Optimal allocation planning of regional water resources with multiple objectives using improved firefly algorithm

Water resource is an important material basis for the development of human society. Its rational utilization and optimal allocation are crucial to maintain the balance of nature, ensuring social and economic development (Chen et al. 2023). Also, it should meet the growing needs of the people. The allocation of water resources refers to the rational allocation of water in different regions under the limited conditions of resources, achieving the optimal water resource utilization effect while minimizing waste and ensuring various water needs (Xiang et al. 2021). Rationalize the allocation of water quality and quantity and solve the problem of water pollution, thereby effectively maintaining the ecological environment and ensuring a stable natural balance (Deng et al. 2020; Wang et al. 2024). In summary, achieving the optimal allocation of water resources holds immense significance, as it contributes to enhancing the efficiency of water resource utilization and driving the advancement of the regional economy and society (Li et al. 2022). In addition, it ensures the healthy and stable ecological environment (Li et al. 2021a; Yao et al. 2023b). To achieve these goals, it is necessary to focus on studying the theories and model solutions of water resource allocation in different regions, promoting the practice of optimizing water resource allocation, and achieving an organic combination of resource guarantee and sustainable development (Ji et al. 2019; Saad & Gamatié 2020; Wu & Wang 2022a, 2022b).

The optimal distribution of water resources has garnered significant attention among scholars on a global scale (Zhang et al. 2020; Wang et al. 2023a). Pal et al. (2020) pioneered a predictive water resource model for the Indus River Basin. Exploring water resource allocation in drought-prone Mexican regions, Chung et al. (2018) intertwined hydrological and economic models. Giuliani et al. (2022) navigated the Ommo-Turkana River basin's complexities with a multi-objective optimal model, while Qiu et al. (2019) championed flood resource allocation alongside ecological benefits. In the Lancang Mekong River Basin, Li et al. (2020a) masterminded a resource allocation model that spanned China, Thailand, and Myanmar. Examining Ordos City's water resource system holistically, Chen et al. (2020) delved into its intricate interplay with energy and food sectors, driving the pursuit of sustainable solutions. This endeavour encompassed strategic water resource allocation within Ordos City, underlining the pivotal significance of efficiency and fairness in this realm. Efficiency, a bedrock of resource management, underscores the economic gains borne from optimal allocation. In tandem, fairness ensures a harmonious supply–demand equilibrium and reaps the societal benefits embedded in water's multifaceted value (Li et al. 2020b). Only by scientifically and systematically allocating water resources to each water conservancy department can economic and social benefits be generated (Di Baldassarre et al. 2019). In terms of efficiency, Xu et al. (2019) pivotally championed economic prosperity as an essential criterion in the objective function for gauging water resource allocation. Jiang et al. (2019) assessed the fairness of allocation by building a simulation model dedicated to optimizing water allocation to irrigation ponds, where spatial dynamics takes an important role in the assessment by scrutinizing the supply and demand of water. Concurrently, Hunt & Shahab (2021) ingeniously tapped into the water scarcity rate as a salient metric, mirroring the social fabric's aspirations for equitable distribution. These methodologies have undoubtedly yielded promising outcomes in regional water resource management (Li & Yu 2023). Yet, their vista remains anchored within present boundaries, as they fall short in the realm of intergenerational equity, warranting further exploration and innovation (Wang et al. 2023b; Wu et al. 2023; Yao et al. 2023a).

With rapid computer technology advancement, intelligent optimization algorithms like particle swarm optimization (PSO) (Bai et al. 2017) and flower pollination algorithm (Bai et al. 2022) solve multi-objective problems, such as nature-inspired algorithms (NIAs), mimicking natural behaviours, and gain traction. Amid complex industrialization, NIA excels in optimizing intricate challenges like water resource allocation. It handles non-deterministic polynomial problems effectively, achieving significant results. Moradikian et al. (2022) used a trial dynamic overdrive pacing trial algorithm to optimize the allocation of water resources for wastewater recovery in urban areas; Ghorbani Mooselu et al. (2019) used non-dominated sorting genetic algorithm II (NSGA-II) to optimize the allocation of water resources in rivers; Thilagavathi & Amudha (2019) used the social spider algorithm to optimize the allocation of water bodies in the Coimbatore region of India. NIAs are versatile and adaptable, showing promise in various applications by effectively addressing complex optimization challenges.

