Application of MOHUS in Multi-Objective Optimal Allocation of Water Resources for the Central Route South-to-North Water Diversion Project in Hebei Province, China

Abstract

With the increasingly severe problem of water shortage and the increasing contradiction between supply and demand, the optimal allocation of water resources has become an essential method for alleviating the water crisis. In this context, natural-element-inspired optimization algorithms have been extensively used to solve water resource optimization problems. The hunting search (HUS) algorithm has a slow convergence speed, and low accuracy, which makes it easy to fall into local optima when solving multi-objective problems, and it is also not easy to apply directly to multi-objective optimization. It is improved by introducing a fast, non-dominated sorting, congestion comparison operator, and elite retention strategy. The improved algorithm was evaluated against popular algorithms using the ZTD series of objective functions, and the results indicate that the improved algorithm performs better in terms of convergence and diversity of solution sets. The improved algorithm is then applied to solve the optimal allocation model for water resources in the receiving area of the South-to-North Water Diversion Project in Hebei Province, which is based on the objective of social and economic benefits. Based on the obtained Pareto optimal frontier, the scheme with a special preference for the minimum shortage of water resources is selected as the ultimate decision-making scheme. The optimization results show that in 2030, the total water demand of water users in the receiving area is 4661.82 × 10⁶ m³, the total water allocation is 4082.88 × 10⁶ m³, and the water deficit is 578.94 × 10⁶ m³. The improved hunting search algorithm offers a new solution method for the multi-objective water resource optimization allocation problem.

1. Introduction

Water resources play an important role in people’s daily lives and production, and are the key strategic resources for humanity’s survival and development, and are responsible for maintaining the sustainable development of the global ecological environment [1]. At present, the frequent occurrence of global extreme climate phenomena, coupled with the continuous rise in population and rapid urbanization, the demand for water resources in various countries is rising, which has led to an increasingly serious water shortage problem [2,3,4]. In China, the total water resources account for only 6% of the world’s water resources, but the population base is relatively large, and the per-capital water resources are only 28% of the world’s per-capital level [5]. The overall distribution of water resources in China is characterized by more in the south and less in the north, more in the mountainous areas and less in the plains, coupled with extremely uneven intra- and inter-annual distribution and a large population base, the problem of water resource scarcity is particularly prominent [6,7]. Therefore, there is an urgent need to optimize the allocation of water resources, change the use of water resources from inefficient use to efficient and higher-valued use, enhance the overall efficiency of water resources, fundamentally alleviate the contradiction between supply and demand, and facilitate the sustained use of regional water resources.

The optimal allocation of water resources is a complex decision-making problem. In the allocation process, not only must water scheduling be considered, but the balance between supply and demand, water quality protection, ecological restoration, socio-economic benefits, and other multiple objectives must be considered [8,9]. In the middle of the 20th century, Masse proposed the optimal scheduling of reservoirs in the storage and supply of water, which opened up the water resource allocation of the research problem prologue [10]. In early studies on the optimal allocation of water resources, single-objective optimization methods, like linear programming [11] and dynamic programming [12], were often used, and the studies were mainly oriented toward maximizing economic benefits [13,14]. However, these methods do not consider social and ecological benefits, which leads to limitations in the optimization results in practical applications. After the 1970s, with the development of multi-objective optimization theory, intelligent algorithms, and computing technology, the problem of optimal allocation of water resources became a complex decision-making question with multiple objectives and constraints [15]. The two main types of methods are a priori [16] and a posteriori [17]. The former transforms the multi-objective problem into a single-objective model by presetting the objective weights [18], and the latter maintains the structure of the multi-objective question, and then the decision-makers further screen the optimal solutions according to the actual needs and priorities after obtaining the corresponding solution sets.

In practical engineering applications, water resources optimization and allocation problems usually have multivariate and multidimensional characteristics [19,20,21,22], which often lead to the traditional optimization algorithms easily falling into the local optimal. Meanwhile, there are the problems with low computational efficiency and high parameter sensitivity. In recent years, nature-inspired metaheuristic algorithms have received considerable attention due to their simple principles and easy implementation. Compared with traditional optimization, these methods have shown superior performance in solving water allocation problems. Therefore, the application of metaheuristic algorithms in multi-objective optimization and allocation has been widely studied by academics [23]. Deb et al. proposed NSGA-II [24] and NSGA-III [25] algorithms by improving the traditional genetic algorithms, which have been used by subsequent scholars in water resource allocation studies [26,27,28,29]. Meanwhile, the multi-objective particle swarm optimization (MOPSO) algorithm [30] has also been applied in the field of water allocation [30,31,32], which simulates the behavior of bird flock foraging and explores the optimal solution set through information sharing and collaboration among particles. Subsequent scholars have also proposed the Multi-objective Artificial Bee Colony Algorithm (MOABC) [33], which solves multi-objective optimization problems by simulating the collaborative mechanism of hiring, observing, and scouting bees in a bee colony and has been successfully used in complex problems such as reservoir scheduling [34,35]. There is also the Multi-Objective Whale Optimization Algorithm (MOWOA) [36], which achieves efficient global optimization by simulating the three feeding strategies of whales and has been used to solve complex multi-objective optimization models for water resources [37,38]. In addition to the above algorithms, there are constantly multi-objective optimization algorithms that have been successfully used in the field of water resource optimization and allocation in recent years, such as the multi-objective gray wolf optimization algorithm (MOGWO) [39,40], multi-objective moth-flame algorithm (MOMFO) [41], multi-objective marine predator algorithm (MOMPA) [42], multi-objective butterfly algorithm (MOBOA) [43], and multi-objective cuckoo search (MOCS) [44].

