## 3 The Cloud-TODIM Method
The expert weight \(v^{t}_{ij}\) can be expressed by  
\[v^{t}_{ij}=\frac{sim\left(C^{t}_{ij},\overline{C}_{ij}\right)}{\sum\limits_{ t=1}^{q}sim\left(C^{t}_{ij},\overline{C}_{ij}\right)} \tag{8}\]  
Step \(1.3\) Compute the decision matrix and normalization matrix for RRS \begin{table}
\begin{tabular}{c c c} \hline
**Symbol** & **Definition** & **Description** \\ \hline \(w=(w_{1},w_{2},\ldots,w_{m})^{T}\) & Criteria weight vector & \(w_{j}\) denotes the weight of criterion \(Z_{j}\). \\  & & \(0\leq w_{j}\leq 1\), \(\sum\limits_{j=1}^{m}w_{j}=1\). \\ \hline \(CR=\left(C_{ij}\right)_{m\times n}\) & Decision matrix & \(C_{ij}\) denotes the evaluation cloud of alternative \\  & & \(O_{i}\) concerning criterion \(Z_{j}\). \\ \hline \(CR^{\prime}=\left(C^{\prime}_{ij}\right)_{m\times n}\) & Normalized decision matrix & \\ \hline \end{tabular}
\end{table} 
Table 2: _Cont._According to Equation (4), the weighted cloud of the RRS of the expert group can be calculated by  
\[\left\{ \begin{array}{l}Ex_{ij}=v_{ij}^{1}Ex_{ij}^{1}+v_{ij}^{2}Ex_{ij}^{2}+\cdots +v_{ij}^{q}Ex_{ij}^{q}\\ \\ En_{ij}=\sqrt{v_{ij}^{1}\Big{(}En_{ij}^{1}\Big{)}^{2}+v_{ij}^{2}\Big{(}En_{ij}^{ 2}\Big{)}^{2}+\cdots v_{ij}^{q}\Big{(}En_{ij}^{q}\Big{)}^{2}}\\ \\ He_{ij}=\sqrt{v_{ij}^{1}\Big{(}He_{ij}^{1}\Big{)}^{2}+v_{ij}^{2}\Big{(}He_{ij}^ {2}\Big{)}^{2}+\cdots v_{ij}^{q}\Big{(}He_{ij}^{q}\Big{)}^{2}}\end{array} \right. \tag{9}\]  
The weighted cloud matrix \(CR=\big{(}C_{ij}\big{)}_{m\times n}\) of the RRS is composed of all the evaluation weighted clouds \(C_{ij}\); the \(CR\) can be expressed by  
\[CR=\left[ \begin{array}{cccc}C_{11}&C_{12}&\cdots&C_{1n}\\ C_{21}&C_{22}&\cdots&C_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ C_{m1}&C_{m2}&\cdots&C_{mn}\end{array} \right] \tag{10}\]  
The weighted cloud matrix can be normalized by  
\[CR_{ij}^{\prime}=\left\{ \begin{array}{l}\Big{(}\frac{C_{ij}-\min\limits_{1 \leq i\leq m}C_{ij}}{\max\limits_{1\leq i\leq m}C_{ij}-\min\limits_{1\leq j \leq m}C_{ij}},En_{ij},He_{ij}\Big{)}&\text{benefit criteria}\\ \\ \Big{(}\frac{\max\limits_{1\leq i\leq m}C_{ij}-C_{ij}}{\max\limits_{1\leq i\leq m }C_{ij}-\min\limits_{1\leq i\leq m}C_{ij}},En_{ij},He_{ij}\Big{)}&\text{ consumption criteria}\end{array} \right. \tag{11}\]  
Step 2 Obtain the relative weight \(w_{jr}\) of the criterion \(Z_{j}\) to the reference criterion \(Z_{r}\). \[w_{jr}=\frac{w_{j}}{w_{r}} \tag{12}\]  
\[w_{r}=\max\{w_{1},w_{2},\ldots,w_{m}\}\]  
Step 3 Calculate the dominance degree \(\varphi_{ik}^{j}\)[15; 17; 19] of the RRS \(O_{i}\) over the RRS \(O_{k}\) regarding criterion \(Z_{j}\). \[\varphi_{ik}^{j}=\left\{ \begin{array}{l}\sqrt{w_{jr}/\sum\limits_{j=1}^{m}w_ {jr}}\cdot d\Big{(}CR_{ij}^{\prime},CR_{kj}^{\prime}\Big{)}&\quad CR_{ij}^{ \prime}>CR_{kj}^{\prime}\\ \\ 0&\quad CR_{ij}^{\prime}=CR_{kj}^{\prime}\\ \\ -\frac{1}{\theta}\sqrt{\sum\limits_{j=1}^{m}w_{jr}/w_{jr}}\cdot d\Big{(}CR_{ij}^ {\prime},CR_{kj}^{\prime}\Big{)}&\quad CR_{ij}^{\prime}<CR_{kj}^{\prime}\end{array} \right. \tag{13}\]  
where \(\theta\) is a parameter denoting the attenuation factor of the losses, usually belonging to [1; 2.5] [14]. Generally, the larger the \(\theta\), the lower the psychological level of loss avoidance of DMs. Step 4 Compute the overall dominance degree \(\delta_{ik}\) of the RRS \(O_{i}\) over the RRS \(O_{k}\). \[\delta_{ik}=\sum\limits_{j=1}^{m}\varphi_{ik}^{j} \tag{14}\]Step 5 Calculate the normalized overall performance \(\tau_{i}\) of RRS \(O_{i}\). \[\tau_{i}=\frac{\sum\limits_{k=1}^{n}\delta_{ik}-\min\limits_{i}\sum\limits_{k=1} ^{n}\delta_{ik}}{\max\limits_{i}\sum\limits_{k=1}^{n}\delta_{ik}-\min\limits_{i }\sum\limits_{k=1}^{n}\delta_{ik}} \tag{15}\]  
Step 6 Determine the ranking order of RRSs according to the obtained normalized overall performance value. Finally, the RRS with the maximum value is the optimal scheme to implement.