## 5 Simulation Experiments
### Influence of Objective Ranking on Skyline Solutions  
In this section, we discuss the influence of the proposed approach on the ranking of solutions based on user prioritization of the objectives. For the sake of brevity, we focus on three cases suggested by experts from the Yunlin Management Office of the Irrigation Agency Verification (shown in Table 3), all under the condition of 25-50% water supply. Table 4 presents the results of ranking: Each solution in the table is one of the obtained skyline solutions. The first ranking combination gives the highest priority to rice yield and the second-highest to sugar cane yield. Last are vegetable yield, economic benefits, and the proportion not requiring compensation. In terms of the yield objectives, the yield objective of rice is near maximum yield (less restriction), whereas the yield objectives of the other crops remain in the middle values (moderate degree of restriction). A portion of the ranked results are presented in Table 4. Note that, to verify the rationality of the proposed ranking method (i.e., incorporating the expected objective scores mentioned in Section 3.2), we also present the results of the conventional calculation method that does not take the expected scores into account (i.e., the method in which larger values are better no matter the objective). In Table 4, we can see that the top three solutions ranked using our proposed approach all reached the objectives for vegetable yield, economic benefits, and compensation for stopping irrigation, then met the objectives of rice yield and sugar cane yield as much as possible, which better fits the government's thinking. The optimal solution output by a traditional top-\(k\) query (i.e., Solution 105) was not any of the top three solutions found by our skyline query, as their overall (integrated) scores were all lower than that of Solution 105. This confirms that the traditional top-\(k\) query method sometimes cannot find the solutions that best meet the decision-maker's needs. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{3}{*}{**Combination**} & **Rice** & **Sugar Cane** & **Vegetable** & **Agricultural** & **Compensation** \\  & **Expected** & **Expected** & **Expected** & **Benefit** & **for Stopping** \\  & **Score/Rank** & **Score/Rank** & **Score/Rank** & **Expected** & **Expected** \\  & & & & **Score/Rank** & **Score/Rank** \\ \hline
1 & 60\%/1 & 40\%/2 & 46\%/3 & 44\%/3 & 43\%/3 \\
2 & 30\%/2 & 30\%/3 & 30\%/1 & 60\%/1 & 30\%/2 \\
3 & 30\%/3 & 50\%/1 & 10\%/2 & 10\%/2 & 30\%/1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Expected scores and rankings adopted in this experiment. Finally, regarding the last skyline solutions ranked by the target algorithm (Solutions 169, 170, and 171), we found that although they reach demands for rice yield and sugar cane yield, performance in vegetable yield and economic benefits is almost nonexistent. This makes it difficult for these solutions to be listed as options regardless of the scoring method. The above arguments show that the ranking method proposed in this paper can provide a set of solutions that are prioritized more in line with user needs than traditional approaches. We also converted Solutions 1, 2, 3, 169, 170, and 171 into branch maps and observed their selection strategies. The results are displayed in Figure 11. A comparison of these figures shows that the greatest difference between these solutions is the selection of upstream and midstream branches. Solutions 1, 2, and 3 (red number in Figure 11) selected a small portion of the upstream branches for irrigation and completely abandoned the midstream branches, whereas Solutions 169, 170, and 171 (purple number in Figure 11) chose a small portion of the midstream branches and completely abandoned the upstream branches. Previous results indicate that the upstream Branches 2, 4, and 5 display better performance in vegetable yield, whereas the midstream Branches 23, 25, 27, 28, 30, 31, 34, and 37 present better performance in rice or sugar cane yield.