## 5 Experimental Study and Discussion
### Nonadditive Measure Calculation
Similarly, \(a_{1}\) and \(a_{3}\), \(a_{2}\) and \(a_{3}\) belong to \(K_{2}\); \(a_{2}\) and \(a_{4}\), \(a_{2}\) and \(a_{5}\) belong to \(K_{3}\); \(a_{1}\) and \(a_{4}\), \(a_{1}\) and \(a_{5}\), \(a_{3}\) and \(a_{4}\), \(a_{3}\) and \(a_{5}\) belong to \(K_{4}\); \(a_{4}\) and \(a_{5}\) belongs to \(K_{5}\). Therefore, \(I_{12}\in[-X_{12},-\frac{3}{5}X_{12}]\) can be obtained, that is, \(I_{12}\in[-0.0233,-0.0140]\). Similarly, we can obtain all the interaction intervals for the possible attribute pairs, as listed in Table 7. \begin{table}
\begin{tabular}{c c c c} \hline \(\mathbf{X_{jk}}\) & **Value** & \(\mathbf{X_{jk}}\) & **Value** \\ \hline \(X_{12}\) & 0.0233 & \(X_{24}\) & 0.0233 \\ \(X_{13}\) & 0.0316 & \(X_{25}\) & 0.0233 \\ \(X_{14}\) & 0.0316 & \(X_{34}\) & 0.0816 \\ \(X_{15}\) & 0.0316 & \(X_{35}\) & 0.0816 \\ \(X_{23}\) & 0.0233 & \(X_{45}\) & 0.1628 \\ \hline \end{tabular}
\end{table} 
Table 6: Values for \(X_{jk}\). \begin{table}
\begin{tabular}{c c c c} \hline \(I_{jk}\) & **Interaction Interval** & \(I_{jk}\) & **Interaction Interval** \\ \hline \(I_{12}\) & \([-0.0233,-0.0140]\) & \(I_{24}\) & \([-0.0047,0.0047]\) \\ \(I_{13}\) & \([-0.0190,-0.0063]\) & \(I_{25}\) & \([-0.0047,0.0047]\) \\ \(I_{14}\) & \([0.0063,0.0190]\) & \(I_{34}\) & \([0.0163,0.0490]\) \\ \(I_{15}\) & \([0.0063,0.0190]\) & \(I_{35}\) & \([0.0163,0.0490]\) \\ \(I_{23}\) & \([-0.0140,-0.0047]\) & \(I_{45}\) & \([0.0977,0.1628]\) \\ \hline \end{tabular}
\end{table} 
Table 7: Interaction intervals. Third, based on Model (10), the value of the interaction \(I_{jk}\) can be calculated by solving the following nonlinear programming:  
\[\max\sum_{j=1}^{5}\sum_{T\subset A\setminus a_{j}}\frac{(4-|A|)! |\frac{|A|!}{5!}}{\rm{}}h\left(I_{j}-\frac{1}{2}\sum_{a_{k}\in A\setminus\{T \cup\{a_{j}\}\}}I_{jk}+\frac{1}{2}\sum_{a_{k}\in T}I_{jk}\right)\] \[s.t.\left\{ \begin{array}{l}I_{1}=0.0632,\ I_{2}=0.0465,\ I_{3} =0.1631,\ I_{4}=0.3255,\ I_{5}=0.4018\\ I_{12}\in[-0.0233,\ -0.0140],\ I_{13}\in[-0.0190,\ -0.0063],\ I_{14},\ I_{15}\in[0.0063,\ 0.0190]\\ I_{23}\in[-0.0140,\ -0.0047],\ I_{24},\ I_{25}\in[-0.0047,\ 0.0047]\\ I_{34},\ I_{35}\in[0.0163,\ 0.0490],\ I_{45}\in[0.0977,\ 0.1628]\end{array} \right.\]  
By using MATLAB programming, we obtain all the values of \(I_{jk}\), as listed in Table 8.  
Fourth, the values of Mobius representation can be obtained using Equation (11), as listed in Table 9. For example, \(M_{1}=I_{1}-\frac{1}{2}(I_{12}+I_{13}+I_{14}+I_{15}+I_{12})=0.0646\). Finally, the 2-order additive measures of all subsets of \(A\) are calculated using Equation (12). The results are listed in Table 10. For example, \(\mu(a_{1},a_{2})=M_{1}+M_{2}+M_{12}=0.1070\), \(\mu(a_{1},a_{2},a_{3})=M_{1}+M_{2}+M_{3}+M_{12}+M_{13}+M_{23}=0.2086\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Subset** & **Nonadditive Measure** & **Subset** & **Nonadditive Measure** \\ \hline \(\{|\}\) & 0 & \(\{a_{1},a_{2},a_{3}\}\) & 0.2086 \\ \(\{a_{1}\}\) & 0.0646 & \(\{a_{1},a_{2},a_{4}\}\) & 0.3360 \\ \(\{a_{2}\}\) & 0.0650 & \(\{a_{1},a_{2},a_{5}\}\) & 0.4119 \\ \(\{a_{3}\}\) & 0.1290 & \(\{a_{1},a_{3},a_{4}\}\) & 0.4560 \\ \(\{a_{4}\}\) & 0.2121 & \(\{a_{1},a_{3},a_{5}\}\) & 0.5326 \\ \(\{a_{5}\}\) & 0.2890 & \(\{a_{1},a_{4},a_{5}\}\) & 0.7624 \\ \(\{a_{1},a_{2}\}\) & 0.1070 & \(\{a_{2},a_{3},a_{4}\}\) & 0.4409 \\ \(\{a_{1},a_{3}\}\) & 0.1785 & \(\{a_{2},a_{3},a_{5}\}\) & 0.5169 \\ \(\{a_{1},a_{4}\}\) & 0.2942 & \(\{a_{2},a_{4},a_{5}\}\) & 0.7260 \\ \(\{a_{1},a_{5}\}\) & 0.3708 & \(\{a_{3},a_{4},a_{5}\}\) & 0.8877 \\ \(\{a_{2},a_{3}\}\) & 0.1815 & \(\{a_{1},a_{2},a_{3},a_{4}\}\) & 0.4855 \\ \(\{a_{2},a_{4}\}\) & 0.2765 & \(\{a_{1},a_{2},a_{3},a_{5}\}\) & 0.5612 \\ \(\{a_{2},a_{5}\}\) & 0.3525 & \(\{a_{1},a_{2},a_{4},a_{5}\}\) & 0.8029 \\ \(\{a_{3},a_{4}\}\) & 0.3890 & \(\{a_{1},a_{3},a_{4},a_{5}\}\) & 0.9721 \\ \(\{a_{3},a_{5}\}\) & 0.4657 & \(\{a_{2},a_{3},a_{4},a_{5}\}\) & 0.9383 \\ \(\{a_{4},a_{5}\}\) & 0.6631 & \(\{a_{1},a_{2},a_{3},a_{4},a_{5}\}\) & 1 \\ \hline \hline \end{tabular}
\end{table}