Application of Improved NSGA-II Multi-Objective Genetic Algorithm in Optimal Allocation of Water Resources in Main Tarim River Basin

Abstract

As the longest inland river in China, the Tarim River is characterized by water shortage and ecological degradation in the basin, and water resources have become the most important factor restricting the sustainable economic and social development of the basin. In this paper, the optimal allocation model of water resources in the main Tarim River is constructed. The Non-dominated Sorting Genetic Algorithm II (NSGA-II) as a classical multi-objective optimization algorithm suffers from the shortcomings of high computational complexity, long time-consuming non-dominated sorting, and difficulty in diversity preservation under high-dimensional objectives. To address these problems, good point set theory is introduced to improve the distributivity of the solution set, and a linear pressure selection mechanism is utilized to improve the convergence speed of the algorithm. The model is solved by using the improved NSGA-II, and the optimal allocation scheme of water resources in the main Tarim River is proposed. The results show that the total regional water supply remains unchanged under the optimal allocation scheme, while the guaranteed rate of agricultural water supply in the ALE-XQM Irrigation District, XQM-YBZ Irrigation District, YBZ-USM Irrigation District, and CAL-DXHZ Irrigation District is increased by 4.36%, 12.11%, 37.70%, and 0.36%, respectively. The guaranteed rate of ecological water supply is increased by 0.19%, 19.05%, 19.29%, and 36.05%, respectively. And the amount of water discharged from Daxihaizi increased by 0.51 billion m³. In addition, under the three typical hydrological frequency scenarios of moderate year, medium dry year, and extreme dry year in 2030, the guaranteed rate of agricultural and ecological water supply and the amount of water discharged from Daxihaizi can better meet the design requirements. In short, the improved algorithm has obvious superiority, which can make full use of the natural incoming water of the dry river to reduce the overall water shortage and improve the water supply guarantee rate of each region. The optimal allocation scheme can provide scientific reference for the rational allocation of water resources in the Tarim River basin and has an important application value for solving the problem of water resources shortage in Northwest Arid Region.

1. Background

Water is an important substance essential for the survival of all living beings, an irreplaceable strategic resource for living and production, economic development and environmental improvement [1], and an important component of a country’s comprehensive national power. However, the uneven distribution of water resources is widespread throughout the world, and according to the United Nations, about 40 countries are at high risk in terms of water resources, especially in the Middle East and North Africa, most of Sub-Saharan Africa, and some parts of South and East Asia. With the rapid development of economic development and substantial growth in population size, the problem of water shortage has become more and more prominent. In 2020, there are already more than 400 cities (about 600 cities in total) suffering from insufficient water supply, and the total water shortage has reached 6 billion m³, which has become a key factor in restricting economic development, destroying the ecological balance, and affecting the harmony of society and sustainable development. Optimizing the allocation of water resources through the use of engineering and non-engineering measures is important to guarantee the rational use of water resources and realize scientific development.

Research on water resource allocation first started in the 1940s, and at first, it was mainly focused on the optimal scheduling of reservoirs, with the methods of system analysis theory and simulation modeling techniques. Since the 21st century, with the rise in computers and development of artificial intelligence technology, intelligent algorithms have been widely used in the optimal allocation of water resources to provide new ideas for solving water resource allocation problems. It provides new ideas for solving the problems of water resource allocation, which are large in scale, complex in structure, with internal interactions and many exchanges with the external environment [1]. For the multiple uncertainties and multi-objectives under the water resource system, scholars have carried out extensive exploratory studies. Chu considered the principle of water resource allocation and the satisfaction of the water users [2]. Duan carried out the study of water resource allocation with the optimization objectives of water quantity and quality [3], and Fu constructed an interval linear model [4]. In terms of the application of intelligent algorithms, Read et al. sought a solution to the problem of multi-objective allocation of water resources by means of a power index allocation method based on economics [5], and Naghdi et al. integrated the simulation-optimization technique of system dynamics and the Nash bargaining theory with the goal to achieve the optimal allocation of water resources [6]. Li et al. used bionic population intelligence optimization algorithm to solve the water resource allocation problem in Handan City by improving the whale algorithm [7]. Liu improved the artificial fish swarm algorithm by adding adaptive adjustments to the sensing range and moving step size [8]. Fang et al. used improved multi-objective quantum genetic algorithm to optimize the water resource scheduling of the Jiangsu section of the South-to-North Water Diversion East Route Project [9], and Wu et al. used the improved mothball algorithm for Fen River downstream for the rational allocation of scarce water resources [10]. In addition, Zhong et al. improved the particle swarm optimization algorithm with inertia weights [11], Du improved the traditional genetic algorithm [12], and Zhu established the spatial equilibrium water resource allocation model [13], all of which expand the ideas for the study of optimal water resource allocation.

China is one of the 13 major water-poor countries in the world [14], with only 2000 m³ of water resources per capita in 2024, which is only 33% of the world’s per capita level. Influenced by geographic location, monsoon climate, and other factors, the spatial distribution of China’s natural water resources is uneven. Precipitation decreases from the southeast coast to the northwest inland, with annual precipitation more than 1600 mm in some parts of Southeast China, while the average annual precipitation in Xinjiang is only 177 mm. Located in the inland of Northwestern China, Xinjiang is an important agricultural and livestock production area in China with the largest provincial administrative region in terms of land area. In 2023, Xinjiang’s surface water resources were 90.09 billion m³ and underground water resources were 56.13 billion m³, with the total water resources amounted to 95.13 billion m³ after deducting the double-calculated amount of 51.09 billion m³ between surface water resources and underground water resources. In the context of global climate change, the water resource system in Xinjiang based on ice and snow melt water recharge is more fragile, which is mainly manifested in the increase in extreme hydrological events, the increase in water resource uncertainty, and the change in the ecological water demand pattern. The Tarim River basin is a typical area in Xinjiang suffering from water resource shortage problems [15].

