## 4 Model Solving
### Application of NSGA-II-SEABODE for the MODRO Problem  
The NSGA-II-SEABODE algorithm was applied to the MODRO problem of Huangjinxia Reservoir, the established objective functions and constraints of the MODRO problem  
Figure 5: Flow chart of NSGA-II-SEACODE.  
were added into the algorithm. This paper selected five typical years for calculation, i.e., extraordinarily dry year (\(p=95\%\)), dry year (\(p=75\%\)), normal year (\(p=50\%\)), wet year (\(p=25\%\)) and extraordinarily wet year (\(p=5\%\)). The basic parameters of the algorithm are set as follows: the population size is \(N\), the maximum iteration times is _Maxgen_, the crossover probability is \(pc\), the mutation probability is \(pm\), the crossover distribution index is \(\eta_{c}\), the mutation distribution index is \(\eta_{m}\), and the operational periods are 12 months. Taking water levels as the decision variables, \(N\) individuals are randomly generated within the feasible ranges of water levels (upper and lower limits of water level). Through a series of steps, mainly including initialization, multi-objective optimization, multi-attribute decision making and stopping criteria, so repeatedly, until the stopping criterion is satisfied. The specific steps of NSGA-II-SEABODE are shown in Algorithm 1.  
```
Input: The MODRO problem, constraints, decision variables (water levels), population size \(N\), maximum iteration times _Maxgen_, crossover probability \(pc\), mutation probability \(pm\), cross distribution index \(\eta_{c}\), mutation distribution index \(\eta_{m}\). Output: \(\left\{X^{1},\cdots,X^{N}\right\}\) and \(\left\{S^{1},\cdots,S^{N}\right\}\leftarrow\) Final water levels and decision-making scheme. Step 1: Initialization
1.1 Setting parameters: \(N=100\), _Maxgen_ = 2000, \(pc=0.9\), \(pm=0.08\), \(\eta_{c}=20\), \(\eta_{m}=20\), \(Gen=0\).
1.2 \(\left\{X^{1},\cdots,X^{N}\right\}\leftarrow\) Initialization population \(P_{0}\) randomly. Step 2: Multi-objective optimization
2.1 Non-dominated sorting of initialized population \(P_{0}\), select individuals with good fitness. Perform genetic operations to generate the first-generation subgroup \(Q_{0}\).
2.2 \(Gen=2\), intermediate population \(R_{t}=P_{t}\cup Q_{t}\leftarrow\) Combine parent population with offspring population.
2.3 New parent population \(P_{t+1}=R_{t}[0:N]\leftarrow\) Fast non-dominated sorting and congestion calculation for \(R_{t}\), select individuals with good fitness.
2.4 New offspring population \(\leftarrow\) Perform genetic operations.
2.5 If \(Gen=Maxgen\), \(A=\left\{A_{1},\cdots,A_{100}\right\}\leftarrow\) Obtain alternative scheme set; else \(Gen=Gen+1\), go to Step 2.2. Step 3: Multi-attribute decision making
3.1 Decision matrix \(\leftarrow\)\(D=\left\{a,\gamma,\
u,MSI\right\}\) and \(A=\left\{A_{1},\cdots,A_{100}\right\}\).
3.2 \(k=4\), identify 4-order effective scheme set in the 4-dimensional attribute space \(\left[\alpha,\gamma,\
u,MSI\right]\).
3.3 \(k=k-1\), identify non-dominated superior schemes in each (\(k-1\))-order subspace; then identify the number \(x\) of (\(k-1\))-order effective schemes obtained by the intersection of all (\(k-1\))-order subspaces.
3.4 If \(x>1\), go to Step 3.3; else if \(x=1\), the scheme is directly output as the final decision-making scheme \(\left\{S^{1},\cdots,S^{N}\right\}\); else \(x=0\), then the scheme that occupies the largest number of subspaces is selected.
3.5 Terminate decision making. Step 4: Stopping criteria

If the stopping criterion is satisfied, stop; else go to Step 2.
```  
**Algorithm 1**NSGA-II-SEABODE