## 2 Literature Review and Preliminaries
### Dual Hesitant Fuzzy Set  
[PERSON] et al. [13], generalized HFS and defined a DHFS in terms of two functions that return two sets of membership values and non-membership values respectively for every element in the domain as follows:  
**Definition 2** ([13]).: _Let \(X\) be a fixed set, and then a DHFS on \(X\) is described by the following mathematical symbol:_  
\[D=\left\{\langle x,h(x),g(x)\rangle\mid x\in X\right\}, \tag{2}\]  
_in which \(h(x)\) and \(g(x)\) are two sets of some possible values in \([0,1]\), denoting the membership degrees and the non-membership degrees, respectively, of the element \(x\in X\) to the set \(D\), satisfying_  
\[0\leq\gamma^{+}+\eta^{+}\leq 1,\]  
_where \(\gamma^{+}=\bigcup_{\gamma\in h(x)}\max\{\gamma\}\), and \(\eta^{+}=\bigcup_{\eta\in g(x)}\max\{\eta\}\)._  
For convenience, \(d(x)=(h(x),g(x))\) is called a dual hesitant fuzzy element (DHFE) denoted by \(d=(h,g)\).  
[PERSON] et al. [23] proposed the following simple operational laws for DHFEs.  
**Definition 3** ([23]).: _Let \(d=(h_{d},g_{d}),d_{1}=(h_{d_{1}},g_{d_{1}})\) and \(d_{2}=(h_{d_{2}},g_{d_{2}})\) be three DHFEs such that \(l(h_{d})=l(h_{d_{1}})=l(h_{d_{2}})\) and \(l(g_{d})=l(g_{d_{1}})=l(g_{d_{2}})\), then:_  
1. \(d\oplus d_{1}=\left(h_{d}\oplus h_{d_{1}},g_{d}\otimes g_{d_{1}}\right),\)__ \(=\left(\{h_{d}^{\sigma(j)}+h_{d_{1}}^{\sigma(j)}-h_{d_{1}}^{\sigma(j)}h_{d_{1 }}^{\sigma(j)}\mid j=1,2,0...,l(h_{d})\},\{g_{d}^{\sigma(t)}g_{d_{1}}^{\sigma( t)}\mid t=1,2,0...,l(g_{d})\}\right),\)__2. \(d\otimes d_{1}=\left(h_{d}\otimes h_{d_{1}},g_{d}\oplus g_{d_{1}}\right),\) \(=\left(\{h_{d}^{\sigma(j)}h_{d_{1}}^{\sigma(j)}\mid j=1,2,0...,l(h_{d})\},\{g_{ d}^{\sigma(t)}+g_{d_{1}}^{\sigma(t)}-g_{d}^{\sigma(t)}g_{d_{1}}^{\sigma(t)}\mid t=1,2,0...,l(g_{d})\}\right),\) 3. \(d^{\alpha}=\left(\{(h_{d}^{\sigma(j)})^{\alpha}\mid j=1,2,0...,l(h_{d})\},\{1-(1 -h_{d}^{\sigma(t)})^{\alpha}\mid t=1,2,0...,l(g_{d})\}\right),\) 4. \(\alpha d=\left(\{1-(1-h_{d}^{\sigma(j)})^{\alpha}\mid j=1,2,0...,l(h_{d})\},\{(g _{d}^{\sigma(t)})^{\alpha}\mid t=1,2,0...,l(g_{d})\}\right),\)  
_where \(\alpha\) is a positive integral, \(l(h_{d})\) and \(l(g_{d})\) denote the number of values in \(h_{d}\) and \(g_{d}\), \(h_{d}^{\sigma(j)}\) is the jth smallest value in \(h_{d}\) and \(g_{d}^{\sigma(t)}\) is the jth smallest value in \(g_{d}\)._  
**Remark 1**.: _Here, it is assumed that all the DHFEs have the same number of values for membership degrees and non-membership degrees. However, if they are different, then we need to equalize them according to a transformation and regulation method [23]._  
**Definition 4**.: _[_24_]_ _Let \(A\) and \(B\) be any dual hesitant fuzzy sets of the same length; then, a dual hesitant normalized Hamming distance between \(A\) and \(B\) is given as:_  
\[d_{dwh}(A,B)=\sum_{i=1}^{n}\left(\frac{1}{n}\left(\frac{1}{l(x_{i})}\left( \sum_{j=1}^{l(h(x_{i}))}\left|\gamma_{A}^{\rho(j)}(x_{i})-\gamma_{B}^{\rho(j)} (x_{i})\right|+\sum_{k=1}^{l(g(x_{i}))}\left|\eta_{A}^{\rho(k)}(x_{i})-\eta_{ B}^{\rho(k)}(x_{i})\right|\right)\right)\right). \tag{3}\]  
\(l(x_{i})=l(h(x_{i}))+l(g(x_{i}));\quad l(h(x_{i}))\) _and \(l(g(x_{i}))\) denote the number of elements in \(h\) and \(g\), respectively, \(\gamma_{A}^{\rho(j)}(x_{i})\) is the jth largest value in \(h_{A}(x_{i})\) and \(\eta_{A}^{\rho(k)}(x_{i})\) is the kth largest value in \(g_{A}(x_{i})\)._