Tarim River is the longest inland river in China and the fifth largest inland river in the world, with a total length of 2486 km and a total basin area of 1.02 million km². In recent years, with global climate change and the increasing tension between the supply and demand of water resources, the rational allocation of water resources in the Tarim River basin has become a hot issue for scholars. Combined with the strategic policy of China’s water conservancy, Chen et al. established a model for the optimal allocation of water resources in the Tarim River basin with the constraints of the ’three red lines’ and used the coefficient matrix method to solve the model, which provided a basic idea for the management of water resources in the main Tarim River [15]. Cai carried out water resource optimization allocation for the oasis in the Tarim River basin, and put forward a new way of thinking for the management of water resources in the Tarim River basin [16]. The results showed that it is necessary to continuously improve the content of agricultural science and technology, improve the utilization rate of water resources, and further strengthen the development of natural vegetation and fruit trees. Combined with virtual water theory, Wang established an optimal allocation model of water resources and put forward suggestions for crop structure adjustment and agricultural water use in each sub-basin of the Tarim River basin [17]. Jia et al. took the gate control area as a water distribution unit and the gate diversion flow as a decision variable for the optimal allocation of water resources in the main Tarim River and put forward the scheduling rules for ecological water use [18]. Ding calculated the ecological water demand of the upper, middle, and lower reaches of the main Tarim River under four different scenarios and estimated the flow rate of each control section by discussing the water resource allocation scheme under different scenarios, which provided a scientific basis for the development and utilization of water resources in the basin [19].

With the development of computer technology and in-depth research, intelligent algorithms have achieved a series of results in the field of water resources optimization and allocation. However, the application of intelligent optimization algorithms in water resource allocation in the arid region of Northwest China, especially in the Tarim River basin, is relatively less. Improving the convergence speed and solution set distributivity of the NSGA-II algorithm has important theoretical and applied values. Improving convergence speed can significantly reduce the computational cost of complex engineering optimization problems, meet the demand of real-time decision-making, and enhance the ability of the algorithm to cope with high-dimensional variables and severe constraints. Optimizing the distribution performance of the solution set provides more uniform and diverse Pareto fronts, which comprehensively reflect the trade-off relationship between objectives and provide robust decision support for scenarios such as medical resource allocation and supply chain risk control. The combination of the two not only promotes efficiency breakthroughs in industrial optimization, energy and environmental protection, and artificial intelligence, but also helps solve complex multi-objective problems in frontier sciences such as genetic analysis and quantum computation, becoming a key bridge connecting algorithmic innovation and interdisciplinary practice. In this paper, we take the main Tarim River basin as the research object and construct the water resources optimal allocation model firstly. Secondly, we improve the traditional NSGA-II algorithm by overcoming its limitation in convergence speed and global search ability. Then, we apply the improved algorithm to solve the main Tarim River water resource optimal allocation model, so as to optimize the regional water resources under different scenarios. The results of the optimized allocation of regional water resources can provide scientific references for the study of optimized allocation of water resources in the Tarim River, as well as in similar arid areas.

2. Study Area

The main Tarim River is located in the north of Tarim Basin, originating from the confluence of Aksu River, Yarkant River, and Hotan River, Xiaojiake, and terminating at Taitama Lake, located over the longitude of 81°51′–88°30′E and the latitude of 39°30′–41°35′ N [20]. The main Tarim River is divided into upstream, middle, and downstream sections according to geomorphological features. The upstream is defined as the 495 km river from Shaojiaok to Yingbazha, with an average longitudinal slope of 1/5400. The upstream channel is relatively straight, and the width of the water surface is 500–1000 m. The middle reaches of the main Tarim River are defined as the 398 km river from Yingbazha to Chala, with an average longitudinal slope of 1/7000. The river channel is curved, and the width of the water surface is 50–500 m [21].

It is worth noting that the main Tarim River does not produce flow by itself. Historically, there have been 144 rivers from 9 major water systems, including Aksu River, Kashgar River, Yarkant River, Hotan River, Kaidu-Konchak River, Dina River, Weigang River, Kriya River, Chelchen River, etc., which converge into the main Tarim River. As a result of human activities and the impact of climate change, after the 1950s, the water supply from the main Tarim River relied entirely on the supply of the Aksu River, Hotan River, Yarkant River, and Kaidu-Konchak River, while the water supply from the Lake Taitama in the tailback was mainly supplied by the main Tarim River and the Chelsea River, resulting in the formation of a pattern of ‘Five sources and One trunk’ in the Tarim River Basin. At present, the ‘Five Sources and One Stem’ basin covers an area of 306,000 km², with a total water resource of 26.96 billion m³, accounting for 71% of the total water resources of the basin. In addition, it is also the main water source area, oasis distribution area, and economic development area of the southern border. Among them, the amount of surface and underground water resources is 25.84 billion m³ and 1.12 billion m³, respectively. The irrigated area is 34.64 million acres, accounting for 59% of the total irrigated area of the basin. And the natural forest and grassland area is 63.27 million acres, accounting for 89% of the whole basin. Figure 1 shows the regional overview of the Tarim River.

