# Multiple Attribute Decision-Making Method Using Linguistic Cubic Hesitant Variables
## 4 MADM Approach Using the WAA and WGA Operators of LCHVs  
In this section, a MADM approach is proposed by using the WAA and WGA operators of the LCHVs in a LCHV setting. For a MADM problem with LCHV information, suppose \(Z\) = \(\{Z_{1},Z_{2},\ldots,Z_{m}\}\) and \(R\) = \(\{R_{1},R_{2},\ldots,R_{n}\}\) are two sets of \(m\) alternatives and \(n\) attributes, respectively. When decision makers are requested to assess the alternative \(Z_{i}\) (\(i\) = 1, 2, \(\ldots\), \(m\)) over the attribute \(R_{j}\) (\(j\) = 1, 2, \(\ldots\), \(n\)), they may assign an interval linguistic value to \(\tilde{y}_{uij}\) and a set of several possible linguistic values to \(\tilde{y}_{hij}\) due to their hesitancy and indeterminacy from the predefined LTS \(Y\) = \(\{y_{l}\,|\,l\in[0,\,q]\}\), where \(q\) is an even number. Thus, the assessed hybrid information of \(\tilde{y}_{uij}\) and \(\tilde{y}_{hij}\) corresponding to each attribute \(R_{j}\) on each alternative \(Z_{i}\) can be represented as a LCHV \(z_{ij}=(\tilde{y}_{uij},\tilde{y}_{hij})=([y_{uij},y_{\tilde{y}_{i}j}],\left\{ y_{\lambda_{ij(1)}},y_{\lambda_{ij(2)}},\ldots,y_{\lambda_{ij(\tilde{y}_{i}j})} \right\})\) (\(j\) = 1, 2, \(\ldots\), \(n\); \(i\) = 1, 2, \(\ldots\), \(m\)). Hence, a LCHV decision matrix \(M\) = \((z_{ij})_{m\times n}\) can be constructed based on all the assessed LCHVs. Then, the weight of each attribute \(R_{j}\) is \(\omega_{j}\) \(\in\) [0,1] and \(\sum_{j=1}^{n}\omega_{j}=1\). Thus, the WAA or WGA operators of the LCHVs and the linguistic score function of the LCHV are utilized to develop a MADM approach with LCHV information, which can be summarized as the following decision steps:  
**Step 1. ** The LCMNs of (\(r_{i1}\), \(r_{i2}\), \(\ldots\), \(r_{in}\)) (\(i\) = 1, 2, \(\ldots\), \(m\)) in \(M\) = \((z_{ij})_{m\times n}\) can be obtained as \(c_{i}\), where \(r_{ij}\) is the number of LVs in \(\tilde{y}_{hij}\) for \(z_{ij}\). Based on the number of occurrences of \(c_{i}/r_{ij}\) in a LCHV \(z_{ij}\) (\(i\) = 1, 2, \(\ldots\), \(m\); \(j\) = 1, 2, \(\ldots\), \(n\)), \(z_{ij}\) is extended to the following form:  
\[ \begin{array}{ll}z_{ij}^{e}&=\left([y_{a_{ij}},y_{\beta_{j}}],\left\{ \tilde{y}_{\lambda_{ij(1)}^{1}},y_{\lambda_{ij(1)}^{2}},\ldots y_{\lambda_{ ij(1)}^{e_{i}/r_{ij}}},y_{\lambda_{ij(2)}^{1}},y_{\lambda_{ij(2)}^{2}},\ldots,y_{ \lambda_{ij(r_{ij})}^{e_{i}/r_{ij}}},\ldots,y_{\lambda_{ij(r_{ij})}^{1}},y_{ \lambda_{ij(r_{ij})}^{2}},\ldots,y_{\lambda_{ij(r_{ij})}^{e_{i}/r_{ij}}}} \right\}\right)\\ &=\left([y_{a_{ij}},y_{\beta_{ij}}],\left\{y_{\lambda_{ij}^{(1)}},y_{ \lambda_{ij}^{(2)}},\ldots,y_{\lambda_{ij}^{(c_{i})}}\right\}\right)\text{.} \end{array} \]  
Thus, the extended decision matrix is constructed as  
\[M^{e}= \begin{array}{ll}Z_{1}&\left(\begin{array}{cccc}z_{11}^{e}&z_{12}^{e}& \cdots&z_{1n}^{e}\\ Z_{21}^{e}&z_{22}^{e}&\cdots&z_{2n}^{e}\\ \vdots&\vdots&\vdots&\vdots\\ Z_{m}&\left(\begin{array}{cccc}z_{m1}^{e}&z_{m2}^{e}&\cdots&z_{mn}^{e}\\ \end{array}\end{array}\right)\end{array} \]  
**Step 2. ** Based on Equations (4) or (12), the aggregation values of \(z_{i}\) for \(Z_{i}\) are calculated using the following formula:  
\[ \begin{split}& z_{i}=F_{LCHV}(z_{11}^{e},z_{22}^{e},\cdots,z_{in}^{e })=\sum\limits_{j=1}^{n}\omega_{j}z_{ij}^{e}\\ &=\left(\begin{bmatrix}y_{q-\{\Pi_{j=1}^{n}-(\frac{z_{ij}}{q})^{ \omega_{j}},y}},y_{q-\{\Pi_{j=1}^{n}-(1-\frac{\beta_{ij}}{q})^{\omega_{j}}\}} \end{bmatrix}\begin{cases}y_{q-\{\Pi_{j=1}^{n}-(\frac{\lambda_{ij}^{(1)}}{q})^{ \omega_{j}},y}},y_{q-\{\Pi_{j=1}^{n}-(1-\frac{\lambda_{ij}^{(2)}}{q})^{\omega_ {j}},\cdots,y}},y_{q-\{\Pi_{j=1}^{n}-(1-\frac{\lambda_{ij}^{(e)}}{q})^{\omega_ {j}}\}}\end{cases}\right)\end{split}  \tag{13}\]  
or  
\[