## 4 An Integrated GBWM-PROMETHEE II Framework for Group Decision Making
### Determine DMs' Weights  
In this section, for deriving reasonable DMs' weights instead of assigning the same proportion to all DMs, an innovative method for solving incomplete weight information for MAGDM is introduced. Inspired by [PERSON] (2018), we constructed a technique to determine the experts' weights based on the similarity degree of q-ROFSs, which has the capability of characterizing DMs' true feelings in larger practicable membership grades. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\alpha\) & \(e_{1}(\cdot)\) & \(e_{2}(\cdot)\) & \(\cdots\) & \(e_{\tau}(\cdot)\) & \(\cdots\) & \(e_{n}(\cdot)\) \\ \hline \(\alpha_{1}\) & \(e_{1}(\alpha_{1})\) & \(e_{2}(\alpha_{1})\) & \(\cdots\) & \(e_{\tau}(\alpha_{1})\) & \(\cdots\) & \(e_{n}(\alpha_{1})\) \\ \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) \\ \(\alpha_{\varsigma}\) & \(e_{1}(\alpha_{\varsigma})\) & \(e_{2}(\alpha_{\varsigma})\) & \(\cdots\) & \(e_{\tau}(\alpha_{\varsigma})\) & \(\cdots\) & \(e_{n}(\alpha_{\varsigma})\) \\ \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) \\ \(\alpha_{m}\) & \(e_{1}(\alpha_{m})\) & \(e_{2}(\alpha_{m})\) & \(\cdots\) & \(e_{\tau}(\alpha_{m})\) & \(\cdots\) & \(e_{n}(\alpha_{m})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Original decision matrix. Figure 2: An integrated GBWM-PROMETHEE II framework for group decision making (GDM). The q-ROF decision matrix with the criteria \(C_{\tau}\) can be depicted as:  
\[X^{(\tau)}= \begin{array}{c}DM_{1}\\ DM_{2}\\ \vdots\\ DM_{\eta}\\ \end{array} \left[ \begin{array}{cccc}\alpha_{1}&\alpha_{2}&\ldots&\alpha_{m} \\ \left(\mu_{11^{\tau}}^{\tau}v_{11}^{\tau}\right)&\left(\mu_{12^{\tau}}^{\tau}v_{ 12}^{\tau}\right)&\ldots&\left(\mu_{1m^{\tau}}^{\tau}v_{1m}^{\tau}\right)\\ \left(\mu_{21^{\tau}}^{\tau}v_{21}^{\tau}\right)&\left(\mu_{22^{\tau}}^{\tau}v_ {22}^{\tau}\right)&\ldots&\left(\mu_{2m^{\tau}}^{\tau}v_{2m}^{\tau}\right)\\ \vdots&\vdots&\ldots&\vdots\\ \left(\mu_{\eta 1^{\tau}}^{\tau}v_{\eta 1}^{\tau}\right)&\left(\mu_{\eta 2^{\tau}}^{ \tau}v_{\eta 2}^{\tau}\right)&\cdots&\left(\mu_{\eta m}^{\tau},v_{\eta m}^{\tau}\right) \end{array} \right],\]  
where \(G=\left\{DM_{1},DM_{2},\ldots,DM_{\eta}\right\}\) with \(\eta\geq 2\). Assume that \(\left(\mu_{\kappa}^{\tau},v_{\kappa\varsigma}^{\tau}\right)\) is the evaluation information in the form of q-ROFN given by the \(x\)th DM concerning the relative alternatives \(\alpha_{\varsigma}(\varsigma=1,2,\ldots,m)\). **Step 1:** Determine the mean evaluation value \(\left(\overline{\mu}^{\tau}{}_{\varsigma},\overline{\
u}^{\tau}{}_{\varsigma}\right)\) for \(\alpha_{\varsigma}\) concerning the criteria \(C_{\tau}\) by all DMs. \[\left(\overline{\mu}^{\tau}{}_{\varsigma},\overline{\
u}^{\tau}{}_{\varsigma} \right)=\frac{\sum\limits_{k=1}^{\eta}\left(\mu_{k\varsigma}^{\tau},v_{k \varsigma}^{\tau}\right)}{\eta},\varsigma=1,2,\ldots,m;\tau=1,2,\ldots,n. \tag{14}\]  
**Step 2:** Based on Equation (6), define the similarity degree between any assessment value \(\left(\mu_{\kappa\varsigma}^{\tau},v_{\kappa\varsigma}^{\tau}\right)\) and the mean value \(\left(\overline{\mu}_{\varsigma}^{\tau},\overline{v}_{\varsigma}^{\tau}\right)\) determined by Equation (14):  
\[z_{\kappa\varsigma}^{\tau}=1-d_{q-ROF}\left(\left(\mu_{\kappa\varsigma}^{\tau},v_{\kappa\varsigma}^{\tau}\right)\right),\left(\left(\overline{\mu}^{\tau}{ }_{\varsigma},\overline{\
u}^{\tau}{}_{\varsigma}\right)\right)\right). \tag{15}\]  
**Step 3:** Calculate the overall similarity degree of each DM concerning the criteria \(C_{\tau}\) by Equation (16):  
\[\delta_{\kappa}^{\tau}=\sum_{\varsigma=1}^{m}z_{\kappa\varsigma}^{\tau}. \tag{16}\]  
**Step 4:** Define the weight of each DM as:  
\[\omega_{\kappa}^{\tau}=\frac{\delta_{\kappa}^{\tau}}{\sum_{k=1}^{\eta}\delta_ {\kappa}^{\tau}}. \tag{17}\]  
We can easily determine that Equation (17) satisfies the conditions \(\omega_{\kappa}^{\tau}\in[0,1]\) and \(\sum_{k=1}^{\eta}\omega_{\kappa}^{\tau}=1\).