# Multiple Attribute Decision-Making Method Using Linguistic Cubic Hesitant Variables
## 3 Weighted Aggregation Operators of LCHVs  
### Weighted Arithmetic Averaging (WAA) Operator of LCHVs  
**Definition 7**. : _Set \(z_{j}=([y_{\alpha_{j}},y_{\beta_{j}}],\left\{y_{\lambda_{j1}},y_{\lambda_{j2}}, \ldots,y_{\lambda_{jr}}\right\})(j=1,\,2,\,\ldots,\,n)\) as a group of LCHVs in \(Y=\{y_{1}\,l\in[\)0, q\(]\}\), along with its weight \(\omega_{j}\in[0,1]\) for \(\sum_{j=1}^{n}\omega_{j}=1\). The corresponding WAA operator of the LCHVs is expressed using_  
\[F_{LCHV}(z_{1},z_{2},\cdots,z_{n})=\sum_{j=1}^{n}\omega_{j}z_{j}. \tag{3}\]  
Based on Definitions 6 and 7, there exists the following theorem:  
It is noted that we must make the number of linguistic values of each HLS in LCHNs equal based on the LCMN extension method before using Equation (4) in some cases so as to reach the suitable aggregation operation. The following example is given to illustrate the real operational process:  
**Example 3**. : _Set \(z_{1}=\) ([\(y_{5}\), \(y_{6}\)], [\(y_{4}\), \(y_{6}\)]), \(z_{2}=\) ([\(y_{4}\), \(y_{6}\)], [\(y_{3}\), \(y_{5}\), \(y_{6}\)]), and \(z_{3}=\) ([\(y_{3}\), \(y_{5}\)], [\(y_{3}\), \(y_{4}\), \(y_{5}\)]) as three LCHNs in the LTS \(Y=\) {\(y_{0}\), \(y_{1}\), \(y_{2}\),..., \(y_{8}\)}. Then, their weight vector is given as \(\omega\) = (0.4, 0.25, 0.35)._  
_To reach the suitable aggregation operation by using Equation (4) for the example, we need to extend the three HLSs {\(y_{4}\), \(y_{6}\)}, {\(y_{3}\), \(y_{5}\), \(y_{6}\)} and {\(y_{3}\), \(y_{4}\), \(y_{5}\)} into the same components by using the LCMN extension method for them._  
_First, the LCMN of the three HFSs {\(y_{4}\), \(y_{6}\)}, {\(y_{3}\), \(y_{5}\), \(y_{6}\)} and {\(y_{3}\), \(y_{4}\), \(y_{5}\)} is obtained as \(c\) = 6. Thus, their extension forms are expressed as \(z_{1}^{c}=\) ([\(y_{5}\), \(y_{6}\)], [\(y_{4}\), \(y_{4}\), \(y_{4}\), \(y_{6}\), \(y_{6}\), \(y_{6}\)]), \(z_{2}^{c}=\) ([\(y_{4}\), \(y_{6}\)], [\(y_{3}\), \(y_{5}\), \(y_{5}\), \(y_{6}\), \(y_{6}\)]), and \(z_{3}^{c}=\) ([\(y_{3}\), \(y_{5}\)], [\(y_{3}\), \(y_{3}\), \(y_{4}\), \(y_{4}\), \(y_{5}\), \(y_{5}\)])._  
_Through Equation (4), their operational result of the WAA operator is given by_  
\[ \begin{array}{l}F_{LCHV}(z_{1}^{c},z_{2}^{c},z_{3}^{c})=\frac{3}{\sum_{j=1}^ {n}}\omega z_{j}^{c}\\ =\left(\left[y_{q-\{\pi\}\|\frac{1}{2}\|_{2}^{c}+(1-\frac{1}{2})^{c}+y^{c}},y_ {q-\{\pi\}\|_{1}^{c}+(1-\frac{1}{2})^{c}}\right],\left\{y_{q-\{\pi\}\|_{1}^{c} +(1-\frac{3}{2})^{c}+y},y_{q-\{\pi\}\|_{1}^{c}+(1-\frac{3}{2})^{c}+y},\ldots,y _{q-\{\pi\}\|_{1}^{c}+(1-\frac{3}{2})^{c}}\right\}\right)\\ =\left(\left[y_{q-\{\pi\}\|_{1}^{c}+(1-\frac{1}{2})^{c}+(1-\frac{1}{2})^{c}+y^ {c}},y_{q-\{\pi\}\|_{1}^{c}+(1-\frac{6}{2})^{c}+(1-\frac{6}{2})^{c}+(1-\frac{ 3}{2})^{c}+y^{c}},y_{q-\{\pi\}\|_{1}^{c}+(1-\frac{3}{2})^{c}+y^{c}},y_{q-\{\pi \}\|_{1}^{c}+(1-\frac{3}{2})^{c}+y^{c}}\right]\right)\\ =\left(\left[y_{q-\{\pi\}\|_{1}^{c}+(1-\frac{1}{2})^{c}+(1-\frac{1}{2})^{c}+( 1-\frac{1}{2})^{c}+y^{c}},y_{q-\{\pi\}\|_{1}^{c}+(1-\frac{6}{2})^{c}+(1-\frac{ 6}{2})^{c}+(1-\frac{6}{2})^{c}+(1-\frac{5}{2})^{c}+(1-\frac{5}{2})^{c}+y^{c}} \right]\right)\\ =\left(\left[y_{q-\{\pi\}\|_{1}^{c}+(1-\frac{1}{2})^{c}+(1-\frac{1}{2})^{c}+y^ {c}},y_{q-\{\pi\}\|_{1}^{c}+(1-\frac{6}{2})^{c}+(1-\frac{6}{2})^{c}+(1-\frac{ 6}{2})^{c}+(1-\frac{6}{2})^{c}+(1-\frac{5}{2})^{c}+y^{c}},y_{q-\{\pi\}\|_{1}^{c }+(1-\frac{6}{2})^{c}+(1-\frac{5}{2})^{c}+y^{c}}\right]\right)\\ =\left(\left[y_{q},y_{q}\right],\) {\(y_{3}\),\(\pm\)69, \(y_{3}\),\(\pm\)69, \(y_{4}\),\(\pm\)776, \(y_{5}\),\(\pm\)690}\}.\end{array} \]  
It is obvious that their aggregated result is still LCHN and all the linguistic values in the aggregated LCHN still belong to \(Y\), and then the WAA operator of the LCHNs based on the LCMN extension method shows its objective extension operation without the subjective extension forms depending on decision makers' preferences, which demonstrate the advantage of its operational rationality. **Theorem 2**. : _Set \(z_{j}\) (\(j\) = 1, 2,..., \(n\)) as a group of LCHVs.