The HUS algorithm, as an efficient and intelligent optimization method, has shown good application results in many fields, such as network communication [45,46], energy storage [45,47], and structural engineering [45,48]. However, the traditional HUS algorithm has obvious limitations when dealing with multi-objective optimization problems, which are mainly manifested in the problems of easily falling into local optimum and poor convergence performance. To address these limitations, many scholars have proposed various improvement schemes, including the optimization of algorithm parameters [49], improvement of the operation process, and coupling with other intelligent optimization algorithms [46]. In this study, an improved hunting search algorithm (MOHUS) is proposed, and its technical improvements are mainly reflected in the following three aspects: first, the introduction of a fast, non-dominated sorting mechanism effectively improves the convergence speed of the Pareto frontier; second, the adoption of a congested distance computation strategy ensures that the optimal solution set is uniformly distributed in the objective space; and lastly, the design of an elite retention mechanism maintains the dominant individual in the population’s genetic characteristics. Subsequently, the performance of the improved algorithm is tested through simulation experiments and compared with that of a popular algorithm. A dual-objective water resource optimization allocation model of economic benefit-social benefit is also constructed and verified with a case study of the Hebei receiving area of the South-to-North Water Diversion Middle Route Project. The configuration results can provide powerful data support and a decision-making basis for water-receiving areas to develop scientific and reasonable water resource utilization planning.

2. Materials and Methods

2.1. Hunting Search (HUS) Algorithm

The HUS algorithm is a metaheuristic optimization algorithm inspired by the cooperative hunting behaviors of social animals [45]. This algorithm is initialized with a population of predators (HG) as the starting point and achieves a global search by simulating the collective predation process. During each iteration, the individuals with the highest fitness values are first selected as the “leader,” while the remaining individuals update their positions by moving toward the leader’s direction in accordance with the following position update formula:

$$x_i^{\prime }=x_i+\mathrm{rand}\times \mathrm{MML}\times (x_i^L-x_i)$$

(1)

where $x_i$ is the current position of the i predator; $x_i^L$ is the current position of the leader; $\mathrm{rand}$ is a random number uniformly distributed in [0, 1]; MML is the maximum allowed step size.

After updating their positions, each predator compares the fitness values of its old and new positions. If the new position yields better fitness, the update is accepted. However, if the leader’s position remains unchanged in the current iteration, the predators perform a position correction by incorporating the positions of other individuals and a random perturbation to enhance the exploration of potentially better solutions. The correction formula is as follows:

$$\begin{array}{l}{x}_i^{\prime }=x_i^{\prime }+\tau \\ \tau =bw\times u(-1,1)\\ bw=a\times e^{-b\times iteration}\end{array}$$

(2)

where $\tau$ is the perturbation term; $bw$ is the bandwidth controlling the perturbation scale; $u(-\mathrm{1,1})$ is a random number in (−1,1); $a$ and $b$ are positive constants determining initial bandwidth and decay rate respectively; $ineration$ is the amount of iterations.

This algorithm evaluates its updated solutions, adjusts the leader’s position, and checks the convergence criteria. If satisfied, an optimal solution is returned. To prevent predator trapping in local optima during the search process, a recombination operation is triggered when the objective function difference between the leader and the worst individual falls below a threshold. This mechanism enhances population diversity and facilitates better position discovery, governed by the recombination formula:

$$x_i^{\prime }=x_i+{\displaystyle \frac{\mathrm{rand}\times [r(\max (x_i),\min (x_i))-x_i]}{\alpha \times E{N}^{\beta }}}$$

(3)

where $r(\max \left(x_i\right),\mathrm{m}\mathrm{i}\mathrm{n}(x_i\left)\right)$ is a randomly selected value uniformly distributed within the current variable’s feasible range; Parameters α and β denote convergence rate regulators for global optimization; $EN$ represents the consecutive generations count of local optima entrapment.

2.2. Multi-Objective Hunting Search Algorithm (MOHUS)

This study focuses on improving the HUS algorithm’s performance in multi-objective optimization, where the original version exhibits three fundamental limitations: (1) slow convergence speed, (2) inadequate solution precision, and (3) frequent stagnation at local optima. To address these issues, we propose three key modifications. First, a fast non-dominated sorting mechanism is introduced to efficiently construct the Pareto frontier. Second, a dynamic crowding distance metric is implemented to preserve population diversity across objective spaces. Third, an elite retention strategy ensures the progressive improvement of the solution quality through generations. These synergistic enhancements collectively improve the convergence characteristics of the algorithm while maintaining solution diversity.

