## 5 Numerical Examples  
In this section, we demonstrate four examples to show how our method ranks imprecise data. The first three examples illustrate the way to rank various imprecise data. The last example demonstrates our approach to extending to the simple additive weighting method. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Properties of Mellin Transform** & **Y=\(\mathbf{h(x)}\)** & **M(s)** \\ \hline Scaling property & \(ax\) & \(a^{-s}M(s)\) \\ Multiplication by \(x^{a}\) & \(x^{a}f(x)\) & \(M(s+a)\) \\ Rising to a real power & \(f(x^{a})\) & \(a^{-1}M(\frac{s}{a}),a>0\) \\ Inverse & \(x^{-1}f(x^{-1})\) & \(M(1-s)\) \\ Multiplication by \(lnx\) & \(lnxf(x)\) & \(\frac{d}{ds}M(s)\) \\ Derivative & \(\frac{d}{ds^{a}}f(x)\) & \(\frac{\Gamma(s)}{\Gamma(s-k)}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The properties of the Mellin transform. The Mellin transform derives the values of the mean and spread, and the preference of objects is obtained by incorporating the variance into the mean value. **Example 1**. : _Let four alternatives using imprecise data be used to measure the preference where we do not know any information about our alternatives (i.e., assuming they have uniform distribution). Their imprecise preferences are \(A_{1}\) = (2, 10), \(A_{2}\) = (4, 8), \(A_{3}\) = (5, 7), and \(A_{4}\) = (6, 9), and these probability density functions are represented in Figure 3._  
According to the Mellin transform, we can obtain:  
\[ \begin{array}{ll}E(A_{1})=6.0,&\sigma_{A_{1}}=2.3094\\ E(A_{2})=6.0,&\sigma_{A_{2}}=1.1547\\ E(A_{3})=6.0,&\sigma_{A_{3}}=0.5774\\ E(A_{4})=7.5,&\sigma_{A_{4}}=0.8660\end{array} \]  
then the differences in the alternatives' total mean can be calculated as:  
\[ \begin{array}{ll}\Psi(A_{1},A_{2})=\mu_{\Delta}(A_{1},A_{2})+\lambda\sigma_ {\Delta}(A_{1},A_{2})=0+0.25\times(1.1547-2.3094)=-0.2887<0\\ \Psi(A_{1},A_{3})=\mu_{\Delta}(A_{1},A_{3})+\lambda\sigma_{\Delta}(A_{1},A_{3} )=0+0.25\times(0.5774-2.3094)=-0.4330<0\\ \Psi(A_{1},A_{4})=\mu_{\Delta}(A_{1},A_{4})+\lambda\sigma_{\Delta}(A_{1},A_{4} )=-1.5+0.25\times(0.8660-2.3094)=-1.8609<0\end{array} \]  
Based on the above result, we can conclude the ranking result is \(A_{4}\succ A_{3}\succ A_{2}\succ A_{1}\). We can set \(A_{1}\) as the status quo alternative and defuzzify other alternatives. The result is compared with CoA and Boa and presented, as shown in Table 3.  
**Example 2**. : _Given two imprecise data with triangular distributions, \(A_{1}\) = (0, 6, 10) and \(A_{2}\) = (3.8, 4.8, 5.8), and a crisp value \(A_{3}\) = 4.6, to present the measurement of the preference in \(A_{1}\), \(A_{2}\), and \(A_{3}\). Their probability density functions are listed in Figure 4._ \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Methods** & \(\mathbf{A_{1}}\) & \(\mathbf{A_{2}}\) & \(\mathbf{A_{3}}\) & \(\mathbf{A_{4}}\) & **Rank** \\ \hline CoA & 6 & 6 & 6 & 7.5 & \(A_{4}\succ A_{3}\approx A_{2}\approx A_{1}\) \\ \hline BoA & 6 & 6 & 6 & 7.5 & \(A_{4}\succ A_{3}\approx A_{2}\approx A_{1}\) \\ \hline Proposed & 6 (status quo) & 6.29 & 6.43 & 7.86 & \(A_{4}\succ A_{3}\succ A_{2}\succ A_{1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The comparison of defuzzifying approaches in Example 1. Figure 3: The probability density function in the first example. By using the Mellin transform,  
\[ \begin{array}{ll}E(A_{1})=5.0,\ \sigma_{A_{1}}=2.0412\\ E(A_{2})=4.8,\ \sigma_{A_{2}}=0.4082\\ E(A_{3})=4.6,\ \sigma_{A_{3}}=0\end{array} \]  
then the differences in the alternative's total mean can be calculated as:  
\[ \begin{array}{ll}\Psi(A_{1},A_{2})=\mu_{\Delta}(A_{1},A_{2})+\lambda\sigma_{ \Delta}(A_{1},A_{2})=0.2+0.25\times(0.4082-2.0412)=-0.2083<0\\ \Psi(A_{2},A_{3})=\mu_{\Delta}(A_{2},A_{3})+\lambda\sigma_{\Delta}(A_{2},A_{3} )=0.2+0.25\times(0-0.4082)=-0.0980<0\end{array} \]  
The above result indicates that the preference relation is \(A_{3}\succ A_{2}\succ A_{1}\), where \(A_{3}\) is the best choice. Then, we can compare the proposed method with CoA and Boa, as shown in Table 4.  
**Example 3**.