## 3 Methodology
### Pythagorean Fuzzy Rough Approximation Construction
where  
\[\overline{\alpha}=\frac{\left(1-\omega\right)\left(\mu_{A}^{2} \left(\lambda_{PN}\right)-\mu_{A}^{2}\left(\lambda_{BN}\right)\right)+\omega \left(\
u_{A}^{2}\left(\lambda_{BN}\right)-\
u_{A}^{2}\left(\lambda_{BN} \right)\right)}{\left(\mu_{A}^{2}\left(\lambda_{BN}\right)-\mu_{A}^{2}\left( \lambda_{BN}\right)+\mu_{A}^{2}\left(\lambda_{BP}\right)-\mu_{A}^{2}\left( \lambda_{BP}\right)\right)+\omega\left(\
u_{A}^{2}\left(\lambda_{BN}\right)- \
u_{A}^{2}\left(\lambda_{BP}\right)\right)+\omega\left(\
u_{A}^{2}\left( \lambda_{BN}\right)-\
u_{A}^{2}\left(\lambda_{BP}\right)\right)}}, \tag{19}\]  
From the above decision rules P1\(\
eg\)N1, it can be divided into two cases. (i) if \(\overline{\beta}<\overline{\alpha}\), then the boundary region exists and satisfies:  
\[ \begin{array}{l}\frac{(1-\omega)\left(\mu_{\alpha}^{2}(\lambda_{BP})-\mu_{ \alpha}^{2}(\lambda_{BP})\right)+\omega\left(\
u_{\alpha}^{2}(\lambda_{BP})-\
u _{\alpha}^{2}(\lambda_{BP})\right)}{(1-\omega)\left(\mu_{\alpha}^{2}(\lambda_{ NP})-\mu_{\alpha}^{2}(\lambda_{BN})\right)+\omega\left(\
u_{\alpha}^{2}(\lambda_{BN})-\
u _{\alpha}^{2}(\lambda_{BN})\right)}\\ <\frac{(1-\omega)\left(\mu_{\alpha}^{2}(\lambda_{NP})-\mu_{\alpha}^{2}(\lambda_ {BP})\right)+\omega\left(\
u_{\alpha}^{2}(\lambda_{BP})-\
u_{\alpha}^{2}( \lambda_{BP})\right)}{(1-\omega)\left(\mu_{\alpha}^{2}(\lambda_{BN})-\mu_{ \alpha}^{2}(\lambda_{NN})\right)+\omega\left(\
u_{\alpha}^{2}(\lambda_{BN})- \
u_{\alpha}^{2}(\lambda_{BN})\right)}\end{array} \]  
In this case, \(0\leq\overline{\beta}<\overline{\gamma}<\overline{\alpha}\leq 1\). The above decision rules P1-N1 is a 3 WD model, then P(11) if \(I\left(X_{\alpha}^{\beta},\theta_{\alpha}^{\beta}(x)\right)\geq\overline{\alpha}\), then \(x\in POS\left(X_{\alpha}^{\beta}\right)\); B(11) if \(\overline{\beta}<I\left(X_{\alpha}^{\beta},\theta_{\alpha}^{\beta}(x)\right)<\overline {\alpha}\), then \(x\in BND\left(X_{\alpha}^{\beta}\right)\); N(11) if \(I\left(X_{\alpha}^{\beta},\theta_{\alpha}^{\beta}(x)\right)\leq\overline{\beta}\), then \(x\in NEG\left(X_{\alpha}^{\beta}\right)\). (ii) if \(\overline{\beta}\geq\overline{\alpha}\), then  
\[ \begin{array}{l}\frac{(1-\omega)\left(\mu_{\alpha}^{2}(\lambda_{BP})-\mu_{ \alpha}^{2}(\lambda_{BP})\right)+\omega\left(\
u_{\alpha}^{2}(\lambda_{BP})- \
u_{\alpha}^{2}(\lambda_{BP})\right)}{(1-\omega)\left(\mu_{\alpha}^{2}( \lambda_{PN})-\mu_{\alpha}^{2}(\lambda_{BN})\right)+\omega\left(\
u_{\alpha}^ {2}(\lambda_{BN})-\
u_{\alpha}^{2}(\lambda_{BN})\right)}\\ \geq\frac{(1-\omega)\left(\mu_{\alpha}^{2}(\lambda_{NP})-\mu_{\alpha}^{2}( \lambda_{BP})\right)+\omega\left(\
u_{\alpha}^{2}(\lambda_{BP})-\
u_{\alpha}^ {2}(\lambda_{NP})\right)}{(1-\omega)\left(\mu_{\alpha}^{2}(\lambda_{BN})-\mu_{ \alpha}^{2}(\lambda_{NN})\right)+\omega\left(\
u_{\alpha}^{2}(\lambda_{NN})- \
u_{\alpha}^{2}(\lambda_{BN})\right)}\end{array} \]  
In this case, \(0\leq\overline{\alpha}<\overline{\gamma}<\overline{\beta}\leq 1\). The above decision rules P1-N1 degenerate into two-way decision models P12 and N12, which are special cases of the 3 WD model. P(12) if \(I\left(X_{\alpha}^{\beta},\theta_{\alpha}^{\beta}(x)\right)\geq\overline{\gamma}\), then \(x\in POS\left(X_{\alpha}^{\beta}\right)\); N(12) if \(I\left(X_{\alpha}^{\beta},\theta_{\alpha}^{\beta}(x)\right)\leq\overline{\gamma}\), then \(x\in NEG\left(X_{\alpha}^{\beta}\right)\). Obviously, the Pyhagorean fuzzy decision rough set model with multiple risk preference information can effectively derive the analytic solutions of probability thresholds \(\overline{\alpha}\), \(\overline{\beta}\) and \(\overline{\gamma}\), and then directly obtain their numerical solutions and three-way decision rules. Therefore, the model fully considers the decision maker's risk preference. It can better describe the decision maker's three-way decision rules under different risk preference information and then make the evaluation decision result of minimizing the decision cost.