3. Methodology

3.1. Improved NSGA-II Multi-Objective Genetic Algorithm

NSGA-II algorithm is improved and proposed on the basis of the NSGA algorithm with certain superiority. It adopts fast a non-dominated sorting algorithm to reduce the computational complexity, and adopts a congestion and congestion comparison operator instead of needing to specify the sharing radius to maintain the diversity of the populations. In addition, it introduces the elite strategy to expand the sampling space and prevent the best individuals from being lost, which improves the algorithm’s computing speed and anti-interference [22]. However, the algorithm of traditional NSGA-II still has the following problems in practical applications: (1) Random generation is used in initializing the population. Random represents uncertainty, and the randomly generated population behaves inhomogeneously over the whole space, which affects the distributivity of the solution set. (2) The traditional NSGA-Ⅱ algorithm’s parent selection is usually a binary tournament selection [23]. As far as the tournament implementation is concerned, the selection of the parents is approximated to random selection, which affects the evolutionary speed of the whole population; thus, the search ability of the evolutionary process is not guaranteed and the convergence of the solution set is affected [24].

This thesis addresses the above two problems and introduces a good point set theory to avoid the problem of uneven spatial distribution when generating the initial population and to optimize the distributivity of the solution set; the parent selection strategy based on linear pressure is used to ensure the search ability of the algorithm in the early stage of the algorithm and expand the search space to avoid falling into the local optimum, and increase the pressure on the parent selection in the later stage of the algorithm to ensure the convergence of the algorithm, so that the algorithm can better approximate the true value and improve the convergence of the solution set. The details are as follows:

3.1.1. Improvement of Generation

The good point set is a set of discrete points with distribution uniformity in high-dimensional space that is significantly better than random uniform sampling and has low variance. The theory of good point sets constructs low-discrepancy point sets by means of a number-theoretic approach, and performs well in high-dimensional space sampling, numerical integration, and optimization algorithms, whose core value is to achieve higher coverage uniformity with fewer sampling points.

Assuming that the spatial dimension where the population is located is n and the number of populations is N, the value of the good point r is calculated according to Equations (1) and (2).

$$\mathrm{r}=(r_1,r_2,r_3,\dots ,r_N)$$

(1)

$${\mathrm{r}}_j=\mathrm{mod}(2\cos ({\displaystyle \frac{2\pi j}{7}})N_i,1)$$

(2)

In the formula, N_i denotes the ith individual in the population.

After completing the r-value calculation, the set of good points of number N is constructed based on the resulting good point values.

$${\mathrm{P}}_N(i)=\{(r_1{i}_1,r_2{i}_2,r_3{i}_3,\dots ,r_N{i}_N)\}$$

(3)

Finally, the constructed set of good points P_N is mapped onto the feasible domain where the population is located.

$$X_i^j=a_j+P_N(i)(b_j-a_j)$$

(4)

In the formula, a_j denotes the lower limit of the current dimension and b_j denotes the upper limit of the current dimension.

3.1.2. Improvement of Selection

Assuming that all individuals X in the population are first ranked non-dominated and then ranked in descending order according to the crowding distance X (x₁, x₂, …, x_n). x₁ is the highest ranked individual with the largest fitness value and x_n is the lowest ranked individual with the smallest fitness value. Assuming that the probability of x₁ becoming a parent individual is p₁ and the probability of x_n becoming a parent individual is p_n, there is the expected number of x₁ after the selection operation.

$$up=N\times p_1$$

(5)

The desired quantity of x_n after the selection operation is as follows:

$$ud=N\times p_n$$

(6)

The probability that the ith individual in the population will be the parent individual is calculated according to the equivariate series such that the higher ranked individual in the population has a higher probability of being the parent.

$$p_i={\displaystyle \frac{1}{N}}\left[ud+\left(up-ud\right){\displaystyle \frac{i-1}{N-1}}\right], i = 1, 2, \dots , N$$

(7)

The sum of individual selection expectations in the population is N, then ud + up = 2, 0 ≤ ud ≤ 1, up = 2 − ud.

3.1.3. Improved Algorithmic Process

Step 1: Set the initial population size to nPop, the number of independent variables is nVar, the maximum number of iterations is maxIt, the crossover proportion is Pc, the variation proportion is Pm, the lower bound of the independent variables is varmin, the upper bound of the independent variables is varmax, and the parent generation selection parameters are ud and up.

Step 2: Generate initial populations based on the good point set theory. The external archive set assigns the empty set.

Step 3: Calculate the target values of individuals in the population, then perform fast non-dominated sorting and calculate the crowding degree.

Step 4: Perform ascending sorting based on dominance rank and then descending sorting based on crowding distance. Calculate the probability of an individual becoming a parent after sorting with the following formula:

$$p_i={\displaystyle \frac{1}{nPop}}\left[ud+(up-ud){\displaystyle \frac{i-1}{nPop-1}}\right]$$

(8)

In the formula, p_i is the selection probability of the ith individual, i = 1, 2, ………, nPop.

Step 5: Perform crossover and mutation operations on the parent population after the selection operation to obtain the offspring population.

