Multi-Objective Decision Support for Irrigation Systems Based on Skyline Query

Abstract

The steady increase in droughts worldwide has compelled many researchers to focus on water allocation. Multi-objective decision support for irrigation systems is a popular topic due to its relevance to the national economy and food supply. However, the majority of researchers have relied on conventional top-k designs for their decision support systems despite their limitations with regard to multi-objective systems. Thus, we propose applying a skyline query to the problem. As the input and output formats of skyline queries differ significantly from those of existing systems, we developed a new genetic algorithm and objective ranking. Qualitative and quantitative experiments using real-world data from Taiwan’s largest irrigated region demonstrate the effectiveness of the proposed approach.

1. Introduction

As climate change accelerates, many regions are beginning to experience less rainfall than their historical average, which means governments worldwide are facing major issues in water resource management [1,2,3,4]. Therefore, researchers have begun designing various decision support systems [5,6] to aid governments in distributing limited water resources; among these, decision support for irrigation systems is the most widely discussed [7,8]. For this topic, scholars usually develop methodology in two stages. They first use the agricultural knowledge of professionals to calculate the objective values of different irrigation solutions and then apply decision support algorithms to obtain a small selection of optimal solutions. For example, Jim’enez et al. [9] used the agronomic standards of each crop, such as crop coefficient, root depth, and soil water balance to evaluate the degrees of water wastage; a top-k query was then applied to select the solution with the least wastage. Cobbenhagen et al. [10] employed various crop-growth models to estimate total crop profit and found the solutions with the most profit using top-k. More recently, Zhang et al. [11] used the agronomic standards of the crops, soil water, and salt balance limits to construct a two-layer multi-objective agricultural water allocation model, which evaluates economic benefits and irrigation water productivity. Solutions were also obtained in this case using a top-k query. These examples illustrate the extensive development of the first stage of irrigation decision support, i.e., adding elements and developing models to obtain the objective values. However, all these examples simply applied a top-k query in the second stage of the algorithm to filter for feasible solutions. The current paper seeks to optimize irrigation decision support further by improving upon the second stage of the algorithm.

In the early stages of development, decision support systems used a single objective value as the selection criterion, which means that the best solution was simply the one with the highest objective value. This was the procedure followed by top-k query algorithms. Later, the concept of multiple objective values was proposed. Multiple target values are integrated into one score, and the one with the highest score is selected as the best solution. However, this approach is flawed in cases where objectives conflict with each other. If the parameters of top-k integration are not adjusted well, the algorithm will only return the best solution for one of the conflicting parties, and the user will lose the opportunity to learn about other solutions that may meet current needs [12,13]. Therefore, in 2001, Borzsony et al. [12] proposed the concept of skyline query, which is able to find good solutions to the target problem with seemingly contradictory components and return them all to the user. This concept overcomes the shortcomings of the top-k query. Many scholars have, thus, abandoned top-k in favor of the skyline query. For example, Kriegel et al. [14], Yang et al. [15], and Yang et al. [16] used skyline queries to support path decisions. Htoo and Ohs [17] and Hsu et al. [18] used the skyline query to assist in travel planning. Peng and Wang [19] and Zaman et al. [20] used the skyline query to assist social networks in critical person selection.

In this paper, we are, thus, prompted to verify the effectiveness of the skyline query for irrigation decision support. The role of this paper is to explore upgrading the second stage of existing irrigation decision support systems, so we do not focus on the first stage of the decision-making process, i.e., evaluating the objective values. Rather, we aim to adapt the skyline query to the field of irrigation, as existing skyline query algorithms cannot be directly applied to the irrigation decision support systems. Conventional skyline query algorithms can only process given values, such as price or distance from the hotel to the beach [12,13]. For more complex problems, users must first perform all necessary calculations before inputting the values to the system. We aimed to remove this time-consuming step by designing a new method. In addition, the computation time of skyline query algorithms increases exponentially with the amount of input data [13,21,22]. An irrigated region is usually composed of many areas, so a method for simplifying the computation involved was necessary. Finally, skyline queries judge each objective fairly and independently but tend to produce too many solutions [23,24,25], particularly when the number of objectives is high. While water allocation problems are complex, if the output is inconvenient for decision-makers, the system will not fulfill its role of decision support. We, therefore, modified our approach to output more reasonable solutions.

This paper contributes two novel algorithms to overcome the difficulties described above: a genetic algorithm (GA) to realize the decision support system and objective ranking to sort the solutions. We anticipate that the target system will produce n feasible solutions for each generation. The values of the decision objectives must be obtained for this generation, and a skyline query is performed to derive the fitness scores. The system must then generate the next generation of suitable solutions based on these fitness scores. As the number of generations increases, the system will identify the final skyline solutions. The logic of a GA is to (i) generate individuals, (ii) calculate fitness scores, and (iii) select outstanding individuals to survive. Thus, it is well suited to the problem, as skyline solutions can be obtained before all solutions are obtained. Furthermore, the GA distributes the skyline query algorithm to various generations for execution, so for each round, the number of solutions that the skyline query needs to process is not high. The second algorithm asks decision-makers to rank each decision objective in order of priority. Then, once the system has obtained all of the skyline solutions, it can calculate the degree to which each solution is likely to be favored. The solutions are, thus, ranked, increasing the ease of use for decision-makers. The effectiveness of the proposed method is verified using data collected from the largest irrigation system in Taiwan, the Cho Main Canal.

The remainder of this paper is arranged as follows. Section 2 discusses related works. Section 3 introduces the dataset and the naive objective values we used in this work. Section 4 describes the details of our algorithms. Section 5 presents our simulation experiments and their results. Finally, we draw our conclusions and discuss future works in Section 6.

