## 3 Model Construction and Solving Algorithm
### Solving Algorithm
#### 3.2.2 Application of NSGA-II-TOPSIS-GCA Algorithm for MOODPR Problem  
The new algorithm was used to solve the MOODPR problem for Jiaoyan-Shimen parallel reservoirs. The parameters of the algorithm include the population size \(N\), maximum iteration times \(Maxgen\), crossover probability \(pc\), mutation probability \(pmn\), crossover distribution index \(\eta_{c}\), and mutation dispersion index \(\eta_{m}\). The operation periods are 36 ten-day periods, and the water level represents the decision variable, which is generated randomly within its feasible ranges. The application of the algorithm includes initialization, multi-objective optimization, attribute index weighting, multi-attribute decision making, etc. The detailed steps are shown in Algorithm 1. ``` Input: The decision variables, objective functions, constraints, \(N\), \(Maxgen\), \(pc\), \(pmn\), \(\eta_{c}\), and \(\eta_{m}\). Output: \(\{\mathcal{S}^{1},\ \cdots,\ S^{N}\}\) and \(\{Z^{1},\ \cdots,\ Z^{T}\}\). Step 1: Initialization
1.1 \(N=100\), \(Maxgen\) = 1500, \(pc\) = 0.9, \(pmn=0.08\), \(\eta_{c}\) = 20, \(\eta_{m}\) = 20, \(Gen\) = 0 \(\leftarrow\) Parameter setting. 1.2 \(\{X^{i},\cdots,X^{N}\}\)\(\leftarrow\) Initialization population \(P_{0}\) randomly. Step 2: Multi-objective optimization
2.1 MOOM construction. 2.2 Initialized population \(P_{0}\) is non-dominated sorted, and individuals with better fitness are selected for genetic operation to generate the first-generation offspring population \(Q_{0}\). 2.3 \(Gen\) = \(R_{i}\) = \(P_{i}\cup Q_{i}\)\(\leftarrow\) Merge offspring population and parent population. 2.4 \(P_{i+1}=R_{i}[0:N]\)\(\leftarrow\) Individuals with better fitness are selected as the parent population. 2.5 Perform genetic operation to obtain new offspring population. 2.6 If \(Gen\) = \(Maxgen\), non-inferior solution set \(A\) = \(\{A_{1},A_{2},\cdots A_{100}\}\) is obtained, otherwise \(Gen\) = \(1+Gen\), return to step 2.3. Step 3: Attribute indexes weighting
3.1 \(F\) = \(\{f_{ij}\}_{100\times 4}\) (\(i=1,2,\ldots,100;j=1,2,3,4\)) \(\leftarrow\) Select attribute indexes \(\{a,\ \gamma,\ \
u,\ MS\}\), establish 4-dimensional original decision matrix. 3.2 \(Z\) = \(\{z_{ij}\}_{100\times 4}\)\(\leftarrow\) The 4-dimensional original decision matrix \(F\) = \(\{f_{ij}\}_{100\times 4}\) is normalized to obtain the 4-dimensional decision matrix. 3.3 Calculate the entropy \(H_{j}\) of attribute indexes: \[H_{j}=-\frac{1}{\ln\sum\limits_{i=1}^{100}c_{ij}\eta_{m}\eta_{ij}},\ \epsilon_{ij}=\frac{z_{ij}}{\frac{1}{\ln \sum\limits_{i=1}^{100}c_{ij}}}\] 3.4 Calculate the objective weight \(\omega^{\omega^{\prime\prime}}\) of attribute indexes: \[\omega^{\omega^{\prime\prime}}=\frac{1+\omega^{2(1-H_{j})}}{\sum\limits_{i=1}^ {4}\frac{1}{(1+\omega^{2(1-H_{j})})}}\] 3.5 Calculate the weighted normalized decision matrix \(R\) = \(\{r_{ij}\}_{100\times 4}\), where \(r_{ij}=z_{ij}\times\omega^{\prime\prime}\). Step 4: Multi-attribute decision making
4.1 Determine the positive ideal solution \(R^{+}\) and negative ideal solution \(R^{-}\) of each scheme: \[R^{+} = \left\{ \begin{matrix}max_{ij},j\\ min_{ij},j\end{matrix} \text{ is a }benefit\text{ }

#### 3.2.3 Attribute Indexes Selection

The attribute indexes are the standard to evaluate the quality of the schemes. In this paper, the 4-dimensional decision matrix is composed of the water supply reliability (\(\alpha\)), water shortage depth (\(\
u\)), water supply recoverability (\(\gamma\)), and water shortage index (\(WSI\)). (1) Reliability (\(\alpha\)) refers to the guaranteed degree of water supply. The calculation formula is as follows:

\[\alpha=\frac{\sum_{t=1}^{T}K_{t}}{T} \tag{8}\]

\[\left\{ \begin{aligned} K_{t}&=1,\sum\limits_{m=1}^{M}q _{z}(m,t)\geq Q_{z}(t)\\ K_{t}&=0,\sum\limits_{m=1}^{M}q_{z}(m,t)<Q_{z}(t) \end{aligned} \right. \tag{9}\]

where \(Q_{z}(t)\) denotes the comprehensive water demand flow of parallel reservoirs during period \(t\); \(K_{t}\) is used to determine whether the water supply satisfies the demand during period \(t\).