## 5 A Group Decision-Making Approach for Proportional Dual Hesitant Fuzzy Information
### Algorithm Based on Proportional Dual Hesitant Fuzzy MAGDM  
The multi attribute group decision making (MAGDM) problem under the proportional dual hesitant fuzzy environment is described as follows:  
Consider the MAGDM problem in which there is a set of \(m\) alternatives \(x=\{x_{1},x_{2},0...,x_{m}\}\) and a set of attributes \(c=\{c_{1},c_{2},0...,c_{n}\}\). Furthermore, let there be '\(l\)' DMs \(m=\{m_{1},m_{2},0...,m_{l}\}\). The DMs offer their evaluation values in the form of classical intuitionistic fuzzy numbers (IFNs) keeping in mind all the alternatives and attributes and thus \(k\) evaluation matrices \(R_{k}=\left[\left\langle{{{\mu}_{ij}^{k}},{{\
u}_{ij}^{k}}}\right\rangle \right]_{m\times n};k=1,2,0..,l\) are constructed with the classical IFNs. Then, the overall evaluation matrix \(R\) is computed as follows:  
\[R=\left[{{{pd}_{ij}}}\right]_{m\times n}=\left[\left\langle{\left\{{{h}_{ij}}, {{p}_{ij}}\right\}},{\left\{{{g}_{ij}},{{q}_{ij}}\right\}}\right\rangle\right], \tag{12}\]  
where  
\[{{h}_{ij}}=\bigcup\limits_{k=1}^{l}{{{\mu}_{ij}^{k}},{{g}_{ij}}}=\bigcup\limits _{k=1}^{l}{{{\
u}_{ij}^{k}}}.\]  
Furthermore, \({{p}_{ij}}\) and \({{q}_{ij}}\) contain the proportional (repetition) values of the members of \({{h}_{ij}}\) and \({{g}_{ij}}\), respectively, for \(i=1,2,0...,m,j=1,2,0...,n\).  
It is noted that the overall values of \(R\) are PDHFEs.  
**Definition 17**.: _Let \(R=\left[{{{pd}_{ij}}}\right]_{m\times n}\) be the overall evaluation matrix. Then, \(R^{+}=\left\{{{{pd}_{1}^{+}},{{pd}_{2}^{+}},0...,{{pd}_{n}^{+}}}\right\}\) is referred as the positive ideal solution (PIS) of alternatives, where_  
\[{{pd}_{j}^{+}}=\left\{{\begin{array}{*{20}{c}}{{\left\langle{{\max}_{1\leq i \leq m}\{{pd}_{ij}}}\right\rangle}}&\mbox{for the benefit criteria }c_{j};j=1,2,0...,n\\ \\ {{\left\langle{{\min}_{1\leq m}\{{pd}_{ij}}}\right\rangle}}&\mbox{for the cost criteria }c_{j};j=1,2,0...,n.\end{array}}\right.\]  
**Definition 18**.: _Let \(R=\left[{{{pd}_{ij}}}\right]_{m\times n}\) be the overall evaluation matrix. Then, \(R^{-}=\left\{{{{pd}_{1}^{-}},{{pd}_{2}^{-}},0...,{{pd}_{n}^{-}}}\right\}\) is referred to as the negative ideal solution (NIS) of alternatives, where_  
\[{{pd}_{j}^{-}}=\left\{{\begin{array}{*{20}{c}}{{\left\langle{{\min}_{1\leq i \leq m}\{{pd}_{ij}}}\right\rangle}}&\mbox{for the benefit criteria }c_{j};j=1,2,0...,n\\ \\ {{\left\langle{{\max}_{1\leq i\leq m}\{{pd}_{ij}}}\right\rangle}}&\mbox{for the cost criteria }c_{j};j=1,2,0...,n.\end{array}}\right.\]  
It is noted that, in the above definitions, max and min are taken based on Definition 7.  
**Definition 19**.: _The separation of each alternative from the PIS is defined with the help of proportional dual hesitant normalized Hamming distance as follows:_  
\[{{s}_{i}^{+}}={{d}_{{pd}th}}\left({{R}^{+}},{{R}_{i}}\right),i=1,2,0...,m, \tag{13}\]  
_where \(R_{i}=\left\{{{{pd}_{i1}},{{pd}_{i2}},0...,{{pd}_{in}}}\right\}\)._  
**Definition 20**.: _The separation of each alternative from the NIS is defined as follows:_  
\[{{s}_{i}^{-}}={{d}_{{pd}th}}\left({{R}^{-}},{{R}_{i}}\right),i=1,2,0...,m, \tag{14}\]  
_where \(R_{i}=\left\{{{{pd}_{i1}},{{pd}_{i2}},0...,{{pd}_{in}}}\right\}\)._  
Clearly, the smaller the distance \({{d}_{{pd}th}}\left({{R}^{+}},{{R}_{i}}\right)\) the better the alternative \(x_{i}\) and the larger distance \({{d}_{{pd}th}}\left({{R}^{-}},{{R}_{i}}\right)\) implies the better alternative \(x_{i}\).  
After obtaining the separations of alternatives, the closeness coefficient are calculated by the following formula to rank all the alternatives:  
\[cl(x_{i})=\frac{s_{i}^{-}}{s_{i}^{+}+s_{i}^{-}};i=1,2,0...,m, \tag{15}\]  
where \(cl(x_{i})\leq 0;(i=1,2,0...,m)\) represents the closeness coefficient of each alternative. The larger the \(cl(x_{i})\), the better the alternative \(x_{i}\). Therefore, all the alternatives can be ranked by using the closeness coefficient and thus the best alternative  
\[x^{b}=\left\{x_{i}\mid\max_{1\leq i\leq m}cl(x_{i})\right\} \tag{16}\]  
can be chosen.