NIA includes evolutionary algorithms (EAs) and swarm intelligence (SI) algorithms. Although these NIAs are powerful, they still have limitations. For example, EA has high computational cost, and it is very easy to be stuck in local optima, as well as too sensitive to problem characteristics and noise. SI relies on iterative optimization, which means that they may require a significant number of iterations to converge to a solution and SI is very sensitive to the tuning of parameters, such as population size, crossover probability, and mutation probability. Firefly algorithm (FA) exhibits several advantages over traditional NIA. Firstly, FA often demonstrates superior global search capabilities, effectively avoiding the pitfalls of local optima. Secondly, it shows better adaptability for addressing complex problems and optimizing high-dimensional spaces. In addition, the simplicity and ease of implementation of the FA algorithm make it more appealing for practical applications. Overall, FA algorithm's performance in certain aspects positions it as a notable contender in the field of NIAs. FA has been used in several fields of engineering and optimization problems. Yang & He (2013) introduced the FA, inspired by flickering silver firebugs seeking food. Attraction strength varies with firefly distance, with high-intensity ones attracting low-intensity ones, reducing distance and updating intensity. Fireflies have high brightness and minimum distance, driving them to the optimal solution for the objective function. The FA has been applied in many situations, such as the minimum computing time for digital image compression (Del Ser et al. 2019), feature selection (Bacanin et al. 2023; Dong et al. 2024), multi-objective scheduling problem (Daş et al. 2020; Chakravarthi et al. 2021; Wang et al. 2024), and optimization of neural network parameters (Zhang et al. 2023). While FA offers simplicity, a clear concept, and strong search capability, it can be prone to local optima, slow convergence, and low accuracy, driving researchers to enhance the standard FA. Li et al. (2021b) introduced an automatic learning mechanism in the standard FA to adjust the parameters of the algorithm, allowing the algorithm to adjust the parameter values at any time according to the environment. Kaveh & Javadi (2019) have improved the individual search mechanism and parameter adaptability of fireflies, and embedded chaotic maps to solve mechanical design optimization problems. Ali et al. (2018) employed fuzzy coefficients to fine-tune the parameters of the FA, achieving a balance between its local and global search capabilities. For the improved performance of FA, many optimization problems for test functions and engineering optimization problems have been verified (Liu et al. 2020). FA has been used on water allocation problems too (Wang et al. 2017, 2021). However, FA also comes with some drawbacks. Firstly, the convergence speed of FA may be slower for specific types of problems, requiring more iterations compared to some other algorithms to reach the optimal solution. Secondly, FA can be sensitive to parameter choices, especially for complex problems, necessitating careful tuning for optimal performance. In addition, FA may demand significant computational resources when dealing with large-scale problems, potentially limiting its practical feasibility in certain applications. Finally, the black-box nature of the FA algorithm might pose challenges in interpretation, making it difficult to explain its decisions in applications that require detailed explanations. Considering these factors, using FA requires a careful balance between its strengths and limitations. To improve FA, we introduce an improved one and apply it to water allocation. Finally, we succeed in achieving ideal results.

This study applies an improved firefly algorithm (IFA) to optimize water resource allocation. The enhanced approach includes adaptive inertia weight logarithm, dynamic step size adjustment, and rotation iteration, promoting balanced exploration, development, and faster convergence. It also incorporates a step size adjustment factor to prevent local optima traps.

This article aims to address some issues in FA and provide the improved one called IFA, using various methods, including those from mathematics and algorithm structures. Then we use IFA to optimize the allocation of water resources, including mathematically modelling the water resources of Shaanxi Province, utilizing algorithms to optimize data. It is structured as follows: Section 2 introduces the mathematical model, including objective functions, constraints, and overall flow, as well as the IFA and simulation experiments to validate IFA's performance. Next, Section 3 presents the research and relevant data. Then, Section 4 describes the results obtained from solving the model from various angles. Subsequently, Section 5 discusses the results accompanied by analysis and policy recommendations for water allocation in the region. Finally, Section 6 concludes this study, along with a glimpse into future research.

Efficient water resource allocation can greatly impact society, economy, and sustainability. This study aims to optimize social welfare, economic growth, and ecological preservation through a multi-objective model. Using a holistic approach, synergies among these goals are targeted. Table 1 outlines key parameters and variable implications.

Table 1

Parameters of objective functions

.	Parameter .	Description .
Fundamental	K	Number of subareas
	I(K)	Number of water sources
	J(K)	Number of water users
		The water quantity obtained by user j in partition k from water source i (⁠⁠)
Social		Water demand of user j in partition k (⁠⁠)
Economic		The unit water supply benefit coefficient (⁠⁠)
		The unit water supply cost coefficient (⁠⁠)
		Weight of partition
		Water supply order coefficient of water source j in partition
		Water equity coefficient for user j in partition
Ecological		Pollutant emission content of user j in partition
		Sewage discharge coefficient of user j in partition

.	Parameter .	Description .
Fundamental	K	Number of subareas
	I(K)	Number of water sources
	J(K)	Number of water users
		The water quantity obtained by user j in partition k from water source i (⁠⁠)
Social		Water demand of user j in partition k (⁠⁠)
Economic		The unit water supply benefit coefficient (⁠⁠)
		The unit water supply cost coefficient (⁠⁠)
		Weight of partition
		Water supply order coefficient of water source j in partition
		Water equity coefficient for user j in partition
Ecological		Pollutant emission content of user j in partition
		Sewage discharge coefficient of user j in partition

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In modern times, urbanization serves as a key measure of societal advancement. With urban growth comes increased water consumption, akin to the development of civilizations relying on ample water resources. Abundant water is essential for thriving urban centres, supporting economies, and enabling harmonious living and work environments. However, insufficient water supply poses a primary constraint on urban expansion.

