## 3 Model Construction and Solving Algorithm
### Solving Algorithm
#### 3.2.2 Application of NSGA-II-TOPSIS-GCA Algorithm for MOODPR Problem
(2) Recoverability (\(\gamma\)) refers to the probability that the water supply return to a normal state from an insufficient state (\(q_{z}(m,t)<Q_{z}(t)\)) during operation periods. The calculation formula is as follows:

\[\gamma=\frac{\sum_{t=1}^{T}(K_{t}=1|K_{t}=0)}{T-\sum_{t=1}^{T}K_{t}} \tag{10}\]

(3) Water shortage depth (\(\
u\)) refers to the maximum ecological water shortage degree during operation periods. The ecological water shortage degree can be used to measure the severity of water shortage. The calculation formula is as follows:

\[\
u=\max\{DR_{1},DR_{2},...,DR_{t}\},\text{DR}_{t}=1-\frac{\sum_{m=1}^{M}q_{ \varepsilon}(m.t)}{\sum_{m=1}^{M}Q_{\varepsilon}(m.t)} \tag{11}\]

where \(DR_{t}\) denotes the degree of ecological water shortage during period \(t\); \(q_{\varepsilon}(m,t)\) and \(Q_{\varepsilon}(m,t)\) denote the ecological water supply and demand flow for reservoir \(m\) during period \(t\). (4) The water shortage index (\(WSI\)) reflects the ecological loss degree of parallel reservoirs. The calculation formula is as follows:

\[\text{WSI}=\frac{100}{T}\sum_{t=1}^{T}{DR_{t}}^{2} \tag{12}\]

## 4 Results and Discussion

### Model and Algorithm Application

The MOODPR proposed in this paper mainly includes four modules: problem identification, model construction, the optimization solution, and scheme evaluation. Figure 6 shows the application flowchart of the method:

1. Problem identification. The relevant data of the research area are systematically collected, the development status of the research area is analyzed, and the existing problems are identified. 2. Input parameters such as reservoirs and hydropower stations. 3. Model construction. The MOOM is constructed according to the objective functions and constraints. 4. Optimization solution. The non-inferior solution set is obtained. 5. Scheme evaluation. The appropriate attribute indexes are selected and the 4-dimensional decision matrix is established. The TOPSIS-GCA is adopted to optimize the non-inferior solution set, so as to obtain the decision schemes. 6. Output results, such as attribute indexes and decision schemes. ### Pareto Solution Set

Figure 7 shows the Pareto non-inferior solution sets of comprehensive water supply and ecological AAPFD value. The comprehensive water supply and ecological AAPFD value showed an obvious positive correlation. That is, comprehensive water supply was significantly negatively correlated with ecology, which further highlighted the competition and contradiction of the two objectives. With the continuous decrease in natural inflow,

Figure 6: Application flowchart of the method. the increase in comprehensive water demand caused the comprehensive water supply to show an increasing trend, the ecological AAPFD value increased, and the ecological environment of river channels deteriorated. For example, the maximum comprehensive water supply and minimum ecological AAPFD value under a 25% inflow frequency were 4.63 \(\times\) 10\({}^{8}\) m\({}^{3}\) and 1.974, respectively, while the corresponding values under a 75% inflow frequency were 5.73 \(\times\) 10\({}^{8}\) m\({}^{3}\) and 4.553, respectively. The comprehensive water supply increased by 1.10 \(\times\) 10\({}^{8}\) m\({}^{3}\) and the ecological AAPFD value increased by 2.579, indicating that the natural inflow had a remarkable influence on the reservoirs' benefits. Due to the decrease in natural inflow under a 95% inflow frequency, the comprehensive water supply and ecological AAPFD value were significantly affected. In addition, the amplification of water supply was less than that of ecology in all non-inferior solution sets, indicating that the ecology objective was more sensitive to the comprehensive water supply objective, which also required the decision makers to comprehensively weigh the pros and cons of the two in actual operation. ### Decision Making Results

The Pareto non-inferior solution sets of the above typical years were taken as the alternative scheme sets.