## 4 The Modifications and Extensions of Classic Hellwig's Method
### Hellwig-Based Methods with Data Represented by Fuzzy Numbers  
The preliminary definitions of fuzzy sets, fuzzy numbers, fuzzy operations, and fuzzy distances are presented below. The fuzzy set has been defined by [PERSON] [162] as follows. Let \(X\) be a universe of discourse of objects. A fuzzy set \(\hat{n}\) in \(X\) is given by  
\[A=\{<x,\mu_{\hat{n}}(x)>|x\in X\}, \tag{23}\]  
where \(\mu_{\hat{n}}\colon X\to[0,1]\) is a function with the condition \(0\leq\mu_{\hat{n}}(x)\leq 1\) for every \(x\in X\). The value \(\mu_{\hat{n}}(x)\) denotes degree of membership of the element \(x\in X\) to the set \(\mu_{\hat{n}}\). A fuzzy number is a fuzzy subset of the universe of discourse \(X\) that is both convex and normal. The membership function of a triangular fuzzy number (TFN) is expressed in the following way [55]:  
\[\mu_{\hat{n}}(x)=\left\{ \begin{array}{c}0\text{ for }x<a\\ \frac{x-a}{b-a}\text{ for }a\leq x\leq\ b\\ \frac{x-c}{b-c}\text{ for }b<x\leq\ c\\ 0\text{ for }x>c\end{array} \right. \tag{24}\]  
Then, the triangular fuzzy number \(\hat{n}\) can be represented by \((a,b,c)\), where \(a\)is the left threshold value, \(b\) is the midpoint, and \(c\) is the right threshold value. Note that both the crisp data \(a\) and the interval data \((a,c)\) can be regarded as triangular fuzzy numbers of the following form \((a,a,a)\) and \(\left(a,\frac{a+c}{2},c\right)\), respectively. A triangular fuzzy number \((a,b,c)\) is positive if \(a>0\). For any two positive triangular fuzzy numbers, \(\hat{m}=(a_{1},b_{1},c_{1})\) and \(\hat{k}=(a_{2},b_{2},c_{2})\) (\(a_{1},a_{2}>0\)), and a positive real number \(r\), the main operations of fuzzy numbers \(\hat{m}\) and \(\hat{k}\) can be expressed as follows [55]. \[\text{Addition: }\hat{m}\oplus\hat{k}=(a_{1}+a_{2},b_{1}+b_{2},c_{1}+c_{2}), \tag{25}\]Multiplication by a scalar: \(\hat{m}\otimes r=\left(a_{1}r,b_{1}r,c_{1}r\right)\), (26) Multiplication: \(\hat{m}\otimes\hat{k}\cong\left(a_{1}a_{2},b_{1}b_{2},c_{1}c_{2}\right)\). (27) Max of TFN:\(max\Big{(}\hat{m},\hat{k}\Big{)}=\left(max(a_{1},a_{2}),max(b_{1},b_{2}),max(c_{1},c_{2})\right)\) (28) Min of TFN: \(min\Big{(}\hat{m},\hat{k}\Big{)}=\left(minx(a_{1},a_{2}),minx(b_{1},b_{2}), minx(c_{1},c_{2})\right)\) (29) Vertex distance: \(d\Big{(}\hat{m},\hat{k}\Big{)}=\sqrt{\frac{1}{3}\Big{(}\big{(}a_{1}-a_{2})^{2} +\left(b_{1}-b_{2}\right)^{2}+\left(c_{1}-c_{2}\right)^{2}\Big{)}}\) (30) Defuzzification formula : \(df(\hat{m})=\frac{a_{1}+b_{1}+c_{1}}{3}\). (31)  
Let \(A=\{A_{1},A_{2},\ldots,A_{m}\},i=1,2,\ldots,m\) be the set of alternatives and \(C=\{C_{1},C_{2},\ldots,C_{n}\}\), \(j=1,2,\ldots,n\) the set of criteria, where \(\widehat{x}_{ij}=\big{(}a_{ij},b_{ij},c_{ij}\big{)}\) is the evaluation of the \(i\)th alternative with respect to the \(j\)th criterion represented by positive triangular fuzzy number (\(i=1,2,\ldots,m\); \(j=1,2,\ldots,n\)). We will now review the fuzzy Hellwig-based methods, focusing on the steps of the algorithm (see Figure 1). Step 1. Definition of a fuzzy decision matrix  
The fuzzy decision matrix has the following form:  
\[FD=\left[\widehat{x}_{ij}\right], \tag{32}\]  
where \(\widehat{x}_{ij}=\big{(}a_{ij},b_{ij},c_{ij}\big{)}\) represents the evaluation of the \(i\)th alternative with respect to the \(j\)th criterion, expressed as a triangular fuzzy number \((i=1,2,\ldots,m\); \(j=1,2,\ldots,n)\). If the performance ratings of alternatives on qualitative criteria are expressed by linguistic terms, using linguistic terms, these terms can be represented by triangular fuzzy numbers (TFNs) as shown in Table 6.  
Step 2. Determination of the system of weights  
The system of weights is represented as shown in Formula (8). The weights can also be expressed as positive triangular fuzzy numbers (see Table 7). Step 2. Determination of the system of weights  
The system of weights is represented as shown in Formula (8). The weights can also be expressed as positive triangular fuzzy numbers (see Table 7).