Therefore, using water scarcity as a measure of social benefits is relatively reliable. The minimum value of water shortage is represented by Equation (1), in tens of thousands. The smaller the water shortage, the more water demand in different cities can be guaranteed, and the faster society can develop.

Water resources offer economic gains in sectors like aquaculture and industrial cooling. To optimize benefits, integrated planning is crucial, aligning industries for economic returns.

Therefore, Equation (2) is used as an economic benefit function to represent the maximum economic benefit after overall planning, in units of yuan. When the economic benefits are greater, various fields can advance faster.

signifies the sequence in which water source i allocates water to users across distinct regions k.

The range of is between 0 and 1. If the number is 1, it represents the highest priority for water supply from source i.

reflects the water usage order of water source i and user j in different regions k. The range of is between 0 and 1, where a value of 1 indicates the highest priority for users to utilize water resources.

Urbanization, including industrial wastewater, sewage, and runoff, contaminates water bodies and ecosystems. Mitigating harm aligns with sustainability for society and urban areas.

Equation (5) assesses ecological benefits via minimal pollutant emissions (in units of mg) in production, enhancing both processes and daily life. These advantages enable further investment in ecological balance, fostering a symbiotic interplay.

In water allocation, meeting user demand forms social benefits, while delivering water to zones with benefits exceeding costs yields economic gains. Discharging sewage from partitions provides ecological advantages.

Water resource allocation must consider constraints beyond benefits, including capacity alignment, demand limits, and transmission feasibility. Four restrictive factors are as follows: supply capacity, resident demand, transmission feasibility, and non-negativity.

Equation (6) shows the water supply amount from water source i to sub-area k, indicating that the water supply amount from sub-area k to water source i exceeds the sum of the water supply amounts from sub-area k to water source i and then finally to all users (in units of ⁠).

In Equation (7), and are the minimum and maximum water demands of user j in sub-zone k, respectively, which indicates that the distribution of water supply from sub-area k to user j must be within a certain range, wherein and (⁠⁠).

In Equation (8), is the maximum water delivery capacity for water source i to user j from sub-district k, which indicates that the water supply cannot exceed the maximum water delivery capacity, otherwise it cannot meet the demand, = min {}(⁠⁠).

In Equation (9), it indicates that the water supply cannot be negative, and otherwise it does not comply with social rules and cannot generate social benefits.

The FA is a nature-inspired optimization method mimicking the signalling behaviour of fireflies (Yang & He 2013). It assumes fireflies lack gender distinctions, enabling any to attract others based on brightness. Brightness determines attraction strength, and dimmer fireflies move towards brighter ones. If no brighter firefly exists, random movement occurs. Brightness relates to the objective function, forming the core of FA. This section explains FA's mathematical basis, equations, and principles. Relevant parameters are listed in Table 2.

Table 2

Parameters of FA

.	Parameter .	Description .
Brightness		The maximum brightness of fireflies
	Rij	The distance between fireflies i and j (defined as Euclidean distance)
Attractiveness	γ,	The light absorption coefficient of a given propagation medium = (l ∼ l/⁠)
		= (⁠∼ l ∼ l/⁠)
		Best attractiveness
Param	Dim	Dimension
	Rangepop	Value range
	M	population size
	Gama	The absorption coefficient of light by the propagating medium
	belta0	Initial attractiveness value
	Alpha	Step disturbance factor
	ite	Iterations

.	Parameter .	Description .
Brightness		The maximum brightness of fireflies
	Rij	The distance between fireflies i and j (defined as Euclidean distance)
Attractiveness	γ,	The light absorption coefficient of a given propagation medium = (l ∼ l/⁠)
		= (⁠∼ l ∼ l/⁠)
		Best attractiveness
Param	Dim	Dimension
	Rangepop	Value range
	M	population size
	Gama	The absorption coefficient of light by the propagating medium
	belta0	Initial attractiveness value
	Alpha	Step disturbance factor
	ite	Iterations

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Each firefly lacks gender bias, making them responsive to the allure of others regardless of gender. The degree of attraction is tied to their brightness; when comparing two fireflies, the dimmer one moves towards the brighter counterpart. Attraction declines as the gap between fireflies increases. If no brighter firefly exists, one moves randomly. Brightness hinges on the objective function landscape, directly correlating in maximization problems.

Within the FA, two pivotal considerations emerge: the modulation of brightness and the mechanism of attraction. For the sake of simplification, we can posit that firefly allure hinges upon their brightness, an attribute intertwined with the encoded objective function. In essence, the degree of attractiveness exhibited by fireflies is intrinsically tied to their brightness, which, in turn, is intricately linked to the specific objective function that is being encoded.

