# Dombi Aggregation Operators of Neutrosophic Cubic Sets for Multiple Attribute Decision-Making
## 3 Some Dombi Operations of NCNs
** By using Equation (5), the score values of \(S(X_{j})\) of the collective NCN for the alternatives \(X_{j}\) (\(j=1\), 2, 3, 4) can be calculated as the following results:  
\[S(X_{1})=0.4966,\,S(X_{2})=0.4626,\,S(X_{3})=0.4880,\,S(X_{4})=0.4685.\]  
**Step 3'. ** According to the above score values, the ranking order of the alternatives is \(X_{1}\succ X_{3}\succ X_{4}\succ X_{2}\) and thus \(X_{1}\) is the best alternative. Further, all the ranking results of alternatives are listed in Tables 1 and 2 when the parameter \(\rho\) is changed from 1 to 5 in the NCDWAA and NCWGA operators. **Example 4** ([29]). : _A customer wishes to buy a mobile phone and needs to evaluate three models (alternatives) \(Q_{k}\) (\(k=1\), 2, 3) according to the following three attributes (specifications): \(H_{1}=\) Processor; \(H_{2}=\) Camera; (3) \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\rho\) & \(S(x_{1})\)\(S(x_{2})\)\(S(x_{3})\)\(S(x_{4})\) & **Ranking Order** & **The Best Alternative** \\ \hline
1 & 0.5928, 0.5576, 0.6206, 0.5942 & \(X_{3}\succ X_{4}\succ X_{1}\succ X_{2}\) & \(X_{3}\) \\
2 & 0.6176, 0.5896, 0.6763, 0.6360 & \(X_{3}\succ X_{4}\succ X_{1}\succ X_{2}\) & \(X_{3}\) \\
3 & 0.6334, 0.6091, 0.7047, 0.6631 & \(X_{3}\succ X_{4}\succ X_{1}\succ X_{2}\) & \(X_{3}\) \\
4 & 0.6441, 0.6210, 0.7215, 0.6802 & \(X_{3}\succ X_{4}\succ X_{1}\succ X_{2}\) & \(X_{3}\) \\
5 & 0.6516, 0.6289, 0.7323, 0.6916 & \(X_{3}\succ X_{4}\succ X_{1}\succ X_{2}\) & \(X_{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Ranking results of the NCDWAA operator for different operational parameters. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\rho\) & \(S(x_{1})\)\(S(x_{2})\)\(S(x_{3})\)\(S(x_{4})\) & **Ranking Order** & **The Best Alternative** \\ \hline
1 & 0.4966, 0.4626, 0.4880, 0.4685 & \(X_{1}\succ X_{3}\succ X_{4}\succ X_{2}\) & \(X_{1}\) \\
2 & 0.4524, 0.4246, 0.4645, 0.4112 & \(X_{3}\succ X_{1}\succ X_{2}\succ X_{4}\) & \(X_{3}\) \\
3 & 0.4238, 0.3980, 0.4483, 0.3781 & \(X_{3}\succ X_{1}\succ X_{2}\succ X_{4}\) & \(X_{3}\) \\
4 & 0.4053, 0.3803, 0.4364, 0.3584 & \(X_{3}\succ X_{1}\succ X_{2}\succ X_{4}\) & \(X_{3}\) \\
5 & 0.3925, 0.3680, 0.4274, 0.3456 & \(X_{3}\succ X_{1}\succ X_{2}\succ X_{4}\) & \(X_{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Ranking results of the NCDWGA operator for different operational parameters. \(H_{3}=\) Battery. The weight vector of the attributes is given by \(\omega_{H}=\left(\frac{1}{2}\,\ \frac{1}{3}\,\ \frac{1}{6}\ \right)\). The decision matrix can be constructed under the NCS environment as follows:  
\[M_{2}=\left[ \begin{array}{l}\left(<[0.2,0.7],[0.3,0.7],[0.3,0.8]>,<0.3,0.4,0.1> \right)\left(<[0.4,0.7],[0.3,0.7],[0.5,0.8]>,<0.2,0.4,0.5>\right)\\ \left(<[0.2,0.7],[0.3,0.7],[0.4,0.6]>,<0.9,0.6,0.2>\right)\left(<[0.2,0.3],[0. 3,0.6],[0.1,0.4]>,<0.6,0.7,0.6>\right)\\ \left(<[0.2,0.5],[0.2,0.7],[0.1,0.2]>,<0.5,0.7,0.2>\right)\left(<[0.1,0.6],[0.2,0.6],[0.3,0.4]>,<0.4,0.5,0.6>\right)\\ \left(<[0.2,0.8],[0.2,0.7],[0.1,0.6]>,<0.1,0.3,0.5>\right)\\ \left(<[0.2,0.7],[0.4,0.7],[0.1,0.3]>,<0.3,0.5,0.7>\right)\\ \left(<[0.2,0.5],[0.3,0.4],[0.3,0.4]>,<0.2,0.4,0.6>\right)\\ \end{array} \right]\]  
Then, we use the NCDWAA operator or the NCDWGA operator to solve the MADM problem with NCN information. By the same steps as that of Example 2, we obtain the ranking results of the alternatives. Tables 3 and 4 list the ranking results of the NCDWAA operator and NCWGA operator, respectively, when the parameter \(\rho\) is changed from 1 to 5.