## 4 An Integrated GBWM-PROMETHEE II Framework for Group Decision Making
### Determining the Criteria Weights Using GBWM Method  
By inspiring the characteristics of the BWM, a novel method named the group best-worst method is proposed to derive the criteria weights with GDM problems. Compared with the original BWM method [24], [PERSON] et al. [26] paid attention to the significance of the DMs for calculating the optimal weights with respect to each criterion. In this section, we suppose that \(\eta\) experts with respect to \(n\) criteria, then the calculation process can be represented as follows:  
**Step 1:** Determine a collection of criteria \(C_{\tau}(\tau=1,2,\ldots,n)\) for decision making. **Step 2:** Identify the best criterion \(C_{B}\) and the worst criterion \(C_{W}\), respectively. In this step, we assume that one specific expert determines \(C_{B}\) and \(C_{W}\). For GDM, \(C_{B}\) denotes the most desirable criterion and \(C_{W}\) is the least desirable criterion. **Step 3:** Utilize the numbers between 1 to 9 to determine the preference of \(C_{B}\) over all the other criteria, simplified as best-to-others (BTO) vector:  
\[\overline{A}_{B}=(\overline{\alpha}_{B1},\overline{\alpha}_{B2},\dots,\overline {\alpha}_{Bn}) \tag{18}\]  
where \(\overline{\alpha}_{B\tau}\) denotes the preference degree of \(C_{B}\) over \(C_{\tau}\). Note that \(\overline{\alpha}_{BB}=1\). **Step 4:** Utilize the numbers between 1 to 9 to determine the preference of all the criteria over \(C_{W}\), simplified as others-to-worst (OTW) vector:  
\[\overline{A}_{W}=(\overline{\alpha}_{1W},\overline{\alpha}_{2W},\dots, \overline{\alpha}_{nW})^{T} \tag{19}\]  
where \(\overline{\alpha}_{1W}\) denotes the preference degree of \(C_{\tau}\) over \(C_{W}\). Note that \(\overline{\alpha}_{WW}=1\). **Step 5:** Obtain the optimal weights \(\omega^{\star}=\left(\omega_{1}^{\star},\omega_{2}^{\star},\dots,\omega_{n}^ {\star}\right)\) utilizing M1 in [21] as follows:  
\[\text{Min}{\sum_{k=1}^{\eta}\omega_{k}\text{Max}}\ \bigg{\{}\bigg{|}\frac{ \omega_{B}}{\omega_{\tau}}-\alpha_{B\tau}^{k}\bigg{|}\bigg{|}\frac{\omega_{ \tau}}{\omega_{W}}-\alpha_{\tau W}^{k}\bigg{|}\bigg{\}}St.{\sum_{\tau=1}^{n} \omega_{\tau}}=1 \tag{20}\]  
\(\omega_{\tau}\geq 0\), for all \  
where \(d_{\tau}(\alpha_{a},\alpha_{b})\) indicates the difference between the performance of the feasible alternatives \(\alpha_{a}\) and \(\alpha_{b}\) concerning each criterion. **Step 3:** Define the preference function \(p_{\tau}(\alpha_{a},\alpha_{b})\):  
\[p_{\tau}(\alpha_{a},\alpha_{b})=F_{\tau}[d_{\tau}(\alpha_{a},\alpha_{b})] \tag{25}\]  
where \(p_{\tau}(\alpha_{a},\alpha_{b})\) indicates the preference of \(\alpha_{a}\) over \(\alpha_{b}\) concerning each criterion, expressed as the function of \(d_{\tau}(\alpha_{a},\alpha_{b})\). Here, to match the characteristics of the criteria better, one predefined shape of the preference function named the usual (type I) [19] is applied. That is, \(p_{\tau}(\alpha_{a},\alpha_{b})=d_{\tau}(\alpha_{a},\alpha_{b})=Sco_{F}(\alpha _{a})-Sco_{F}(\alpha_{b})\). **Step 4:** Determine the global preference index \(\Pi(\alpha_{a},\alpha_{b})\) by Equation (26):  
\[\Pi(\alpha_{a},\alpha_{b})=\sum_{\tau=1}^{n}p_{\tau}(\alpha_{a},\alpha_{b}) \omega_{\tau} \tag{26}\]  
where \(\Pi(\alpha_{a},\alpha_{b})\) measures how much \(\alpha_{a}\) over \(\alpha_{b}\) in view of the overall criteria, and the associated weights \(\omega_{\tau}\) are determined by utilizing the GBWM method in Section 4.2. **Step 5:** Aggregate \(\Pi(\alpha_{a},\alpha_{b})\) into the positive \(\varphi^{+}(\alpha_{a})\) and negative \(\varphi^{-}(\alpha_{a})\) outranking flows by the following equations:  
\[\varphi^{+}(\alpha_{a})=\frac{1}{m-1}\sum_{\alpha_{b}\in A}\Pi(\alpha_{a}, \alpha_{b}) \tag{27}\]  
\[\varphi^{-}(\alpha_{a})=\frac{1}{m-1}\sum_{\alpha_{b}\in A}\Pi(\alpha_{b}, \alpha_{a}) \tag{28}\]  
where \(\varphi^{+}(\alpha_{a})\) and \(\varphi^{-}(\alpha_{a})\) depict that the preference degree of \(\alpha_{a}\) outranks all other alternatives and is outranked by them, respectively.