Step 6: Perform fast non-dominated sorting on the offspring population and calculate the crowding degree and derive the dominance relation.

Step 7: Check whether the population satisfies the convergence condition or whether the number of genetic evolutionary generations reaches the maximum number of iterations. If it satisfies the termination condition, then stop the iteration and output the result, otherwise go to Step 4.

The flow of the improved algorithm is shown in Figure 2.

3.2. The Water Resources Optimal Allocation Model of the Main Tarim River Basin

The main Tarim River basin is located in the north of the Tarim Basin, bordering the plains under the southern foothills of the Tianshan Mountains, and extends to the Alar Reclamation Area of the First Agricultural Division in the west, and to the east of the Peacock River and its tailing Lop Nor Depression. In recent decades, due to the large-scale development of land resources, rough water resource management, unsupported water conservancy facilities, and unreasonable water resource distribution in the main Tarim River, the amount of water entering the main Tarim River is decreasing. In this paper, the main Tarim River basin is divided into six sub-irrigation districts according to the location of six control cross sections. The six sub-irrigation districts are the Alaer–Xinqiman (ALE-XQM) Irrigation District, Xinqiman–Yingbazha (XQM-YBZ) Irrigation District, Yingbazha–Usman (YBZ-USM) Irrigation District, Usman–Aqike (USM-AQK) Irrigation District, Aqike–Chala (AQK-CAL) Irrigation District, and Chala–Daxihaizi (CAL-DXHZ) Irrigation District. Figure 3 shows the generalization of the main Tarim River system. In this paper, we take the six sub-irrigation districts in the main Tarim River basin as the object and construct an optimal allocation model of water resources. Then, we employ the improved NSGA-II multi-objective genetic algorithm to solve the model and propose the optimal water resource allocation strategy for the main Tarim River basin.

3.2.1. Objective Function

The optimal allocation of water resources in the Tarim River basin is an important guarantee for the healthy and sustainable development of the region and even Xinjiang’s socio-economic ecology. On the one hand, the year-round climate of the basin is arid with scarce precipitation and large evaporation. And the irrational water resource allocation makes the problem of water shortage in the basin more prominent; thus, the water shortage in the basin should be an important indicator to measure the effect of optimal allocation of water resources. On the other hand, the main Tarim River basin is the core ecological function area of the Tarim River for desertification control, and guaranteeing its ecological water consumption is the key to restoring and protecting the ecological environment of the basin. Therefore, the optimal allocation model in this paper is established by considering the minimum overall regional water shortage and the maximum ecological water supply in the region during non-flood season as objective functions.

Objective 1: Minimize overall regional water shortages

$$\min f_1(x,y)={\displaystyle \sum _{i=1}^6\left[(W_{xi}-x_i)+(W_{yi}-y_i)\right]}$$

(9)

In the formula, W_xi is the agricultural water demand of the ith irrigation district, i = 1, 2, …, 6. x_i is the agricultural water supply of the ith irrigation area. W_yi is the ecological water demand of the ith irrigation area. y_i is the ecological water supply of the ith irrigation area.

Objective 2: Maximize the rate of guaranteed ecological water supply in the irrigation districts

$$\max f_2(y)={\displaystyle \sum _{i=1}^6{y}_i/}{W}_{yi}$$

(10)

3.2.2. Constraint Condition

In this paper, four constraints are considered in the model. The constraints are shown as follows:

a.

Total water resource constraints:

$${\displaystyle \sum _{i=1}^6(x_i+y_i})\le W_u=W_i-W_e$$

(11)

In the formula, W_u is water availability. W_i is the incoming water from Alaer cross section. W_e is the ecological base flow rate of the main stream of Tarim River.

b.

Maximum agricultural water supply constraints:

$$x_i\le W_{xi}$$

(12)

c.

Minimum ecological water supply constraints:

$$y_i\ge \alpha W_{yi}$$

(13)

In the formula, α is a human-set parameter, which can be taken as 0~1, and in this study, it is taken as 0.1 [25].

d.

Non-negative constraints on variables:

$$x_i\ge 0; y_i\ge 0$$

(14)

3.3. Pearson-III Frequency Curve

The probability distribution curves used in hydrological analysis and calculation are commonly known as hydrological frequency curves, and there are many types of these curves in mathematics. The probability distribution curves commonly used in Chinese engineering hydrology are normal curve, lognormal curve, and Pearson-III frequency curve.

According to the ‘Water Conservancy and Hydropower Engineering Design Flood Calculation Code’, the line type of the frequency curve is generally chosen as the Pearson-III frequency curve. The Pearson-III frequency curve is a positively skewed curve with an asymmetric single peak that is finite at one end and infinite at the other, with the following probability density function:

$$f(x)={\displaystyle \frac{{\beta }^{\alpha }}{\mathrm{\Gamma }(\alpha )}}{(x-a_0)}^{(\alpha -1)}{e}^{-\beta (x-a_0)}$$

(15)

In the formula, Γ(α) is the gamma function of α, and α, β, and a₀ are three parameters characterizing the shape, scale, and position of the curve, α > 0, β > 0.