(1)

Fast non-dominant sorting. Non-dominant ranking is a hierarchical division of individuals based on dominance relationships, which is used to classify populations and prioritize individuals with better performance. Assuming that the predator population size is N, the dominance $n_i$ and the dominance set $T_i$ for each individual i are first calculated, where $n_i$ is the number of individuals that dominate the individual i and $T_i$ is the set of individuals dominated by the individual i. All individuals with a dominance of 0 are deposited into the front set $T_1$. Subsequently, all individuals are dominated by individual k in $T_1$ are deposited into the ensemble $T_k$, and $n_k$−1 is performed on each individual in $T_k$. If the $n_k$ of an individual is updated to 0, it is stored in the set $T_p$. Then, with $T_1$ as the first layer of non-dominant leading edges and $T_p$ as the new leading edge is set, the above steps are continued until all individuals are sorted. This method can effectively distinguish different levels in the population and ensure that the best individuals are prioritized and selected for subsequent optimization. A schematic of fast, non-dominated sorting is shown in Figure 1.

(2)

Crowding. Crowding is a measure of how sparse a solution is in the target space, which helps maintain the diversity of the solution set. First, the solutions of the objective function are ranked, and then the crowding of each solution is calculated as follows:

$$d_i={\displaystyle \sum _{k=1}^n\left(\frac{f_k(i+1)-f_k(i-1)}{f_k^{\max }-f_k^{\min }}\right)}$$

(4)

where $f_k(i+1)$ and $f_k(i-1)$ is the target value of the solution before and after the solution on the objective k, $f_k^{max}$ and $f_k^{min}$ is the maximum and minimum value of the objective function. Thus, more crowded solutions are selected to maintain the diversity of the solutions. A schematic diagram of the crowding distance is shown in Figure 2.

In this paper, we calculated the crowding distances of individuals within the same non-dominated frontier layer in MOHUS to quantify their distribution in the target space. When selecting leaders, hunters with higher crowding distances are prioritized.

(3)

Elite retention strategies. Firstly, the current population was merged with the original population, and the non-dominant rank of individuals was divided by rapid non-dominant ranking. Then, the new population was filled in ascending order to be close to the original size, and the individuals in the critical level were screened from high to low according to crowding degree d_i. The new population was placed into the new population in order until the original size was reached. Finally, individuals with the lowest non-dominance level and highest congestion were selected as leaders from the new population, and the remaining individuals were guided to update their positions to strengthen the global search.

3. Case Study

3.1. Background of Study Area

The water-receiving area of the Central Route of the South-to-North Water Diversion Project in Hebei Province is located in the core region of the North China Plain, south of Beijing and Tianjin, spanning approximately 114°23′ E to 116°42′ E and 36°02′ N to 39°30′ N [50], as shown in Figure 3. This area includes seven major cities: Handan, Xingtai, Shijiazhuang, Baoding, Cangzhou, Hengshui, and Langfang. Geographically, it is bounded by the Taihang Mountains to the west, and its central and eastern parts consist of flat terrain within the North China Plain, extending eastward to the Bohai Sea. The region adjoins the Beijing-Tianjin metropolitan area to the north and borders Henan and Shandong Provinces to the south, making it a densely populated and economically vital zone with intensive agricultural activity. Climatically, the area experiences a semi-humid to semi-arid continental monsoon climate with marked seasonal variations and the average annual precipitation ranges from 500 mm to 600 mm. The river systems primarily belong to the Haihe River Basin, encompassing five major tributaries: the Yongding River, Daqing River, Ziya River, South Canal, and Tuhai-Majia River [7].

The receiving areas along the Central Route of the South-to-North Water Diversion Project in Hebei Province face significant challenges due to water shortages and unequal spatial distribution. While the northern part of the region has relatively sufficient water resources, the southern part suffers from acute water shortages. Additionally, the area experiences pronounced seasonal and inter-annual variability in water availability, posing serious challenges to the reliable water supply. Currently, the region’s water supply system relies on multiple sources, including water transferred from the South-to-North Water Diversion Project, local surface water, groundwater, and reclaimed water. However, these sources differ considerably in terms of the available quantity, water quality, and supply stability, making it difficult to meet future demand through any single source. Moreover, long-term overdependence on groundwater in certain areas has led to severe overexploitation, threatening the continued use of water resources. To address these challenges, it is imperative to develop a water resource optimization model for the region. Such a model would enable integrated multi-source water allocation, improve water use efficiency, and enhance water supply security. By optimizing the distribution of available water resources, the model would help reduce reliance on local groundwater extraction and promote sustainable water management, which is a critical step toward addressing the region’s water challenges under current resource and environmental constraints.

3.2. Data Sources

This study takes the water supply area of the South-to-North Water Diversion Middle Line Project in Hebei Province as the research scope, and the water use units covered include urban domestic water, rural domestic water, industrial water, and ecological water (excluding ecological water in the river). The main sources of water supply include south-to-north water transfer, surface water, groundwater, and reclaimed water. The study uses 2022 as the base year and 2030 as the planning target year. The relevant data are mainly from the “Hebei Provincial Water Resources Bulletin [51]”, the “Statistical Yearbook [52]” of Hebei Province and the “Hebei Provincial Water Quota [53]”, and other sources.

