## 3 Model Construction and Solving Algorithm
### Solving Algorithm
#### 3.2.2 Application of NSGA-II-TOPSIS-GCA Algorithm for MOODPR Problem
The maximum ten-day water shortage period of comprehensive and ecological water supply was 2.17 \(\times\) 10\({}^{7}\) m\({}^{3}\) and 3.6 \(\times\) 10\({}^{6}\) m\({}^{3}\), respectively. In the dry year, the comprehensive water demand could be satisfied in July, August, September, and October, and the ecological water demand could be satisfied in July and

Figure 9: Satisfaction degree of ecological water supply per ten-day period from May to April of the next year: (**a**) P = 25%; (**b**) P = 50%; (**c**) P = 75%; (**d**) P = 95%. August. Due to the lack of upstream inflow, it is difficult to fully satisfy the water demand in other months. Among them, in early and late May, late November, early December, early February, mid-March, and early and late April, the inflow of Jiaoyan reservoir was low, and Shimen reservoir undertook the main water supply task. In the flood season, the satisfaction degree of the ecological water demand was 90.14%, and the number of water-shortage periods was 6. In the non-flood season, the satisfaction degree was 76.73%, and the number of water-shortage periods was 11 ten-day periods. The maximum ten-day water-shortage period of comprehensive and ecological water supply was 2.41 \(\times\) 10\({}^{7}\) m\({}^{3}\) and 1.9 \(\times\) 10\({}^{6}\) m\({}^{3}\), respectively. In the extraordinarily dry year, due to the increase in upstream inflow in early and middle May, early July, late August, middle and late September, October, early and middle November, early and late March, and middle and late April, parallel reservoirs could meet the ecological and comprehensive water demand. In other periods, the ecological and comprehensive water supply were below the water demand line. In late November, early December, and late February, Shimen Reservoir undertook the main water supply task. In the flood season, the satisfaction degree of the ecological water demand was 81.48%, and the number of water-shortage periods was 9. In the non-flood season, the satisfaction degree was 72.44%, and the number of water-shortage periods was 12 ten-day periods. The maximum ten-day water-shortage period of comprehensive and ecological water supply was 4.46 \(\times\) 10\({}^{7}\) m\({}^{3}\) and 2.3 \(\times\) 10\({}^{6}\) m\({}^{3}\), respectively. The new method proposed in this paper is applicable to the multi-objective optimal operation decision of a single reservoir and s reservoir group and has a certain universality. It should be noted that when constructing the model, the conflict degree among various objectives should be considered to avoid redundant objectives, resulting in a large number of false non-dominated solutions. In addition, when evaluating the schemes, it is necessary to consider the reservoir operation mode, operation objectives, attribute indexes, and decision-makers' preferences, and also take into account many objective factors such as actual water conditions and operating conditions. By consulting relevant literature and similar studies, it was found that there are few studies on the multi-objective optimal operation decision of Jiaoyan-Shimen parallel reservoirs, and the application of TOPSIS-GCA in multi-attribute decision-making of reservoir operation is also rare. Reference [44] introduced the multi-attribute decision-making method based on GCA and TOPSIS to construct a multi-attribute risk decision-making model for flood control operations. Reference [45] adopted the NSGA-II-SEABODE algorithm to solve the multi-stakeholder coordinated operation model of the reservoir, and the relationship between irrigation benefit and ecological benefit was explored. Compared with previous studies, the NSGA-II-TOPSIS-GCA algorithm proposed in this paper can not only adapt to multi-objective problem solving and improve search efficiency but also consider the correlation among attribute indexes and the implicit relationship among the data, making the decision-making scheme more reasonable and feasible.