After calculating the mean value $\overline{\mathrm{x}}$, the coefficient of divergence C_v, and the coefficient of skewness C_s of the resulting sequence, the following relationship can be observed:

$$\alpha ={\displaystyle \frac{4}{C_s^2}}$$

(16)

$$\beta ={\displaystyle \frac{2}{{\overline{x}\mathrm{C}}_{\mathrm{s}}{C}_v}}$$

(17)

$$a_0=\overline{x}(1-{\displaystyle \frac{2C_v}{C_s}})$$

(18)

4. Results and Analysis

4.1. Testing Algorithm Improvement

To verify the generalizability and superiority of the improved algorithm, the improved NSGA-II algorithm is tested using ZDT test functions (ZDT1, ZDT2, ZDT3, and ZDT6) in this paper. The testing process sets the population size as 100, the number of iterations as 50, the crossover probability as 0.80, and the variance probability as 0.05. The iterative process of the algorithm with the decision variable as the X-coordinate, the objective function as the Y-coordinate, and the number of iterations as the Z-coordinate are shown in Figure 4.

Comparison experiments between the improved NSGA-II algorithm and the traditional algorithm with the same parameter settings show that (Figure 5, horizontal coordinates are the input values of the decision variables and vertical coordinates are the output values of the multi-objective), based on the Pareto curve analysis of the ZDT test function, the improved algorithm shows significant advantages with the same number of iterations: its solution set approaches the theoretical Pareto frontier faster, and its distribution is denser and more even, and has a higher degree of overlap with the known values. The overlap of the optimal solution is higher. For ZDT3, as a segmented discontinuous function, the traditional algorithm is easy to fall into local optimization, and the solution set cannot be uniformly distributed in the five ideal intervals. The improved algorithm solves this problem well. This result verifies the effectiveness of the two improvement strategies proposed in this paper, the good point set theory and the linear pressure selection mechanism, in accelerating the convergence and improving the quality of the solution set—not only shortening the cost of computation time, but also enhancing the ability of the algorithm to portray the complex trade-off relationship, which is useful for the high-precision and high-efficiency multi-objective optimization needs in practical engineering (such as real-time system design and resource constraint scenarios); it provides a more reliable solution tool, and at the same time lays a performance validation foundation for subsequent water resource allocation studies.

In addition, this paper adopts Euclidean distance (ED) and Spacing (SP) metrics to quantitatively evaluate the improved NSGA-II algorithm [26]. The statistics of the metrics computed before and after the algorithm improvement are shown in

Table 1. Comparison of performance evaluation indexes between traditional and improved NSGA-II.

Function	Pre-Improved ED	Improved ED	Performance Improvement	Pre-Improved SP	Improved SP	Performance Improvement
ZDT1	0.7613	0.2302	69.76%	0.8917	0.5135	42.41%
ZDT2	1.1551	0.4143	64.14%	1.0595	0.6971	34.20%
ZDT3	0.6548	0.2244	65.73%	0.9976	0.6169	38.16%
ZDT6	0.5770	0.0013	99.77%	0.8742	0.5712	34.66%

. In general, ED reveals the difference between the iteration result and the desired result. The smaller the ED is, the better the iteration result will be. SP can reflect the uniformity of the distribution of the population. The smaller the SP is, the more uniform the distribution of the population will be. The results in Table 1 show that the performance of the improved algorithm is enhanced, with the ED of the improved algorithm shrinking by 64.14–99.77% and SP shrinking by 34.20–42.41%.

Overall, the improved NSGA-II algorithm proposed in this thesis has excellent distributivity and convergence. The distributivity is benefitted from the good point set theory for generating the initial population, and the convergence is benefitted from the parent selection strategy based on linear pressure.

4.2. Comparisons Between Optimal Water Resource Allocation Schemes and Actual Water Resource Allocation Schemes in Year of 2020

The NSGA-II algorithm considers the conflicting objectives of economy and ecology in the optimal allocation of water resources through the framework of multi-objective optimization, screens the high-quality solution sets by non-dominated sorting, maintains the diversity of the solution sets by the congestion distance, and finally generates the balanced Pareto-optimal allocation scheme, which provides decision makers with a space for choosing the water resource allocation strategy that takes into account the multi-dimensional needs.

In this paper, the improved NSGA-II algorithm is applied to the optimal allocation of water resources in the main Tarim River with 2020 as the study period. And the optimized water resource allocation scheme is compared with the actual water resource allocation scheme. In 2020, the amount of incoming water from the Alaer section of the main Tarim River is 4.106 billion m³, and the amount of groundwater that can be extracted is 107.52 million m³. After fulfilling the domestic and industrial water demand of 369.8 million m³, the remaining groundwater of 70.54 million m³ can be used for agricultural recharge.

The actual water resource allocation scheme of the main Tarim River basin in 2020 is shown in

Table 2. Actual allocation of water resources in main Tarim River in 2020. Unit: billion m³.