3.3. Supply and Demand Water Forecasting

The water demand forecast was intended to predict the water demand of cities in the receiving area by 2030. It uses 2022 as the base year and adopts the quota method combined with the development plans of cities in the receiving areas. The water demand of water users in 2030 will be about 4661.82 × 10⁶ m³. The results are presented in

Table 1. Prediction results of water demand in the water-receiving region in 2030 (10⁶ m³).

Subregions	Domestic			Industry	Ecology	Total
	Urban Domestic	Rural Domestic	Subtotal
Handan	203.37	90.21	293.59	221.50	190.20	705.28
Xingtai	146.34	78.45	224.79	132.48	162.89	520.16
Shijiazhuang	450.29	78.21	528.49	276.04	241.65	1046.18
Hengshui	94.45	42.86	137.32	78.04	91.75	307.11
Cangzhou	162.04	104.06	266.10	239.65	175.68	681.44
Baoding	285.36	97.61	382.97	172.87	222.23	778.06
Xiong’an	42.78	10.24	53.02	20.60	29.28	102.91
Langfang	183.13	66.63	249.76	137.31	133.61	520.68
Total	1567.76	568.28	2136.03	1278.49	1247.29	4661.82

.

Projections of the available water supply in 2030 were conducted based on comprehensive assessments of the resource conditions of the water-receiving region. Surface water and groundwater supply capacities were determined using official water resource evaluation reports from each municipality within the region. For the water supply from the South-to-North Water Diversion Middle Line Project, the planned allocation volumes from Hebei Province’s municipal supporting projects were adopted. The reclaimed water supply potential was estimated according to municipal wastewater reuse development reports across the receiving area, with detailed projection results presented in

Table 2. Forecast results of available water supply in the receiving area in 2030 (10⁶ m³).

Subregions	South-to-North Transferred Water	Ground Water	Reservoir Water	Recycled Water
Handan	352.02	631.16	√	161.53
Xingtai	333.35	597.69		111.63
Shijiazhuang	781.54	1401.28	√	320.98
Hengshui	310.12	556.03	√	70.77
Cangzhou	453.02	812.25	√	139.50
Baoding	521.15	934.01	√	203.06
Langfang	258.43	463.36		134.82
Xiong’an	30.00	53.78	√	29.53
Total	3039.63	5449.60	1040	1171.82

. The total projected water supply for the receiving region in 2030 is approximately 10.70 billion m³. It should be noted that both surface water and groundwater supplies in this study include agricultural water use within the receiving area, which accounts for the relatively large projected values. However, in practical applications, groundwater resources are prioritized for domestic use, while surface water is primarily allocated for industrial and ecological purposes. This allocation strategy ensures that the substantial supply volumes reported herein do not affect the subsequent water resource configuration analysis in this study.

3.4. Optimal Water Resource Allocation Model

3.4.1. Objective Functions

In this study, social and economic benefits are chosen as the goals of optimal allocation, and the solution model is constructed and optimized for analysis. The social benefit function is designed to minimize regional water shortages, and the economic benefit function is designed to maximize regional economic benefits. The calculation formulas are as follows:

$$\max f_1(x)=-\min \left\{{\displaystyle \sum _{k=1}^K{\displaystyle \sum _{j=1}^{J(k)}\left[D_j^k-\left({\displaystyle \sum _{i=1}^{I(k)}{x}_{ij}^k}+{\displaystyle \sum _{m=1}^{M(k)}{x}_{mj}^k}\right)\right]}}\right\}$$

(5)

where: k and j are sub-districts and water users; $D_j^k$ is the water demand of households in District K; i, m is an independent, public water source; $x_{ij}^k$ is the amount of water supplied by an independent water source to households in District k. $x_{mj}^k$ is the amount of water supplied by public water sources to households in District k and j.

$$\max f_2(x)=\max \left\{\sum _{k=1}^K\sum _{j=1}^{J\left(\mathrm{k}\right)}\left[\sum _{i=1}^{I(k)}\left(b_{ij}^k-c_{ij}^k\right)x_{ij}^k{\alpha }_i^k+\sum _{m=1}^{M(k)}\left(b_{mj}^k-c_{mj}^k\right)x_{mj}^k{\alpha }_m^k\right]{\lambda }_j^k\right\}$$

(6)

where ${\lambda }_j^k$ is the fair coefficient of water use, $b_{ij}^k,c_{ij}^k$ and ${\alpha }_i^k$ is water supply efficiency, cost efficiency, and water supply sequence for the independent water source to supply water to households in K area; $b_{mj}^k,c_{mj}^k$ and ${\alpha }_m^k$ is water supply efficiency, cost efficiency and water supply sequence for the public water sources M to the water supply to household J in K area.

3.4.2. Constraint Conditions

This study establishes constraint conditions based on three key aspects: available water supply, water transfer capacity, and water demand.

(1)

Water demand constraints:

$$D_{j\min }^k⩽({\displaystyle \sum _{i=1}^{I(k)}{x}_{ij}^k}+{\displaystyle \sum _{m=1}^{M(k)}{x}_{mj}^k})⩽D_{j\max }^k$$

(7)

where $D_{jmin}^k$ is the minimum water demand of households in Zone K, $D_{jmax}^k$ is the maximum water demand of households in District K.

