## 3 Algorithm Improvement and Scheme Evaluation
### Improvement Strategy for the NSGA-III Algorithm
#### 3.1.4 Algorithm Implementation Process:
The values of \(a_{j}^{k}\) and \(\beta_{j}^{k}\) were determined based on China's National Standard GB 50318-2017 _Code for Urban Danriage Engineering Planning Standards_[39] and, considering the actual sewage discharge situation in Jinzhong City, ecological water use is not included in the river pollution discharge calculation. Therefore, the COD concentration in the sewage discharged by domestic, agricultural, secondary industry, tertiary industry, and ecological water use for 2030 is set to 400, 200, 300, 500 mg/L, and 0 mg/L. The sewage discharge coefficient is set to 0.8, 0.2, 0.3, 0.8, 0. (6) Minimum Water Supply Guarantee Rate \(c_{j}^{k}\): Based on the actual water supply and demand situation in the study area and the water supply and demand forecast values for the planning year, the minimum water supply guarantee rates for domestic, agricultural, secondary industry, tertiary industry, and ecological water users in 2030 are set to 0.87, 0.7, 0.7, 0.80, and 0.80, respectively. ## 5 Results and Analysis

### Evaluation of Scheme Effectiveness

Using MATLAB R2022b programming, the improved I-NSGA-III algorithm was employed to solve the water resource optimization allocation model for Jinzhong City. To identify the optimal parameters, data from the 2030 drought scenario were selected. Multiple crossover probabilities Pc and mutation probabilities Pm were configured to solve the developed water resources optimization allocation model. Each parameter combination was executed 20 times, and the Hypervolume (HV) indicator was used to evaluatethe resulting Pareto front. Based on previous studies [30; 40], commonly used crossover probabilities include \(\mathrm{Pc}=0.7,0.8\), and \(0.9\), while mutation probabilities are typically set as \(\mathrm{Pm}=0.05\) or \(\mathrm{Pm}=1/\mathrm{D}\), where \(\mathrm{D}\) denotes the number of decision variables in the model. The computational results are summarized in Table 7. Ultimately, the crossover probability was set to \(\mathrm{Pc}=0.8\), and the mutation probability was set to \(\mathrm{Pm}=1/\mathrm{D}\). Relevant data were input, and the algorithm parameters were set as follows: population size \(N=200\), maximum number of iterations \(t_{\mathrm{max}}=200\), crossover probability \(P_{c}=0.8\), mutation probability \(P_{\mathrm{m}}=1/D\) (where \(D\) is the number of decision variables in the model). The algorithm solves the model to obtain a Pareto set containing multiple solutions, which must be filtered to obtain a suitable configuration scheme. Here, TOPSIS is used to select the relatively optimal solution from the set of options. For the model's three objective functions, no special adjustments are made, and their weights are considered equal, each being \(1/3\). In practical applications, decision-makers can set different objective weights based on personal preferences or use other evaluation methods to obtain a suitable water resource allocation scheme according to specific needs. The results of the water resource allocation for the planning year obtained using TOPSIS are shown in Figure 5.

The scheme with the highest comprehensive benefits was evaluated using a multi-system coupling coordination degree evaluation model (Section 3.2). The weights of the \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{\(\mathbf{P_{m}}\) = 0.05} & \multicolumn{2}{c}{\(\mathbf{P_{m}}\) = 1/D} \\ \cline{2-5}  & **HV (Median)** & **HV (Std.)** & **HV (Median)** & **HV (Std.)** \\ \hline \(\mathrm{P_{c}}=0.7\) & 0.42 & 2.56 \(\times\) 10\({}^{-5}\) & 0.48 & 3.15 \(\times\) 10\({}^{-5}\) \\ \(\mathrm{P_{c}}=0.8\) & 0.46 & 4.32 \(\times\) 10\({}^{-5}\) & 0.54 & 1.76 \(\times\) 10\({}^{-5}\) \\ \(\mathrm{P_{c}}=0.9\) & 0.37 & 6.32 \(\times\) 10\({}^{-5}\) & 0.45 & 1.58 \(\times\) 10\({}^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Computational Results of Hypervolume for Different Crossover and Mutation Probabilities in the 2030 Drought Scenario. Figure 5: Water resources allocation results for each administrative region in planning year.