In the context of a basic maximum optimization problem, the firefly's brightness at a specific position can be represented as ∝. However, attraction is relative and changes according to observers or other fireflies. Thus, it adjusts with the distance among fireflies. Also, light intensity diminishes with distance and is affected by medium absorption. This interaction is governed by the inverse square law, which represents the intensity at the light source. For a constant medium absorption coefficient, light intensity shifts with distance, including the initial intensity.

For fixed γ, the feature length becomes τ = ⁠. When m⁠, on the contrary, for the given length scale in the optimization problem τ, parameter γ can be used as a typical initial value. That is, γ = ⁠.

The second term corresponds to attractiveness, while the third term represents randomization, governed by a randomization parameter Rand, which follows a uniform distribution in [0,1]. In most cases of our implementation, the sum of these terms suffices. Furthermore, the randomization term can be readily adapted to follow a normal distribution or other distributions. Moreover, when significant scale disparities exist across dimensions, such as for in one dimension or for in another, it is advisable to use instead, with scale parameters in dimension D tailored to the actual interests’ scale of the fireflies.

Now, the parameter represents the change in attractiveness, and its value is crucial for determining convergence speed and the behaviour of the FA, in theory. However, in reality, it is determined by the feature length of the system to be optimized. Therefore, in most situations, it typically begins at 0.

Based on the mathematical principles of the FA, the following pseudo-code is provided, as shown in Algorithm 1.

Algorithm 1 Pseudo-code of the FA

Input: brightness ⁠, population of fireflies ⁠, absorption coefficient γ, iteration Max-Generation

Output: best brightness ⁠, re-ranked fireflies set

1: Objective function f (⁠⁠), =

2: Generate initial population of fireflies

3: Light intensity at is determined by f (⁠⁠)

4: Define light absorption coefficient γ

5: While (⁠ < Max-Generation)

6:    For = 1: n all in fireflies

7:     Forj in 1: all n fireflies

8:      If (⁠⁠), Move firefly I towards j in d-dimension;

9:      End If

10:       Attractiveness varies with distance r via exp [-]

11:       Evaluate new solutions and update light intensity

12:     End for

13:    End for

14:    Rank the fireflies and find the current best

15: End while

FA has not only strengths but also limitations, relying on outstanding individuals within range for effective search. This reliance slows convergence. Near peaks, step size challenges can cause oscillation. This study proposes three enhancements, detailed below. Parameters are outlined in Table 3:

Table 3

Parameters of IFA

.	Parameter .	Description .
Brightness		The maximum brightness of fireflies
	Rij	The distance between fireflies i and j (defined as Euclidean distance)
Attractiveness	γ, I	The light absorption coefficient of a given propagation medium = (l ∼ l/⁠)
		= (⁠∼ l ∼ l/⁠)
		Best attractiveness
Param	Dim	Dimension
	Rangepop	Value range
	M	Population size
	Gama	The absorption coefficient of light by the propagating medium
	belta0	Initial attractiveness value
	Alpha	Step disturbance factor
	ite	Iterations
New Param	a	Step size regulator
	T	Rotation iteration parameters

.	Parameter .	Description .
Brightness		The maximum brightness of fireflies
	Rij	The distance between fireflies i and j (defined as Euclidean distance)
Attractiveness	γ, I	The light absorption coefficient of a given propagation medium = (l ∼ l/⁠)
		= (⁠∼ l ∼ l/⁠)
		Best attractiveness
Param	Dim	Dimension
	Rangepop	Value range
	M	Population size
	Gama	The absorption coefficient of light by the propagating medium
	belta0	Initial attractiveness value
	Alpha	Step disturbance factor
	ite	Iterations
New Param	a	Step size regulator
	T	Rotation iteration parameters

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During each iteration, every individual is drawn towards the brightest one, causing the brightest individual to remain static until other even brighter individuals emerge. To counteract this, we introduce an element of randomness into the quest for the brightest individual. This entails exploring the vicinity and relocating to a more recent position if fitness levels advance, otherwise, adhering to the initial position is maintained.

In the context of algorithmic evaluation, this study employs a comprehensive selection of seven algorithms, which includes both the FA and the IFA, for rigorous testing. Each test function undergoes 100 iterations with a population size of 50 for each algorithm. The convergence curve, often referred to as the fitness curve, serves as a pivotal benchmark for meticulously assessing algorithmic performance.

Figure 3 and Table 6 show that IFA consistently outperforms original FA in nine test functions under uniform conditions. IFA exhibits better statistical results and faster convergence. While trailing SMA in to ⁠, IFA surpasses other algorithms. Particularly in and ⁠, IFA excels, finding theoretical optimal values. This stems from IFA's step size adaptability reduction and rotation iteration, enhancing search speed and accuracy. IFA proves highly feasible.