(2)

Public Water Constraints:

$$\left\{\begin{array}{c}{\displaystyle \sum _{j=1}^{J(k)}{x}_{mj}^k}⩽W_m^k\\ {\displaystyle \sum _{k=1}^K{W}_m^k}⩽W_m\end{array}\right.$$

(8)

$${\displaystyle \sum _{k=1}^K{W}_m^k}⩽Q_m^k$$

(9)

where $W_m^k$ is the maximum amount of water that can be supplied from a public water source to Area K, $Q_m^k$ is the maximum amount of water delivered from a public water source in Zone K.

(3)

Independent Water Constraints:

$${\displaystyle \sum _{j=1}^{J(k)}{x}_{ij}^k}⩽Q_i^k$$

(10)

$${\displaystyle \sum _{j=1}^{J(k)}{x}_{ij}^k}⩽W_i^k$$

(11)

where $W_i^k$ is the maximum amount of water that can be supplied to Area K from an independent water source, $Q_i^k$ is the maximum amount of water transported by an independent water source in Zone K.

3.5. Model Parameters Are Determined

3.5.1. Coefficient of Water Supply Efficiency $b_{ij}^k$

In this study, the water supply efficiency coefficient of each city is determined according to the output value of unit water consumption, which fully considers the practical situation of each water industry and the requirements of regional economic development. First, it is essential to guarantee the use of water for domestic use and assign a large value to the water supply efficiency coefficient of urban and rural life, with the benefit of urban domestic water set at 800 CNY/m³ and the benefit of rural domestic water set at 750 CNY/m³. For industrial water use, the inverse of water use per 10,000 CNY of industrial value added is used to calculate the benefit of the water supply of water resources. This indicator represents the economic efficiency and resource utilization efficiency of industrial water and is a significant indicator for the management and utilization of water resources in the industrial sector. Secondly, with regard to the accounting of ecological water and water resource income, combined with the experience of predecessors, a unified standard of 650 CNY/m³ is used for the calculation. The income from water resources used by various industries is calculated as the comprehensive income value of water use in each district and county, and the detailed results are displayed in

Table 3. Revenue calculation of water use units in various industries in the water-receiving area in 2030 (CNY/m³).

Subregions	Unit Benefit
	Urban Domestic	Rural Domestic	Industry	Ecology
Handan	800	750	540	650
Xingtai	800	750	520	650
Shijiazhuang	800	750	600	650
Hengshui	800	750	515	650
Cangzhou	800	750	550	650
Baoding	800	750	580	650
Xiong’an	800	750	580	650
Langfang	800	750	590	650

.

3.5.2. Coefficient of Water Supply Cost $c_{ij}^k$

Based on the newest water supply costs, the water supply coefficients for urban, rural, industrial, and ecological water use are listed in

Table 4. Water tariff structure by category for cities in the water-receiving area (CNY/m³).

Subregions	Urban Domestic	Rural Domestic	Industry	Ecology
Handan	3.6	3.5	6.3	2.0
Xingtai	2.8	2.5	5.2	1.5
Shijiazhuang	2.9	2.7	5.3	1.2
Hengshui	3.0	2.7	5.5	1.6
Cangzhou	3.5	3.5	6.0	1.5
Baoding	3.3	3.2	5.8	2.5
Xiong’an	3.5	3.5	6.0	1.5
Langfang	3.6	3.5	6.2	2.8

.

3.5.3. Coefficient of Water Supply Sequence ${\alpha }_i^k$

The water supply priority factor reflects the preferential order among different water sources, which is determined by Equation (12). In this study, the water sources included inter-basin transferred water (SNWDP supply), reclaimed water, surface water, and groundwater. The corresponding priority coefficients for the different water sources serving various user types are presented in

Table 5. Water allocation priority coefficients for different water sources and user types.

Water Supply	Urban Domestic	Rural Domestic	Industry	Ecology
Groundwater	0.33	0.33	0.1
Reservoir water			0.2	0.33
South-to-North transferred water	0.67	0.67	0.3
Recycled water			0.4	0.67

.

$${\alpha }_i^k={\displaystyle \frac{1+n_{kimax}-n_{ki}}{{\displaystyle \sum _i^n(1+n_{kimax}-n_{ki})}}}$$

(12)

where $n_{ki}$ is the amount of water supplied by source $i$ in subarea $k$ and $n_{k\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum amount of water supplied.

3.5.4. Coefficient of Water Supply Fairness ${\lambda }_i^k$

The water allocation equity coefficient shows the prioritization hierarchy among water users and can be calculated using Equation (12), as follows: The water supply priority is established as urban domestic use > rural domestic use > ecological use > industrial use, with corresponding equity coefficients of 0.4, 0.3, 0.2, and 0.1, respectively.

