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The standard cuckoo search algorithm (SCSA) is an intelligent population optimization algorithm, which is also a heuristic search algorithm. The advantages of the SCSA (such as its convenient operation, heuristic searching, etc.) make it easy to find an optimal solution and maintain a wide searching range. However, the SCSA also has some drawbacks, such as long searching time, and the ease of falling on a local optimum. In order to solve the problems existing with SCSA, in this paper, an improved standard cuckoo search algorithm (ISCSA) was studied, which includes chaotic initialization and a Gaussian disturbance algorithm. As a case study, taking economic, social and ecological benefits as the objective function, multi-objective water resources optimal allocation models were constructed in Xianxiang Region, China. The ISCSA was applied to solve the water allocation models and a multi-objective optimal water supply scheme for Xinxiang region was obtained. Water resources optimal allocation schemes for the planning level year (2025) for 12 water supply sub-regions were predicted. A desirable eco-environment and other benefits were achieved using the studied methods. The results show that the ISCSA has obvious advantages in the solution of water resources optimal allocation and planning.

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  • The improved standard cuckoo search algorithm (ISCSA) was studied to overcome the disadvantages of the standard cuckoo search algorithm (SCSA);

  • The superiority of the ISCSA algorithm was tested and comparisons were made among different functions;

  • A typical region of China was selected and multi-objective water resources optimal allocation models were constructed and solved using ISCSA;

  • Desirable results were achieved by using the studied methods.

Graphical Abstract

Graphical Abstract
Graphical Abstract
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multi-objective optimization model, improved standard cuckoo search algorithm (ISCSA), standard cuckoo search algorithm (SCSA), water resources optimal allocation scheme
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With the development of the social economy and population, problems of water supply and demand are becoming more and more prominent. Due to climate change, drought and flood disasters occur alternately, which causes great losses to the social economy. Therefore, optimal water resources allocation has become an important research topic in China and around the world (Chang et al. 2016).

During recent years, in order to find an effective water resources allocation algorithm, a lot of new algorithms have been studied, such as particle swarm optimization (Civicioglu & Besdok 2013), the social emotion optimization algorithm (SEOA) (Fallah et al. 2019), self-adaptive multi-objective harmony search (Choi et al. 2017), the fast network multi-objective design algorithm (Creaco & Franchini 2012), the genetic algorithm (Deb et al. 2002), the shuffled frog leaping algorithm (Eusuff & Lansey 2003), and harmony search optimization (Geem et al. 2002), etc. The above algorithms are intelligent algorithms, which have been used to solve optimal water resources allocation models, and good results have been obtained. However, most intelligent algorithms have the following disadvantages (Kaveh & Bakhshpoori 2016): they have many uncertainty parameters and complex determinations, as well as poor local searching ability, etc. The optimal solution for these other intelligent algorithms can easily wander near a final scheme and lead to a fast convergence rate. Compared with other intelligent algorithms, the advantages of the standard cuckoo search algorithm (SCSA) are its relatively few parameters, simple programming, few revised parameters in practical problems, and fast convergence (Montalvo et al. 2010; Mohammad rezapour et al. 2019), etc.

However, the SCSA algorithm has some drawbacks, such as long searching time, and the ease of falling on a local optimum. In the process of high dimensional function optimization, there exist problems of slow convergence and low optimization precision (Prasad & Park 2004). During recent years, the SCSA algorithm has been applied in many fields including water resources optimal allocation, but the effect is still unsatisfactory (Saleh & Tanyimboh 2014).

Therefore, in order to solve the problems with the SCSA algorithm, in this paper, an improvement on SCSA was studied. The main contributions in this paper are as following: (1) chaos initialization conditions were introduced. In the initial stage, we tried to keep the position of the bird nests in the solution space as evenly spaced as possible to improve operation efficiency. (2) Gaussian disturbance was added to make the algorithm maintain the ability to jump out of the local optimum. (3) As a case study, Xinxiang water supply region of China was selected. Optimal water resources allocation models were constructed based on water distribution systems in the region. (4) The improved standard cuckoo searching algorithm (ISCSA) was applied to solve the optimal water resources allocation models in the region, and desirable schemes were achieved. The study results will support the reasonable utilization of local water resources and social economic development.

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Standard cuckoo search algorithm (SCSA)

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The SCSA is an intelligent population optimization algorithm, which is also a heuristic search algorithm (Solihin & Zanil 2016).

The SCSA algorithm has the advantages of convenient operation, ease of finding the optimal solution and a wide searching range path (Vasan & Simonovic 2010). The main idea of SCSA is based on two concepts, i.e. the breeding and feeding behavior of the cuckoo and the host birds' flight mechanism.

Improved standard cuckoo search algorithm (ISCSA)

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Chaotic Initialization

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Because of the uneven distribution of bird nests and influence of the initialization settings, there exists strong randomness in the SCSA. Therefore, chaotic cubic mapping was put forward to initialize the bird nests and, in this way, the distribution of the SCSA in the specific solution space can be maintained with a certain rule so as to establish a foundation for effective global searching.

