## 3 The Notion of the Proportional Dual Hesitant Fuzzy Set  
Inspired by the idea behind proportional hesitant fuzzy set, this section puts forward a novel concept called PDHFS to model and compute the group decision making (GDM) problems with more care and produce more accurate results. The basic operations and properties are also studied in this section. **Definition 8**. : _Let \(X\) be a universe of discourse, and a proportional dual hesitant fuzzy set (PDHFS) on \(X\) is a triplet described as_  
\[S=\left\{\langle x,ph(x),qg(x)\rangle\mid x\in X\right\}=\left\{\langle x, \left\{\left(h(x),p(x)\right)\right\},\left\{\left(g(x),q(x)\right)\right\} \rangle\mid x\in X\right\}, \tag{7}\]  
_in which \(h(x)=\left\{\gamma_{1},\gamma_{2},0...,\gamma_{n}\right\}\) and \(g(x)=\left\{\eta_{1},\eta_{2},0...,\eta_{m}\right\}\) are two sets in \([0,1]\), presenting \(n\) kinds of possible membership and \(m\) kinds of possible non-membership degrees of the element \(x\in X\) to the set \(S\), respectively, and \(p(x)=\left\{\tau_{1},\tau_{2},0...,\tau_{n}\right\}\), \(q(x)=\left\{\tau_{1}^{\prime},\tau_{2}^{\prime},0...,\tau_{m}^{\prime}\right\}\) are sets in \([0,1]\), presenting the proportion of membership degrees and non-membership degrees respectively, with the conditions:_  
\[\sum_{i}^{n}\tau_{i}=1,\;\sum_{j}^{m}\tau_{j}^{\prime}=1;\]  
\[0\leq\gamma^{+}+\eta^{+}\leq 1;\]  
_where \(\gamma^{+}=\bigcup_{\gamma\in h(x)}\max\{\gamma\}\) and \(\eta^{+}=\bigcup_{\eta\in g(x)}\max\{\eta\}\)._  
For convenience and simplicity, we call the pair \(pd(x)=\langle ph(x),qg(x)\rangle\) as a proportional dual hesitant fuzzy element (PDHFE) and also will be written \(pd=\langle ph,qg\rangle\) in short. For clarification, we provide the following example. **Example 1**. : _Let \(X=\left\{x_{1},x_{2},x_{3}\right\}\). Consider_  
\[S=\bigg{\{}\left\langle x_{1},\left\{\left(0.4,0.8\right),\left(0.5,0.2\right) \right\},\left\{\left(0.5,1\right)\right\}\right\rangle,\langle x_{2},\left\{ \left(0.4,0.3\right),\left(0.6,0.5\right),\left(0.7,0.2\right)\right\},\left\{ \left(0.3,0.6\right),\left(0.2,0.4\right)\right\}\rangle\right.\]  
\[\left.,\langle x_{3},\left\{\left(0.4,0.3\right),\left(0.5,0.4\right),\left(0. 3,0.3\right)\right\},\left\{\left(0.5,0.4\right),\left(0.4,0.6\right)\right\} \rangle\right.\bigg{\}}.\]  
_Then, it is easy to check the conditions of PDHFS and_  
\(\langle x_{1},\left\{\left(0.4,0.8\right),\left(0.5,0.2\right)\right\},\left\{ \left(0.5,1\right)\right\}\rangle,\)  
\(\langle x_{2},\left\{\left(0.4,0.3\right),\left(0.6,0.5\right),\left(0.7,0.2 \right)\right\},\left\{\left(0.3,0.6\right),\left(0.2,0.4\right)\right\}\rangle\)_,_  
\(\langle x_{3},\left\{\left(0.4,0.3\right),\left(0.5,0.4\right),\left(0.3,0.3 \right)\right\},\left\{\left(0.5,0.4\right),\left(0.4,0.6\right)\right\}\rangle\) _are PDHFEs._  
**Remark 2**. : _Particularly, if the proportion values of all terms in membership part and non-membership part become equal, then PDHFS reduces to DHFS. Then, in that case, PDHFS reduces to HFS. Thus, the definition of PDHFS encompasses these fuzzy set extensions._  
In order to illustrate the PDHFS more straightforwardly, in the following, a practical life example is provided to depict the difference between PDHFS and DHFS:  
**Example 2**. : _Take the evaluation of a university teacher based on the attribute/criteria like research skill as an example. Three experts provide the following evaluation values to measure his performance._  
\[\left(\mu_{1},\
u_{1}\right)=\left(0.6,0.4\right),\left(\mu_{2},\
u_{2} \right)=\left(0.3,0.4\right)\text{and}\left(\mu_{3},\
u_{3}\right)=\left(0.6,0.2\right)\]in terms of DHFS the above evaluation data can be expressed as_  
\[E=\left\{\{0.6,0.3\}\{0.4,0.2\}\right\}.\]  
_The proportion of the membership value \(0.6\) is \(\frac{2}{3}\). Similarly, we can find the proportion of all other membership and non-membership values.