4. Results and Discussion

4.1. Simulation Experiments

To evaluate the stability and solution accuracy of the proposed algorithm, we conducted a comprehensive performance assessment using four well-established, bi-objective benchmark functions. Moreover, the improved algorithm is compared with two high-performing multi-objective optimization algorithms: MOPSO and NSGA-II. The performance evaluation employed two critical metrics to assess the solution quality: convergence and diversity. Convergence was measured using Generational Distance (GD), which quantifies how close the set of solutions obtained is to the true Pareto frontier. Diversity was evaluated through spread (Δ), indicating the homogeneity of the distribution of solutions across the Pareto frontier. The mathematical formulations of these evaluation metrics are as follows:

• Generational Distance

$$GD={\displaystyle \frac{1}{N_P}}{\displaystyle \sum _{i=1}^{N_P}{d}_i}$$

(13)

where $N_p$ represents number of computed Pareto optimal solutions, with $d_i$ represent the minimum distances of solution $x_i$ to ideal Pareto optimal frontier.

• Spread

$$∆={\displaystyle \frac{d_{\min }+d_{\max }+{\displaystyle \sum _{c=1}^{N_P-1}|}{d}_c-\overline{d}|}{d_{\min }+d_{\max }+(N_P-1)\overline{d}}}$$

(14)

where $d_{min}$ is the distance between the solution with minimum objective value in the obtained Pareto set and its counterpart on the theoretical Pareto front; $d_{max}$ is the distance between the solutions with maximum objective values in both sets; $\overline{d}$ denotes the average distance between adjacent solutions and $d_c$ indicates the distance between consecutive solutions.

The experimental results of the algorithm evaluation are shown in

Table 6. The performance test results.

Test Function	Algorithm	Generational Distance	Spread
ZDT1	NSGA-II	0.033	0.463
	MOPSO	0.058	0.681
	MOHUS	0.035	0.439
ZDT2	NSGA-II	0.072	0.436
	MOPSO	0.089	0.639
	MOHUS	0.068	0.429
ZDT3	NSGA-II	0.114	0.576
	MOPSO	0.391	0.832
	MOHUS	0.106	0.550
ZDT6	NSGA-II	0.043	0.467
	MOPSO	0.061	0.738
	MOHUS	0.039	0.418

, which compares the performance of the proposed Augmented Hunting Search (MOHUS) algorithm with NSGA-II and MOPSO using two key metrics: Generational Distance (GD) for convergence measurement and Spread (Δ) for diversity assessment. The comparative analysis revealed that while NSGA-II demonstrated slightly better convergence on the ZDT1 test function, the MOHUS algorithm outperformed both benchmark algorithms in terms of convergence and solution diversity across the remaining test functions (ZDT2, ZDT3, and ZDT6). The superior metric values for each test case are marked in bold in Table 6 for clear identification. These results demonstrate that the improved algorithm has significant advantages in solving multi-objective optimization problems. As visually confirmed in Figure 4, the Pareto front solutions obtained for MOHUS show excellent agreement with the true Pareto optimal solutions, clearly illustrating its outstanding performance in terms of both convergence accuracy and solution distribution uniformity. The comprehensive evaluation validates MOHUS as a competitive approach for multi-objective optimization tasks.

4.2. Results Analysis

This study employs the Multi-Objective Hunting Search Algorithm (MOHUS) to solve the multi-objective optimal allocation of water resources in the Hebei section of the Central Route of the South-to-North Water Diversion Project. The algorithm parameters were established as follows: the population size is 600, and the maximum number of iterations is 500. Numerical solutions were obtained using the MATLAB (R2019b version) platform, yielding the corresponding Pareto optimal solution sets. Figure 5 illustrates the convergence process of the average individual values of the objective functions during the algorithm optimization. As shown in the figure, the iterative curves of the average values of both objective functions demonstrate convergence at approximately 250 iterations. After 250 iterations, although minor fluctuations persisted, the results exhibit a stable trend.

The study gained a Pareto optimal solution set for water allocation in the area through multi-objective optimization modeling, providing decision-makers with multidimensional alternatives. It is significant to note that all the solutions in Pareto’s frontier are non-dominant, which means that there is no absolutely optimal solution. Decision-makers must make trade-offs among competing goals, like economic benefits and social benefits, based on practical requirements. Considering the more prominent contradiction between the supply and demand of water resources in the Hebei section of the South-to-North Water Diversion Central Route, we selected an optimization scheme that prioritizes social benefits as the final solution. A detailed analysis was subsequently conducted for the selected scheme.

From the water supply source perspective, the total allocated water volume reached 4082.88 × 10⁶ m³. The breakdown by source shows that surface water contributed 227.00 × 10⁶ m³, groundwater 196.29 × 10⁶ m³, South-to-North Water Diversion supply 2610.34 × 10⁶ m³, and reclaimed water 1023.99 × 10⁶ m³. The proportional distribution among these sources was calculated as 6.18:4.81:63.93:25.08, respectively. The comprehensive water supply quantities from different sources across the receiving area are presented in

Table 7. The findings of water distribution in the receiving areas by water source in 2030 (10⁶ m³).

