## 4 Model Solving
### Evaluation Indexes Selection  
In order to seek the optimal decision-making scheme, four evaluation indexes are selected to construct the decision-making space with 4-dimensional attributes and evaluate the alternative schemes in different typical years, which includes reliability of water diversion (\(\alpha\)), recoverability of insufficient water diversion (\(\gamma\)), water shortage depth (\(\
u\)) and water shortage index (\(WSI\)). The calculation method of each index is as follows:  
(1) Reliability of water diversion (\(\alpha\)): The index represents the ratio of the number of periods in which the water diversion meets the user's needs to the total number of operational periods, and reflects the guaranteed degree of water diversion. The index is a maximized type index. The greater the value, the better. The calculation formula is as follows in Equations (11) and (12):  
\[\alpha=\frac{\sum_{t=1}^{T}K_{t}}{T} \tag{11}\]  
\[ \begin{cases}K_{t}=1,Q_{g}(t)\geq Q_{need}(t)\\ K_{t}=0,Q_{g}(t)<Q_{need}(t)\end{cases}  \tag{12}\]  
where \(\alpha\) is the reliability of water diversion; \(Q_{g}(t)\) is the water diversion during each period; \(Q_{need}(t)\) is the water demand during each period; \(K_{t}\) is the discrimination coefficient of whether the water supply meets the water demand. When \(Q_{g}(t)\geq Q_{need}(t)\), \(K_{t}=1\); Otherwise, \(K_{t}=0\). (2) Recoverability of insufficient water diversion (\(\gamma\)): The index indicates the average probability of the water diversion recovering from the failure state (\(Q_{g}(t)<Q_{need}(t)\)) to the normal state during the reservoir operational periods, which is a maximized type index. The greater the value, the better. The calculation formula is as follows in Equation (13):  
\[\gamma=\frac{\sum_{t=1}^{T}(K_{t}=1|K_{t-1}=0)}{T-\sum_{t=1}^{T}K_{t}} \tag{13}\]  
where \(\gamma\) is the recoverability of insufficient water diversion. (3) Water shortage depth (\(\
u\)): The index indicates the degree of water shortage during the periods of insufficient water supply, and is represented by the maximum of relative water shortage in each period. The index is a minimized type index. The smaller the value, the better. The calculation formula is as follows in Equation (14):  
\[\
u=\max\{DR_{1},DR_{2},\ldots,DR_{t}\},\ DR_{t}=1-\frac{Q_{g}(t)}{Q_{need}(t)}. \tag{14}\]  
where \(\
u\) is the depth of water shortage; \(DR_{t}\) is the relative water shortage during each period. (4) Water shortage index (\(WSI\)): The index reflects the loss degree of reservoir water supply benefits, which is a minimized type index. The smaller the value, the better. The calculation formula is as follows in Equation (15):  
\[WSI=\frac{100}{T}\sum_{t=1}^{T}{DR_{t}}^{2} \tag{15}\]  
where \(WSI\) is the water shortage index. The method of multi-objective optimization and multi-attribute decision making for reservoir operation includes four steps: objectives extraction, model construction, optimization solution and decision-making scheme selection. The specific process is shown in Figure 6.  
Step 1: Data input. The basic parameters of the reservoir, hydropower station, pump station and monthly inflow data are input. Step 2: Objectives extraction and correlation analysis. The objectives are extracted according to the reservoir operation task. By analyzing the correlation of every two objectives, the competitive objectives are obtained. Step 3: Model construction. The multi-objective optimal operation model is constructed through competitive objectives obtained by Step 2 and related constraints. Step 4: Optimization solution. The Pareto solution set or the alternative scheme set is obtained by the NSGA-II. Step 5: Decision-making scheme selection. The evaluation indexes are selected to form a decision-making space with 4-dimensional attributes. The approach of SEABODE is used to optimize the scheme set obtained by Step 4, and the final decision-making scheme is obtained. Step 6: Output the Pareto solution set, the variable values, the evaluation indexes and the comparison results.