# Multiple Attribute Decision-Making Method Using Linguistic Cubic Hesitant Variables
## 2 Linguistic Cubic Hesitant Variables (LCHVs)
: _Let us consider that \(z_{1}=([y_{4},y_{6}],[y_{4},y_{5}])\) and \(z_{2}=([y_{5},y_{6}],[y_{4},y_{5},y_{6}])\) are two linguistic cubic hesitant numbers (LCHNs) in the LTS \(Y=\{y_{0},y_{1},y_{2},\ldots,y_{8}\}\)._  
_Then, the LCMN obtained from \(r_{1}=2\) and \(r_{2}=3\) in \(z_{1}\) and \(z_{2}\) is \(c=6\). Through Equation (1), the two LCHNs \(z_{1}\) and \(z_{2}\) can be extended to the following forms:_  
\[z_{1}^{\epsilon}=([y_{4},y_{6}],\{y_{4},y_{4},y_{4},y_{5},y_{5},y_{5}\})\text { and }z_{2}^{\epsilon}=([y_{5},y_{6}],\{y_{4},y_{4},y_{5},y_{5},y_{6},y_{6}\})\]  
**Definition 3**. : _Let \(z_{1}=(\widetilde{y}_{u1},\widetilde{y}_{h1})=([y_{\alpha_{1}},y_{g_{1}}], \{y_{\lambda_{11}},y_{\lambda_{12}},\ldots,y_{\lambda_{1r}}\})\) and \(z_{2}=(\widetilde{y}_{u2},\widetilde{y}_{h2})=([y_{\alpha_{2}},y_{g_{2}}], \{y_{\lambda_{21}},y_{\lambda_{22}},\ldots,y_{\lambda_{2}}\})\) be two LCHVs in the LTS \(Y=\{y_{l}\,l\in[0,\,q]\}\). Then, we define_  
1. \(z_{1}=z_{2}\Leftrightarrow\widetilde{y}_{u1}=\widetilde{y}_{u2}\) _and_ \(\widetilde{y}_{h1}=\widetilde{y}_{h2}\)_, i.e.,_ \(y_{\alpha_{1}}=y_{\alpha_{2}}\)_,_ \(y_{\beta_{1}}=y_{\beta_{2}}\)_, and_ \(y_{\lambda_{1k}}=y_{\lambda_{2k}}\) _for_ \(k\) _= 1, 2,..., \(r\);_
2. \(z_{1}\subseteq z_{2}\Leftrightarrow\widetilde{y}_{u1}\subseteq\widetilde{y}_{u2}\) _and_ \(\widetilde{y}_{h1}\subseteq\widetilde{y}_{h2}\)_, i.e.,_ \(y_{\alpha_{1}}\leq y_{\alpha_{2}}\)_,_ \(y_{\beta_{1}}\leq y_{\beta_{2}}\)_, and_ \(y_{\lambda_{1k}}\leq y_{\lambda_{2k}}\) _for_ \(k\) _= 1, 2,..., \(r\)._  
For a convenient comparison in a linguistic setting, we present the linguistic score function of the LCHV and the ranking method of the LCHVs below. **Definition 4**. : _Let \(z=([y_{u},y_{\beta}],\{y_{\lambda_{1}},y_{\lambda_{2}},\ldots,y_{\lambda_{r}}\})\) be a LCHV in the LTS \(Y=\{y_{l}\,l\in[0,\,q]\}\). Then, its linguistic score function is defined as the following:_  
\[y_{L(z)}=y_{\frac{1}{2}(\frac{1}{2}(\alpha+\beta)+\frac{1}{2}\lambda_{k})}\text { for }L(z)\in[0,\,q]. \tag{2}\]  
**Definition 5**. : _Let \(z_{1}=([y_{\alpha_{1}},y_{\beta_{1}}],\{y_{\lambda_{11}},y_{\lambda_{12}},\ldots,y_{ \lambda_{1r}}\})\)and \(z_{2}=([y_{\alpha_{2}},y_{\beta_{2}}],\{y_{\lambda_{21}},y_{\lambda_{22}},\ldots,y_ {\lambda_{2r}}\})\) be two LCHVs in the LTS \(Y=\{y_{1}\,l\in[\)0, q\(]\}\). Then, their linguistic scores are \(y_{L(z_{1})}\) and \(y_{L(z_{2})}\). Thus, the ranking relations are defined as follows:_  
1. _If_ \(y_{L(z_{1})}>y_{L(z_{2})}\)_, then_ \(z_{1}>z_{2}\)_;_
2. _If_ \(y_{L(z_{1})}<y_{L(z_{2})}\)_, then_ \(z_{1}<z_{2}\)_;_
3. _If_ \(y_{L(z_{1})}=y_{L(z_{2})}\)_, then_ \(z_{1}=z_{2}\)_._  
**Example 2**. : _Let us consider that \(z_{1}=(\{y_{5},\,y_{6}\},\,\{y_{4},\,y_{6}\})\) and \(z_{2}=(\{y_{4},\,y_{6}\},\,\{y_{3},\,y_{5},\,y_{6}\})\) are two LCHNs in the LTS \(Y=\{y_{0},\,y_{1},\,y_{2},\,\ldots,\,y_{8}\}\)._  
_Then, through Equation (2), we calculate the linguistic score values of the two LCHNs below:_  
\[y_{L(z_{1})}=y_{\frac{1}{2}(\frac{1}{2}(\alpha_{1}+\beta_{1})+\frac{1}{2}\sum_ {k=1}^{2}\lambda_{1k})}=y_{\frac{1}{2}(\frac{1}{2}(5+6)+\frac{1}{2}(4+6))}=y_{ 5.25}\]  
\[y_{L(z_{2})}=y_{\frac{1}{2}(\frac{1}{2}(\alpha_{2}+\beta_{2})+\frac{1}{2}\sum_ {k=1}^{3}\lambda_{2k})}=y_{\frac{1}{2}(\frac{1}{2}(4+6)+\frac{1}{3}(3+5+6))}= y_{4.8333}\]  
_Since there is \(y_{L(z_{1})}>y_{L(z_{2})}\), they are ranked as \(z_{1}>z_{2}\)._  
**Definition 6**. : _Let \(z_{1}=([y_{\alpha_{1}},y_{\beta_{1}}],\{y_{\lambda_{11}},y_{\lambda_{12}}, \ldots,y_{\lambda_{1r}}\})\) and \(z_{2}=([y_{\alpha_{2}},y_{\beta_{2}}],\{y_{\lambda_{21}},y_{\lambda_{22}}, \ldots,y_{\lambda_{2r}}\})\) be two LCHVs in the LTS \(Y=\{y_{1}\,l\in[\)0, q\(]\}\). Then, their operational relations are defined as below:_  
1.