# Multiple Attribute Decision-Making Method Using Linguistic Cubic Hesitant Variables
## 5 Illustrative Example
\begin{array}{l}\left[y_{8-8\times(1-4/8)^{6\times 3}(1-4/8)^{ 6\times 3}(1-4/8)^{6\times 3}(1-4/8)^{6\times 3}(1-4/8)^{6\times 3}(1-6/8)^{6 \times 3}(1-7/8)^{6\times 2}}\right],\\ \left(\begin{array}{l}y_{8-8\times(1-5/8)^{6\times 3}(1-4/8)^{6\times 3}(1-5/8)^{6 \times 3}(1-4/8)^{6\times 3}(1-4/8)^{6\times 3}(1-5/8)^{6\times 3}(1-5/8)^{6 \times 3}(1-6/8)^{6\times 3}(1-5/8)^{6\times 3}}\\ y_{8-8\times(1-6/8)^{6\times 3}(1-6/8)^{6\times 3}(1-6/8)^{6\times 3}(1-7/8)^{6 \times 3}(1-6/8)^{6\times 3}(1-6/8)^{6\times 3}(1-6/8)^{6\times 3}} \end{array}\right)\end{array} \right)\] \[=(\left[y_{4,000},y_{6,2589}\right],\left\{y_{4,6822},y_{4,6822},y_ {5,3969},y_{6,000},y_{6,4308},y_{6,4308}\}).\]  
Using a similar calculation, we can obtain the aggregated LCHNs \(z_{i}\) for \(Z_{i}\) (\(i=2,3,4\)):  
\[z_{2}=([y_{4,011},y_{6,1167}],\,y_{4,8617},\,y_{5}2427,\,y_{5} 957,\,y_{6}2983,\,y_{6}2983)),\] \[z_{3}=([y_{5,3969},y],\,\{y_{4,8223},\,y_{5}2337,\,y_{6},\,y_{6} 2589,\,y_{6}2589),\,\},\] \[z_{4}=([y_{5,3523},y_{6}8813],\,\{y_{5}2337,\,y_{5}2337,\,y_{5}695, \,y_{6}340,\,y_{7},\,y\}).\]  
**Step 3. ** Through Equation (2), the linguistic score value of \(y_{L(z_{1})}\) is given below:  
\[y_{L(z_{1})}=y_{\frac{1}{2}(\frac{1}{2}(a_{1}+\beta_{1})+\frac{1}{6}\sum_{k=1}^ {6}\lambda_{1k})}=y_{\frac{1}{2}(\frac{1}{2}(4+6.2589)+\frac{1}{6}(4.6822+4.6822+ 5.3969+6+6.4308+6.4308))}=y_{5,3666}\]  
Through a similar calculation, we can obtain the linguistic score values of \(y_{L(z_{i})}\)(\(i=2,3,4\)):  
\[y_{L(z_{2})}=y_{5,3228},\,y_{L(z_{3})}=y_{5,8822},\,y_{L(z_{4})}=y_{6,1173}.\]  
**Step 4. ** The four candidates are ranked as \(Z_{4}\)\(\succ\)\(Z_{3}\)\(\succ\)\(Z_{1}\)\(\succ\)\(Z_{2}\) based on the linguistic score values. Thus, \(Z_{4}\) is the best one among them. For another application, we can also use the presented MADM method based on the WGA operator of LCHVs for the example and indicate the following decision steps:  
**Step 1. ** The same as Step 1. **Step 2. ** Through Equation (14), the aggregated LCHN \(z_{1}\) for \(Z_{1}\) is yielded as follows:  
\[z_{1}=G_{LCHV}(z_{11}^{e},z_{12}^{e},z_{13}^{e})=\prod_{j=1}^{3}z_{1j}^{\,\,\, \omega_{j}}\]  
\[\left( \begin{array}{l}\left[y_{8\times(4/8)^{0.45}\times(4/8)^{0.35}\times(4/8)^{0. 22},y_{8\times(6/8)^{0.45}\times(6/8)^{0.35}\times(7/8)^{0.2}}\right],\\ \left\{\begin{array}{l}y_{8\times(5/8)^{0.45}\times(4/8)^{0.35}\times(5/8)^{0. 22}},y_{8\times(5/8)^{0.45}\times(4/8)^{0.35}\times(5/8)^{0.22}},y_{8\times(5/8)^ {0.45}\times(6/8)^{0.35}\times(5/8)^{0.2}},\\ y_{8\times(6/8)^{0.45}\times(6/8)^{0.35}\times(6/8)^{0.22}},y_{8\times(6/8)^{0.45} \times(7/8)^{0.35}\times(6/8)^{0.2}},y_{8\times(6/8)^{0.45}\times(7/8)^{0.35} \times(6/8)^{0.2}}\end{array}\right\}\end{array} \right)\]  
Using a similar calculation, we can obtain the aggregated LCHNs \(z_{i}\) for \(Z_{i}\) (\(i=2,3,4\)):  
\[z_{2}=([y_{3,3798},y_{5,8383}],\,\{y_{4,6099},y_{4,6099},y_{5,0068},y_{5,6}249,y_{ 6,109},y_{6,109})\},\] \[z_{3}=([y_{5,3295},y],\,\{y_{4,781},\,y_{4,7818},\,y_{5}1857,\,y_{6},y_{6}1859),\,\},\] \[z_{4}=([y_{5,109},y_{6,787}],\,\{y_{5}1857,\,y_{5}1857,\,y_{5}2591,\,y_{ 5}309,\,y,\,y\,y\}).\]  
**Step 3. ** Through Equation (2), the linguistic score value of \(y_{L(z_{1})}\) is given below:  
\[\mathcal{Y}_{L(z_{1})}=y_{\frac{1}{2}\left[\frac{1}{2}(\alpha_{1}+\beta_{4})+ \frac{1}{6}\sum_{k\
eq 4}^{k}\lambda_{4}\right]}=\mathcal{Y}_{\frac{1}{2}\left[ \frac{1}{2}(4+6.1879)+\frac{1}{6}(4.6244+4.6244+5.3295+6+6.3326+6.3326)\right]} =y_{5,3172}\]  
Using a similar calculation, we can obtain the linguistic score values of \(y_{L(z_{1})}\) (\(i=2\), \(3\), \(4\)):  
\[\mathcal{Y}_{L(z_{2})}\ =y_{5,0879},\ \ \mathcal{Y}_{L(z_{3})}\ =y_{5, \text{s.s.ess}},\ \text{and}\ \ \mathcal{Y}_{L(z_{4})}\ =y_{6,0388}.\]  
**Step 4. ** The four candidates are ranked as \(Z_{4}\succ Z_{3}\succ Z_{1}\succ Z_{2}\). Thus, \(Z_{4}\) is still the best one among them. Clearly, the above two ranking orders based on the WAA and WGA operators of LCHVs and the best candidate are identical in this MADM problem.