	Agricultural Water Demand	Agricultural Water Supply	Water Deficit	Ecological Water Demand	Ecological Water Supply	Water Deficit
ALE-XQM	3.44	2.25	1.19	2.53	1.05	1.48
XQM-YBZ	4.49	2.26	2.23	3.36	0.45	2.91
YBZ-USM	1.87	0.3	1.57	7.93	1.27	6.66
USM-AQK	0.28	1.5	−1.22	0.85	1.05	−0.2
AQK-CAL	0.27	1.23	−0.96	0.13	1.19	−1.06
CAL-DXHZ	2.77	2.05	0.72	1.78	0.06	1.72
total	13.12	9.59	3.53	16.58	5.07	11.51

. According to the recent Comprehensive Management Plan for the Tarim River Basin issued in 2001, the guaranteed rate of agricultural water supply is required to be no less than 75%, the guaranteed rate of ecological water supply is required to be no less than 50%, and the amount of water discharged from Daxihaizi is required to be no less than 350 million m³. The results show that the guaranteed rate of agricultural water supply in the main Tarim River basin is 16.04–535.71%, and the guaranteed rate of ecological water supply is 3.37–915.38%. The ALE-XQM irrigation district, XQM-YBZ irrigation district, YBZ-USM irrigation district, and CAL-DXHZ irrigation district do not meet the agricultural and ecological water supply design assurance requirements. It is obvious that the USM-AQK irrigation district’s and AQK-CAL irrigation district’s water resources are seriously wasted with the problem of excessive water use. The discharge volume of Daxihaizi is 299 million m³, which does not meet the design requirements of 350 million m³.

The improved NSGA-II was used to solve the multi-objective optimal allocation model of water resources in the main Tarim River. To ensure the global search ability of the algorithm, the initial population size is set to 100. To ensure the quality of the solution set, the number of iterations is set to 500, and the algorithm ends when an individual satisfies all the constraints or reaches the preset number of iterations. The results of the water resource optimal allocation scheme in the main Tarim River basin in 2020 are shown in

Table 3. Optimized allocation of water resources in main Tarim River in 2020. Unit: billion m³.

	Agricultural Water Demand	Agricultural Water Supply	Water Deficit	Ecological Water Demand	Ecological Water Supply	Water Deficit
ALE-XQM	3.44	2.40	1.04	2.53	1.00	1.53
XQM-YBZ	4.49	2.80	1.69	3.36	1.09	2.27
YBZ-USM	1.87	1.00	0.87	7.93	2.88	5.05
USM-AQK	0.28	0.18	0.10	0.85	0.30	0.55
AQK-CAL	0.27	0.18	0.09	0.13	0.06	0.07
CAL-DXHZ	2.77	2.06	0.71	1.78	0.70	1.08
total	13.12	8.63	4.49	16.58	6.03	10.55

. It can be seen that the guaranteed rate of agricultural water supply under the optimal allocation scheme is 53.74% to 74.37%. The guaranteed rate of ecological water supply is 32.56% to 44.23%. Compared with the actual configuration, the optimized configuration improves the guaranteed agricultural water supply and ecological water supply in the ALE-XQM irrigation district, XQM-YBZ irrigation district, YBZ-USM irrigation district, and CAL-DXHZ irrigation district. It also avoids water wastage in the USM-AQK irrigation district and AQK-CAL irrigation district.

A comparison of the water supply guarantee rate between the optimized water resource allocation and the actual allocation scheme in the main Tarim River basin in 2020 is shown in Figure 6. Compared with the actual allocation scheme, the total water supply in the region remains unchanged, while the optimization scheme improves the problem of excess water supply in the USM-AQK and AQK-CAL irrigation districts, and the problem of water wastage in the region is alleviated. At the same time, the guaranteed rate of agricultural and ecological water supply in the ALE-XQM irrigation district, the XQM-YBZ irrigation district, the YBZ-USM irrigation district, and the CAL-DXHZ irrigation district has increased, and the amount of water discharged from Daxihaizi has increased by 0.051 billion m³. In conclusion, the water shortage problem of the main Tarim River basin was effectively relieved by adopting the improved NSGA-II algorithm, and the water supply guarantee rate of each irrigation area has increased. On the basis of making full use of the incoming water from the Alaer section, the optimized allocation scheme fully guarantees the ecological base flow and the water discharge of Daxihaizi, so that the water resources are reasonably distributed.

4.3. The Optimal Water Resource Allocation Schemes Under Different Hydrological Frequency Scenarios in the Year of 2030

Taking 2030 in the future as the study period, this paper carries out a study on the water resource optimal allocation of the main Tarim River basin under different scenarios. Three hydrological frequency scenarios, corresponding to three typical hydrological years, are considered, namely, moderate year (50% incoming water frequency), medium dry year (75% incoming water frequency), and extreme dry year (90% incoming water frequency). The number of iterations is set to 500 in the process of solving the water resource allocation model using the improved NSGA-Ⅱ.

The runoff data of the main Tarim River from 1960 to 2020 were analyzed based on the Pearson-III frequency curve [27]. By analyzing the long timeseries of incoming water data, the parameter of the Pearson-III frequency curve was calculated with $\overline{\mathrm{x}}$ equal to 45.40, C_v equal to 0.26, and C_s equal to −0.13. The results show that the water inflow of the Alaer section in 2030 is 4539 million m³, 3634 million m³, and 2769 million m³ under 50%, 75%, and 90% incoming water frequency, respectively. Meanwhile, according to the water demand prediction based on the quota method [28], the total agricultural water demand of the main Tarim River basin will be 1.045 billion m³, the total ecological water demand will be 1.658 billion m³, and the domestic and industrial water demand will be 0.4553 billion m³ in 2030. In addition, considering that the groundwater recoverable amount of the main Tarim River basin in 2030 will not be more than the current status in 2020 of 1075.2 million m³, the remaining 0.6199 billion m³ will be used for agricultural water supply after meeting the domestic and industrial water demand.

The optimal allocation scheme of water resources under the 50%, 75%, and 90% incoming water frequency in 2030 are shown in

Table 4. The optimized water resource allocation scheme under the 50% hydrological frequency scenario.

