## 3 Preliminaries  
### Pythagorean Uncertain Linguistic Sets  
The PULSs were proposed by [PERSON] et al. [26] based on uncertain linguistic term sets [36] and Pythagorean fuzzy sets [37] to character the fuzziness and uncertainty in practical situations. **Definition 1**[38]. : _Let \(S=\left\{s_{0},s_{1},\ldots,s_{g}\right\}\) be a finite and ordered linguistic term set, in which \(s_{i}\) indicates a possible value for a linguistic variable and \(g\) is an even number. In general, the linguistic term set has the following properties:_  
1. _The set is ordered:_ \(s_{i}>s_{j}\)_, if_ \(i>j\)_;_
2. _Negation operator:_ \(neg(s_{i})=s_{g-i}\)_;_
3. _Max operator:_ \(max\big{(}s_{i},s_{j}\big{)}=s_{i}\)_, if_ \(i\geq j\)_;_
4. _Min operator:_ \(min\big{(}s_{i},s_{j}\big{)}=s_{i}\)_, if_ \(i\leq j\)_._  
To preserve all given linguistic information, the discrete linguistic term set S was extended by [PERSON] [39] to a continuous form \(\bar{S}=\left\{s_{i}|i\in[0,\xi]\right\}\), in which \(\xi(\xi>g)\) is a sufficiently large positive inte  
**Definition 2**[40]. : _Let \(\widetilde{s}=[s_{\theta},s_{\tau}]\), \(\theta\leq\tau\) and \(s_{\theta},s_{\tau}\in\widetilde{S}\), where \(s_{\theta}\) is the lower limit of \(\widetilde{s}\), and \(s_{\tau}\) is the upper limit of \(\widetilde{s}\), so \(\widetilde{s}\)is called an uncertain linguistic variable._  
**Definition 3**[26]. : _Let \(X\) be a nonempty set of the universe, then a PULS \(A\) in \(X\) is expressed by_  
\[A=\left\{\left\langle x_{i}\mid\left(\widetilde{s}(x_{i}),\left(\mu_{p}(x_{i}),v_{p}(x_{i})\right)\right)\right\rangle\mid x_{i}\in X\right\} \tag{1}\]  
_where \(\widetilde{s}(x_{i})=\left[s_{\theta(x_{i})},s_{\tau(x_{i})}\right]\) is an uncertain linguistic variable, \(\mu_{p}(x)\) and \(v_{p}(x)\) represent the degrees of membership and nonmembership of the element \(x_{i}\in X\) to \(\widetilde{s}(x_{i})\), respectively; \(\mu_{p}(x)\in[0,1]\), \(v_{p}(x)\in[0,1]\), and for every \(x_{i}\in X\), we have \(0\leq u_{p}^{2}(x_{i})+v_{p}^{2}(x_{i})\leq 1\). The degree of indeterminacy can be computed by \(\pi_{p}(x_{i})=\sqrt{1-\left(u_{p}(x_{i})\right)^{2}-\left(v_{p}(x_{i})\right) ^{2}}\). For convenience, \(\widetilde{\alpha}=\left\langle\left[s_{\theta(x)},s_{\tau(x)}\right],\left( \mu(\alpha),v(\alpha)\right)\right\rangle\) is named as a Pythagorean uncertain linguistic variable (PULV)._  
**Definition 4**[41]. : _For any two PULVs \(\widetilde{\alpha}=\left\langle\left[s_{\theta(\alpha)},s_{\tau(\alpha)} \right],\left(\mu(\alpha),v(\alpha)\right)\right\rangle\) and \(\widetilde{\beta}=\left\langle\left[s_{\theta(\beta)},s_{\tau(\beta)}\right], \left(\mu(\beta),v(\beta)\right)\right\rangle\), the basic operations of \(\widetilde{\alpha}\) and \(\widetilde{\beta}\) are defined as follows:_  
(1) \(\widetilde{\alpha}\oplus\widetilde{\beta}=\left\langle\left[s_{\theta(\alpha )+\theta(\beta)},s_{\tau(\alpha)+\tau(\beta)}\right],\left(\sqrt{\mu^{2}( \alpha)+\mu^{2}(\beta)-\mu^{2}(\alpha)\mu^{2}(\beta)},v(\alpha)v(\beta)\right)\right\rangle\)_;_  
(2) \(\widetilde{\alpha}\otimes\widetilde{\beta}=\left\langle\left[s_{\theta(\alpha )\times\tau(\beta)}\right],\left(\mu(\alpha)\mu(\beta),\sqrt{v^{2}(\alpha)+v ^{2}(\beta)-v^{2}(\alpha)v^{2}(\beta)}\right)\right\rangle\)_;_  
(3) \(\lambda\widetilde{\alpha}=\left\langle\left[s_{\lambda\times\theta(\alpha)},s _{\lambda\times\tau(\beta)}\right],\left(\sqrt{1-\left(1-\mu^{2}(\alpha) \right)^{\lambda}},\left(v(\alpha)\right)^{\lambda}\right)\right\rangle\)_;_  
(4) \(\widetilde{\alpha}^{\lambda}=\left\langle\left[s_{\left(\theta(\alpha)\right) ^{\lambda}},s_{\left(\tau(\widetilde{\beta})\right)^{\lambda}}\right],\left( \left(\mu(\alpha)\right)^{\lambda},\sqrt{1-\left(1-v^{2}(\alpha)\right)^{ \  
**Definition 8**[42]. : _Let \(\widetilde{\alpha}_{i}=\Big{\langle}\Big{[}s_{\theta(a_{i})},s_{\tau(a_{i})}\Big{]},\big{(}\mu(\alpha_{i}),v(\alpha_{i})\big{)}\big{\rangle}(i=1,2,\ldots,n)\) be a collection of PULVs. Then, the Pythagorean uncertain linguistic weighted averaging (PULWA) operator is defined as:_  
\[