{\min}x_{ij}&\text{if}\quad x_{ij}\in N.\end{cases}  \tag{2}\]  
for \(j=1,2,\dots,n\). Step 3. Construction of the normalized decision matrix  
\[\overline{D}=\left[\overline{x}_{ij}\right] \tag{3}\]  
using the standardization formula  
\[\overline{x}_{ij}=\frac{x_{ij}-\overline{x}_{j}}{S_{j}}, \tag{4}\]  
where \(\overline{x}_{j}=\frac{1}{m}\sum_{i=1}^{m}x_{ij}\), \(S_{j}=\sqrt{\frac{1}{m}\sum_{i=1}^{m}\left(x_{ij}-\overline{x}_{j}\right)^{2} },i=1,2,\dots,m;\ j=1,2,\dots,n\). Step 4. Calculation of the distance of the \(i\)th object from the pattern of development using the Euclidean distance  
\[d_{i}^{+}(O_{i},I)=\sqrt{\sum_{j=1}^{n}\left(\overline{x}_{ij}-\overline{x}_{ j}^{+}\right)^{2}}, \tag{5}\]where \(\overline{x}_{j}^{+}=\frac{x_{j}^{+}-\overline{x}_{j}}{5_{j}}\) is the normalized values of the pattern of development, \(i=1,2,\ldots,m\); \(j=1,2,\ldots,n\). Step 5. Computation of the relative closeness of the \(i\)th object to the pattern of development using the following formula  
\[H(O_{i})=1-\frac{d_{i}^{+}}{d_{0}}, \tag{6}\]  
where \(d_{0}=\overline{d}+2S\), \(\overline{d}=\frac{1}{m}\sum_{i=1}^{m}d_{i}^{+}\), \(S=\sqrt{\frac{1}{m}\sum_{i=1}^{m}\left(d_{i}^{+}-\overline{d}\right)^{2}}i=1, 2,\ldots,m\). Step 6. Ranking the objects according to the decreasing values of \(H(O_{i})\)  
The synthetic measure \(H\) usually takes the values from the interval \([0,1]\). The higher the values of the measure, the closer the object is to the pattern of development. In the subsequent stage, the objects can be categorized into four distinct classes. Class 1 consists of objects within the range: \([\overline{H}+S_{H};\max\{H_{i}\}]\) which indicates a very high level of the TMD measure. Class 2 includes objects that fall within the interval: \([\overline{H};\overline{H}+S_{H})\), representing a medium-high level of the TMD measure. Class 3 is defined by the range \([\overline{H}-S_{H};\overline{H})\), indicating a medium-low level of the TMD measure. Lastly, Class 4 encompasses objects within the interval: \([\min\{H_{i}\};\overline{H}-S_{H})\), which signifies a very low level of the TMD measure. Here, \(\overline{H}\) represents the average value, and \(S_{H}\) denotes the standard deviation of the synthetic measure.