Subregions	Water Demand	Water Allocation
		Surface Water	Ground Water	South-to-North Transferred Water	Recycled Water	Total
Handan	705.28	70.38	6.73	349.38	162.80	589.30
Xingtai	520.16	0.00	4.44	297.14	104.44	406.03
Shijiazhuang	1046.18	110.19	5.60	651.79	278.61	1046.18
Hengshui	307.11	38.31	10.39	189.79	68.62	307.11
Cangzhou	681.44	0.00	15.02	430.31	113.80	559.14
Baoding	778.06	10.30	24.41	459.29	195.98	689.98
Xiong’an	102.91	23.10	23.02	30.00	22.99	99.12
Langfang	520.68	0.00	106.66	202.63	76.73	386.02
Total	4661.82	252.28	196.29	2610.34	1023.99	4082.88

.

From the water user’s perspective, the total allocated water volume was 4082.88 × 10⁶ m³, with a water shortage of 578.94 × 10⁶ m³, resulting in a deficiency rate of 12.42%. A detailed analysis by region shows that Handan City experienced no shortages in either urban or rural domestic water supply but had industrial and ecological deficits of 68.06 × 10⁶ m³ and 47.92 × 10⁶ m³, respectively. Xingtai City similarly maintained an adequate domestic water supply while showing industrial and ecological shortfalls of 47.71 × 10⁶ m³ and 66.43 × 10⁶ m³. Both Shijiazhuang and Hengshui Cities reported no water shortages across all sectors. Cangzhou City demonstrated sufficient domestic water availability but industrial and ecological deficits of 64.53 × 10⁶ m³ and 57.76 × 10⁶ m³. Baoding City showed the same pattern with industrial and ecological shortages of 53.41 × 10⁶ m³ and 34.67 × 10⁶ m³. Langfang City reported industrial and ecological water deficits of 74.98 × 10⁶ m³ and 59.68 × 10⁶ m³ while maintaining domestic water security. Xiongan New Area only showed a minor ecological water shortage of 3.79 × 10⁶ m³, with all other sectors meeting demand. Sectoral water allocation details across the receiving area are presented in

Table 8. The findings of water distribution in Receiving Areas by user type in 2030 (10⁶ m³).

Subregions	Water Demand	Water Allocation
		Urban Domestic	Rural Domestic	Industry	Ecology	Total
Handan	705.28	203.37	90.21	155.54	140.18	589.30
Xingtai	520.16	146.34	78.45	76.80	104.44	406.03
Shijiazhuang	1046.18	450.29	78.21	276.04	241.65	1046.18
Hengshui	307.11	94.45	42.86	78.04	91.75	307.11
Cangzhou	681.44	162.04	104.06	179.23	113.80	559.14
Baoding	778.06	285.36	97.61	120.71	186.30	689.98
Langfang	520.68	183.13	66.63	59.53	76.73	386.02
Xiong’an	102.91	42.78	10.24	20.60	25.49	99.12
Total	4661.82	1567.76	568.28	966.49	980.35	4082.88

.

Figure 6a shows the urban domestic water allocation across cities in the water-receiving region. The total urban domestic water demand and allocated supply both equaled 1567.76 × 10⁶ m³, indicating full satisfaction of water requirements in all sub-districts. Urban water supply is jointly provided by South-to-North Water Diversion (SNWD) sources (94.46%) and groundwater (5.54%), with SNWD serving as the primary water source for urban domestic needs in the receiving areas.

Figure 6b shows rural domestic water allocation across municipalities in the water-receiving zone. The total rural domestic water demand and allocated supply both reached 568.28 × 10⁶ m³, demonstrating complete fulfillment of water requirements in all sub-districts. The rural water supply system combines South-to-North Water Diversion (SNWD) sources (79.79%) with groundwater (20.21%), establishing SNWD as the principal water source for rural domestic needs in the service area.

Figure 6c shows the industrial water allocation to municipalities in the receiving zone. The total industrial water demand in the receiving zone is 1278.49 × 10⁶ m³, the total water allocation is 966.43 × 10⁶ m³, the water deficit is 312.06 × 10⁶ m³, and the water deficit rate is 24.41%. The cities with industrial water shortages are Handan, Xingtai, Cangzhou, Baoding, and Langfang, with water shortage rates of 29.78%, 42.03%, 25.21%, 30.17%, and 56.66%, respectively. The industrial water in the receiving areas is mainly supplied by water transferred from the South-to-North Water Diversion (SNWD), recycled water, and surface water, accounting for 69.40%, 6.85%, and 23.75% of the total allocated water, respectively, of which the water transferred from the SNWD is the main source of water supply.

Figure 6d shows the water allocation for the ecosystem in the municipalities of the receiving water area. The total amount of ecological water demand in the receiving area is 1247.29 × 10⁶ m³, the total amount of distribution is 984.30 × 10⁶ m³, and the total amount of water shortage is 262.99 × 10⁶ m³, with a water shortage rate of 21.09%. The cities with ecological water shortages are Handan, Xingtai, Cangzhou, Baoding, and Langfang, with water shortage rates of 26.30%, 35.88%, 35.22%, 16.17%, and 45.57%, respectively. The ecological water in the receiving areas was supplied by recycled water and surface water, accounting for 91.00% and 9.00% of the total water allocated, respectively, with recycled water being the main source of water supply.

