## 2 Preliminaries
_In condition_ \(s(u_{1})=s(u_{2})\) _and_ \(a\ (u_{1})>a\ (u_{2})\) _, then_ \(u_{1}\) _is greater than_ \(u_{2}\) _, denoted by_ \(u_{{}_{1}}>u_{2}\)_;_
3. _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 superior to_ \(u_{2}\) _, denoted by_ \(u_{{}_{1}}>u_{2}\)_;_
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 8 [21]. **_Consider P as a universal set. Then, the bipolar fuzzy set (BFS) is stated below:_  
\[A_{NS}=\{(\chi,\Im(\chi),I(\chi),f(\chi))|\chi\in P\}\]6  
_where \(\Im_{F}^{+}(\chi):F\to L^{+}\) represents a positive membership function and the negative membership function is \(f_{{}_{F}}^{-}(\chi):F\to K^{-}\), where \(L^{+}=[0,1]\) and \(K^{-}=[-1,0]\)._  
**Definition 9 [25]. **_Suppose \(A\) is a bipolar neutrosophic set contained by universal set P, stated as:_  
\[A_{NS}=\{(\chi,\Im(\chi),I(\chi),f(\chi))|\chi\in P\}\]7  
_Let \(\Im^{+}(\chi),I^{+}(\chi),f^{+}(\chi)=BN^{+}\) as well as \(\Im^{-}(\chi),I^{-}(\chi),f^{-}(\chi)=BN^{-}\)._  
_Here, \(\Im^{+}(\chi),I^{+}(\chi),f^{+}(\chi)\) are positive membership functions representingtrue, indeterminate and false for \(\chi\in P\), and \(\Im^{-}(\chi),I^{-}(\chi),f^{-}(\chi)\) represent true, indeterminate and false for negative membership functions. Now, when mapping \(BN^{+}:A\to L^{+}\) and \(BN^{-}:A\to L^{-}\), here, \(L^{+}=[0,1]\) and \(L^{-}=[-1,0]\). This applies with the condition \(0\leq\Im^{+}(\chi)+I^{+}(\chi)+f^{+}(\chi)+\Im^{-}(\chi)+I^{-}(\chi)+f^{-}( \chi)\leq 6\)._  
**Example 1. **_Let \(P=\{\mathrm{ch}\,\mathrm{I}_{1},\chi_{2},\chi_{3}\}\), then_  
\[A=\left\{ \begin{array}{l}(\mathrm{ch}\,\mathrm{I}_{1},0.1,0.5,0.3,-0.4,-0.5,- 0.6),\\ (\chi_{2},0.2,0.7,0.4,-0.3,-0.6,-0.2),\\ (\chi_{3},0.4,0.6,0.7,-0.2,-0.3,-0.1)\end{array} \right\}\]  
_is a subset of universal set P that is BNS._Some fundamental operations [12] used for BNSs are stated below:  
Let \(A_{1}=\big{\{}\big{(}\chi,\Im_{1}^{+}(\chi),I_{1}^{+}(\chi),f_{1}^{+}(\chi), \Im_{1}^{-}(\chi),I_{1}^{-}(\chi),f_{1}^{-}(\chi)\big{)}|\chi\in P\big{\}}\) and \(A_{2}=\big{\{}(\chi,\Im_{2}^{+}(\chi),I_{2}^{+}(\chi),f_{2}^{+}(\chi),\Im_{2}^ {-}(\chi),I_{2}^{-}(\chi),f_{2}^{-}(\chi))\big{|}\chi\in P\big{\}}\) be two BNSs. 1. Then,\(A_{1}\subseteq A_{2}\) if and only if \[\Im_{1}^{+}(\chi)\leq\Im_{2}^{+}(\chi),I_{1}^{+}(\chi)\leq I_{2}^{+}(\chi),f_ {1}^{+}(\chi)\geq f_{2}^{+}(\chi),\] and \[\Im_{1}^{-}(\chi)\geq\Im_{2}^{-}(\chi),I_{1}^{-}(\chi)\geq I_{2}^{-}(\chi),f_ {1}^{-}(\chi)\leq f_{2}^{-}(\chi)\] ii. \(A_{1}=A_{2}\) if and only if \[\Im_{1}^{+}(\chi)=\Im_{2}^{+}(\chi),I_{1}^{+}(\chi)=I_{2}^{+}(\chi),f_{1}^{+}( \chi)=f_{2}^{+}(\chi),\] and \[\Im_{1}^{-}(\chi)=\Im_{2}^{-}(\chi),I_{1}^{-}(\chi)=I_{2}^{-}(\chi),f_{1}^{- }(\chi)=f_{2}^{-}(\chi)\] iii. The union is defined as below: \[(A_{1}\cup A_{2})=\bigg{\{}\bigg{(}\max(\Im_{1}^{+}(\chi),\Im_{2}^{+}(\chi)),\frac{I_{1}^{+}(\chi)+I_{2}^{+}(\chi)}{2},\min(f_{1}^{+}(\chi),f_{2}^{+}(\chi )),\min(\Im_{1}^{-}(\chi),\Im_{2}^{-}(\chi)),\frac{I_{1}^{-}(\chi)+I_{2}^{-}( \chi)}{2},\max(f_{1}^{-}(\chi),f_{2}^{-}(\chi))\bigg{)}\bigg{\}},\] iv. The intersection is defined as: \[(A_{1}\cap A_{2})=\bigg{\{}\bigg{(}\min(\Im_{1}^{+}(\chi),\Im_{2}^{+}(\chi)),\frac{I_{1}^{+}(\chi)+I_{2}^{+}(\chi)}{2},\max(f_{1}^{+}(\chi),f_{2}^{+}(\chi )),\max(\Im_{1}^{-}(\chi),\Im_{2}^{-}(\chi)),\frac{I_{1}^{-}(\chi)+I_{2}^{-}( \chi)}{2},\min(f_{1}^{-}(\chi),f_{2}^{-}(\chi))\bigg{)}\bigg{\}},\] v. Let \(A=\big{\{}(\chi,\Im^{+}(\chi),I^{+}(\chi),f^{+}(\chi),\Im^{-}(\chi),I^{-}( \chi),f^{-}(\chi))|\chi\in P\big{\}}\) be BNSs.