## 3 The Cloud-TODIM Method  
In the traditional TODIM method, on the basis of different criteria, the dominance degree of each scheme over the others is calculated. Then the ranking of the schemes can be computed based on the overall dominance degrees. However, the traditional TODIM method can only solve a multi-attribute decision-making problem, where the criteria are represented in the format of crisp numbers. As for linguistic evaluation information, it is usually the characteristics of uncertainty, randomness, and fuzziness that cannot be described by a definite number. The introduced cloud model (Section 2) can effectively handle uncertain, random, and fuzzy information and be organically integrated with the TODIM method. Consequently, the cloud-TODIM model is developed to quantitatively rank the RRSs and obtain the optimal scheme. As shown in Figure 3, a decision-making framework for RRS selection of TSF failure is displayed to make the calculation processing clear and straightforward. The implementation of the cloud-TODIM method is briefly described below [15; 36; 37]. Suppose the problem of RRS selection has \(n\) schemes (alternatives) and \(m\) criteria, and \(q\) experts form an expert group. The main symbols and their description are displayed in Table 2. \begin{table}
\begin{tabular}{c c c} \hline
**Symbol** & **Definition** & **Description** \\ \hline \(O=\{O_{1},O_{2},\ldots O_{n}\}\) & Feasible RRS (alternative) set & \(O_{i}\) denotes the \(i\)th RRS, \(i=1,2,\ldots,n\). \\ \hline \(Z=\{Z_{1},Z_{2},\ldots Z_{m}\}\) & Finite set of evaluation criteria for the RRS problem & \(Z_{j}\) denotes the \(j\)th criterion, \(j=1,2,\ldots,m\). \\ \hline \(C_{ij}^{t}=\left(Ex_{ij^{\prime}}^{t},En_{ij^{\prime}}^{t},H\epsilon_{ij}^{t}\right)\) & Evaluation cloud & The evaluation cloud of expert \(E_{t}\) for alternative \(O_{i}\) under the criterion \(Z_{j}\). \\ \hline \end{tabular}
\end{table} 
Table 2: Main symbols within the cloud–TODIM method. Figure 3: Framework of decision-making for RRS selection. Step \(1\) Normalize the decision matrix \(CR=\left(C_{ij}\right)_{m\times n}\) into matrix \(CR^{\prime}=\left(C^{\prime}_{ij}\right)_{m\times n}\). Step \(1.1\) Convert evaluation information into a cloud  
The procedure of the conversion of the evaluation information into clouds can be summarized as follows: First, determine the alternatives and evaluation criteria. Then, collect initial evaluation language values from experts. Finally, transform the expression of evaluation criteria into a cloud according to Section 2.3. Step \(1.2\) Calculate the expert weight  
The expert weight can be determined by the similarity between the evaluation cloud and the AAC, provided by the single expert and the expert group, respectively. The smaller the similarity, the smaller the corresponding weight, and vice versa. In terms of the scheme \(O_{i}\) with respect to the criteria \(Z_{j}\), based on Equation (3), the AAC \(\overline{C_{ij}}\) of the expert group can be calculated by  
\[\left\{ \begin{array}{l}\overline{E}x_{ij}=\frac{Ex_{ij}^{1}+Ex_{ij}^{2}+ \cdots+Ex_{ij}^{q}}{q}\\ \\ \overline{E}n_{ij}=\sqrt{\frac{\left(En_{ij}^{1}\right)^{2}+\left(En_{ij}^{2} \right)^{2}+\cdots\left(En_{ij}^{q}\right)^{2}}{q}}\\ \\ \overline{H}e_{ij}=\sqrt{\frac{\left(He_{ij}^{1}\right)^{2}+\left(He_{ij}^{2} \right)^{2}+\cdots\left(He_{ij}^{q}\right)^{2}}{q}}\end{array} \right. \tag{6}\]  
Then, the similarity between the evaluation cloud \(C^{t}_{ij}\) of the single expert and the AAC \(\overline{C_{ij}}\) of the expert group can be calculated by  
\[sim\left(C^{t}_{ij},\overline{C}_{ij}\right)=1-\frac{d\left(C^{t}_{ij}, \overline{C}_{ij}\right)}{\sum\limits_{t=1}^{q}d\left(C^{t}_{ij},\overline{C }_{ij}\right)} \tag{7}\]  
where \(d\left(C^{t}_{ij},\overline{C}_{ij}\right)\) can be computed according to Section 2.4.