# Dombi Aggregation Operators of Neutrosophic Cubic Sets for Multiple Attribute Decision-Making
## 2 Preliminaries  
In this section, we firstly present some concepts of interval neutrosophic sets, single-valued neutrosophic sets, cubic sets and NCSs and then introduce some ranking methods of NCSs based on their score, accuracy and certainty functions. **Definition 1** ([6]). : _Let \(Z\) be a non-empty set. An interval neutrosophic sets \(G\) in \(Z  
**Definition 3** ([26]). : _Let \(Z\) be a non-empty set, then a cubic set \(C\) in \(Z\) is constructed as the following form:_  
\[C=\{v,\,A(v),\,\bar{a}(v)\,\,|\,\,v\in Z\}, \tag{3}\]  
_for \(v\in Z\). It can be noted by \(C=\{A,\,\bar{a}\}\). Then, \(C=\{A,\,\bar{a}\}\) in \(Z\) is called an internal cubic set if \(\,A^{-}(v)\leq a(v)\leq A^{+}(v)\) for \(v\in Z\) and \(C=\{A,\,\bar{a}\}\) in \(Z\) is called an external cubic set if \(\,a(v)\
otin\left[A^{-}(v),\,\,A^{+}(v)\right]\) for \(v\in Z\)._  
[PERSON] et al. [27] and [PERSON] et al. [28] extended cubic sets to the neutrosophic sets and proposed the concept of a NCS as follows. **Definition 4** ([27; 28]). : _Let \(Z\) be a universe of discourse, then a neutrosophic cubic set \(X\) in \(Z\) is denoted as the following form:_  
\[X=[v,\,<\)T(v),\,I(v),\,F(v)\!\!>,\,<\!\!t(v),\,i(v),\,f(v)\!\!>\,\,|\,\,v\in Z], \tag{4}\]  
_where \(<\)T(\(v\)), I(\(v\)), F(\(v\))\(>\) is an interval neutrosophic set [6] in \(Z\) and the intervals \(T(v)=\left[T^{-}(v),\,\,T^{+}(v)\right]\subseteq\left[0,1\right]\), I(\(v)=\left[I^{-}(v),\,\,I^{+}(v)\right]\subseteq\left[0,1\right]\), and \(F(v)=\,\,\left[F^{-}(v),\,\,F^{+}(v)\right]\subseteq\left[0,1\right]\) for \(v\in Z\) represent the truth, indeterminacy and falsity membership degrees, respectively; then \(<\)t(\(v\)), i(\(v\)), f(\(v\))\(>\) is a single valued neutrosophic set [5;7] in \(Z\) and t(\(v\)), i(\(v\)), f(\(v\))\(\in\left[0,1\right]\) for \(v\in Z\) represent the membership degrees of truth, indeterminacy and falsity, respectively._  
Then, a neutrosophic cubic sets \(X=\{v,\,<\)T(\(v\)), I(\(v\)), F(\(v\))\(>\), \(<\)t(\(v\)), i(\(v\)), f(\(v\))\(>\)\(\,|\,\,v\in Z\}\) is called an internal NCS if \(\,T^{-}(v)\leq t(v)\leq T^{+}(v)\), \(I^{-}(v)\leq i(v)\leq I^{+}(v)\), and \(F^{-}(v)\leq f(v)\leq F^{+}(v)\), for \(v\in Z\); and a NCS \(X\) is called an external NCS if \(t(v)\
otin\left(T^{-}(v),\,\,T^{+}(v)\right)\), i(\(v\))\(\
otin\left(I^{-}(v),\,\,I^{+}(v)\right)\) and \(f(v)\
eq\left(F^{-}(v),\,\,F^{+}(v)\right)\) for \(v\in Z\)[27]. For convenient expression, a basic element (\(v\), \(<\)T(\(v\)), I(\(v\)), F(\(v\))\(>\), \(<\)t(\(v\)), i(\(v\)), f(\(v\))\(>\)) in a NCS \(X\) is denoted by \(x=\,\,\left(<\)\(T^{-},\,\,T^{+}\right]\), \(\left[I^{-},\,\,I^{+}\right]\), \(\left[F^{-},\,\,F^{+}\right]\)\(>\), \(<\)\(t\), \(f\)\(>\)), which is called a neutrosophic cubic number (NCN) [31], where \(\left[T^{-},\,\,T^{+}\right]\), \(\left[I^{-},\,\,I^{+}\right]\), \(\left[F^{-},\,\,F^{+}\right]\subseteq\left[0,1\right]\) and \(t\), \(i,f\in\left[0,1\right]\) satisfy the condition \(0\leq T^{+}+\,\,I^{+}+\,\,F^{+}\leq\,3\) and \(0\leq t+i+f\leq 3\). For any neutrosophic cubic number, we provide the following score, accuracy and certainty functions. **Definition 5** ([54]). : _Let \(x=\,\,\left(<\left[T^{-},\,\,T^{+}\right]\), \(\left[I^{-},\,\,I^{+}\right]\), \(\left[F^{-},\,\,F^{+}\right]\)\(>\), \(<\)\(t\), \(i\), \(f\)\(>\)) be a neutrosophic cubic number. Then, its score, accuracy and certainty functions are defined as follows:_  
\[S(x)=\left[\left(4+T^{-}+\,\,T^{+}-\,\,I^{-}-\,\,I^{+}-\,\,F^{-}-\,\,F^{+} \right)\right.+\left(2+t-i-f\right)\right]/9 \tag{5}\]  
\[A(x)=\left[\left(T^{-}+\,\,T^{+}-\,\,F^{-}-\,\,F^{+}\right)\right.+\left(t-f \right)\right]/3 \tag{6}\]  
\[C(x)=\left(T^{-}+\,\,T^{+}+\,\,t\right)/3 \tag{7}\]  
_where, \(S(x)\), \(A(x)\) and \(C(x)\) represent the score, accuracy and certainty functions of the NCNs, respectively._  
The score function \(S(x)\) is a useful index in ranking NCNs. For a NCN, the bigger the truth-membership is, the greater the NCN is.