	Agricultural Water Demand	Agricultural Water Supply	Water Deficit	Ecological Water Demand	Ecological Water Supply	Water Deficit
ALE-XQM	2.59	2.55	0.04	2.53	1.91	0.62
XQM-YBZ	3.18	3.15	0.03	3.36	2.61	0.75
YBZ-USM	1.44	1.39	0.05	7.93	5.56	2.37
USM-AQK	0.24	0.23	0.01	0.85	0.63	0.22
AQK-CAL	0.23	0.22	0.01	0.13	0.09	0.04
CAL-DXHZ	2.77	2.67	0.10	1.78	1.27	0.51
total	10.45	10.21	0.24	16.58	12.07	4.51

In the year of 2030. Unit: billion m³.

,

Table 5. The optimized water resource allocation scheme under the 75% hydrological frequency scenario.

	Agricultural Water Demand	Agricultural Water Supply	Water Deficit	Ecological Water Demand	Ecological Water Supply	Water Deficit
ALE-XQM	2.59	1.97	0.62	2.53	1.42	1.11
XQM-YBZ	3.18	2.44	0.74	3.36	1.71	1.65
YBZ-USM	1.44	1.14	0.30	7.93	3.97	3.96
USM-AQK	0.24	0.20	0.04	0.85	0.43	0.42
AQK-CAL	0.23	0.18	0.05	0.13	0.07	0.06
CAL-DXHZ	2.77	2.14	0.63	1.78	1.06	0.72
total	10.45	8.07	2.38	16.58	8.66	7.92

In the year of 2030. Unit: billion m³.

and

Table 6. The optimized water resource allocation scheme under the 90% hydrological frequency scenario.

	Agricultural Water Demand	Agricultural Water Supply	Water Deficit	Ecological Water Demand	Ecological Water Supply	Water Deficit
ALE-XQM	2.59	1.76	0.83	2.53	0.32	2.21
XQM-YBZ	3.18	2.01	1.17	3.36	0.47	2.89
YBZ-USM	1.44	1.09	0.35	7.93	1.24	6.69
USM-AQK	0.24	0.16	0.08	0.85	0.23	0.62
AQK-CAL	0.23	0.17	0.06	0.13	0.03	0.10
CAL-DXHZ	2.77	2.11	0.66	1.78	0.20	1.58
total	10.45	7.30	3.15	16.58	2.49	14.09

In the year of 2030. Unit: billion m³.

. The results show that under 50%, 75%, and 90% hydrological frequency scenarios, the guaranteed rate of agricultural water supply in the six irrigation districts in 2030 will be 95.65–99.06%, 76.06–83.33%, and 63.21–76.17%, respectively, which all meet the designed requirement of the 75% guaranteed rate of agricultural water supply. And at the same time, the amount of water discharged by Daxihazi will meet the requirement of 350 million m³. Under the 50% and 75% hydrological frequency scenarios, the ecological water supply guarantee rates of the six irrigation districts in 2030 were 69.23–77.68%, and 50.06–59.55%, which all meet the designed guarantee rate of 50%. However, the guarantee rate of ecological water supply in the six irrigation districts under the 90% hydrological frequency scenario range from 11.24% to 27.06%, which did not meet the 50% guarantee rate, are higher than the 10% minimum water demand requirement for ecological vegetation in the basin.

5. Discussion

The convergence and distribution of the solution set is an important indication of the performance of the algorithm [29,30]. Limited by the distribution of the solution set, in some complex optimization problems, the traditional NSGA-II may fall into local optimal solutions. Limited by the convergence of the solution set, the traditional NSGA-II algorithm may not converge as fast as some other multi-objective optimization algorithms in reaching the optimal solution. For the improvement of the NSGA-II algorithm, most of the current research focuses on coupling the NSGA-II algorithm with other heuristic algorithms and combining the advantages of the two algorithms. For example, the Cauchy Mutation method (CM) is used for optimization to improve the convergence speed of the algorithm [31]; the Simulated Annealing algorithm (SA) is used to generate a new subpopulation to increase the diversity of the population [32]; and the projection method is used to retain the historical optimization information to improve the convergence of the solution set [33].

This paper addresses the shortcomings of the NSGA-II algorithm itself. For the distribution of the solution set, the deviation is generally O(n^(1-/2)) using the random method to take points, while using the method proposed by Hua Luogeng with the ‘best consistent distribution of the good set of points’ to take points, the deviation is only O(n^(−1+ε)). There is a difference of 10² times [34]. It follows that using the theory of the good point set is much less biased than the random method. At the same time, for the convergence of the solution set, this paper proposes a parent selection strategy based on linear pressure. Setting ud = up = 1 at the beginning of the algorithm, when the parent population is chosen in a randomized manner, ensures the global search ability of the algorithm and the diversity of the offspring population. Setting up to increase and ud to decrease with the number of iterations in the later stages of the algorithm, makes the probability of an excellent individual to become a parent increase, as well as improves the algorithm’s optimization-seeking efficiency.

