# Multiple Attribute Decision-Making Method Using Linguistic Cubic Hesitant Variables
## 2 Linguistic Cubic Hesitant Variables (LCHVs)  
This section presents the concept of the LCHV, along with internal and external LCHV concepts, the linguistic score function of the LCHV for ranking LCHVs, and the operational relations of LCHVs. Based on the hybrid idea of both an interval/uncertain LV and a hesitant LV, we propose the concept of LCHV, including an internal LCHV and an external LCHV, as below. Definition 1: Set a LTS as \(Y\) = \(\{y_{1}\,l\in\,[0,\,q]\}\), where \(q\) is an even number. A LCHV \(z\) in \(Y\) is constructed using \(z=(\widetilde{y}_{u},\widetilde{y}_{h})\), where \(\widetilde{y}_{u}=[y_{a},y_{g}]\) for \(\beta\geq\alpha\) and \(y_{a}\), \(y_{\beta}\in Y\) is an interval/uncertain LV and \(\left\{y_{\lambda_{k}}\middle|y_{\lambda_{k}}\in Y,k=1,2,\ldots,r\right\}\) is a set of r possible LVs (i.e., a hesitant linguistic variable (HLV)) ranked in an ascending order. Especially when r = 1, LCHV is reduced to LCV introduced in [19]. **Definition 2**. : _Let \(z=(\widetilde{y}_{u},\widetilde{y}_{h})\)be a LCHV, with \(\widetilde{y}_{u}=[y_{u},y_{g}]\)for \(y_{u}\), \(y_{g}\)\(\in\)Y and \(\widetilde{y}_{h}=\left\{y_{\lambda_{k}}\middle|y_{\lambda_{k}}\in Y,k=1,2, \ldots,r\right\}\). Then, we call_  
1. \(z=(\widetilde{y}_{u},\widetilde{y}_{h})=([y_{u},y_{g}],\{y_{\lambda_{1}},y_{ \lambda_{2}},\ldots,y_{\lambda_{r}}\})\) _an internal LCHV if every_ \(\lambda_{k}\in[\alpha,\beta]\) _(_\(k\) _= 1, 2,..., _r_) for_ \(\alpha\)_,_ \(\beta\in[0,q]\)_;_
2. \(z=(\widetilde{y}_{u},\widetilde{y}_{h})=([y_{u},y_{g}],\{y_{\lambda_{1}},y_{ \lambda_{2}},\ldots,y_{\lambda_{r}}\})\) _an external LCHV if every_ \(\lambda_{k}\
ot\in[\alpha,\beta]\) _(_\(k\) _= 1, 2,..., _r_) for_ \(\alpha\)_,_ \(\beta\)_,_ \(\lambda_{k}\in[0,q]\)_._  
For the two LCHVs, \(z_{1}=(\widetilde{y}_{u1},\widetilde{y}_{h1})\) and \(z_{2}=(\widetilde{y}_{u2},\widetilde{y}_{h2})\), the number of LVs in \(\widetilde{y}_{h1}\) and \(\widetilde{y}_{h2}\) may imply difference. To realize the suitable operations of different LCHVs, we can extend the two HLVs \(\widetilde{y}_{h1}\) and \(\widetilde{y}_{h2}\) until both reach the same number of LVs on the basis of the LCMN extension method, which shows its objective advantage instead of the subjective extension methods depending on personal preference [31; 32]. Suppose \(z_{j}=(\widetilde{y}_{uj},\widetilde{y}_{hj})=([y_{aj},y_{g_{j}}],\left\{y_{ \lambda_{1}},y_{\lambda_{2}},\ldots,y_{\lambda_{j_{j_{j}}}}\right\})\) (\(j\) = 1, 2,..., _n_) is a group of LCHVs and the LCMN of \((r_{1},r_{1},\ldots,r_{n})\) for \(\widetilde{y}_{hj}\) (\(j\) = 1, 2,..., _n_) is \(c\). Then, one can extend them to the same number of LVs based on the following extension forms:  
\[ \begin{split}& z_{1}^{\epsilon}=\left([y_{\alpha_{1}},y_{g_{1}}], \left\{y_{\lambda_{11}^{\epsilon}},y_{\lambda_{11}^{\epsilon}},\ldots y_{ \lambda_{11}^{\epsilon/r_{1}}},y_{\lambda_{12}^{\epsilon}},y_{\lambda_{12}^{ \epsilon}},\ldots,y_{\lambda_{12}^{\epsilon/r_{1}}},\ldots,y_{\lambda_{11}^{ \epsilon/r_{1}}},y_{\lambda_{11}^{\epsilon}},\ldots,y_{\lambda_{1r_{1}}^{ \epsilon/r_{1}}}\right\}\right),\\ & z_{2}^{\epsilon}=\left([y_{\alpha_{2}},y_{\beta_{2}}],\left\{y _{\lambda_{21}^{\epsilon}},y_{\lambda_{21}^{\epsilon}},\ldots y_{\lambda_{21}^ {\epsilon/r_{2}}},y_{\lambda_{12}^{\epsilon}},y_{\lambda_{22}^{\epsilon}}, \ldots,y_{\lambda_{22}^{\epsilon/r_{2}}},\ldots,y_{\lambda_{22}^{\epsilon/r_{ 2}}},y_{\lambda_{22}^{\epsilon/r_{2}}},\ldots,y_{\lambda_{22}^{\epsilon/r_{2}}} \right\}\right),\\ &\ldots,\\ & z_{n}^{\epsilon}=\left([y_{\alpha_{n}},y_{g_{n}}],\left\{y_{ \lambda_{n1}^{\epsilon}},y_{\lambda_{n1}^{\epsilon}},\ldots y_{\lambda_{n1}^{ \epsilon}},y_{\lambda_{n2}^{\epsilon}},y_{\lambda_{n2}^{\epsilon}},\ldots,y_{ \lambda_{n2}^{\epsilon/r_{n}}},\ldots,y_{\lambda_{n2}^{\epsilon/r_{n}}}, \ldots,y_{\lambda_{n2}^{\epsilon/r_{n}}},\ldots,y_{\lambda_{n2}^{\epsilon/r_{ n}}},\ldots,y_{\lambda_{n2}^{\epsilon/r_{n}}},\ldots,y_{\lambda_{n2}^{\epsilon/r_{n}}} \right\}\right).\end{split}  \tag{1}\]  
**Example 1**.