2. Related Work

2.1. Decision Support Systems

The purpose of decision support systems is to simulate human logical reasoning to process information automatically and, thus, swiftly and accurately provide recommendations to decision-makers. This enhances decision-making quality, shortens decision-making time, and reduces decision-making costs [26]. Decision support methods are, thus, widely applied in various fields to establish an efficient expert system to solve optimization problems such as product design [27,28,29], recommendation systems [30,31,32], and system improvement [33,34,35]. Below, we introduce the two most common approaches to decision support systems: numerical optimization and heuristic algorithms. We also discuss the development of these two solutions in the field of water allocation.

Numerical optimization methods mainly solve optimization problems by constructing a mathematical model. The decision-makers must present one or more functions to define the objectives, input variables and limits, and then optimal solutions are obtained using mathematical methods. For instance, Ogbolumani et al. [36] proposed a food-energy-water-nexus multi-objective optimization framework based on mixed-integer programming. This framework considers the relationships between the energy, water, and food sectors; includes social, economic, and environmental restrictions; and ultimately identifies optimal land use allocation to maximize income and minimize costs and environmental impact. Schoonen et al. [37] developed a model predictive control framework based on mixed-integer bilinear programming. This framework integrates the short-term influence of crop growth with the long-term influence of profit and optimizes water allocation and irrigation machinery deployment to maximize profits. Zhang et al. [38] presented a mathematical model combining interval linear programming and interval multi-objective programming. Their model takes the characteristics of various irrigation branches in the irrigated area into account to coordinate three objectives: net benefit maximization, water resource allocation maximization in important irrigation areas, and seepage loss minimization.

Heuristic algorithms are a type of random search algorithm that uses special pruning mechanisms to reduce search branches, thereby increasing computational efficiency. Thus, heuristic algorithms can find near-optimal solutions that satisfy the objective conditions within a reasonable amount of time. For example, Ngo et al. [39] proposed compromise programming to approach the multi-objective problem by decomposing the original multi-objective vehicle routing problem into a minimized distance problem. They then used a hybrid version of the genetic algorithm with a local search algorithm to obtain solutions. Ngo et al. [40,41,42] also published several follow-up papers discussing how to combine compromise programming with the transformation of genetic algorithms to solve educational scheduling problems. Daqaq et al. [43] recently designed a multi-objective backtracking search algorithm to find the best compromise in multi-condition decision-making for power system configuration. Many scholars have also applied the concept of Electre to such problems. Aiello et al. [44] extended the Electre procedure to solve the difficult problem of unequal area facility layout. Asgharizadeh et al. [45] integrated Electre TRI and particle swarm optimization to solve multi-criteria inventory classification. Modjtahedi and Daneshvar [46] extended the Electre TRI approach to solving credit risk classification. López et al. [47] used the hierarchical Electre III method to solve the problems of university ranking under multiple criteria. Recently, Alla et al. [48] integrated the concepts of Electre III and differential evolution to assist cloud computing systems in task scheduling. In water allocation, Arif et al. [49] investigated the influence of soil moisture and water depth on rice growth stages under dry, flooded, and moderate irrigation procedures and then employed a GA to identify the most suitable irrigation water depths for each rice growth stage so as to maximize yield. Nguyen et al. [50] proposed an optimization framework based on the Max-Min Ant System algorithm, which integrates the growth models of crops to estimate crop yield and evaluate irrigation management strategies. Then, they used dynamic decision variable options to increase search efficiency and ultimately identify optimal irrigation and fertilization plans. Wang et al. [51] developed a water allocation model combining a simulated annealing algorithm and particle swarm optimization. This model simultaneously considers economic, social, and environmental goals for Yinchuan City in simulation scenarios with 50%, 75%, and 90% precipitation.

All of the methods described above effectively identify water allocation solutions, but all are subject to limitations. Most combine multiple features into a single score, which tends to produce solutions dominated by strong features, overlooking the needs of weaker features. In addition, they only provide the optimal solution and neglect to present other next-best solutions. Thus, in the event that users are dissatisfied with the optimal solution, they can only obtain other options by modifying the parameters of the model.

2.2. Skyline Queries

A skyline query is a multi-condition query algorithm [12,13,52,53,54,55]. If users give the algorithm multiple conditions and a database, they will be able to identify all of the data points that meet at least one of their conditions from the database. Skyline queries are superior to top-k algorithms, which can produce solutions that meet none of the users’ conditions.

Past studies have mainly adopted two approaches to realize skyline queries: block nested loop algorithms [12] and branch-and-bound skyline algorithms [13]. The former [12] are the most intuitive method, starting with a set of candidate skyline results and then performing dominance checks for each data point p. If p is found to be dominated by no other points, then p is likely to be a skyline point. Thus, p is placed in the skyline result set. If p is dominated by another point, then it is not a skyline point and must be removed. The algorithm does not terminate until dominance checks between all point pairs in the database have been performed and every point has been checked. The points within the skyline result set in the end are then the results of the skyline query. The greatest shortcoming of this approach is that dominance checks must be performed for every pair of points in the database, so when the amount of data increases, computation time increases exponentially. However, the algorithm is effective, simple, and suitable for databases with a limited number of data points or for a large number of appropriately grouped data points.

Branch-and-bound skyline algorithms are based on data trees [13]. Before executing the query, this algorithm first constructs a tree structure to store the data points. Next, it uses an approximate feature to represent the data points in the same node in the tree. If this approximate feature is dominated by another point, then it is impossible for any of the data points in this node to become the skyline query results, and they are all deleted. With this method, the algorithm can swiftly remove large quantities of points that cannot be skyline points, making it much more efficient than block nested loop algorithms. However, an index tree structure of the dataset can only be established when all of the data point values have been calculated. Thus, it is not suitable for cases where the data point values require additional calculation. The selection of an appropriate skyline method depends on the characteristics of the target application.