It is worth noting that although the NSGA-II algorithm is widely used in multi-objective optimization, its limitations significantly affect the performance in specific scenarios [35]. For example, the high-dimensional objective space is under-pressurized for selection due to the surge in the proportion of non-dominated solutions, and the congestion distance fails. Complex shaped Pareto fronts lead to the uneven distribution of solution sets and the omission of key trade-off solutions. Dynamic environments lack real-time responsiveness and are prone to falling into historical optimality. Tight constraints and large-scale variables lead to convergence difficulties or inefficiencies. Discrete/mixed-variable problems generate invalid solutions due to restricted coding mechanisms. High computational cost and multimodal frontiers face resource overruns and local convergence risks, respectively. These limitations need to be combined with improved strategies to extend the applicability and avoid blind application in complex engineering and scientific problems leading to optimization failure [36].

It is worth mentioning that in recent years, scholars have conducted a lot of research on heuristic algorithms based on intelligent optimization rules, and different kinds of bionic swarm intelligent algorithms with different advanced intelligence have appeared continuously and have been gradually applied to the problem solving of various aspects such as combinatorial optimization and resource scheduling. Bionic swarm intelligence refers to the process and characteristics of many simple individuals exhibiting complex intelligent behaviors by means of mutual synergy. The main swarm intelligence algorithms are ant colony optimization (ACO) [37], particle swarm optimization (PSO) [38], artificial fish swarm algorithm (AFSA) [39], glow worm swarm optimization (GSO) [40], and artificial bee colony (ABC) [41]. The flexibility and versatility of bionic intelligence algorithms allow them to be adapted to specific needs and combined with other algorithms to further enhance performance. The main drawbacks of NSGA-II are centered on the low efficiency of high-dimensional target space, parameter sensitivity, and weak adaptation to complex constraints and dynamic environments. The combination of bionic intelligence algorithm and NSGA-II algorithm can be considered in the subsequent research to achieve the purpose of improving NSGA-II algorithm.

6. Conclusions

In this paper, the traditional NSGA-II algorithm is improved by applying the good point set theory and linear pressure selection strategy. The improved NSGA-II algorithm is proposed and applied to solve the optimal allocation model of water resources in the main Tarim River basin, which significantly improves the distribution and convergence of the algorithm. The water resources optimal allocation model of the main Tarim River basin was constructed by taking the minimize overall regional water shortage and the maximize guaranteed rate of regional ecological water supply during non-flood season as the objectives, and the optimal allocation schemes under different scenarios are put forward. The results show that the Euclidean distance is reduced by 60–90% and the distributivity index is increased by 30–40% by applying the improved NSGA-II algorithm to solve the ZDT functions, and the convergence and distribution of the algorithm are greatly improved. Compared with the actual water allocation scheme, the optimized allocation scheme can make full use of the natural water from the river, reduce the total regional water shortage, and improve the water supply guarantee rate of each irrigation district. In 2020, compared with the traditional allocation scheme, the optimized allocation scheme improves the wastage of water resources in the basin and effectively increases the guaranteed rate of agricultural and ecological water supply in each irrigation district. In 2030, the optimal allocation scheme can satisfy the designed requirements of agricultural and ecological water supply in both the moderate year and medium dry year scenarios. On the basis of fulfilling the water discharge volume of Daxihaizi under extreme dry year scenarios, the optimal allocation scheme can basically meet the designed requirements of agricultural water supply and meet the minimum water demand of ecological vegetation in the watershed. An increase in the guaranteed rate of agricultural water supply means more stable and reliable water for agriculture. This can reduce crop yield reductions due to water shortages and increase yields, which in turn improves the efficiency of agricultural water use. At the same time, ecosystem recovery requires sufficient water, and guaranteed ecological water can reduce ecosystem degradation, which plays an important role in solving the problems of the Tarim River’s outflow and biodiversity reduction. The results can provide valuable references for study design of similar area and are of great significance for the study of regional water resources optimal allocation.

Improving the convergence speed and solution set distribution of the NSGA-II algorithm reduces the computational resource consumption and time cost of complex engineering optimization problems by reducing the number of iterations to quickly approach the Pareto frontier (linear pressure selection), and provides decision makers with diversified solutions that take into account the economy and security by uniformly covering the solution set of objective trade-off relationships (good point set strategy). For example, it can be used to screen out feasible configurations that reduce investment costs in energy system planning, or to generate stable solutions with load ratios lower than the safety threshold in power scheduling to avoid the risk of overloading [42]. At the performance optimization level, the improved algorithms not only enhance multi-objective synergy, but also enhance dynamic adaptability. Improved algorithms can be widely used in industry, AI, and other fields to promote the core value of the technology to the ground with lower cost, higher safety standards, and better overall performance, providing a new generation of solutions with both efficiency and robustness for the optimization of complex systems.

The article breaks through the inhomogeneity of random initialization by proposing an initialization strategy based on the theory of good point sets. The design of dynamic pressure selection mechanism solves the problems of randomness and slow convergence of traditional binary tournament selection. It provides a novel population evolution framework for multi-objective evolutionary algorithms and extends the practical dimension of Pareto dominance theory. It also enhances the practical value of the algorithm in complex engineering optimization and provides fast response solutions for dynamic multi-objective problems. This research provides a new technical path for the theoretical development and engineering application of multi-objective evolutionary algorithms, which have important academic value and industrial application prospects. In the future, combining intelligent bionic thinking with the NSGA-II algorithm is an inventive way to improve the performance of the NSGA-II algorithm. And the distribution and convergence of the solution set can be further improved by improving the mutation operation of the algorithm and adding an integrated selection strategy, which can dynamically adjusting the objective weights.