## 5 Numerical Examples
: _Assume three alternatives (\(A_{1}\), \(A_{2}\), and \(A_{3}\)) using interval data to measure the preference rating, and of these alternatives, \(A_{2}\) and \(A_{3}\) are known to have trapezoidal and triangular distributions, respectively. Their scores are \(A_{1}\) = (4, 7), \(A_{2}\) = (4, 5, 6, 7), and \(A_{3}\) = (2, 7, 8), respectively, and the probability density functions are described in Figure 5._  
By using the Mellin transform, the mean and the spread can be calculated as:  
\[ \begin{array}{ll}E(A_{1})=5.5,\ \sigma_{A_{1}}=0.8660\\ E(A_{2})=5.5,\ \sigma_{A_{2}}=0.4167\\ E(A_{3})=5.67,\ \sigma_{A_{3}}=1.2329\end{array} \]  
then the differences in the alternatives' total mean can be calculated as:  
\[ \begin{array}{ll}\Psi(A_{1},A_{2})=\mu_{\Delta}(A_{1},A_{2})+\lambda\sigma_{ \Delta}(A_{1},A_{2})=0+0.25\times(0.4167-0.8660)=-0.1378<0\\ \Psi(A_{1},A_{3})=\mu_{\Delta}(A_{1},A_{3})+\lambda\sigma_{\Delta}(A_{1},A_{3 })=-0.1+0.25\times(1.2329-0.8660)=-0.008<0\\ \Psi(A_{2},A_{3})=\mu_{\Delta}(A_{2},A_{3})+\lambda\sigma_{\Delta}(A_{2},A_{3 })=-0.17+0.25\times(1.2329-0.4167)=0.0341>0\end{array} \]  
Figure 4: The probability density function in the second example. \begin{table}
\begin{tabular}{c c c c c} \hline
**Methods** & \(A_{1}\) & \(A_{2}\) & \(A_{3}\) & **Rank** \\ \hline CoA & 5.33 & 4.59 & 4.6 & \(A_{1}\succ A_{3}\succ A_{2}\) \\ \hline BoA & 5.5 & 4.6 & 4.6 & \(A_{1}\succ A_{3}\approx A_{2}\) \\ \hline Proposed & 4.6 & 4.8 (status quo) & 4.9 & \(A_{3}\succ A_{2}\succ A_{1}\) \\ \hline \end{tabular}
\end{table} 
Table 4: The comparison of defuzzifying approaches in Example 2. The above result indicates that \(A_{2}\succ A_{3}\succ A_{1}\), and we can compare the proposed method with CoA and Boa, as shown in Table 5.  
**Example 4**. : _In order to extend to further applications in MADM, we demonstrate a way to employ the SAW method using our concept. The SAW method can be expressed as:_  
\[V(A_{i})=V_{i}=\sum_{j=1}^{n}w_{j}r_{ij}(x_{ij}),\quad i=1,\cdots,m;j=1,\ldots,n \tag{20}\]  
_where \(V(A_{i})\) is the value function of alternative \(A_{i}\), and \(w_{j}\) and \(v_{j}(\cdot)\)are weight and value functions of attribute \(j\), respectively. After a normalization process, the value of alternative \(A_{i}\) can be rewritten as:_  
\[V_{i}=\sum_{j=1}^{n}w_{j}r_{ij},\quad i=1,\cdots,m \tag{21}\]  
_Next, we can apply our concept to the simple additive weighting method according to follow equations:_  
\[\Delta V_{ik}=V_{i}-V_{k}=\sum_{j=1}^{n}w_{j}r_{ij}-\sum_{j=1}^{n}w_{j}r_{kj}= \sum_{j=1}^{n}w_{j}\Delta r_{ikj},\;i,k=1,\cdots,m;j=1,\cdots,n \tag{22}\]  
_and let:_  
\[\Delta r_{ikj}=\mu_{\Delta}(A_{ij},A_{kj})+\lambda\sigma_{\Delta}(A_{ij},A_{kj}) \tag{23}\]  
_Then,_  
\[\Psi(A_{ij},A_{kj})=\sum_{j=1}^{h}w_{j}\mu_{\Delta}(A_{ij},A_{kj})+\sum_{j=1}^ {h}w_{j}\lambda\sigma_{\Delta}(A_{ij},A_{kj}) \tag{24}\]  
_when \(\Psi(A_{ij},A_{kj})>0\) means \(A_{i}\succ A_{k}\) where \(i,k\in[1,\ldots,m]\), and vice versa._  
Figure 5: The probability density function in the third example. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Methods** & \(A_{1}\) & \(A_{2}\) & \(A_{3}\) & **Rank** \\ \hline CoA & 5.5 & 5.5 & 5.67 & \(A_{3}\succ A_{1}\approx A_{2}\) \\ \hline BoA & 5.5 & 5.5 & 5.9 & \(A_{3}\succ A_{1}\approx A_{2}\) \\ \hline Proposed & 5.5 (status quo) & 5.64 & 5.51 & \(A_{2}\succ A_{3}\succ A_{1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The comparison of defuzzifying approaches in Example 3. Assume there are three alternatives described in Table 6, where each alternative is measured using interval data. In addition, we assume these alternatives satisfy the uniform distributions. Next, we need to normalize all alternatives, and linear normalization is most often used in the SAW method [20]. Therefore, in this example, we normalize the interval data according to the following equation:  
\[r_{ij}=[\frac{r^{-}}{r^{*}},\frac{r^{+}}{r^{*}}] \tag{25}\]  
where \(r^{-}\) and \(r^{+}\) are the interval boundary, and \(r^{*}\) is the maximum value in \(j\)th attribute.