## 2 Preliminaries
Then, the complement \(A^{\prime}\) is defined as: \[\Im_{A^{\prime}}^{+}(\chi)=\big{\{}1^{+}\big{\}}-\Im_{A}^{+}(\chi),I_{A^{\prime }}^{+}(\chi)=\big{\{}1^{+}\big{\}}-I_{A}^{+}(\chi),f_{A^{\prime}}^{+}(\chi)= \big{\{}1^{+}\big{\}}-f_{A}^{+}(\chi),\] and \[\Im_{A^{\prime}}^{-}(\chi)=\big{\{}1^{-}\big{\}}-\Im_{A}^{-}(\chi),I_{A^{\prime }}^{-}(\chi)=\big{\{}1^{-}\big{\}}-I_{A}^{-}(\chi),f_{A^{\prime}}^{-}(\chi)= \big{\{}1^{-}\big{\}}-f_{A}^{-}(\chi),\]  
**Definition 10 [25]. **_Consider two bipolar neutrosophic numbers (BNNs) \(u_{1}=\big{(}\Im_{1}^{+},I_{1}^{+},f_{1}^{+},\Im_{1}^{-},I_{1}^{-},f_{1}^{-} \big{)}\) and \(u_{2}=\big{(}\Im_{2}^{+},I_{2}^{+},f_{2}^{+},\Im_{2}^{-},I_{2}^{-},f_{2}^{-} \big{)}\). The following are the basic operations between two BNNs:_  
\[\text{i.u}_{1}+u_{2}=\big{(}\Im_{1}^{+}+\Im_{2}^{+}-\Im_{1}^{+}\Im_{2}^{+},I_{ 1}^{+}I_{2}^{+},f_{1}^{+}I_{2}^{+},-\Im_{1}^{-}\Im_{2}^{-},-\big{(}-I_{1}^{-} -I_{2}^{-}-I_{1}^{-}I_{2}^{-}\big{)},-\big{(}-f_{1}^{-}-f_{2}^{-}-f_{1}^{-}f_{2} ^{-}\big{)},\]  
\[\text{ii.u}_{1}\cdot u_{2}=\big{(}\Im_{1}^{+}\Im_{2}^{+},I_{1}^{+}+I_{2}^{-} \Im_{1}^{+}I_{2}^{+},f_{1}^{+}+I_{2}^{+}-f_{1}^{+}I_{2}^{+},-\big{(}-\Im_{1}^{-} \Im_{2}^{-}-\Im_{1}^{-}\Im_{2}^{-}\big{)},-\mathrm{I}_{1}^{-}I_{2}^{-},-\mathrm{ I}_{1}^{-}I_{2}^{-}\big{)};\]  
\[\Im_{1}^{-}(\chi)\geq\Im_{2}^{-}(\chi),I_{1}^{-}(\chi)\geq I_{2}^{-}(\chi),f_ {1}^{-}(\chi)\leq f_{2}^{-}(\chi)\]0  
\[\Im_{1}^{-}(\chi)\geq\Im_{2}^{-}(\chi),I_{1}^{-}(\chi)\geq I_{2}^{-}(\chi),f_ {1}^{-}(\chi)\leq f_{2}^{-}(\chi)\]1  
_where \(\lambda>0\)._  
**Definition 11 [25]. **_The score function for a bipolar neutrosophic number \(u=(\Im^{+},I^{+},f^{+},\Im^{-},I^{-},f^{-})\) denoted by \(S(u)\) is:_  
\[\Im_{1}^{-}(\chi)\geq\Im_{2}^{-}(\chi),I_{1}^{-}(\chi)\geq I_{2}^{-}(\chi),f_ {1}^{-}(\chi)\leq f_{2}^{-}(\chi)\]2  
**Definition 12 [25]. **_The accuracy function defined for a BNNu \(=(\Im^{+},I^{+},f^{+},\Im^{-},I^{-},f^{-})\) is \(a(u)\), as below:_\[\Im_{1}^{-}(\chi)\geq\Im_{2}^{-}(\chi),I_{1}^{-}(\chi)\geq I_{2}^{-}(\chi),f_ {1}^{-}(\chi)\leq f_{2}^{-}(\chi)\]3  
**Definition 13**[25]. : _Consider \(u=(\Im^{+},I^{+},f^{+},\Im^{-},I^{-},f^{-})\) to represent the neutrosophic number. Then, theretainty function is:_  
\[\Im_{1}^{-}(\chi)\geq\Im_{2}^{-}(\chi),I_{1}^{-}(\chi)\geq I_{2}^{-}(\chi),f_ {1}^{-}(\chi)\leq f_{2}^{-}(\chi)\]4  
**Definition 14**[25]. : _Consider \(u_{1}=\left(\Im^{+}_{2},I^{+}_{1},I^{+}_{1},\Im^{-}_{1},I^{-}_{1},f^{-}_{1}\right)\) and \(u_{2}=\left(\Im^{+}_{2},I^{+}_{2},f^{+}_{2},\Im^{-}_{2},I^{-}_{2},f^{-}_{2}\right)\) to represent two BNNs. Then, acomparison method among BNNs is:_  
1. _In condition_ \(S(u_{1})\succ S(u_{2})\) _then_ \(u_{1}\) _is greater than_ \(u_{2}\) _, denoted by_ \(u_{1}\succ u_{2}\)__
2. _In condition_ \(S(u_{1})=S(u_{2})\) _and_ \(a(u_{1})\succ a(u_{2})\) _then_ \(u_{1}\) _is greater than_ \(u_{2}\) _, denoted by_ \(u_{1}\succ u_{2}\)__
3. _In condition_ \(S(u_{1})=S(u_{2})\)_,_ \(a(u_{1})=a(u_{2})\) _and_ \(c(u_{1})\succ c(u_{2})\) _in that case,_ \(u_{1}\) _is superior to_ \(u_{2}\)_, denote by_ \(u_{1}\succ u_{2}\)_;_
4. _In condition_ \(S(u_{1})=S(u_{2})\)_,_ \(a(u_{1})=a(u_{2})\) _and_ \(c(u_{1})=c(u_{2})\) _in that case,_ \(u_{1}\) _is equal to_ \(u_{2}\) _, denoted by_ \(u_{1}=u_{2}\)_._  
**Definition 15**. : _Consider \(u=(\Im^{+},I^{+},f^{+},\Im^{-},I^{-},f^{-})\), \(u_{1}=\left(\Im^{+}_{1},I^{+}_{1},f^{+}_{1},\Im^{-}_{1},I^{-}_{1},f^{-}_{1}\right)\) and \(u_{2}=\left(\Im^{+}_{2},I^{+}_{2},f^{+}_{2},\Im^{-}_{2},I^{-}_{2},f^{-}_{2}\right)\) to be BNNs, and \(\lambda\succ 0\) any real value.