## 3 Model Construction and Solving Algorithm
### Solving Algorithm
#### 3.2.2 Application of NSGA-II-TOPSIS-GCA Algorithm for MOODPR Problem
The scheme sets under 25%, 50%, 75%, and 95% inflow frequencies were denoted as A, B, C, and D, respectively. By calculating the attribute indexes of each scheme, the 4-dimensional original decision matrix of each scheme set was constructed. Figure 7: Pareto non-inferior solution sets: (**a**) P = 25%; (**b**) P = 50%; (**c**) P = 75%; (**d**) P = 95%. Table 2 shows the statistical results of attribute indexes for four scheme sets. Each scheme set had different 4-dimensional attribute indexes. In order to determine the decision schemes, it is necessary to use TOPSIS-GCA to evaluate the scheme sets. The 4-dimensional original decision matrix was normalized and the objective weights of 4-dimensional attribute indexes for the new 4-dimensional decision matrix in different typical years were calculated as follows: \begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Scheme Sets**} & \multicolumn{4}{c}{**Attribute Indexes**} \\ \cline{2-5}  & & \(\mathbf{\alpha}\) & \(\mathbf{\gamma}\) & \(\mathbf{\
u}\) & **WSI** \\ \hline \multirow{2}{*}{A} & Variation range & [0.750, 0.806] & [0.556, 0.857] & [0.743, 0.843] & [5.073, 11.603] \\  & Standard deviation & 0.0249 & 0.133 & 0.0279 & 2.095 \\ \hline \multirow{2}{*}{B} & Variation range & [0.588, 0.689] & [0.494, 0.667] & [0.827, 0.892] & [21.826, 29.463] \\  & Standard deviation & 0.0161 & 0.0486 & 0.0166 & 1.256 \\ \hline \multirow{2}{*}{C} & Variation range & [0.533, 0.639] & [0.400, 0.538] & [0.829, 0.990] & [23.758, 30.795] \\  & Standard deviation & 0.0170 & 0.0509 & 0.0407 & 1.413 \\ \hline \multirow{2}{*}{D} & Variation range & [0.250, 0.528] & [0.111, 0.389] & [0.862, 0.995] & [37.566, 57.470] \\  & Standard deviation & 0.0912 & 0.0594 & 0.0335 & 5.705 \\ \hline \hline \end{tabular} 

The 4-dimensional original decision matrix was normalized and the objective weights of 4-dimensional attribute indexes for the new 4-dimensional decision matrix in different typical years were calculated as follows: \begin{tabular}{c c} \hline \hline \multicolumn{1}{c}{**Scheme Sets**} & \multicolumn{1}{c}{**Positive Ideal Solutions**} & \multicolumn{1}{c}{**Negative Ideal Solutions**} \\ \hline A & \(R+=\) (0.0262, 0.0326, 0.0233, 0.0157) & \(R^{-}=\) (0.0243, 0.0211, 0.0265, 0.0358) \\ B & \(R+=\) (0.0272, 0.0294, 0.0240, 0.0210) & \(R^{-}=\) (0.0225, 0.0218, 0.0259, 0.0282) \\ C & \(R+=\) (0.0264, 0.0285, 0.0227, 0.0215) & \(R^{-}=\) (0.0241, 0.0212, 0.0273, 0.0279) \\ D & \(R+=\) (0.0320, 0.0378, 0.0222, 0.0192) & \(R^{-}=\) (0.0152, 0.0108, 0.0276, 0.0294) \\ \hline \hline \end{tabular} 

Table 4 shows the hybrid closeness degree of schemes and their sorts. The scheme with the largest hybrid closeness degree was the optimal scheme, and the scheme with the smallest hybrid closeness degree was the worst scheme. Table 5 shows the attribute indexes of final decision schemes for four scheme sets. The attribute indexes of different scheme sets varied greatly. The reliability \(\alpha\) in the wet year was 0.806, indicating that there were 29 ten-day periods to meet the comprehensive water demand. The reliability \(\alpha\) under 50%, 75%, and 95% inflow frequencies was 0.611, 0.556, and 0.467, which was 0.195, 0.250, and 0.339 lower than that of the 25% inflow frequency, indicating that only 22, 20, and 18 ten-day periods met the requirement of comprehensive water, and the other periods were affected. In comparison with the other inflow frequencies, under a 25% inflow frequency, the recoverability \(\gamma\) increased by 0.391 on average, the water shortage depth \(\
u\) decreased by 0.187 on average, and the _WSI_ decreased by 25.459 on average. On the whole, the attribute indexes in the wet year were better than those of other inflow frequencies.