New skyline calculation frameworks are constantly being proposed, such as the multiple skyline layers algorithm, group-based skyline query, and representative G-skyline query. The multiple skyline layers algorithm can more swiftly divide the original dataset into the first i skyline layers than past methods. The group-based skyline framework divides multiple skyline layers into primary groups and subgroups. The representative G-skyline query remedies the shortcomings of too many groups in the group-based skyline framework [56]. Recently, Zheng et al. [57] proposed a skyline query methodology based on user preferences. This approach reduces redundant dimensions by (1) eliminating the dimensions that the user is not interested in and (2) prioritizing dimensions based on the degree of user interest. This ultimately reduces the size of the candidate set and makes the search results closer to user needs. Finally, the dynamic skyline query was proposed for online medical diagnostic systems [58]. This approach ensures the efficacy and privacy of the skyline method while searching through encrypted data and increases the applicability of the skyline query approach.

3. Dataset and Calculation of Objective Values

In this section, we introduce the dataset used to verify the proposed methods and define the objective values used in our experiments. We also present an example to highlight the difference between applications of a skyline query and a top-k query for irrigation decision support.

3.1. Dataset

The datasets used in our experiments come from the Jhuoshuei River irrigated region in the Chiayi-Tainan Plain of Taiwan. We chose this region because it is the largest agricultural area in Taiwan, with the highest agricultural yield and the largest number of irrigation canals. The target region has a total of 55 irrigation branches that belong to three major branch systems. The relationships between them are shown in Figure 1. The data formats for each branch and the relationships between them are presented in Appendix A (

Table A1. All data fields and details for the Jhuoshuei River irrigated system.

BranchIndex	Level of the Branch	Rice Cultivation Area (Ha)	Sugarcane Cultivation Area (Ha)	Fruit Cultivation Area (Ha)	Water Transportation Loss (Ton/Day)
1	Main	672.812	323.275	10.733	80,438.4
2	Secondary	39.336	14.383	0.223	21,859.2
3	Third	13.813	12.888	1.218	7430.4
4	Fourth	12.122	15.344	86.672	26,784
5	Fifth	147.002	198.879	79.775	23,414.4
6	Sixth	33.256	162.446	751.906	18,748.8
7	Seventh	46.523	74.284	10.638	65,664
8	Eighth	80.765	16.445	2.257	2073.6
9	Ninth	54.170	8.246	1.811	3456
10	Tenth	11.702	41.582	4.460	3456
11	Eleventh	28.435	27.794	0.000	0
12	Twelfth	3.753	38.689	3.607	0
13	Thirteenth	3.371	96.550	3.514	0
14	Fourteenth	25.543	16.810	5.971	4147.2
15	Fifteenth	87.234	285.317	43.294	35,337.6
16	Sixteenth	4.192	12.444	85.320	4060.8
17	Seventeenth	1.255	26.508	0.993	3628.8
18	Eighteenth	52.105	135.147	9.597	8121.6
19	Nineteenth	0.000	15.869	0.858	3888
20	Twentieth	1.826	55.617	8.344	4665.6
21	Twenty-First	3.342	44.632	13.456	2851.2
22	Twenty-Second	196.293	244.900	4.369	98,150.4
23	Twenty-Third	29.312	52.098	0.798	20,390.4
24	Twenty-Fourth	2.893	1.613	0.000	5443.2
25	Twenty-Fifth	8.335	21.794	0.603	2592
26	Twenty-Sixth	54.010	184.233	10.751	9504
27	Twenty-Seventh	12.083	26.985	1.927	1615.68
28	Twenty-Eighth	5.521	37.843	2.818	2332.8
29	Twenty-Ninth	6.539	58.847	0.000	5443.2
30	Thirtieth	42.948	49.289	119.655	7516.8
31	Thirty-first	20.041	62.005	13.662	28,598.4
32	Thirty-Second	12.322	183.894	240.109	10,627.2
33	Thirty-Third	14.912	122.325	16.456	13,478.4
34	Thirty-Fourth	7.673	153.188	8.202	37,238.4
35	Thirty-Fifth	0.000	9.438	0.000	14,169.6
36	Thirty-Sixth	13.095	103.040	2.281	5616
37	Thirty-Seventh	1.452	102.279	2.746	13,132.8
38	Thirty-Eighth	121.679	226.800	10.147	125,280
39	Thirty-Ninth	218.694	190.622	3.928	10,972.8
40	Fortieth	73.351	130.311	41.617	15,552
41	Forty-First	122.740	531.748	0.885	30,499.2
42	Forty-Second	55.070	85.936	19.303	3715.2
43	Forty-Third	30.574	106.576	0.835	6393.6
44	Forty-Fourth	29.973	70.183	18.646	7603.2
45	Forty-Fifth	373.423	614.132	1.278	21,513.6
46	Forty-Sixth	41.354	204.913	10.792	10,627.2
47	Forty-Seventh	43.544	164.834	3.998	4924.8
48	Forty-Eighth	34.225	126.595	7.902	2419.2
49	Forty-Ninth	8.043	44.074	2.953	2851.2
50	Fiftieth	5.460	8.949	125.072	14,428.8
51	Fifty-First	0.000	2.022	88.407	4233.6
52	Fifty-Second	3.190	0.373	1.059	0
53	Fifty-Third	1.296	16.369	1.823	3369.6
54	Fifty-Fourth	13.197	4.987	5.339	4147.2
55	Terminal	0.655	2.399	0.234	17,280

,

Table A2. The relationships between all the branches (Part 1).

		Downstream
		1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35
upstream	1	0	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0
	2	0	0	1	1	0	0	1	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	3	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	4	0	0	0	0	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	5	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	6	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	7	0	0	0	0	0	0	0	1	1	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	8	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	9	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	10	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	11	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	12	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	13	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	14	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	15	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	16	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	17	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	18	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	19	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	20	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	21	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	22	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0
	23	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1	1	1	0	0	0	0	0	0	0	0
	24	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	25	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	26	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	27	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0
	28	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1	1	0	0	0	0
	29	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	30	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	31	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1	1	0
	32	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	33	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	34	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1
	35	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	36	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	37	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	38	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	39	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	40	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	41	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	42	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	43	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	44	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	45	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	46	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	47	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	48	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	49	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	50	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	51	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	52	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	53	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	54	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	55	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0

and

Table A3. The relationships between all the branches (Part 2).