4.3. Discussion

In this paper, according to the forecast of socio-economic development indicators in the planning target year of 2030, the quota method is adopted to forecast the water demand of eight receiving units and 32 water users. The total water demand of the receiving area in 2030 will be 4661.82 × 10⁶ m³, including the urban domestic water demand of 1567.76 × 10⁶ m³, rural domestic water demand of 568.28 × 10⁶ m³, industrial water demand of 1278.49 × 10⁶ m³, and ecological water demand of 1247.29 × 10⁶ m³. The water allocation results indicate that the south-to-north water transfer is 2610.34 × 10⁶ m³, surface water is 252.28 × 10⁶ m³, groundwater is 196.29 × 10⁶ m³, and recycled water is 1023.99 × 10⁶ m³, resulting in a deficit of 578.94 × 10⁶ m³ and a deficit rate of 12.42%, which is mainly reflected in the demand for water for industrial and ecological purposes. Through the analysis, the proportion of south-to-north water transfer in the urban water supply is 94.46%, while the proportion of rural water supply is only 79.79%, which indicates that the receiving area still exploits groundwater to a certain extent, and the amount of rural groundwater exploitation is high. Therefore, in order to decrease the continuous exploitation of groundwater sources in receiving areas, the water supply of the South-to-North Water Diversion Project can be increased through measures such as increasing the South-to-North Water Diversion and Storage Project or recharging the water supply through the Diversion of the Yangtze River to the Han River Project. In this way, it can effectively alleviate the problem of overexploitation of groundwater, reduce the occurrence of disasters such as ground subsidence, and, at the same time, help the gradual recovery of the groundwater level and improve the regional water cycle. In addition, the south-to-north water transfer can supplement ecological water, restore wetlands, rivers, and other ecosystems, and alleviate the ecological degradation caused by groundwater depletion.

The HUS algorithm is a metaheuristic optimization algorithm based on group intelligence that addresses issues by mimicking the cooperative strategies employed by predator groups during the hunting process. This algorithm features a straightforward structure, few parameters, and excellent global exploration capabilities. However, the traditional HUS algorithm has limitations, such as insufficient convergence accuracy and poor stability, when solving complex multi-objective optimization challenges. To improve the performance of the HUS algorithm in water resource allocation optimization, an improved HUS algorithm is proposed in this study. The improvement strategies include (1) introducing a non-dominated sorting mechanism to divide the solution set into Pareto classes to clarify the dominant relationship of the solutions in the objective space; (2) adopting a congestion distance assessment method to maintain the distributivity of the solution set to ensure an evenly distributed population across the Pareto front; and (3) incorporating an elite retention strategy to avoid the loss of high-quality individuals during iteration, thus effectively improving the algorithm convergence stability and search accuracy.

To validate the performance of the enhanced algorithms, this research selects two classical multi-objective optimization methods as comparative benchmarks. The results of the simulation experiments show that the improved HUS algorithm has significant advantages in terms of the two key aspects of convergence and diversity of solution sets. This study confirms that the improved HUS algorithm offers a novel methodology for addressing complex water resource optimization challenges.

5. Conclusions

This study presents an enhanced hunting search algorithm (MOHUS) for solving the multi-objective water allocation model that simultaneously maximizes economic benefits and minimizes water shortages. By solving this model, the Pareto optimal front for this multi-objective optimization problem is obtained. Single-objective optimization problems tend to directly compare the values of a single-objective function to obtain a solution. However, in multi-objective optimization problems, due to the competitive relationship between the objectives, solutions cannot be compared using a simple merit relationship. Therefore, the decision-maker must conduct objective prioritization based on priority criteria to determine the most operationally feasible solution. In this study, since the supply and demand of water resources in the receiving area are relatively tense, the optimal allocation of different water sources in the receiving area of the South-to-North Water Diversion Project in Hebei Province in 2030 is obtained by using the minimum water deficit as the decision criterion, and the one with the smallest water deficit is screened from the Pareto optimal frontier as the final allocation scheme. The study shows that the total regional water demand under this scheme is 4661.82 × 10⁶ m³, the total water supply is 4082.88 × 10⁶ m³, and the water deficit is 578.94 × 10⁶ m³, with a total water deficit rate of 12.42%. The allocation scheme not only conforms to the basic principles of water resource management in the receiving area but also realizes multi-objective synergistic optimization, providing a reliable decision basis and technological assistance for the scientific positioning of the receiving area.

In addition, to verify the performance of the improved algorithm, four standard multi-objective test functions (ZDT series) are selected in this study, and comparative experiments are carried out for the improved algorithm, NSGA-II, and MOPSO. The results indicate that the MOHUS algorithm exhibits better performance in both convergence and diversity: the Pareto frontier it obtains is not only closer to the true frontier, but also the solution set is more evenly distributed. This result proves the effectiveness and superiority of the MOHUS algorithm in solving multi-objective optimization problems.

This paper uses the water users of the South-to-North Water Diversion Project as a starting point and, therefore, does not consider agricultural water units when constructing the model. However, agricultural water use is also a key factor in regional water allocation. Future research can further expand the scope of this study to build a more comprehensive water allocation model and provide more scientific and reasonable water allocation strategies for various water-using sectors.