## 3 Bipolar Neutrosophic Einstein Average AOs
### BN Einstein HybridGeometric Aggregation Operators  
**Definition 21**. : _The BN Einstein hybrid geometric (BNEHG) aggregation operator is stated as follows:_  
\[BNEHG_{w,\
u}(u_{1},u_{2},u_{3},\ldots,u_{n-1},u_{n})=\mathop{ \otimes}\limits_{l=1}^{n}\left(\
u_{l}\dot{u}_{\sigma(l)}\right)=\
u_{1}\dot {u}_{\sigma(1)}\otimes\
u_{2}\dot{u}_{\sigma(2)}\otimes\
u_{3}\dot{u}_{\sigma( 3)}\otimes\ldots\otimes\
u_{l}\dot{u}_{\sigma(l)}, \tag{12}\]  
_where \(w=(w_{1},w_{2},w_{3},\ldots,w_{n-1},w_{n})\) is a weighting vector of \(u_{l}\), such that\(w_{l}\in[0,1]\), \(\mathop{\sum}\limits_{l=1}^{n}w_{l}=1\) and \(\dot{u}_{\rho(l)}\) represent the \(l\)-th biggest component of the BN argument \(\dot{u}_{l}\big{(}\dot{u}_{l}=(nv_{l})u_{l},l=1,2,\ldots,n\big{)}\), and \(\
u=(\
u_{1},\
u_{2},\ldots,\
u_{n})\) is the weighting vector of BN arguments \(u_{l}\)along with \(\
u_{l}\in[0,1]\), \(\mathop{\sum}\limits_{l=1}^{n}v_{l}=1\)._  
Note that BNEHG reduces to BNEWG if \(w=\left(\frac{1}{n},\frac{1}{n},\ldots,\frac{1}{n}\right)^{T}\) or the BNEOWG operator if \(\
u=\left(\frac{1}{n},\frac{1}{n},\ldots,\frac{1}{n}\right)\). **Theorem 20**. : _The (BNEHG) operator gives in return a BNV with_  
\[BNEHG_{w,\
u}(u_{1},u_{2},u_{3},\ldots,u_{n})=\] \[\left(\frac{2_{\bar{1}}}{\frac{2_{\bar{1}}}{\bar{1}^{+}\bar{1}^{-} \bar{2}^{-}_{\bar{2}}}},\frac{\bar{f}_{\bar{1}^{-}\bar{2}}}{\bar{1}^{+}\bar{2} ^{-}_{\bar{2}}},\frac{-\bar{3}^{-}\bar{4}}{\bar{3}^{+}\bar{4}^{+}},\frac{- \bar{2}^{-}\bar{2}^{-}_{\bar{3}}}{\bar{4}^{+}\bar{3}^{-}_{\bar{3}}},\frac{- \bar{2}^{-}\bar{2}_{\bar{3}}}{\bar{4}^{+}\bar{3}^{-}_{\bar{3}}}\right) \tag{13}\]  
_where_  
\[ \begin{array}{l}\bar{1}=\mathop{\prod}\limits_{j=1}^{n}\left(\hat{\Im}_{ \sigma(j)}^{+}\right)^{\omega_{j}},\bar{2}=\mathop{\prod}\limits_{j=1}^{n} \left(2-\hat{\Im}_{\sigma(j)}^{+}\right)^{\omega_{j}},\bar{3}=\mathop{\prod} \limits_{j=1}^{n}\left(1+\left|\hat{\Im}_{\sigma(j)}^{-}\right|\right)^{\omega_{ j}},\bar{4}=\mathop{\prod}\limits_{j=1}^{n}\left(1+\hat{\Im}_{\sigma(j)}^{-} \right)^{\omega_{j}},\\ \dot{\bar{1}}_{\bar{1}}=\mathop{\prod}\limits_{j=1}^{n}\left(1+\hat{i}_{\sigma( j)}^{+}\right)^{\omega_{j}},\dot{5}_{\bar{2}}=\mathop{\prod}\limits_{j=1}^{n}\left(1- \hat{i}_{\sigma(j)}^{+}\right)^{\omega_{j}},\dot{5}_{\bar{3}}=\mathop{\prod} \limits_{j=1}^{n}\left|\hat{i}_{\sigma(j)}^{-}\right|^{\omega_{j}},\dot{5}_{ \bar{4}}=\mathop{\prod}\limits_{j=1}^{n}\left(2+\hat{i}_{\sigma(j)}^{-}\right)^{ \omega_{j}},\\ \dot{f}_{\bar{1}}=\mathop{\prod}\limits_{j=1}^{n}\left(1+\hat{i}_{\sigma(j)}^{+} \right)^{\omega_{j}},\dot{f}_{\bar{2}}=\mathop{\prod}\limits_{j=1}^{n}\left(1- \hat{i}_{\sigma(j)}^{+}\right)^{\omega_{j}},\dot{7}_{\bar{3}}=\mathop{\prod} \limits_{j=1}^{n}\left|\hat{i}_{\sigma(j)}^{-}\right|^{\omega_{j}},\dot{7}_{ \bar{4}}=\mathop{\prod}\limits_{j=1}^{n}\left(2+\hat{i}_{\sigma(j)}^{-}\right)^{ \omega_{j}}.\end{array} \]_where \(w=(w_{1},w_{2},w_{3},\ldots,w_{n-1},w_{n})\) is a weighting vector of \(u_{l}\), such that \(w_{l}\in[0,1]\), \(\sum\limits_{l=1}^{n}w_{l}=1\) and \(\dot{u}_{\rho(l)}\) represent the \(l\)-th biggest component of the BN argument \(\dot{u}_{l}(\dot{u}_{l}=(nv_{l})u_{l},l=1,2,\ldots,n)\), and \(v=(v_{1},v_{2},\ldots,v_{n})\) is the weighting vector of BN arguments \(u_{l}\),along with \(v_{l}\in[0,1]\), \(\sum\limits_{l=1}^{n}v_{l}=1\). Here, \(n\) represents a balancing coefficient._  
**Proof. ** The theorem is straightforward.\(\square\)  
**5. Multi-Criteria Group DM Problem Based on BN Einstein Aggregation Operators**  
This section includes a multi-criteria application based on BN Einstein aggregation operators (AOs), along with crisp numbers serving as attributes' weights and BN numbers serving as attributes' values. _5.1. Algorithm_  
Consider \(G=\{G_{1},G_{2},\ldots,G_{m}\}\) to represent a collection of finite \(m\) alternatives, \(L=\{L_{1},L_{2},L_{