		Downstream
		36	37	38	39	40	41	42	43	44	45	46	47	48	49	50	51	52	53	54	55
upstream	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	3	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	5	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	6	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	7	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	8	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	9	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	10	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	11	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	12	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	13	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	14	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	15	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	16	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	17	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	18	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	19	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	20	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	21	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	22	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	23	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	24	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	25	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	26	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	27	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	28	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	29	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	30	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	31	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	32	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	33	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	34	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	35	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	36	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	37	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	38	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	39	0	0	0	0	1	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0
	40	0	0	0	0	0	1	0	0	1	0	0	0	0	0	0	0	0	0	0	0
	41	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0
	42	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0
	43	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	44	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0
	45	0	0	0	0	0	0	0	0	0	0	0	1	1	1	0	0	0	0	0	0
	46	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	47	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	48	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	49	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	50	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1	1	1	1
	51	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	52	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	53	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	54	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	55	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0

). The black text presents the numbers of the branches, and the thickness of the lines represents the width of the branches. According to the irrigation data of 2017 provided by the Yunlin Management Office of the Irrigation Agency, the main crops in this irrigation area are rice, sugar cane, and vegetables, with total cultivation areas of 2926.456 has, 5568.742 has, and 1893.242 has, respectively. The total yield per hectare of the three types of crops was 5776 kg, 47,836 kg, and 18,367 kg, respectively, and their daily standard water demand was 183.6, 44.2, and 52.41 m³, respectively. Their farm prices per kilogram were NTD (new Taiwan dollar) 22.13, 0.96, and 28.78, and the government ultimately set the compensation amounts per hectare for the three types of crops at NTD 93,000, 82,000, and 75,440. Most papers use data from multiple years to ensure the accuracy of the objective values; however, as the focus of the current paper is the methodology of filtering solutions and not the accuracy of prediction, we use data from only a single year. We verify the proposed approach by measuring its performance in selecting a small set of solutions among many. Performance is, thus, related to the number of solutions and not the amount of input data.

3.2. Calculation of Objective Values

We consider five objective values: rice yield, sugar cane yield, vegetable yield, overall agricultural economic benefits, and compensation for stopping irrigation. For convenience in interpretation, we converted the objective values into percentages and considered the proportion of farmers not requiring compensation to make all objective values follow the larger-the-better criterion. We use these five relatively intuitive target values to create a simple model for the first stage, as our focus lies on the development of the second stage of decision support. The proposed algorithm accepts the output of the first stage as input, so more sophisticated models could be substituted for the first stage without any change to the performance of our filtering approach. The five selected objective values are calculated as follows.

(1)

Total cultivation area of each crop is calculated as follows:

$$O_i={\sum }_{j=1}^M\left(A_i^j\times C^j\right),$$

(1)

where i and j indicate crop type and serial number of irrigated areas, respectively; O_i denotes the total cultivation area of crop i in the current solution; M represents the total number of irrigated areas; A^j_i is the cultivation area of crop i along irrigated area j; and C^j indicates whether irrigated area j is selected in the current solution, equaling 1 if it is selected and 0 if it is not.

(2)

Total yield of each crop is P_i = O_i × U_i, where i indicates the crop type; P_i is the total yield of crop i in the current solution; O_i denotes the total cultivation area of crop i in the current solution, and U_i is the average yield of crop i per unit of area based on agronomic standards.

(3)

Total output value of each crop is V_i = P_i × W_i, where i indicates the crop type; V_i is the total output value of crop i in the current solution; P_i is the total yield of crop i in the current solution; and W_i denotes the average output value of crop i per unit of weight recorded in agronomic standards.

While directly using agronomic standards for the second and third objectives may not be the most accurate approach, it is beyond the scope of this study. If the readers need accurate, objective values, other approaches may be applied to predict relevant values without any adverse impact on the proposed algorithm.

(4)

Total compensation for stopping irrigation is calculated as follows:

$$T={\sum }_{i=1}^N\left(\left({\sum }_{j=1}^M{A}_i^j\right)-O_i\right)\times Q_i,$$

(2)

where i and j indicate crop type and a serial number of irrigated area, respectively; T denotes the total compensation for stopping irrigation under the current solution; N represents the number of crop types; M represents the total number of irrigated areas; A^j_i is the cultivation area of crop i along irrigated area j; O_i denotes the total cultivation area of crop i in the current solution; and Q_i is the compensation for stopping irrigation for crop i per unit of area.

(5)

Total demand of irrigation water resources is calculated as follows:

$$S=\left({\sum }_{i=1}^N{O}_i\times R_i\right)+{\sum }_{j=1}^M{D}^j,$$

(3)

where i and j indicate crop type and serial number of irrigated area, respectively; S denotes the total demand of water resources under the current solution; N represents the number of crop types; M represents the total number of irrigated areas; O_i denotes the total cultivation area of crop i in the current solution; R_i is the demand of water resources of crop i per unit of area; and D^j is the amount of water resources wasted by each area.

3.3. Comparison of Skyline Query and Top-k Query for Irrigation Decision Support

In this section, we compare the results of the proposed algorithm to those of a top-k query. We consider the irrigation system depicted in Figure 2, and the six feasible solutions under consideration by the government are presented in

Table 1. Skyline and top-k results, for example, in Figure 2.

