## 2 Basic Definition of Type-2 Hesitant Pythagorean Fuzzy Sets  
### Basic Concepts of Type-2 Hesitant Pythagorean Fuzzy Sets  
**Definition 1**: ([34; 35] Hesitant fuzzy set (HFS))_. \(X=\{x_{1},x_{2},\cdots,x_{n}\}\) is set as a domain, and the probability hesitant fuzzy set on this domain is defined as follows:_  
\[A=\{\langle x,\mu_{A}(x)\rangle|x\in X\} \tag{1}\]  
_where \(\mu_{A}(x)\) is the non-empty finite subset in interval \([0,1]\), indicating the possible membership of \(x\in X\) about set \(A\). \(\mu=\mu_{A}(x)\) is referred to as the hesitant fuzzy element._  
**Definition 2**: ([36; 37] Pythagorean hesitant fuzzy set (PHFS))_. \(X=\{x_{1},x_{2},\cdots,x_{n}\}\) is set as a domain, and the \(q\)-order hesitant fuzzy set on this domain is defined as below:_  
\[A=\{\langle x,\Gamma_{A}(x),\Psi_{A}(x)\rangle|x\in X\} \tag{2}\]  
_where \(\Gamma_{A}(x)\) and \(\Psi_{A}(x)\) denote the non-empty finite subsets in interval \([0,1]\), \(\Gamma_{A}(x)\) and \(\Psi_{A}(x)\) stands for the possible membership and non-membership of element \(x\in X\), and \(0\leq\mu^{2}+\
u^{2}\leq 1\) for \(\forall x\in X\), \(\forall\mu\in\Gamma_{A}(X)\), and \(\forall\
u\in\Psi_{A}(X)\)._  
**Definition 3**: _Type-2 hesitant Pythagorean fuzzy set (T2 HPFS): \(X=\{x_{1},x_{2},\cdots,x_{n}\}\) is set as a domain, and the Type-2 hesitant Pythagorean fuzzy set on this domain is defined as follows:_  
\[A=\Big{\{}\Big{\langle}x,(\Gamma_{1},\Psi_{1}),(\Gamma_{2},\Psi_{2}),\ldots,( \Gamma_{k_{(1)}},\Psi_{k_{(1)}})\Big{\rangle}\Big{|}x\in X\Big{\}} \tag{3}\]  
_where \(\Gamma_{m}\) and \(\Psi_{m}\) are non-empty finite subsets in interval \([0,1]\), and \(m=1,2,\ldots,k_{(1)}\). \(\mu_{m}^{2}+v_{m}^{2}\leq 1\) for \(\forall x\in X\), \(\forall m\in\Big{\{}1,2,\ldots k_{(1)}\Big{\}}\), \(\forall\mu_{m}\in\Gamma_{m}\), and \(\forall\
u_{m}\in\Psi_{m}\). \(h=<(\Gamma_{1},\Psi_{1}),(\Gamma_{2},\Psi_{2}),\ldots\)\((\Gamma_{k_{(1)}},\Psi_{k_{(1)}})>\) is referred to as a Type-2 hesitant Pythagorean fuzzy element (T2 HPFE). For \(f_{m}=(\Gamma_{m},\Psi_{m})\) and \(m=1,2,\ldots,k_{(1)}\), \(h=<f_{1},f_{2},\ldots f_{k_{(1)}}>_{2}\). \(f_{m}\) is referred to as the first-layer hesitant fuzzy elements of element \(x\) for set \(A\). \(\Gamma_{m}\) and \(\Psi_{m}\) are the sets of possible membership and non-membership of element \(x\) to hesitant fuzzy element \(f_{m}\), respectively. For \(f^{(2)}=\Big{\{}f_{1},f_{2},\ldots,f_{k_{(1)}}\Big{\}}\), \(f^{(2)}\) is called the second-layer hesitant fuzzy element of element \(x\) for set \(A\)._  
**Definition 4**: _Type-n hesitant Pythagorean fuzzy set (TnHPFS): \(X=\{x_{1},x_{2},\cdots,x_{n}\}\) is set as a domain, and the Type-n hesitant Pythagorean fuzzy set on this domain is defined as below:_  
\[A=\Big{\{}x,<f_{1}^{(n-1)},f_{2}^{(n-1)},\ldots,f_{k_{(n-1)}}^{(n-1)}>|x\in X \Big{\}} \tag{4}\]_where \(f_{l}^{(n-1)}\) is the \(n\)-first-layer hesitant fuzzy element of element \(x\) for set \(A\), and \(l=1,2,\ldots,k_{(n-1)}\). \(h=<f_{1}^{(n-1)},f_{2}^{(n-1)},\ldots,f_{k_{(n-1)}}^{(n-1)}>_{q}\) is referred to as the type \(n\) hesitant Pythagorean fuzzy element (TnHPFE), and all ThHPFEs are denoted as TuHPFE (X). For \(f^{(n)}=\left\{f_{1}^{(n-1)},f_{2}^{(n-1)},\ldots,f_{k_{(n-1)}}^{(n-1)}\right\}\), \(f^{(n)}\) is referred to as the \(n\)-layer hesitant fuzzy element of element \(x\) for set \(A\)._