# Decision-Making Approach Based on Neutrosophic Rough Information
## 2 Neutrosophic Rough Information
\begin{array}{c|cccc}\hline R&p&q&r&s\\ \hline p&(1,1,0)&(0.1,0.9,0.8)&(0.7,0.5,0.8)&(0.1,0.9,0.8)\\ q&(0.9,0.8,0.1)&(1,1,0)&(0.1,0.9,0.8)&(0.4,0.3,0.9)\\ r&(0.7,0.5,0.8)&(0.1,0.9,0.8)&(1,1,0)&(0.1,0.9,0.8)\\ s&(0.1,0.9,0.8)&(0.4,0.3,0.9)&(0.1,0.9,0.8)&(1,1,0)\\ \end{array} \]  
_Let \(X_{1}=\{(p,0.1,0.4,0.8),(q,0.2,0.3,0.9)\,,(r,0.1,0.6,0.8),(s,0.9,0.6,0.3)\}\) be a neutrosophic set on \(X^{*}\). The lower and upper approximations of \(X_{1}\) are given as:_  
\[ \begin{array}{c|cccc}\underline{R}X_{1}&=\{(p,0.1,0.3,0.8),(q,0.2,0.3,0.9), (r,0.1,0.3,0.8),(s,0.8,0.4,0.4)\},\\ \overline{R}X_{1}&=\{(p,0.1,0.6,0.8),(q,0.4,0.6,0.8),(r,0.1,0.6,0.8),(s,0.9,0.6,0.3)\}.\\ \end{array} \]  
_Let \(Y^{*}=\{pq,qr,rs,sp\}\subseteq X^{*}\times X^{*}\) and \(S\) a neutrosophic tolerance relation on \(Y^{*}\) which is given as:_  
\[ \begin{array}{c|cccc}\hline S&pq&qr&rs&sp\\ \hline pq&(1,1,0)&(0.1,0.9,0.8)&(0.4,0.3,0.9)&(0.1,0.9,0.8)\\ qr&(0.1,0.9,0.8)&(1,1,0)&(0.1,0.9,0.8)&(0.4,0.3,0.9)\\ rs&(0.4,0.3,0.9)&(0.1,0.9,0.8)&(1,1,0)&(0.1,0.9,0.8)\\ sp&(0.1,0.9,0.8)&(0.4,0.3,0.9)&(0.1,0.9,0.8)&(1,1,0)\\ \end{array} \]  
_Let \(Y_{1}=\{(pq,0.1,0.3,0.8),(qr,0.1,0.3,0.3),(rs,0.1,0.3,0.8),(sp,0.1,0.3,0.8)\}\) be a neutrosophic set on \(Y^{*}\). The lower and upper approximations of \(Y_{1}\) are given as:_  
\[ \begin{array}{c|cccc}\underline{SY}_{1}&=\{(pq,0.1,0.3,0.8),(qr,0.1,0.3,0.3), (rs,0.1,0.3,0.8),(sp,0.1,0.3,0.8)\},\\ \overline{SY}_{1}&=\{(pq,0.1,0.3,0.8),(qr,0.1,0.3,0.3),(rs,0.1,0.3,0.8),(sp,0.1,0.3,0.8)\}.\\ \end{array} \]  
_Thus, \(G_{1}=(\underline{G}_{1},\overline{G}_{1})\) is a regular neutrosophic rough digraph as shown in Figure 3._  
**Example 8**. : _Let \(X^{*}=\{p,q,r,s\}\) be a neutrosophic tolerance relation on \(X^{*}\). The lower and upper approximations of \(X_{1}\) are given as:_  
\[\begin{array}{c|cccc}\underline{R}X_{1}&=\{(p,0.1,0.3,0.8),(q,0.2,0.3,0.9),(r,0.1,0.3,0.8),(s,0.8,0.4,0.4)\},\\ \overline{R}X_{1}&=\{(p,0.1,0.6,0.8),(q,0.4,0.6,0.8),(r,0.1,0.6,0.8),(s,0.9,0.6,0.3)\}.\\  
**Definition 8**. : _Let \(G_{1}=(\underline{G}_{1},\overline{G}_{1})\) and \(G_{2}=(\underline{G}_{2},\overline{G}_{2})\) be two neutrosophic rough digraphs. Then the direct sum of \(G_{1}\) and \(G_{2}\) is a neutrosophic rough digraph \(G=G_{1}\oplus G_{2}=(\underline{G}_{1}\oplus\underline{G}_{2},\overline{G}_ {1}\oplus\overline{G}_{2})\), where \(\underline{G}_{1}\oplus\underline{G}_{2}=(\underline{R}X_{1}\oplus \underline{R}X_{2},\underline{S}Y_{1}\oplus\underline{S}Y_{2})\) and \(\overline{G}_{1}\oplus\overline{G}_{2}=(\overline{R}X_{1}\oplus\overline{R} X_{2},\overline{S}Y_{1}\oplus\overline{S}Y_{2})\) are neutrosophic digraphs._  
\[T_{\underline{R}X_{1}\oplus\underline{R}X_{2}}(x)=\left\{ \begin{array}{ll}T_{ \underline{R}X_{1}}(x),&\text{if }\ x\in\underline{R}X_{1}-\underline{R}X_{2}\\ T_{\underline{R}X_{2}}(x),&\text{if }\ x\in\underline{R}X_{2}-\underline{R}X_{1}\\ \max(T_{\underline{R}X_{1}}(x),T_{\underline{R}X_{2}}(x)),&\text{if }\ x\in \underline{R}X_{1}\cap\underline{R}X_{2}\\ \end{array} \right.\] \[I_{\underline{R}X_{1}\oplus\underline{R}X_{2}}(x)=\left\{ \begin{array}[] {ll}I_{\underline{R}X_{1}}(x),&\text{if }\ x\in\underline{R}X_{1}-\underline{R}X_{2}\\ I_{\underline{R}X_{2}}(x),&\text{if }\ x\in\underline{R}X_{2}-\underline{R}X_{1}\\ \max(I_{\underline{R}X_{1}}(x),I_{\underline{R}X_{2}}(x)),&\text{if }\ x\in \underline{R}X_{1}\cap\underline{R}X_{2}\\ \end{array} \right.\] \[F_{\underline{R}X_{1}\oplus\underline{R}X_{2}}(x)=\left\{ \begin{array} []{ll}F_{\underline{R}X_{1}}(x),&\text{if }\ x\in\underline{R}X_{1}-\underline{R}X_{2}\\ F_{\underline{R}X_{2}}(x),&\text{if }\ x\in\underline{R}X_{2}-\underline{R}X_{1}\\ \min(F_{\underline{R}X_{1}}(x),F_{\underline{R}X_{2}}(x)),&\text{if }\ x\in \underline{R}X_{1}\cap\underline{R}X_{2}\\ \end{array} \right. \tag{1}\]  
\[T_{\underline{S}Y_{1}\oplus\underline{S}Y_{2}}(x,y)=\left\{ \begin{array}[] {ll}T_{\underline{S}Y_{1}}(x,y),&\text{if }\ (x,y)\in\underline{S}Y_{1}\\ T_{\underline{S}Y_{2}}(x,y),&\text{if }\ (x,y)\in\underline{S}Y_{2}\\ \end{array}