## 5 Simulation Experiments
### Rationality of Using a GA to Identify Skyline Solutions  
We motivate our selection of a GA to identify the skyline solutions using a quantitative discussion and a qualitative discussion. Table 2 exhibits the number of skyline solutions found by the proposed approach in the three water supply rate conditions; we can see that as the water supply rate increases, the number of generations that the GA needs and the number of solutions it finds also increase. This seems reasonable because a greater water supply means more branches in the irrigated region receive water and more combinations of possible solutions. Of course, when the number of skyline solutions increases, the GA needs more rounds of evolution to find the final results. Figures 6-8 present the evolution of the number of temporary skyline solutions in different generations of the GA in the three irrigation conditions with water supply rates of 0-25%, 25-50%, and 50-75%. Note that each figure contains 10 sub-graphs. The target experiments must consider five decision objectives, but each sub-graph can only present the conditions of two decision objectives at the same time. Thus, we created graphs for each pair of objectives. The points on each sub-figure represent the skyline results of a specific generation under two objectives. For example, in Figure 6g, the red crosses represent the skyline solutions of the 100 th generation under a water supply rate of 0-25%. The green circles represent the skyline solutions of the 300 th generation. The line connecting the points of the same generation is called that generation's skyline. According to the definition of the skyline query, all feasible solutions fall to the lower left of this boundary line. This line, thus, represents an upper bound on the performance of feasible solutions at multiple objective values. In the majority of the sub-figures, this skyline will gradually move to the upper right corner as the generations increase. This is because as the number of generations evolves, the algorithm can find feasible solutions that perform better at multiple objective values simultaneously. We can also see that as the number of generations increases in the three graphs, the results found by the GA also gradually become fixed, meaning that they are approaching the optimal solutions. In all figures, there are some exceptional cases. For example, Figure 6h,i have only one point and no skyline. This means that under a specific \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **0–25\%** & **25–50\%** & **50–75\%** \\ \hline Epoch & 313 & 1715 & 4751 \\ Skyline number & 10 & 171 & 484 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Numbers of GA evolutions and skyline solutions under different water supply rates. combination of objective values, only one solution meets user requirements. Further, the skylines of some figures, such as Figure 6a-d, do not move as generations increase. This means that the skyline found by the GA in the first generation is the same as the one found in the last generation. In both cases, the solution of this pair of objective value combinations is relatively simple, so other solutions cannot be found. Figure 6, which presents the results for a water supply rate of 0-25%, has more exceptional cases than the other figures. This means that obtaining a skyline solution under multiple objectives becomes more difficult and complex as the water supply rate increases. This conclusion is supported by the results presented in Table 2.  
We can understand the competitive relationships among the five decision objectives through observation of Figures 6-8. We first discuss the case of 0-25% water supply rates. Figure 6a,b,e show that a competitive relationship exists among the water demands of the crops. When the irrigated area of one type of crop increases, the irrigated areas of the other crops decrease. From Figure 6b,c,e,f, we can see that when the irrigated area of rice or sugar cane decreases, the irrigated area and agricultural economic benefits of the vegetables increase substantially.