## 4 The Modifications and Extensions of Classic Hellwig's Method
### The Hellwig-Based Methods with Data Represented by Real Numbers or Linguistic Evaluation  
First, we posit the representation of data through real numbers or linguistic terms with a numerical scale. The modification of [PERSON]'s method extends across all aspects delineated in Figure 1. To illustrate the universality of these methods in solving various complex problems, we utilized the terminology of MCDM analyses. Let \(A=\{A_{1},A_{2},\ldots,A_{m}\}\), \(i=1,2,\ldots,m\) be the set of alternatives, \(C=\{C_{1},C_{2},\ldots,C_{n}\}\), \(j=1,2,\ldots,n\) the set of criteria, and \(C=P\cup N\), where \(P\) is the set of benefit criteria and \(N\), is the set of cost criteria. The alternative \(A_{i}\) is represented as a vector, \(A_{i}=[x_{i1},x_{i2},\ldots,x_{im}]\), \(i=1,2,\ldots,m\). The general framework rooted in [PERSON]'s principles involves the following steps. Step 1. Definition of a decision matrix  
We assume that alternatives are represented by real numbers or assessed using an ordinal scale. Let \(OS=\left\{1,2,\ldots,k\right\}\) represent the numerical values of the linguistic scale, where a higher number indicates a greater preference. The linguistic scale should include 5, 7, 9, 11, or 13 distinct levels. As highlighted by [PERSON] [23] and [PERSON] [120], the scale must be sufficiently granular to effectively differentiate the performances of the evaluated options. An example of a seven-point scale is shown in Table 1.  
The decision matrix \(D\) is defined as  
\[D=\left[x_{ij}\right], \tag{7}\]  
where \(x_{ij}\) represents the assessment of the \(i\)th alternative with respect to the \(j\)th criterion \((i=1,2,\ldots,m\;;j=1,2,\ldots,n)\). The value \(x_{ij}\) can be either a real number \(\left(x_{ij}\in R\right)\) or a numerical representation of the linguistic scale. Step 2. Determination of the system of weights  
The system of weights is represented as  
\[w=\left[w_{1},\ldots,w_{n}\right], \tag{8}\]  
where \(w_{j}>0\) (\(j=1,\ldots,n\)) denotes the weight of the criterion \(C_{j}\), and \(\sum_{j=1}^{n}w_{j}=1\). The original [PERSON]'s approach did not assign specific weights to the criteria. [PERSON] and [PERSON] [56] observed that equal weights were commonly used in many applications. [PERSON] et al. [50] noted that equal weights are often preferred due to reasons such as the simplicity of construction, lack of a theoretical basis for a differential weighting system, disagreements among decision-makers, and insufficient statistical or empirical evidence. The literature presents various methods for determining weights [57; 59; 121; 122], each with its own advantages and limitations. These methods are typically categorized into three main types [58]: subjective, objective, and integrated. Subjective weights are assigned based on expert judgment. Objective weights are derived from the statistical properties of the available data, employing theoretical or mathematical statistical methods. Integrated weight systems combine elements of both subjective and objective approaches. Table 2 outlines the objective weighting methods used in [PERSON]'s approach, including those based on the coefficient of variation, correlation matrix, and entropy [43; 121; 122; 123]. A higher weight coefficient is assigned to criteria with a higher average coefficient of variation or those that show a stronger correlation with the values of other criteria. The entropy-based weighting method operates on the principle that criteria with higher entropy (greater variability) are considered more important, while criteria with lower entropy are perceived as less influential in the decision-making process. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Numerical Representation Linguistic Scale (Rating)** & **Linguistic Term** & **Label** \\ \hline
1 & Very poor & VP \\
2 & Poor & P \\
3 & Medium-poor & MP \\
4 & Fair & F \\
5 & Medium-good & MG \\
6 & Good & G \\
7 & Very-good & VG \\ \hline \hline \end{tabular} Source: see [23; 120].  
\end{table}