Table 6

Simulation results

Statistic .	Algorithm .	.	.	.	.	.	.	.	.	.
Convergency value	FA	169,392	166,606.3	−3,608.5	83.45	0.94	3.19	3.765	169,392	3.6
	IFA	0.19	9.33	9.10	2.14	4.72	1.93	1.38	0.19	9.33
	ABC	−3,828.4	−3,783.7	−3,828.4	−3,828.4	−3,828.4	−3,828.4	−3,828.4	−3,828.4	−3,783.7
	GOA	83.46	1.62	7.37	2.5	1.21	2.02	0.15	83.45789	1.62
	SMA	210,278.5	0.013	1.56	0.49	0.37	4.79	4.35	210,278.5	1.35
	BOA	17,849.6	2.5	0.10	0.13	2.64	3.71	0.37	17,849.6	2.5
	MFO	18,785.9	0.012	0.067	0.09	1.27	0.23	1.67	18,785.91	1.18
Stable generation	FA	0.38	0.19	−3,828.4	3.37	6.4		−78.71	3.8	2.64
	IFA	2.76	9.33	−3,783.7	2.27	2.66	9.34	−78.90	0	1.14
	ABC	3.54	−9.10	−3,828.4	8.46	0.049	1.14	−7.89	1.07	1.48
	GOA	5.22	2.14	−3,828.4	1.72	1.24	7.19	−7.89	7.4	4.31
	SMA	8.31	4.72	−3,828.4	3.89	6.6	7.02	−7.89	0	1.28
	BOA	0.94	1.93	−3,828.4	1.78	0.97	3.26	−7.40	1.11	0.19
	MFO	1.92	1.4	−3,828.4	3.85	4.10	2.25	−7.89	2.94	2.08

Statistic .	Algorithm .	.	.	.	.	.	.	.	.	.
Convergency value	FA	169,392	166,606.3	−3,608.5	83.45	0.94	3.19	3.765	169,392	3.6
	IFA	0.19	9.33	9.10	2.14	4.72	1.93	1.38	0.19	9.33
	ABC	−3,828.4	−3,783.7	−3,828.4	−3,828.4	−3,828.4	−3,828.4	−3,828.4	−3,828.4	−3,783.7
	GOA	83.46	1.62	7.37	2.5	1.21	2.02	0.15	83.45789	1.62
	SMA	210,278.5	0.013	1.56	0.49	0.37	4.79	4.35	210,278.5	1.35
	BOA	17,849.6	2.5	0.10	0.13	2.64	3.71	0.37	17,849.6	2.5
	MFO	18,785.9	0.012	0.067	0.09	1.27	0.23	1.67	18,785.91	1.18
Stable generation	FA	0.38	0.19	−3,828.4	3.37	6.4		−78.71	3.8	2.64
	IFA	2.76	9.33	−3,783.7	2.27	2.66	9.34	−78.90	0	1.14
	ABC	3.54	−9.10	−3,828.4	8.46	0.049	1.14	−7.89	1.07	1.48
	GOA	5.22	2.14	−3,828.4	1.72	1.24	7.19	−7.89	7.4	4.31
	SMA	8.31	4.72	−3,828.4	3.89	6.6	7.02	−7.89	0	1.28
	BOA	0.94	1.93	−3,828.4	1.78	0.97	3.26	−7.40	1.11	0.19
	MFO	1.92	1.4	−3,828.4	3.85	4.10	2.25	−7.89	2.94	2.08

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In Equation (19), F (x) represents the all-encompassing benefit function, wherein mth signifies the weight coefficient assigned to the mth objective function. The entropy weight method is chosen due to its inherent capacity to profoundly reveal the discriminative prowess of indicators, leading to improved weight determinations that are more objective. This method is underpinned by a solid theoretical foundation, enjoying elevated credibility, and boasts a straightforward algorithm with practical applications. Thus, this study employs the entropy weight method to ascertain weight coefficients. Initially, the calculation involves the independent assessment of the values corresponding to the three distinct objective functions.

In Equation (21), the parameter m takes values of 1, 2, and 3. Here, signifies the minimum value within the mth objective function, while represents the maximum value in the mth objective function.

In Equation (22), N represents the total number of evaluated objects, m = 1, 2, 3.

In Equation (23), satisfies the condition ⁠.

In Equation (24), the larger value of is, more attention should be paid to the role of this indicator in the comprehensive evaluation index system.

In Equation (25), is the final weight coefficient of each indicator.

The principal techniques employed to address constraint conditions in water resource optimization configuration models encompass rejecting infeasible solutions, utilizing penalty function methods, and employing the Lagrange multiplier method, among others. Rejecting infeasible solutions consistently during the iterative process, particularly in scenarios where the feasible region is constrained and small, can detrimentally impact solution convergence speed. The Lagrange multiplier method, while useful, introduces substantial computational complexity, notably when handling multi-objective solutions, potentially resulting in slower problem-solving.

