# Dombi Aggregation Operators of Neutrosophic Cubic Sets for Multiple Attribute Decision-Making
## 3 Some Dombi Operations of NCNs
### Comparison Analysis  
From Tables 1-4, we see that the ranking orders corresponding to the NCDWAA and NCDWGA operators show obvious difference in the MADM problem. In Example 3, Table 1 indicates that the different parameters of \(\rho\) may not influence the ranking orders corresponding to the NCDWAA operator; while Table 2 shows the different parameters of \(\rho\) can change the ranking orders based on the NCDWGA operator. In Table 2, when \(\rho=1\), the best alternative is \(X_{1}\), while the worst alternative is \(X_{2}\); when \(\rho=2\), \(\rho=3\), \(\rho=4\) and \(\rho=5\), the ranking order is changed and the best alternative is \(X_{3}\) and the worst alternative is \(X_{4}\). In Example 4, Tables 3 and 4 indicate that the different values of \(\rho\) can change the ranking orders based on the NCDWGA and NCDWGA operators. In Table 3, when \(\rho=1\) and \(\rho=2\), \(Q_{1}\) is the worst alternative; when \(\rho=3\), \(\rho=4\) and \(\rho=5\), the ranking order is changed and \(Q_{3}\) is the worst alternative. In Table 4, when \(\rho=1\), \(Q_{1}\) is the worst alternative; when \(\rho=2\), \(\rho=3\), \(\rho=4\) and \(\rho=5\), the ranking order is changed and \(Q_{2}\) is the worst alternative.  
From the results of Tables 1-4, we can say that the NCDWAA and NCDWGA operators are sensitive to \(\rho\). Hence, decision makers can specify some parameter \(\rho\) according to actual requirements and/or their preference.  
Compared with the existing MADM method for NCSs introduced in [(29)], Table 5 lists the MADM results using NCDWAA and NCDWGA operators proposed in this paper and the weighted average operator (\(\mathcal{A}_{\text{W}}\)) of NCSs in the relevant paper [(29)], respectively. From Table 5, we see that the ranking orders based on the Dombi operators proposed in this paper and the weighted average operator (\(\mathcal{A}_{\text{W}}\)) of  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(\rho\) & \(\text{\bf{S(Q_{1})}}\) \(\text{\bf{S(Q_{2})}}\) \(\text{\bf{S(Q_{3})}}\) & **Ranking Order** & **The Best Alternative** \\ \hline
1 & 0.4760, 0.4856, 0.5095 & \(Q_{3}\succ Q_{2}\succ Q_{1}\) & \(Q_{3}\) \\
2 & 0.4604, 0.4502, 0.4883 & \(Q_{3}\succ Q_{1}\succ Q_{2}\) & \(Q_{3}\) \\
3 & 0.4509, 0.4300, 0.4737 & \(Q_{3}\succ Q_{1}\succ Q_{2}\) & \(Q_{3}\) \\
4 & 0.448, 0.4176, 0.4637 & \(Q_{3}\succ Q_{1}\succ Q_{2}\) & \(Q_{3}\) \\
5 & 0.4406, 0.4093, 0.4567 & \(Q_{3}\succ Q_{1}\succ Q_{2}\) & \(Q_{3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Ranking results of the NCDWGA operator for different operational parameters.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(\rho\) & \(\text{\bf{S(Q_{1})}}\) \(\text{\bf{S(Q_{2})}}\) \(\text{\bf{S(Q_{3})}}\) & **Ranking Order** & **The Best Alternative** \\ \hline
1 & 0.5241, 0.5739, 0.5437 & \(Q_{2}\succ Q_{3}\succ Q_{1}\) & \(Q_{2}\) \\
2 & 0.5410, 0.5934, 0.5474 & \(Q_{2}\succ Q_{3}\succ Q_{1}\) & \(Q_{2}\) \\
3 & 0.5534, 0.6041, 0.5513 & \(Q_{2}\succ Q_{1}\succ Q_{3}\) & \(Q_{2}\) \\
4 & 0.5626, 0.6109, 0.5547 & \(Q_{2}\succ Q_{1}\succ Q_{3}\) & \(Q_{2}\) \\
5 & 0.5697, 0.6158, 0.5574 & \(Q_{2}\succ Q_{1}\succ Q_{3}\) & \(Q_{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Ranking results of the NCDWAA operator for different operational parameters.  
NCSs have obvious difference since different aggregation operators may be result in different ranking orders. Due to no parameter selected in [29], the proposed MADM based on Dombi aggregation operators is more flexible than the approach provided in [29].  
For further comparison, the existing related decision-making approaches [51; 52; 53] based on some Dombi operations cannot deal with the decision-making problem with NCSs. However, the decision-making method presented in this paper can describe attributes with interval neutrosophic sets and single valued neutrosophic sets information simultaneously. Therefore, the paper provides a new effective way for decision makers to deal with MADM problems under neutrosophic cubic environment.