## 4 The Comparison Notions between PDHFSs Based on PDHFEs  
A distance measure is designed for determining the degree of distance between two sets in the fuzzy set theory, which has received attention from scholars [25; 26; 27]. This section first defines the distance between two PDHFSs. Based on it, the comparison law is developed for PDHFSs. Based on Definition 4, we come up with the following definition. **Definition 14**. : _Let \(A\) and \(B\) be any two PDHFSs of equal length on \(X=\left\{x_{1},x_{2},0...,x_{n}\right\}\); then, the proportional dual hesitant normalized Hamming distance is_  
\[\begin{array}{l}d_{pathh}\left(A,B\right)=\sum_{i=1}^{n}\left(\frac{1}{n} \left(\frac{1}{2I(x_{i})}\left(\sum_{j=1}^{l(kx_{i})}\left(\left|\tau_{A}^{ \rho(j)}(x_{i})\tau_{A}^{\rho(j)}(x_{i})-\tau_{B}^{\rho(j)}(x_{i})\tau_{B}^{ \rho(j)}(x_{i})\right|+\left|\tau_{A}^{\rho(j)}(x_{i})-\tau_{B}^{\rho(j)}(x_{i} )\right|\right)\right.\right.\\ \left.\left.+\sum_{k=1}^{l(gx_{i})}\left(\left|\eta_{A}^{\rho(k)}(x_{i})\tau_{ A}^{\rho(k)}(x_{i})-\eta_{B}^{\rho(k)}(x_{i})\tau_{B}^{\rho(k)}(x_{i})\right|+ \left|\tau_{A}^{\rho(k)}(x_{i})-\tau_{B}^{\rho(k)}(x_{i})\right|\right)\right) \right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \right) \tag{9}\]  
_where \(l(x_{i})=l(hx_{i})+l(gx_{i})\), \(l(hx_{i})\) and \(l(gx_{i})\) represent the number of elements in \(h\) and \(g\), respectively, \(\gamma_{A}^{\rho(j)}(x_{i})\tau_{A}^{\rho(j)}(x_{i})\) is the \(j\)th largest product value in \(ph_{A}(x_{i})\) and \(\eta_{A}^{\rho(k)}(x_{i})\tau_{A}^{\rho(k)}(x_{i})\) is the \(k\)th largest product value in \(qg_{A}(x_{i})\)._  
One can check easily that the following properties hold for \(d_{pathh}\). \(0\leq d\left(A,B\right)\leq 1\);  
\(d\left(A,B\right)=0\) if and only if \(A=B\);  
\(d\left(A,B\right)=d\left(B,A\right)\). For the purpose of ranking alternatives, in the following, we develop the comparison method for PDHFSs. **Definition 15**. : _Let pd be a PDHFE on \(X=\left\{x_{1},x_{2},0...,x_{n}\right\}\); then, the score function of pd is defined as follows:_  
\[s\left(pd\right)=\sum_{\left(\gamma,\tau\right)\in ph}\gamma\tau-\sum_{\left( \eta,\tau^{\prime}\right)\in qg}\eta\tau^{\prime} \tag{10}\]  
_and the deviation function of pd is defined as follows:_  
\[\sigma\left(pd\right)=\left(\sum_{\left(\gamma,\tau\right)\in ph}\tau\left( \gamma-s\left(pd\right)\right)^{2}+\sum_{\left(\eta,\tau^{\prime}\right)\in qg }\tau^{\prime}\left(\eta-s\left(pd\right)\right)^{2}\right)^{\frac{1}{2}}. \tag{11}\]  
Combining with the distance measure, the comparison method for PDHFEs can be defined as follows:  
**Definition 16**. : _Let pd\({}_{l}\)\((l=1,2)\) be two PDHFEs, \(s\left(pd_{l}\right)\)\((l=1,2)\) and \(\sigma\left(pd_{l}\right)\)\((l=1,2)\) be their score and deviation values, respectively._  
1. _if_ \(s\left(pd_{l}\right)>s\left(pd_{2}\right)\)_, then pd_\({}_{l}>pd_{2}\)_; on the contrary, there is pd_\({}_{l}<pd_{2}\)_._
2. _if_ \(s\left(pd_{l}\right)=s\left(pd_{2}\right)\) _and_ \(\sigma\left(pd_{l}\right)<\sigma\left(pd_{2}\right)\)_, then pd_\({}_{l}>pd_{2}\)_._
3. _if_ \(s\left(pd_{l}\right)=s\left(pd_{2}\right),\sigma\left(pd_{l}\right)=\sigma \left(pd_{2}\right)\)_,_  
_(a) and_ \(d_{pathh}\left(\left\{pd_{1},\digamma\right\}\right)=d_{pathh}\left(\left\{ pd_{2},\digamma\right\}\right)\)_, then pd_\({}_{1}=pd_{2}\)_,_  
_(b) and_ \(d_{pathh}\left(\left\{pd_{1},\digamma\right\}\right)<d_{pathh}\left(\left\{ pd_{2},\digamma\right\}\right)\)_, then pd_\({}_{1}>pd_{2}\)_,_  
_where \(\digamma\) is the fully proportional dual hesitant fuzzy set and \(d_{pathh}\left(A,B\right)\) is the distance between \(A\) and \(B\)._  
**Example 3**. : _Let pd\({}_{1}=\left\langle\left\{\left(0.5,0.2\right),\left(0.4,0.8\right)\right\}, \left\{\left(0.5,0.6\right),\left(0.4,0.4\right)\right\}\right\rangle\) and pd\({}_{2}=\left\langle\left\{\left(0.4,0.6\right),\left(0.6,0.4\right)\right\}, \left\{\left(0.4,1\right)\right\}\right\rangle\) be two PDHFEs. Then, \(s(pd_{1})=-0.04\) and \(s(pd_{2})=0.08\).