## 2 Basic Concepts  
[PERSON] [1; 2] proposed Neutrosophic sets (NSs). [PERSON] et al. [6; 7] further proposed the SVNSs. **Definition 1** [6; 7]. : _Let \(X\) be a space of points (objects) with a generic element in fix set \(X\), named by \(x\). An SVNS \(A\) in \(X\) is depicted as the following:_  
\[A=\{(x,T_{A}(x),I_{A}(x),F_{A}(x))|x\in X\}\oplus, \tag{1}\]  
_where \(T_{A}(x)(0\leq T_{A}(x)\leq 1)\) is truth-membership function, \(I_{A}(x)(0\leq I_{A}(x)\leq 1)\) is indeterminacy-membership and \(F_{A}(x)(0\leq F_{A}(x)\leq 1)\) is falsity-membership function, and \(0\leq T_{A}(x)+I_{A}(x)+F_{A}(x)\leq 3\)._  
[PERSON] et al. [14] gave the order between two SVNNs. **Definition 2** [14]. : _Let \(A_{1}=\big{(}T_{A_{1}},I_{A_{1}},F_{A_{1}}\big{)}\) and \(A_{2}=\big{(}T_{A_{2}},I_{A_{2}},F_{A_{2}}\big{)}\) be two SVNNs, \(s(A_{1})=\frac{\big{(}2+T_{A_{1}}-I_{A_{1}}-F_{A_{1}}\big{)}}{3}\) and \(s(A_{2})=\frac{\big{(}2+T_{A_{2}}-I_{A_{2}}-F_{A_{2}}\big{)}}{3}\) be the scores of \(A_{1}\) and \(A_{2}\), respectively, and let \(H(A_{1})=T_{A_{1}}-F_{A_{1}}\) and \(H(A_{2})=T_{A_{2}}-F_{A_{2}}\) be the accuracy degrees of \(A_{1}\) and \(A_{2}\), respectively, then if \(S(A_{1})<S(A_{2})\), \(A_{1}<A_{2}\); if \(S(A_{1})=S(A_{2})\), then (1) if \(H(A_{1})=H(A_{2})\), \(A_{1}=A_{2}\); (2) if \(H(A_{1})<H(A_{2})\), \(A_{1}<A_{2}\)._  
**Definition 3** [6]. : _Let \(A=(T_{A_{1}},I_{A_{1}},F_{A_{1}})\) and \(A_{2}=(T_{A_{2}},I_{A_{2}},F_{A_{2}})\) be two SVNNs and \(\lambda\) be a positive real number, some operations of SVNNs are defined:_  
1. \(A_{1}\oplus A_{2}=(T_{A_{1}}+T_{A_{2}}-T_{A_{1}}T_{A_{2}},I_{A_{1}}I_{A_{2}},F _{A_{1}}F_{A_{2}})\)_;_
2. \(A_{1}\otimes A_{2}=(T_{A_{1}}T_{A_{2}},I_{A_{1}}+I_{A_{2}}-I_{A_{1}}I_{A_{2}}, F_{A_{1}}+F_{A_{2}}-F_{A_{1}}F_{A_{2}})\)_;_
3. \(\lambda A_{1}=(1-(1-T_{A_{1}})^{\lambda},(I_{A_{1}})^{\lambda},(F_{A_{1}})^{ \lambda}),\lambda>0\)_;_
4. \((A_{1})^{\lambda}=((T_{A_{1}})^{\lambda},1-(1-I_{A_{1}})^{\lambda},1-(1-F_{A_{ 1}})^{\lambda}),\lambda>0\)_._  
[PERSON] et al. [34] develop the dual generalized WBM (DGWBM) operator and dual generalized WGBM (DGWGBM) operator. **Definition 4** [34]. : _Let \(b_{i}(i=1,2,\ldots,n)\) be a set of nonnegative crisp numbers with the weight \(w=(w_{1},w_{2},\ldots,w_{n})^{T},w_{i}\in[0,1]\)\((i=1,2,\ldots,n)\) and \(\sum_{i=1}^{n}w_{i}=1\), if_  
\[\mathrm{DGWBM}_{w}^{R}(b_{1},b_{2},\ldots,b_{n})=\left(\sum_{i_{1},i_{2}, \ldots,i_{n}=1}^{n}\left(\prod_{j=1}^{n}w_{i_{j}}b_{i_{j}}^{r_{j}}\right) \right)^{1/\sum_{j=1}^{n}r_{j}}, \tag{2}\]  
_where \(R=(r_{1},r_{2},\ldots,r_{n})^{T}\) is the parameter vector with \(r_{i}\geq 0(i=1,2,\ldots,n)\)._  
Several special cases can be obtained given the change of the parameter vector. If \(R=(\lambda,0,0,\ldots,0)\), then we obtain  
\[\mathrm{DGWBM}^{(\lambda,0,0,\ldots,0)}(b_{1},b_{2},\ldots,b_{n})=(\sum_{i=1}^ {n}w_{i}b_{i}^{\lambda})^{1/\lambda}, \tag{3}\]  
which is the generalized weighted averaging operator. If \(R=(s,t,0,0,\ldots,0)\), then we obtain  
\[\text{DGWBM}^{(s,t,0,0,\ldots,0)}(b_{1},b_{2},\ldots,b_{n})=(\sum_{i,j=1}^{n}w_{ i}w_{j}b_{i}^{s}b_{j}^{t})^{1/(s+t)}, \tag{4}\]  
which is the weighted BM. If \(R=(s,t,r,0,0,\ldots,0)\), then we obtain  
\[\text{DGWBM}^{(s,t,r,0,0,\ldots,0)}(b_{1},b_{2},\ldots,b_{n})=(\sum_{i,j,k=1}^{ n}w_{i}w_{j}w_{k}b_{i}^{s}b_{j}^{t}b_{k}^{r})^{1/(s+t+k)}. \tag{5}\]  
**Definition 5**[34]. : _Let \(b_{i}(i=1,2,\ldots,n)\) be a set of nonnegative crisp numbers with weight being \(w=(w_{1},w_{2},\ldots,w_{n})^{T},w_{i}\in[0,1]\)\((i=1,2,\ldots,n)\) and \(\sum_{i=1}^{n}w_{i}=1\), if_  
\[\text{DGWBM}^{R}(b_{1},b_{2},\ldots,b_{n})=\frac{1}{\sum_{j=1}^{n}r_{j}}\Bigg{(} \prod_{1,i_{2},\ldots,i_{n}=1}^{n}\Bigg{(}\sum_{j=1}^{n}\Big{(}r_{j}b_{i_{j}} \Big{)}\Bigg{)}\Bigg{)}^{\prod_{j=1}^{n}w_{j}}, \tag{6}\]  
_where \(R=(r_{1},r_{2},\ldots,r_{n})^{T}\) is the parameter vector with \(r_{i}\geq 0(i=1,2,\ldots,n)\)._  
Similar to the DGWBM, we can consider some special cases given the change of the parameter vector. 1.