Logistic mapping is a typical chaotic system, the expressions of which are as follows:
(1)
in which, μ is a control variable, and when μ = 4, through logistic mapping, an entire chaotic status could be realized, then, the initial cuckoo nest position generated by chaotic iteration is optimized to make the next search pair, in the following steps.

For N bird's nest locations in D dimension space, a D dimension vector is randomly generated and recorded as the first nest position, i.e. ⁠, in which, ⁠.

If N − 1 iterations are carried out for each dimension in each nest, then N − 1 bird's nests could be generated, i.e. ⁠.

After all the bird's nest iterations are finished, the results are mapped to the solution space using Equation (2):
(2)
where, is the logistic mapping of the dth dimension in the ith nest, Ud and Ld represent the upper and lower bounds of the dth dimension in the searching space, yid is the dth dimension of the ith nest in the searching space, xid represents the coordinates of the ith nest at the dth dimension searching space.

First, chaotic logistic mapping is used to initialize the bird nests' positions so that they can be evenly distributed in the solution space.

Secondly, the position of the first bird nest is used to adjust the adaptive step size. When it is far from the initial bird's nest position, a larger step size will be selected to search the optimum solution. The step size calculation is shown in Equation (3):
(3)
in which, rand is a random number between 0 and 1, xi is a bird location in the solution space, Xlocation represents the current location of the initial nest.

Thirdly, in the process of cuckoo bird nest selection, the maximum number of cuckoo nests selected and the change of the first nest location are used as the end conditions. The recognition distance method in the original algorithm is no longer used, which could improve the operation speed of the algorithm.

Gaussian Disturbance

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In SCSA, it is easy to fall on a local optimum; in fact, most intelligent algorithms find it difficult to jump out of the local optimal solution, which results in an undesirable conclusion (Yasar 2016). Therefore, in this paper, Gaussian micro-disturbance is performed on the former nest position for each iteration, as follows,
(4)
(5)
where, Gaussian(.) is the Gaussian function is the adaptive value of the former nest, that is, the adaptation value of the former nest before disturbance is the adaptive value of the former bird nest after disturbance.

The micro-disturbance for could jump it out of the local optimum domain to achieve the desired improved systematic analysis, and the efficiency and accuracy of the entire algorithm, which could be helpful to increase the diversity of bird nest and improve the accuracy of solution (Razmkhah et al. 2010).

Algorithm validation

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In order to test the superiority of the ISCSA algorithm, six standard functions (see Table 1) were selected as benchmark functions (Zheng et al. 2013; Omid Bozo et al. 2014; Wang et al. 2015), in which f1 ∼ f3 are single peak functions, f4 ∼ f6 are multi-peak functions.

Table 1

Test functions

FunctionFormulaScopeTheoretical optimal solution
Sphere  [−100,100] 0 
Schwefel's 2.22  [−10,10] 0 
Schwefel's 1.2  [−100,100] 0 
Rastrigin  [−5.12,5.12] 0 
Ackley  [−32,32] 0 
Griewank  [−600,600] 0 
FunctionFormulaScopeTheoretical optimal solution
Sphere  [−100,100] 0 
Schwefel's 2.22  [−10,10] 0 
Schwefel's 1.2  [−100,100] 0 
Rastrigin  [−5.12,5.12] 0 
Ackley  [−32,32] 0 
Griewank  [−600,600] 0 
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The performance of the ISCSA algorithm was tested using Matlab. Two intelligent algorithms, the genetic algorithm (GA) and particle swarm optimization (PSO), were used to compare with the ISCSA algorithm. The parameters of the algorithm were set as follows: population base number, maximum iteration number and function dimensions. Each test function was optimized for 20 times, and according to the test results (best, worst, average, standard deviation), the ISCSA algorithm could be evaluated. The comparisons and test results can be seen in Table 2.

