## 3 The Notion of the Proportional Dual Hesitant Fuzzy Set
The overall evaluation can be represented by PDHFE as follows:_  
\[pd=\left\langle\left\{(0.6,\frac{2}{3}),(0.3,\frac{1}{3})\right\},\left\{(0.2, \frac{1}{3}),(0.4,\frac{2}{3})\right\}\right\rangle.\]  
_In the DHFE, a value repeated more times has no more importance than less repeated values, which is unrealistic because a higher repeated value should be more important in aggregating DMs' opinions than the other values. Thus, DHFE fails to express the evaluation of a teacher more reasonably. Therefore, to get more reasonable results, it is better that the experts provide their evaluations in terms of PDHFEs because PDHFE gives importance to the repeated values._  
**Definition 9**. : _Let \(X\) be a universe of discourse, for any \(x\in X\), we call:_  
1. \(pd(x)=\left\langle\left\{\left(0,1\right)\right\},\left\{\left(1,1\right)\right\}\right\rangle\) _as the empty proportional dual hesitant fuzzy element (EPDHFE)._ _A PDHFS whose all PDHFEs are EPDHFEs is called an empty proportional dual hesitant fuzzy set (EPDHFS), denoted by_ \(\Theta\)_;_
2. \(pd(x)=\left\langle\left\{\left(1,1\right)\right\},\left\{\left(1,1\right)\right\}\right\rangle\) _as the fully proportional dual hesitant fuzzy element (FPDHFE)._ _A PDHFS whose all PDHFEs are FPDHFEs is called fully proportional dual hesitant fuzzy set (FPDHFS), denoted by_ \(F\)_._  
**Definition 10**. : _For a given PDHFE \(pd=\left\langle\left\{ph\right\},\left\{qg\right\}\right\rangle\), its complement is defined as follows:_  
\[pd^{c}=\left\{ \begin{array}{ll}\bigcup_{(\gamma,\tau_{\tau})\in ph,(\eta, \tau_{\eta}^{\prime})\in qg}\left\langle\left\{\left(\eta,\tau_{\eta}^{ \prime}\right)\right\},\left\{\left(\gamma,\tau_{\eta}\right)\right\}\right\rangle& \text{if }h\
eq\emptyset,g\
eq\emptyset\\ \bigcup_{(\gamma,\tau_{\tau})\in ph}\left\{\left(1-\gamma,\tau_{\gamma} \right),\emptyset\right\rangle&\text{if }h\
eq\emptyset,g=\emptyset\\ \bigcup_{(\eta,\tau_{\eta}^{\prime})\in qg}\left\langle\emptyset,\left\{\left( 1-\eta,\tau_{\eta}^{\prime}\right)\right\}\right\rangle&\text{if }h=\emptyset,g\
eq\emptyset.\end{array} \right. \tag{8}\]  
**Proposition 1**. : _The complement is involuative, i.e., \(\left(pd^{c}\right)^{c}=pd\)_  
Proof. : It trivially follows from the Definition 10. Next, the union and intersection of PDHFSs are defined which are based on the corresponding lower and upper bounds of membership and non-membership PHFSs. For any two PDHFSs say, \(pd_{1}\) and \(pd_{2}\), one should note that \(ph^{-}=\bigcup_{(\gamma,\tau)\in ph}\min\left\{\gamma\cdot\tau\right\}\), \(ph^{+}=\bigcup_{(\gamma,\tau)\in ph}\max\left\{\gamma\cdot\tau\right\}\), \(qg^{-}=\bigcup_{(\eta,\tau)\in qg}\min\left\{\eta\cdot\tau^{\prime}\right\}\) and \(qg^{+}=\bigcup_{(\eta,\tau^{\prime})\in qg}\max\left\{\eta\cdot\tau^{\prime}\right\}\) represent the corresponding lower and upper bounds to \(ph\) and \(qg\), respectively. **Definition 11**. : _Let \(X\) be a universe of discourse and \(pd_{1}\), \(pd_{2}\) be two PDHFEs. Then,_  
1. \(pd_{1}\bigcup pd_{2}=\)__ \(\left\langle\left\{ph\in(ph_{1}\bigcup ph_{2})\mid ph\geq\max\left(ph_{1}^{-}, ph_{2}^{-}\right)\right\},\left\{qg\in(qg_{1}\bigcap qg_{2})\mid qg\leq\min \left(qg_{1}^{+},qg_{2}^{+}\right)\right\}\right\rangle\)_;_
2. \(pd_{1}\bigcap pd_{2}=\)__ \(\left\langle\left\{ph\in(ph_{1}\bigcap ph_{2})\mid ph\leq\min\left(ph_{1}^{+}, ph_{2}^{+}\right)\right\},\left\{qg\in(qg_{1}\bigcup qg_{2})\mid qg\geq\max \left(qg_{1}^{-},qg_{2}^{-}\right)\right\}\right\rangle\)_._ \(ph_{1}\bigcup ph_{2}\) _and_ \(qg_{1}\bigcap qg_{2}\) _are computed according to Definition 6._Let \(l_{h}(pd)\) and \(l_{g}(pd)\) represent the number of values in \(ph\) and \(qg\), respectively, of a PDHFE \(pd=(ph,qg)\). To prove this, \(l(pd)=\left(l_{h}(pd),l_{g}(pd)\right)\) in most cases \(l(pd_{A})\
eq l(pd_{B})\) i.e., \(l_{h}(pd_{A})\
eq l_{h}(pd_{B}),l_{g}(pd_{A})\
eq l_{g}(pd_{B})\).