• (a)

  The process entails acquiring pertinent water resource data from relevant geographical regions. Subsequently, key parameters such as water supply, water demand, and equity coefficients are identified and established.

• (b)

  The overall process involves a series of steps as follows, as illustrated in Figure 4.

Data Collection: Gather pertinent water resource data from relevant regions.

Parameter Determination: Identify key parameters including water supply, water demand, and equity coefficients.

Benefit Calculation: Utilize Equations (1), (2), and (5) to compute the values of social, economic, and ecological benefits.

Weight Coefficient Computation: Calculate weight coefficients for the three benefits using Equations (20)–(25).

Comprehensive Benefit Calculation: Determine comprehensive benefits by applying the weight coefficients to the values of individual benefits according to Equation (19).

Constraints Integration: Combine constraints established by Equations (6)–(9) with the constraint set in Equation (26).

Optimization Process: Employ the optimized FA to solve for the optimal comprehensive benefit value.

Output and Reporting: Present the optimal planning scheme along with corresponding values of the three benefits.

The holistic workflow illustrated in Figure 4 represents the seamless progression through these steps, resulting in the attainment of an optimized planning scheme and the associated values of social, economic, and ecological benefits.

However, due to the large population and rapid economic development in Shaanxi Province, water supply resources have brought pressure. According to statistics, in 2019, the per capita water resources in Shaanxi Province was 1,329 m³, lower than the national average. Groundwater is the main source of water intake in Shaanxi Province. Nevertheless, the past few years have witnessed a notable exacerbation of groundwater level decline and compromised groundwater quality, predominantly driven by excessive extraction. Concurrently, intermittent drought conditions have intensified the immediacy of water resource challenges.

The primary water sources within the study area encompass surface water, groundwater (including shallow and deep aquifers), as well as alternative sources such as treated sewage reuse and rainwater harvesting. Water consumption is categorized into three distinct domains based on user attributes: production, domestic use, and ecological preservation. Within the production category, water is sub-divided into primary industry water, secondary industry water, and tertiary industry water. For the purpose of this analysis, the focus primarily rests on primary industry water and secondary industry water, which pertain to industrial and agricultural water usage. Based on the prevailing conditions in the study area, details are presented in Tables 7 and 8.

Shaanxi Province holds significant cultural and industrial importance, securing the fourth spot in terms of industrial profits nationally. In alignment with the prevailing conditions and the region's social development blueprint, the hierarchy of water supply sources is established as follows: surface water, followed by groundwater, and supplemented by other alternative water supplies. Correspondingly, the order of water allocation is outlined as follows: agricultural water takes precedence, followed by domestic water, then industrial water, and finally, ecological water considerations.

Considering the tangible circumstances surrounding water supply sources and the requisites of diverse water-consuming sectors, the water supply order coefficient for this study is determined to be 0.61 for surface water, 0.34 for groundwater, and 0.05 for other water supplies. The water fairness coefficient is determined as 0.61 for agricultural water, 0.21 for domestic water, 0.12 for industrial water, and 0.06 for ecological water.

Leveraging the compiled data and existing planning, additional parameters for the water resource allocation model in the region are ascertained. The specific outcomes regarding demand are detailed in Table 9, with the pollutant emission concentration being derived from the cumulative sum of chemical oxygen demand (COD) and ammonia nitrogen.

Table 9

Model parameters from the demand side

User .	.	.	.	.	.
Domesticity	500	3.4	0.21	449.38	0.83
Industry	1,200	5.7	0.12	730.37	0.80
Agriculture	26	7.34	0.61	89.5	0.35
Ecology	500	2.62	0.06	0	0

User .	.	.	.	.	.
Domesticity	500	3.4	0.21	449.38	0.83
Industry	1,200	5.7	0.12	730.37	0.80
Agriculture	26	7.34	0.61	89.5	0.35
Ecology	500	2.62	0.06	0	0

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For ⁠, the unit is yuan/m³, given that ecology encompasses the intricate interplay and interdependence between organisms and their immediate environment, the ecological environment referenced in the article pertains to an urban ecological milieu, analogous to the living environment. Consequently, for the ‘domestic’ and ‘ecological’ categories, the two values are identical.

For ⁠, the unit is yuan/⁠, implementing a tiered pricing system for diverse water usages based on their inherent characteristics.

For ⁠, allocate water according to the government's policy planning for Shaanxi Province.

For ⁠, the unit is mg/L.

For ⁠, allocation is based on local sewage discharge standards, and ecological water is not considered.

For ⁠, allocate water based on water inflow and logistical feasibility, as outlined in Table 10.

Table 10

Model parameters from the supply side

Supplier .	.
Surface water	0.61
Ground water	0.34
Other water	0.05

Supplier .	.
Surface water	0.61
Ground water	0.34
Other water	0.05

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Following the process of dimensionless transformation, the data units are standardized, and the weight distribution is determined using the entropy weight calculation method discussed in Section 2. The resultant weight allocation for social, economic, and ecological benefits are found to be 0.206, 0.535, and 0.259, respectively.