Index Numberof Plan	Selected Branch	Rice Yield	Cane Yield	Vegetable Yield	Skyline Result	Summation of All Yields	Top-k Result
1	1, 2, 3	35	21	17	v	73	v
2	1, 2	25	16	14		55	v
3	1, 4	16	22	10	v	48
4	1, 3	25	17	12		54	v
5	1, 5	16	13	19	v	48
6	1	15	12	9		36

. In a skyline query, each solution is regarded as a data point. Thus, we selected only three objective values (rice yield, cane yield, and vegetable yield, which were calculated as discussed in the previous section) so that we could plot the results in a three-dimensional (3D) format. A skyline query checks each objective value for the solutions in pairs, seeking solutions with at least one objective value that is not poorer than that in the other solutions (i.e., solutions that are not dominated by other solutions). For example, in Table 1, all of the objective values (i.e., rice yield, cane yield, and vegetable yield) in Solution 2 are poorer than those in Solution 1, so the system considers Solution 2 to be dominated by Solution 1 and, thus, removes it. Visually, in Figure 3a, Solution 2 is below and to the left of Solution 1. Solution 1 has a smaller cane yield than Solution 3 but performs better than Solution 3 in the other two objectives. Thus, Solutions 1 and 3 do not dominate each other, meaning that both are feasible. Overall, Solutions 1, 3, and 5 are not dominated by any of the other solutions: Solution 1 is the best solution for rice yield, and Solutions 3 and 5 are the best solutions for cane and vegetable yields, respectively. Clearly, all three of these solutions meet government needs.

To apply a top-k query, we must first integrate multiple objective values into a single value, as shown in the summary column of Table 1. The top three solutions are Solutions 1, 2, and 4. However, these solutions only rank the highest for rice yield; all objective values in Solutions 2 and 4 are poorer than those in Solution 1, as shown in Figure 3b. The best solutions for cane yield (i.e., Plan 3) and vegetable yield (i.e., Plan 5) are not selected. These results, thus, lack the validity of those presented in Table 1. This example represents a classic shortcoming of the top-k query: only one among conflicting solutions is selected, as the weights of different objective values are not adjusted well. In contrast, the skyline query can find a set of solutions in which all objective values are optimized by at least one solution.

4. Algorithms

Figure 4 presents a flow chart of the proposed algorithm. First, the user designates the areas that the system must consider in the target region and the total water resources that can be used and ranks each decision objective. The system, then, randomly generates the first generation of possible solutions and begins the process of generation evolution. At the beginning of each generation, the system performs a skyline query: the target of the query is the union set of all of the solutions in this generation and the temporary skyline solutions identified from previous generations. Next, the system regards these query results as the temporary skyline solutions of this generation and calculates the fitness scores of each solution. Based on these fitness scores, the system, then, performs crossover and mutation operations to produce the next generation of solutions. When the system has evolved to the number of generations set by the user, it regards the temporary skyline solutions of this generation as the final results. As there may be many solutions, the system arranges them based on the provided ranking [59,60].

4.1. Generation of Initial Solutions

To enhance the learning efficiency of GAs, researchers usually use random functions to generate the initial generation to ensure the individuals are evenly spread throughout the feasible region. This is an effective approach because the values under consideration are usually independent [61]. However, this approach can produce (1) solutions with discontinuous water supply conditions and (2) solutions with total water usage outside of the range set by the user. The first case is illustrated in Figure 5a, where the 1 and 0 on each branch represent values generated by a random function. We can clearly see that water should be allocated to Branch 5, but water cannot reach Branch 5 from upstream (Branch 1). The second case is illustrated in Figure 5b,c, where we assume that decision-makers allocate 30~50% of the water allocated in previous years. Again, the 1 and 0 on each branch indicate whether water is allocated. In Figure 5b, the values on all of the branches are 1, which means that the total water demand is far greater than the available water amount. In contrast, the values on all of the branches in Figure 5c are 0, so the total water demand is again not within the set range.

To solve the problems described above, we designed the following approach to generate initial solutions, where the amount of available water [α, α + σ] is set by the user:

Step 1.

Calculate the total water demand β when all of the considered areas are given water and set the total water cutoff γ of the current irrigated region as 0.

Step 2.

Select the area most likely to have its water cut off, area e, and calculate its total water demand downstream, δ.

Step 3.

Calculate the value of β – α – γ − δ, which equals η, and determine the next step based on η, which may fit one of three cases: η > σ, σ > η > 0, or η < 0. First, if η > σ, it means that after the water supply to e has been cut off, the total water usage of the system will still be greater than the upper limit; in this case, we cut off the water to e, update the total water cutoff from γ to γ + δ, find the next area most likely to have its water cut off, and return to Step 2 to repeat the cutoff action. Next, if σ > η > 0, it means that after the water supply to e has been cut off, the total water usage of the system falls within the set range; in this case, we cut off the water to e, consider the current result as an initial solution, and end the selection operation. Finally, if η < 0, it means that after the water supply to e has been cut off, the total water usage of the system is below the lower limit. Thus, this cannot be one of the solutions. The system then finds the next area most likely to have its water cut off and returns to Step 2 to repeat the cutoff action.

Note that if the range set for the water supply is too small, the system may check all areas but be unable to find any suitable initial solutions. In this case, a prompt is given to the user to reset the water supply range. Next, the objective values for each solution are calculated as described in Section 3.2.

4.2. Realization of Skyline Query

A skyline query is performed at the beginning of each generation. Suppose the algorithm is performing the evolution of the nth generation of solutions; the subject of the skyline query is the union of the two following sets: (1) all of the solutions in generation n and (2) all of the temporary skyline solutions from generations 1 to n − 1. If n is 1, then the temporary skyline solutions are an empty set. The branch-and-bound search [13] algorithm is selected for most applications. However, this algorithm must construct a tree data structure for the query subject; given the small quantity of dynamic data (i.e., data that often change and requires additional calculations) discussed in this study, the block nested loops algorithm [12] is more appropriate, as no data preprocessing is required for this algorithm [62]. Only two solutions need to be compared, and this can be achieved directly in each dimension. In other words, the algorithm of this system can parallelize computation, that is, obtain the values of the decision objectives in every solution in the target generation while comparing the actual solutions with the earlier calculated solutions. Its greatest shortcoming is that for large quantities of data, computation time increases exponentially. However, for our purposes, only a small amount of data needs to be compared each time (only the solutions of one generation and the few temporary skyline solutions identified in the previous generations). Below, we outline the application of the block nested loop algorithm to the target algorithm. To facilitate our explanation, we refer to all of the solutions in this generation as P = {p₁, p₂, ..., p_m} and the temporary skyline solutions as Q = {q₁, q₂, ..., q_n}. If the solution currently being checked is p_i, then p_i will be compared to all of the solutions in P∪Q aside from itself. Suppose the solution being compared to p_i is r_j, and 0 < j < m + n. Then, the comparison result must be one of the three following cases:

Case 1:

If all of the objective values in p_i are poorer than those in r_j (i.e., p_i is dominated by r_j), then we add 1 to the number of times p_i is dominated.

