## 3 Algorithm Improvement and Scheme Evaluation
### Improvement Strategy for the NSGA-III Algorithm
#### 3.1.4 Algorithm Implementation Process:

Table 2: Convergence result of three algorithms. The test results are shown in Table 3. As observed, the IGD-based tests indicated that I-NSGA-III outperformed the original algorithm on test problems DTLZ1-3 (\(p<0.05\)) but did not show significant superiority on DTLZ4. In terms of HV, I-NSGA-III demonstrated significantly better performance across all test problems, indicating its overall superiority compared to the original algorithm. The reason behind these outcomes may lie in the reference-point enhancement strategy, which enables I-NSGA-III to explore previously unreachable regions in the objective space. These newly identified solutions increase the volume covered by the solution set, thus improving the HV metric. However, this strategy might also compromise convergence in certain regions or cause uneven solution distributions, resulting in inferior IGD values compared to the original algorithm. ### Evaluation of Water Resource Allocation Plans

#### 3.2.1 Establishment of an Indicator System

To assess the feasibility and rationality of water resource allocation plans, a multi-system coupling coordination degree evaluation model was developed from the perspectives of economy, society, and ecological environment, in conjunction with the current water resource status in Jinzhong City [36]. The specific indicator evaluation system is outlined in Table 4. If an indicator has a negative attribute, a smaller value is better for the subsystem; if an indicator has a positive attribute, a larger value is better for the subsystem. #### 3.2.2 Multi-System Coupling Coordination Degree Evaluation Model

The Analytic Hierarchy Process (AHP) and the entropy method are used to calculate the subjective and objective weights of the indicators, respectively. These results are then combined based on a specific formula (12,13) [37]. For a multi-level indicator system, the sum of the weights of all indicators at the same level equals 1. \[\lambda_{u}=\frac{\theta_{\text{u}}W_{\text{u}}}{\sum_{u}\theta_{\text{u}} \text{u}} \tag{12}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**First-Level Indicator** & **Second-Level Indicator** & **Unit** & **Attributes** \\ \hline \multirow{3}{*}{Economic} & per capita GDP & CNY & Positive \\  & Proportion of secondary industry & \% & Positive \\  & Proportion of tertiary industry & \% & Positive \\ \hline \multirow{3}{*}{Social} & per capita water supply & m\({}^{3}\)/person & Positive \\  & Water shortage rate & \% & Negative \\ \hline \multirow{2}{*}{Ecological} & per capita pollutants Emissions & \% & Negative \\  & Ecological water shortage rate & \% & Negative \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Multi-system indicator evaluation system. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Function** & **Model** & **IGD (Median)** & **IGD (Std)** & **IGD \(p\)-Value** & **HV(Median)** & **HV (Std)** & **HV \(p\)-Value** \\ \hline DTLZ1 & I-NSGA-III & \(6.08\times 10^{-4}\) & \(4.12\times 10^{-3}\) & — & \(8.31\times 10^{-1}\) & \(8.04\times 10^{-3}\) & — \\  & NSGA-III & \(1.19\times 10^{-3}\) & \(7.45\times 10^{-3}\) & 0.0001 & \(8.13\times 10^{-1}\) & \(2.89\times 10^{-1}\) & 0.0004 \\ \hline DTLZ & I-NSGA-III & \(2.76\times 10^{-4}\) & \(2.47\times 10^{-5}\) & — & \(5.71\times 10^{-1}\) & \(1.06\times 10^{-5}\) & — \\  & NSGA-III & \(2.83\times 10^{-4}\) & \(1.85\times 10^{-5}\) & 0.0021 & \(5.51\times 10^{-1}\) & \(1.88\times 10^{-5}\) & 0.0001 \\ \hline DTLZ3 & I-NSGA-III & \(1.15\times 10^{-2}\) & \(1.24\times 10^{-2}\) & — & \(5.41\times 10^{-1}\) & \(8.12\times 10^{-3}\) & — \\  & NSGA-III & \(1.78\times 10^{-2}\) & \(1.13\times 10^{-2}\) & 0.0348 & \(5.36\times 10^{-1}\) & \(1.54\times 10^{-2}\) & 0.0266 \\ \hline DTLZ4 & I-NSGA-III & \(3.09\times 10^{-4}\) & \(3.02\times 10^{-1}\) & — & \(5.50\times 10^{-1}\) & \(4.93\times 10^{-2}\) & — \\  & NSGA-III & \(3.79\times 10^{-4}\) & \(3.39\times 10^{-1}\) & 0.1559 & \(3.39\times 10^{-1}\) & \(1.33\times 10^{-1}\) & 0.0348 \\ \hline \hline \end{tabular}
\end{table}