## 3 Methodology
### Pythagorean Fuzzy Rough Approximation Construction
The lower approximation \(\underline{\vartheta}^{\overline{\alpha}}\left(X_{\alpha}^{\beta}\right)\) and upper approximation \(\overline{\vartheta}^{\overline{\alpha}}\left(X_{\alpha}^{\beta}\right)\) of \(\left(\overline{\alpha},\overline{\beta}\right)\) under the Pythagorean fuzzy concept \(X_{\alpha}^{\beta}\subseteq U\) are defined as_  
\[\underline{\vartheta}^{\overline{\alpha}}\left(X_{\alpha}^{\beta}\right)= \left\{x\in U\Big{|}I\left(X_{\alpha}^{\beta},\vartheta_{\alpha}^{\beta}(x) \right)\geq\overline{\alpha}\right\} \tag{9}\]  
\[\overline{\vartheta}^{\overline{\beta}}\left(X_{\alpha}^{\beta}\right)= \left\{x\in U\Big{|}I\left(X_{\alpha}^{\beta},\vartheta_{\alpha}^{\beta}(x) \right)>\overline{\beta}\right\} \tag{10}\]  
_where \(I\left(X_{\alpha}^{\beta},\vartheta_{\alpha}^{\beta}(x)\right)=\left|\vartheta _{\alpha}^{\beta}(x)\cap X_{\alpha}^{\beta}\Big{|}\Big{/}\Big{|}\vartheta_{ \alpha}^{\beta}(x)\right|\) and \(|X|\) represents the cardinal numbers of set \(X\)._  
The thresholds \(\overline{\alpha}\) and \(\overline{\beta}\) divide the universe into the positive region \(POS^{\overline{\alpha}}\left(X_{\alpha}^{\beta}\right)\) (accept decision), negative region \(NEG^{\overline{\beta}}\left(X_{\alpha}^{\beta}\right)\) (reject decision), and boundary region \(BND^{(\overline{\alpha},\overline{\beta})}\left(X_{\alpha}^{\beta}\right)\) (delay decision). The rules are as follows. \[POS^{\overline{\alpha}}\left(X_{\alpha}^{\beta}\right)= \underline{\vartheta}^{\overline{\alpha}}\left(X_{\alpha}^{\beta}\right)= \left\{x\in U\Big{|}I\left(X_{\alpha}^{\beta},\vartheta_{\alpha}^{\beta}(x) \right)\geq\overline{\alpha}\right\},\] \[NEG^{\overline{\beta}}\left(X_{\alpha}^{\beta}\right)=U-\overline {\vartheta}^{\overline{\beta}}\left(X_{\alpha}^{\beta}\right)=\left\{x\in U \Big{|}I\left(X_{\alpha}^{\beta},\vartheta_{\alpha}^{\beta}(x)\right)\leq \overline{\beta}\right\},\] \[BND^{(\overline{\alpha},\overline{\beta})}\left(X_{\alpha}^{\beta} \right)=\overline{\vartheta}^{\overline{\beta}}\left(X_{\alpha}^{\beta}\right) -\underline{\vartheta}^{\overline{\alpha}}\left(X_{\alpha}^{\beta}\right)= \left\{x\in U\Big{|}\overline{\beta}<I\left(X_{\alpha}^{\beta},\vartheta_{ \alpha}^{\beta}(x)\right)<\overline{\alpha}\right\}.\]  
**Property 4**. : _Let \((U,\vartheta)\) be a Pythagorean fuzzy approximation universe, \(X_{\alpha_{1}}^{\beta_{1}},X_{\alpha_{2}}^{\beta_{2}}\subseteq U\), where \(0\leq\alpha_{1}+\beta_{1}\leq 1\) and \(0\leq\alpha_{2}+\beta_{2}\leq 1\). The \(\left(\overline{\alpha},\overline{\beta}\right)\)-lower approximation \(\underline{\vartheta}^{\overline{\alpha}}\left(X_{\alpha_{1}}^{\beta_{1}}\right)\) and the \(\left(\overline{\alpha},\overline{\beta}\right)\)-upper approximation \(\overline{\vartheta}^{\overline{\beta}}\left(X_{\  
risk losses by the three different actions, respectively, when the object does not belong to \(X_{\mathbf{a}}^{\beta}\); the detailed information is shown in Table 1.  
Different decision-makers tend to adopt different risk attitudes in the actual decision-making process, which is inseparable from their personality characteristics. Some are optimistic, some are pessimistic, and some are neutral. The above risk loss matrix by decision-makers with three risk attitudes can be unified into one model. Therefore, we introduce the risk factor \(\omega\) to represent different risk attitudes of decision-makers, where \(0\leq\omega\leq 1\). The definition is as follows:  
**Definition 10**. : _In Table 1, let \(A\left(\lambda_{ij}\right)=\left(\mu_{A}(\lambda_{ij}),\
u_{A}(\lambda_{ij})\right)\) be the Pythagorean fuzzy loss values for decision-makers taking different actions in two states, where \(i=P,B,N;j=P,N\). \(\omega\) is the risk factor of decision-makers.