Case 2:

If all of the objective values in p_i are better than those in r_j (i.e., p_i dominates r_j), then we record that r_j is dominated in this generation.

Case 3:

If p_i and r_j each have better and poorer objective values than the other (i.e., p_i and r_j do not dominate each other), then no action needs to be taken.

After p_i has been compared to all of the other solutions P∪Q aside from itself, we implement the block nested loops algorithm [12]: (1) record dom, the number of times that p_i was dominated (this number serves as the foundation of the fitness function during the crossover operation in the next stage), and (2) keep all of the solutions that were not dominated in P∪Q as the temporary skyline solutions of this generation.

4.3. Production of Next Generation of Solutions

The next generation of solutions is produced using the classic steps of a genetic algorithm [63,64]: (1) convert the solutions into chromosomes, (2) calculate the fitness function of each solution, (3) perform crossover and mutation of chromosomes, and (4) fine-tune the chromosomes.

Suppose the target region contains a total of n areas. We use n bits to represent each solution. If one bit is 0, irrigation to the corresponding area has been stopped. If not, then water is still supplied. Note that upstream–downstream continuity is considered during the production of the initial solutions and in their crossover and mutation with subsequent chromosomes. The water supply must also fall within the set range.

The fitness function of each solution is designed to equal (1/(dom + 1)), where dom denotes the number of times each solution is dominated; this was calculated in the previous section. The more times a solution is dominated, the further away it is from the final skyline solutions; the fewer times it is dominated, the closer it is.

An issue unique to irrigation decision support is the preservation of upstream–downstream relationships among waterways. Applying conventional GAs crossover and mutation rules would result in (1) solutions with discontinuous water supply conditions and (2) solutions with total water demand outside of the set range. We, therefore, designed new chromosome crossover and mutation rules. Suppose the chromosomes crossing over are p and q: The algorithm will first randomly select a bit as the crossover point and exchange all of the chromosomes in p and q downstream of the point. Note that if an exchanged area in a chromosome does not connect upstream, all of the bits in the area linking upstream to the canal are changed to 1. Next, the algorithm randomly selects a bit in the chromosome as the mutation point. If the bit is 0, the algorithm checks whether the point is connected upstream. If so, then the bit is changed from 0 to 1. If not, then the bit is not mutated, and another bit is selected for mutation. If the bit is 1, then the bit at this point and all the bits downstream are changed from 1 to 0.

Finally, when the new generation of chromosomes has been produced, we check whether the total water demand of the newly produced chromosomes falls within the set range. If a new chromosome fits, it directly becomes a member of the next generation. If not, it is fine-tuned. The first step of fine-tuning is generating a random value for all of the areas. If the total water demand of the new chromosome is below the set range, the areas are checked, beginning with the one with the highest random value. If the bit value of this area is 0 and it is connected upstream, we change the bit value of this branch to 1, meaning that it is given water. In contrast, if the total water demand of the new chromosome is greater than the set range, we check the areas, beginning with the one with the lowest random value. If the bit value of this area is 1, we change the bit value of this area and all of the bit values downstream to 0, meaning that no water is given to this area or those downstream. The above process continues until the total water demand of this chromosome falls within the set range.

4.4. Ranking of Final Skyline Solutions

Once the algorithm reaches the number of generations set by the user, the temporary skyline solutions of the last generation are regarded as the final results. Note that under the instruction of experts, we used two types of ranking standards in this study, including a priority score and an expected score. For the priority score imp_i, suppose the imp_i of decision objective i is greater than the imp_j of decision objective j. This means that the government attaches more importance to decision objective i than to decision objective j. The expected score exp_i is the value the government expects the decision objective i to reach. While objective values are often considered to be either larger the better or smaller the better, in practice, governments generally have an acceptable range, and neither values above this range nor below are acceptable. These two scores are applied to the proposed algorithm as follows.

Suppose the priority scores and expected scores for l decision objectives are (imp₁, imp₂, ..., imp_l) and (exp₁, exp₂, ..., exp_l): the k final skyline solution results are {f₁, f₂, ..., f_k}, and the values of solution f_i in each objective are (o_i1, o_i2, ..., o_il). Then, the solutions are ranked as follows.

Step 1:

First, compare the objective values of f_i (o_i1, o_i2, ..., o_il) to the expected scores (exp₁, exp₂, ..., exp_l). If the value of objective j is better than the expected value, then this value is replaced by the expected score. If the value of objective j is equal to or poorer than the expected score, then the value of objective j is retained. After the comparisons are completed, the adjusted values of f_i are defined as (c_i1, c_i2, ..., c_il).

Step 2:

The adjusted values are used to calculate the score of f_i: Sc_i=${\displaystyle {\sum }_{j=1}^l{c}_{ij}}\times im{p}_j$.

Step 3:

After the scores of all of the solutions (i.e., f₁~f_k) are calculated, they are ranked from largest to smallest. Note that if the scores of two solutions are the same, we compare the weighted raw data of the two solutions using their priority scores, i.e., ${\displaystyle {\sum }_{j=1}^l{o}_{ij}}\times im{p}_j$, as the basis of their ranking. If they have the same weighting, we compare the values of each decision objective, beginning with that with the highest priority score, until we find the optimal solution.