The entire optimization process is divided into three stages:

• (a)

  The IFA is applied to optimize the individual social, economic, and ecological objectives independently.

• (b)

  Subsequently, a dimensionless transformation is conducted on the values of the three objectives.

• (c)

  The entropy weight method is then employed on the dimensionless processed data to derive comprehensive benefit outcomes.

In Figure 6, the horizontal axis signifies the number of iterations, while the vertical axis depicts the objective function value. The optimization curve demonstrates an initial swift convergence of the objective function value, followed by a gradual stabilization in the intermediate stage. Eventually, in the later stages of iteration, the curve exhibits a slow and steady change, indicating that the IFA has successfully identified a favourable allocation strategy for each objective as the iteration progresses.

To safeguard against surpassing predefined thresholds, constraints have been imposed on the target values, factoring in the specific quantities and extreme ranges prevalent within the population. Given the relatively randomized nature of initial solution settings, which may not adhere to constraint conditions and might yield significant function values, the swift convergence of the curve in the initial stages corroborates the gradual approach of each condition towards the constraint range. Moreover, the minor changes observed in the intermediate and later stages substantiate that each condition has effectively settled within the constraint range. The ultimate optimal allocation of water resources in this region is systematically presented in Table 11.

In the diagram, the right rectangle denotes three distinct sources of water supply, while the left rectangle signifies five diverse sectors of the water demand. The width of the right rectangle corresponds to the volume of available water resources, while the width of the left rectangle represents the cumulative water requirements of all entities. The width of the intermediary link graph encapsulates the origin, destination, and flow of water.

Analysing Figure 7, it becomes evident that surface water constitutes the primary source of water supply, distributed across all water consumers. Simultaneously, groundwater and recycled water are employed as supplementary resources, catering to the remaining water demands of each respective user category.

From Figure 8, it becomes apparent that surface water is predominantly allocated to domestic and agricultural water categories, constituting proportions of 36.05 and 38.01%, respectively. In contrast, groundwater finds primary allocation in the agricultural and industrial water sectors, accounting for proportions of 45.37 and 38.19%. In addition, recycled water is chiefly distributed to the agricultural and industrial water sectors, representing proportions of 45.60 and 34.99%, respectively.

To delve deeper into the assessment of the IFA and its overarching influence on the entire model, a multipronged analysis is conducted. The initial step involves juxtaposing the outcomes of the IFA solution against the results derived from the local government's strategic plan and the optimization algorithm employed in the SMA. Subsequently, a comprehensive series of comparisons ensue, encompassing economic benefits, social benefits, ecological benefits, and holistic comprehensive benefits. The specific allocation outcomes for the three scenarios are meticulously presented in Table 12, providing a comprehensive overview for comprehensive evaluation and comparison.

Table 13 comprehensively displays the computed values for the three objective functions, which are calculated using Equations (1)–(3) in the second section across three distinct allocation outcomes. The table adopts a standardized unit representation, with social benefits denoted in units of ⁠, economic benefits in units of yuan, and ecological benefits in units of mg. Notably, for social and ecological goals, lower values indicate better outcomes (negative indicators), while for economic objectives, higher values signify superior performance (positive indicators).

The computation method employed for comprehensive goals involves multiplying the corresponding weight of positive indicators by subtracting the weight of the corresponding negative indicators. The specific weight coefficients assigned for social, economic, and ecological benefits are 0.206, 0.535, and 0.259, respectively. Due to the disparate units in the four targets, a method involving corresponding percentages after summation is introduced for effective comparison.

In summary, when evaluating fitness, the outcomes yielded by the IFA optimization algorithm surpass both the initial allocation and results from the SMA. This underscores the efficacy of the IFA in efficiently furnishing optimal solutions for intricate water resource system optimization and configuration predicaments.

Furthermore, the optimal solution offered by IFA not only outperforms the local government's initial allocation but also demonstrates marked advantages over outcomes derived from alternative algorithms. This favourable performance has had a notable positive impact on the overall water resource allocation model, reinforcing the potency of the IFA approach.

The results for 2020 are analysed in conjunction with Table 14 and Figure 10. Firstly, while ensuring domestic water use, the water shortage rate is 0%, and the water shortage rate for industrial water is also 0%. However, the water shortage in the tertiary industry is severe, with a water shortage rate of 10.03%. Analysis suggests that this is because the economic benefits in the tertiary industry and water supply rate are too low. In 2020, the total water demand is 8,592.00⁠, and the total water supply is 8,189.04⁠. The overall water shortage rate has reached 1.72%, and with a guarantee rate of 98.28%, the water shortage rate is relatively severe.

