## 3 Methodology  
### Pythagorean Fuzzy Rough Approximation Construction  
To address the problems of counter intuitive situations and ignoring practical semantics of membership and non-membership degrees, we introduced the Pythagorean fuzzy rough approximation construction method. The similarity measure values of PFSs and PFNs are a real number between 0 and 1. Then, a new Pythagorean fuzzy closeness measure was developed by combining the similarity measures of PFNs with divergence measures, which can overcome the above-mentioned shortcomings of the existing Pythagorean fuzzy similarity measures. Further, it was introduced into the Pythagorean fuzzy decision system. Firstly, the definitions of PFNs and their similarity and divergence measures are as follows. Definition 1 ([36]): Let \(U\) be a given universe, \(A=\{(x,\mu_{A}(x),\
u_{A}(x))|x\in U\}\) be a Pythagorean fuzzy set (PFS) on \(U\), where \(\mu_{A}:U\mapsto[0,1]\) and \(\
u_{A}:U\mapsto[0,1]\) are the membership value and non-membership value of \(A\), respectively. For \(\forall x\in U,0\leq\mu_{A}^{2}(x)+\
u_{A}^{2}(x)\leq 1\), \(\pi_{A}(x)=\sqrt{1-\mu_{A}^{2}(x)-\
u_{A}^{2}(x)}\) is the hesitance degree of the PFS \(A\). Otherwise, the elements of PFS are called Pythagorean fuzzy numbers (PFNs) [37]. For convenience, the PFN can be abbreviated as \(a=(\mu_{a},\
u_{a})\). Definition 2 ([36]): Let \(S(A,B)\) be the similarity measure of the two PFSs \(A\) and \(B\), and the similarity measure \(S(A,B)\) satisfies:  
(1) \(0\leq S(A,B)\leq 1\);  
(2) \(S(A,B)=1\Leftrightarrow A=B\);  
(3) \(S(A,B)=S(B,A)\);  
(4) If \(A\subseteq B\subseteq C\), then \(S(A,C)\leq min\{S(A,B),S(B,C)\}\). Definition 3 ([36]): Let \(D(A,B)\) be the divergence measure of the two PFSs \(A\) and \(B\), and the divergence measure \(D(A,B)\) satisfies:  
(1) \(D(A,B)=D(B,A)\);  
(2) \(D(A,A)=0\);  
(3) \(D(A\cap C,B\cap C)\leq D(A,B)\);  
(4) \(D(A\cup C,B\cup C)\leq D(A,B)\). The closeness measure of PFS is defined as follows. Definition 4: Let \(\vartheta(A,B)\) be the closeness measure of the two PFSs \(A\) and \(B\), and the closeness measure \(\vartheta(A,B)\) satisfies:  
(1) \(\vartheta(A,B)\) is a PFN;  
(2) \(\vartheta(A,B)=(1,0)\), if and only if \(A=B\);  
(3) \(\vartheta(A,B)=\vartheta(B,A)\);  
(4) If \(A\subseteq B\subseteq C\), then \(\vartheta(A,C)\subseteq\vartheta(A,B)\) and \(\vartheta(A,C)\subseteq\vartheta(B,C)\). The similarity of PFNs is defined as follows. Definition 5 ([38]): Let \(a_{1}=(\mu_{1},\
u_{1})\) and \(a_{2}=(\mu_{2},\
u_{2})\) be two PFNs, and the similarity measure between \(a_{1}\) and \(a_{2}\) is defined as  
\[s(a_{1},a_{2})=1-\frac{1}{2}\Big{(}\Big{|}\mu_{1}^{2}-\mu_{2}^{2}\Big{|}+\Big{|} \
u_{1}^{2}-\
u_{2}^{2}\Big{|}+\Big{|}\pi_{1}^{2}-\pi_{2}^{2}\Big{|}\Big{)} \tag{1}\]  
where \(\pi_{1}\) and \(\pi_{2}\) are the hesitance degrees of the two PFNs \(a_{1}\) and \(a_{2}\). The above similarity measure of PFNs is based on the Hamming distance measure of PFNs, and the similarity measure satisfies the following properties. **Property 1. **_Let \(s(a_{1},a_{2})\) be the similarity measure of two PFNs \(a_{1}\) and \(a_{2}\), and then_  
_(1) \(0\leq s(a_{1},a_{2})\leq 1\);_  
_(2) \(s(a_{1},a_{2})=1\Leftrightarrow a_{1}=a_{2}\);_  
_(3) \(s(a_{1},a_{2})=s(a_{2},a_{1})\);_  
_(4) If \(a_{1}\subseteq a_{2}\subseteq a_{3}\), then \(s(a_{1},a_{3})\leq min\{s(a_{1},a_{2}),s(a_{2},a_{3})\}\)._  
Let \(A=\{(x,\mu_{A}(x),\
u_{A}(x))|x\in U\}\) and \(B=\{(x,\mu_{B}(x),\
u_{B}(x))|x\in U\}\), which are two PFSs on the universe \(U\), then the similarity measure and divergence measure are defined as follows. **Theorem 1.