5. Simulation Experiments

5.1. Introduction to Experimental Parameters

The number of chromosomes, generation size, crossover rate, and mutation rate in the GA in our experiments were set at 55, 100, 80%, and 10%, respectively. To accelerate the experiments, we set the target algorithm to terminate whenever the results do not change for 200 consecutive generations. Because this paper is the first to apply GAs to obtain skyline query solutions for agricultural irrigation systems, there are no numerical values from previous studies on which to base our parameters. We, thus, applied a trial-and-error method to determine which set of parameters would obtain results as quickly as possible. For the sake of brevity, we do not detail this trial-and-error process here. To verify the model’s performance in different situations, we consulted with experts from the Yunlin Management Office of the Irrigation Agency, who advised three water supply rates that are commonly considered: 0~25%, 25~50%, and 50~75%. Experiments were run using MATLAB R2022a.

5.2. 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. Numbers of GA evolutions and skyline solutions under different water supply rates.

	0~25%	25~50%	50~75%
Epoch	313	1715	4751
Skyline number	10	171	484

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.

Figure 6, Figure 7 and Figure 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 100th generation under a water supply rate of 0–25%. The green circles represent the skyline solutions of the 300th 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 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 Figure 6, Figure 7 and Figure 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. This demonstrates that a close relationship exists between the irrigated area and the agricultural economic benefits of vegetables. Finally, Figure 6d,g,i,j show that the irrigated areas of the three types of crops do not have a clear impact on the compensation for stopping irrigation. This is because, with such a small water supply, almost all of the farms must stop farming, resulting in little differences in the compensation.

Next, we discuss the case of 25~50% water supply rates. From Figure 7a, we can see that the irrigated areas of rice and sugar cane do not significantly influence each other. However, Figure 7b,e show that any decrease in the irrigated areas of rice or sugar cane greatly increases the irrigated area of vegetables. Figure 7c,f,h similarly indicate a close relationship between the irrigated area and the agricultural economic benefits. However, in Figure 7d, the amount of compensation for stopping irrigation for rice, which was the highest among the three crops, dropped by approximately 16%; the proportion not requiring compensation unexpectedly increased. This means that with water supply rates of 25~50%, the strategy of reducing the irrigated area of the thirsty rice crop exerts varying degrees of positive influence on the irrigated areas of sugar cane and vegetables, agricultural economic benefits, and compensation for stopping irrigation. Lastly, there is the case of 50~75% water supply rates. Based on Figure 8, we arrive at the same conclusions that we did for Figure 7: reducing the irrigated areas of rice and sugar cane increases the irrigated area and agricultural economic performance of vegetables.

These examples confirm that the skyline query concept introduced in this paper can find solutions when the objective values are conflicting (e.g., irrigated area and economic considerations), thereby overcoming the problems encountered by traditional top-k queries. They also show that the final result found by the proposed algorithm aligns with the intuitive choice. This verifies that the skyline query is often more suitable than the traditional top-k query.

We compiled statistics of the branches selected in all of the skyline solutions to understand the degree of importance of different branches in the irrigated region. Figure 9 displays the number of the branches and the number of times they are chosen under different water supply rates. The black numbers are the numbers of the branches, and the numbers in other colors represent the number of times they appear in the skyline results under different water supply rates. For instance, Branch 2 in Figure 10 was chosen 9, 95, and 484 times under the water supply rates of 0~25%, 25~50%, and 50~75%, respectively. From Table 2, we can see that there were 10, 171, and 484 skyline solutions under these conditions; this means that Branch 2, being a crucial branch, was almost always chosen in the shown experiments.

Under the 50~75% water supply rates, the algorithm chose to maintain irrigation to the three main branches (Branches 1, 22, and 38) and gradually reduce irrigation to the branches so as to ensure the most efficient allocation of water resources. Interestingly, over half of the solutions chosen by the target approach do not prioritize irrigation to the wider upstream branches but rather focus on the downstream branches as the main subjects of irrigation (Branches 23, 26, 27, 30, 39, 40, 41, 42, 45, 46, etc.). This contrasts strongly with traditional approaches to prioritize upstream over downstream to reduce water resource divergence. We did not input water resource divergence data, nor did we list it as an objective, so naturally, the algorithm did not take this into consideration. The result underlines the efficacy of the decision support provided by the algorithm, which objectively identified the most important branches and directed resources accordingly. Under the 25~50% water supply rates, the prioritization of downstream over upstream was even more apparent: except for the three main branches (Branches 1, 22, and 38), irrigation to almost all of the most upstream branches was stopped, a portion of the midstream branches were kept, and nine of the downstream branches (Branches 39, 40, 42, 46, 47, 48, 49, 50, and 52) were included in most of the solutions. Under the 0~25% water supply rates, the target algorithm suggested only supplying water to one main branch (Branch 1) and its branches downstream. Under these very low supply rates, irrigation to most of the downstream branches was not possible; thus, most of the objectives were not met. Under these conditions, the target algorithm recommended reserving most of the water resources for the first main branch to achieve the objectives as much as possible.

Based on these experiment results, we observed three phenomena: (1) the traditional approach of prioritizing irrigation based on branch width and upstream/downstream position made it almost impossible to obtain optimal solutions, (2) the imbalance of the selected irrigation branches in the skyline solutions shows that the cultivation areas of the three crop types upstream and downstream are uneven, and (3) factors such as upstream/downstream position and crop type increased the risk of no irrigation in the event of water shortages.

5.3. Comparison of Results of Skyline and Top-k Queries for Irrigation Decision Support

In this section, we compare the results of the skyline query obtained in the previous section to those of a top-k query. We selected k to match the number of solutions found by the skyline query, as shown in Table 2. That is, underwater supply rates of 0~25%, 25~50%, and 50~75%, k was set to 10, 171, and 484, respectively. For the weight of each top-k target value, we adopted the most intuitive method and set the weights of the five objective values uniformly (i.e., all are weighted 20%).