The analysis of water shortage in 2021 is outlined as follows:

Initially, when juxtaposed with the year 2020, it becomes evident that both domestic and industrial water sectors have successfully achieved a commendable zero water shortage rate. Furthermore, a marked improvement is observed in the water scarcity scenario of the tertiary industry, with its water shortage rate plummeting to a mere 0.42% when contrasted with the preceding year. This reduction can be construed as a significant stride towards fulfilling the water requisites of the tertiary sector. Notably, the water scarcity predicaments facing the agricultural and ecological sectors have also undergone amelioration. The water scarcity rates for agriculture and ecology have undergone a decline from 8.57 and 9.98% to 4.33 and 2.06%, respectively. Despite this advancement, it remains evident that the agricultural domain still grapples with a substantial water scarcity challenge, indicative of prevailing pressure on agricultural water utilization.

Nevertheless, a comprehensive analysis of the data for the year 2021 demonstrates a notable amelioration in the overall water scarcity rate by 2.97% in comparison to the preceding year, 2020. Particularly noteworthy is the marked enhancement in water scarcity mitigation within the tertiary industry, reflecting significant strides towards resolving water insufficiency concerns. However, it is important to acknowledge that despite the observed improvements, the agricultural sector's water scarcity remains a formidable issue, necessitating further attention and intervention.

In summation, the local water scarcity conundrum has exhibited improvement, primarily characterized by a notable reduction in the overall water scarcity rate. Nonetheless, the agricultural sector remains a focal point of concern, warranting continued efforts to address and alleviate its persistent water scarcity challenges.

Shaanxi Province is presently situated within a phase of rapid and dynamic development, a trajectory propelled by robust economic expansion that, in turn, engenders a corresponding upsurge in resource requisites. Evident in this context is a discernible escalation in the aggregate water demand, surging from 8,592 in the year 2020 to 8,822 in 2021, manifesting as an augmentation of 230⁠. A concomitant observation lies in the amplification of the maximal water supply, which has surged from a figure of 8,189.04 to a substantial 8,670.01⁠. The statistics lucidly illustrate the consequential augmentation in water supply, thus effectively mitigating the overarching water scarcity scenario in the confines of Shaanxi Province. This augmentation in the quantum of water supply has engendered a discernible enhancement in the overall water security, as underscored by amelioration in the assurance rate. This tangible progression towards a more steadfast and stable supply of water resources is poised to play a pivotal role in fostering a sustainable and resilient water resource framework within the region.

In 2020, Shaanxi Province experienced severe water scarcity in the industrial sector. However, in 2021, the industrial sector witnessed a significant improvement due to focused industrial expansion efforts. Notably, the agricultural sector also saw a 50% reduction in water scarcity, reflecting effective governmental policies. The tertiary sector and ecological preservation demonstrated substantial growth in water demand, indicating the government's commitment to strengthening these sectors. This emphasizes the province's dedication to developing a robust service industry, promoting tourism, and fostering a sustainable green economy. With water allocation for agriculture and industry secured, substantial investments are anticipated in these growing sectors.

For the current situation of water resources in Shaanxi Province, the following suggestions are proposed: (1) increase publicity efforts for scientific water use and water conservation, (2) strengthen water resource management and improve the integrated construction of water affairs, and (3) long-term plan and rich configuration options.

The effective safeguarding, rational exploitation, and optimized allocation of water resources within Shaanxi Province have yielded a continuous enhancement of river water resources and utilization efficiency. This has contributed to a notable mitigation of the water resource supply–demand balance in the region, leading to an improved utilization rate of the limited water resources available. The outcomes of this endeavour offer valuable insights into the proficient management and allocation of water resources within Shaanxi Province.

Building upon these achievements, this study formulates a comprehensive water resource optimization and allocation model tailored to the unique context of Shaanxi Province. By integrating social benefits, emergency considerations, and ecological factors into the framework, a holistic approach is adopted to ensure the balanced and sustainable distribution of water resources. To navigate the complexities of this model, the IFA is harnessed, enabling efficient solution exploration. By leveraging pertinent data from the year 2020 as the foundation for parameter refinement, a comparative analysis is conducted against the allocation outcomes from 2021, providing a dynamic perspective on the evolving water resource landscape. Through this iterative process, a robust and comprehensive water resource allocation strategy emerges, offering a strategic roadmap for addressing water scarcity challenges while fostering harmonious coexistence between human activities and ecological integrity.

Indeed, while this study has made significant strides in optimizing water resource allocation, it acknowledges several areas that warrant further exploration and refinement. The scope of this research is primarily limited to the years 2020 and 2021, with a focus on essential dimensions such as livelihood, economy, and industry. However, a more comprehensive analysis would require extending the timeframe to encompass future years and incorporating a broader spectrum of critical indicators.

This study was supported by the Open Fund of Key Laboratory of Sediment Science and Northern River Training, the Ministry of Water Resources, China Institute of Water Resources and Hydropower Research (Grant No. IWHR-SEDI-2023-10).

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.