As for the comparison results between the two, we list them on the box-and-whisker plot in Figure 10. Figure 10a–c represent the two queries at 0~25%, 25~50%, and 50~75%, respectively, and each subgraph presents the distribution range of all solutions found by the query on the five objective values. In these figures, we can see that no matter the water supply rate or objective value, the distribution range of the top-k query results is smaller than that of the skyline query, and they are usually concentrated on one side of its range. This verifies the arguments put forward by previous scholars [12,13] that in multi-criteria decision-making, the top-k query is biased towards one side when the objective values conflict. The skyline query overcomes this shortcoming. This experiment verified that this is also the case for the field of irrigation decision-making.

5.4. 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. Expected scores and rankings adopted in this experiment.

Combination	Rice Expected Score/Rank	Sugar Cane Expected Score/Rank	Vegetable Expected Score/Rank	Agricultural Economic Benefit Expected Score/Rank	Compensation for Stopping Irrigation Expected Score/Rank
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

), all under the condition of 25~50% water supply.

Table 4. Results based on first ranking under 25~50% water supply rates.

Solution Ranking	Rice Yield	Sugar Cane Yield	Vegetable Yield	Economic Benefits	Compensation for Stopping Irrigation	Total Score without Expected Scores	Total Score with Expected Scores
1	56.244	37.091	52.322	50.816	45.567	391.619	375.914
2	53.782	39.783	49.234	48.784	45.703	384.633	373.912
3	55.026	37.911	51.447	50.13	45.474	387.951	373.9
…	…	…	…	…	…	…	…
105	48.663	37.333	73.72	62.233	46.833	403.441	353.655
…	…	…	…	…	…	…	…
169	65.529	48.903	6.017	26.425	47.067	373.902	335.442
170	65.573	48.799	5.654	26.196	46.967	373.134	334.85
171	64.861	49.529	5.052	25.779	47.028	371.5	333.831

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.

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. This underlines the unevenness in crop distribution upstream and downstream. Furthermore, all six solutions tended to select the downstream branches for irrigation.

In our second experiment, we sought to verify the relationship between economic benefits and vegetable yield when processing solutions with the same score. Thus, we raised the priority of economic benefits and vegetable yield; set lower objective yields for rice, sugar cane, and vegetables; and reduced the amount of compensation not needed. Finally, we set the limit of economic benefits close to the maximum so as to ensure that the ranking algorithm would encounter solutions with the same score.

Table 5. Results based on second ranking under 25~50% water supply rates.

Solution Ranking	Rice Yield	Sugar Cane Yield	Vegetable Yield	Economic Benefits	Compensation for Stopping Irrigation	Total Score without Expected Objective Scores	Total Score with Expected Objective Scores
1	47.907	35.105	77.396	63.961	46.032	492.262	360
2	48.663	37.333	73.72	62.233	46.833	488.744	360
3	44.161	35.397	74.077	61.115	44.472	470.085	360
…	…	…	…	…	…	…	…

displays a portion of the ranked results. As can be seen, the conditions of the top three solutions with the highest scores were all higher than the objective values. This means that the objective yield limits were too low, causing many of the skyline solutions to exceed the objective yield. We can, therefore, see that the three solutions have the same total priority scores. To rank these solutions, the algorithm used the original data to re-calculate the ranking scores to successfully separate them. This verifies that the proposed method can also rank solutions with the same score. Next, we relaxed only the limits of economic benefits, so crops whose yield increases by a greater degree are considered to be more strongly correlated with economic benefits. Under these conditions, the objective yield limits with regard to vegetables varied the most widely, thereby demonstrating that economic benefits are indeed associated with vegetable yield.

We once again converted the first three solutions into branch maps to observe their selection strategies. The results are presented in Figure 12. The skyline solutions prioritizing economic benefits selected a large portion of the upstream branches for irrigation, and the upstream branches mainly serve vegetable production. In other words, economic benefits are indeed related to vegetable yields.

In the third experiment, we sought to confirm the uneven distribution of the crops and verify whether the proposed algorithm can generate a reasonable ranking when crop yield limits exceed the maximum or fall below the minimum. We, thus, increased the priority of sugar cane yield and set lower objectives for rice yield, vegetable yield, economic benefits, and the amount of compensation not needed. The limit of sugar cane yield was set over the maximum to ensure that the algorithm would ignore the limit. Below,

Table 6. Results based on third ranking under 25~50% water supply rates.

Solution Ranking	Rice Yield	Sugar Cane Yield	Vegetable Yield	Economic Benefits	Compensation for Stopping Irrigation	Total Score without Expected Scores	Total Score with Expected Scores
1	59.458	47.52	13.373	29.324	45.652	424.368	302.56
2	59.586	48.738	7.901	26.192	45.433	410.285	302.016
3	59.397	46.994	13.802	29.49	45.428	423.247	300.982
…	…	…	…	…	…	…	…

displays a portion of the ranked results. As intended, the top three solutions all presented high sugar cane yields. However, none had the highest sugar cane yield, as this would have negatively affected vegetable yield and economic benefits, causing loss by leaving these important objectives out of consideration.

Once again, we converted Solutions 1, 2, and 3 into branch maps to observe their selection strategies. The results are presented in Figure 13. These three solutions prioritizing sugar cane yield completely abandoned the upstream branches, selecting only midstream and downstream branches. Furthermore, these four solutions displayed varying performance in rice yield and vegetable yield, indicating that the cultivation areas of rice and sugar cane are not evenly distributed; indeed, their cultivation areas are concentrated along the midstream and downstream branches, causing the target algorithm to prioritize the downstream over the upstream.

6. Conclusions and Directions for Future Work

This paper applies the popular skyline query concept to improve multi-condition irrigation decision support in the agricultural field. We verified the proposed method using real-world data from Taiwan’s irrigation system. A complete irrigation decision-making system usually includes first-stage target value prediction modeling and a second-stage decision algorithm. This paper focuses on the second stage. In the future, we plan to develop a new prediction model to evaluate the objective values based on generative Artificial Intelligence. We plan to use this generative Artificial Intelligence model to produce a variety of reasonable water distribution conditions for specific water supply conditions and government needs. The skyline query proposed in this paper will then be applied to find the best solutions among conflicting water allocation conditions.