Table 2

Comparison of test results

FunctionAlgorithmBestWorstAverageStandard Deviation
Sphere GA 8.23 × 10−06 3.90 × 10−02 7.00 × 10−03 9.00 × 10−03 
PSO 1.46 × 10−08 3.51 × 10−04 4.92 × 10−05 8.68 × 10−05 
SCSA 7.65 × 10−43 4.70 × 10−39 2.08 × 10−39 2.26 × 10−39 
ISCSA 4.32 × 10−130 7.72 × 10−128 7.18 × 10−129 2.62 × 10−128 
Schwefel's 2.22 GA 2.30 × 10−03 3.14 × 10−01 1.02 × 10−01 8.30 × 10−02 
PSO 2.58 × 10−05 6.51 × 10−01 6.40 × 10−02 2.37 × 10−01 
SCSA 3.20 × 10−24 9.18 × 10−23 1.95 × 10−23 2.25 × 10−23 
ISCSA 3.06 × 10−74 2.05 × 10−71 3.22 × 10−72 5.70 × 10−71 
Schwefel's 1.2 GA 3.08 × 10−06 2.57 × 10−02 8.60 × 10−03 1.90 × 10−02 
PSO 5.02 × 10−08 2.88 × 10−04 5.15 × 10−05 5.82 × 10−05 
SCSA 3.23 × 10−45 2.88 × 10−42 5.03 × 10−43 6.13 × 10−43 
ISCSA 2.47 × 10−141 2.53 × 10−135 1.46 × 10−136 4.86 × 10−136 
Rastrigin GA 3.09 × 10−04 9.56 × 10−01 3.17 × 10−01 4.25 × 10−01 
PSO 7.03 × 10−05 4.09 × 10+01 1.40 × 10+01 1.82 × 10+01 
SCSA 4.26 × 10−14 3.07 × 10−11 3.48 × 10−12 6.70 × 10−12 
ISCSA 0 0 0 0 
Ackley GA 6.30 × 10−04 1.15 × 10−01 4.10 × 10−02 4.30 × 10−02 
PSO 1.13 × 10−05 8.20 × 10−02 1.50 × 10−02 2.60 × 10−02 
SCSA 1.21 × 10−05 1.72 × 10−04 4.83 × 10−05 6.82 × 10−05 
ISCSA 1.13 × 10−15 1.13 × 10−15 1.13 × 10−15 1.08 × 10−15 
Griewank GA 3.20 × 10−08 7.30 × 10−02 1.60 × 10−02 2.10 × 10−02 
PSO 1.41 × 10−10 1.11 × 10+00 4.77 × 10−01 5.38 × 10−01 
SCSA 7.65 × 10−12 1.73 × 10−08 2.91 × 10−09 4.52 × 10−09 
ISCSA 0 0 0 0 
FunctionAlgorithmBestWorstAverageStandard Deviation
Sphere GA 8.23 × 10−06 3.90 × 10−02 7.00 × 10−03 9.00 × 10−03 
PSO 1.46 × 10−08 3.51 × 10−04 4.92 × 10−05 8.68 × 10−05 
SCSA 7.65 × 10−43 4.70 × 10−39 2.08 × 10−39 2.26 × 10−39 
ISCSA 4.32 × 10−130 7.72 × 10−128 7.18 × 10−129 2.62 × 10−128 
Schwefel's 2.22 GA 2.30 × 10−03 3.14 × 10−01 1.02 × 10−01 8.30 × 10−02 
PSO 2.58 × 10−05 6.51 × 10−01 6.40 × 10−02 2.37 × 10−01 
SCSA 3.20 × 10−24 9.18 × 10−23 1.95 × 10−23 2.25 × 10−23 
ISCSA 3.06 × 10−74 2.05 × 10−71 3.22 × 10−72 5.70 × 10−71 
Schwefel's 1.2 GA 3.08 × 10−06 2.57 × 10−02 8.60 × 10−03 1.90 × 10−02 
PSO 5.02 × 10−08 2.88 × 10−04 5.15 × 10−05 5.82 × 10−05 
SCSA 3.23 × 10−45 2.88 × 10−42 5.03 × 10−43 6.13 × 10−43 
ISCSA 2.47 × 10−141 2.53 × 10−135 1.46 × 10−136 4.86 × 10−136 
Rastrigin GA 3.09 × 10−04 9.56 × 10−01 3.17 × 10−01 4.25 × 10−01 
PSO 7.03 × 10−05 4.09 × 10+01 1.40 × 10+01 1.82 × 10+01 
SCSA 4.26 × 10−14 3.07 × 10−11 3.48 × 10−12 6.70 × 10−12 
ISCSA 0 0 0 0 
Ackley GA 6.30 × 10−04 1.15 × 10−01 4.10 × 10−02 4.30 × 10−02 
PSO 1.13 × 10−05 8.20 × 10−02 1.50 × 10−02 2.60 × 10−02 
SCSA 1.21 × 10−05 1.72 × 10−04 4.83 × 10−05 6.82 × 10−05 
ISCSA 1.13 × 10−15 1.13 × 10−15 1.13 × 10−15 1.08 × 10−15 
Griewank GA 3.20 × 10−08 7.30 × 10−02 1.60 × 10−02 2.10 × 10−02 
PSO 1.41 × 10−10 1.11 × 10+00 4.77 × 10−01 5.38 × 10−01 
SCSA 7.65 × 10−12 1.73 × 10−08 2.91 × 10−09 4.52 × 10−09 
ISCSA 0 0 0 0 
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According to Table 2, in single peak function f1 ∼ f3, the optimization ability of the different algorithms is ISCSA > SCSA > PSO > GA, with the optimal values increased by more than 50%, which indicates that the optimization ability could be greatly improved by using the ISCSA. This is because the optimization ability of the GA algorithm is very limited, which could only be improved through a mutation operation to create a new solution space. In multi-peak function f4 ∼ f6, the optimization ability of the algorithms is, in turn, ISCSA > SCSA > GA > PSO. This is because with the PSO algorithms it is easy to fall into the local optimal solution; in fact, the optimal value of the PSO algorithm is better than that of the GA algorithm, but the overall level of the PSO keeps low compared with the GA algorithm. The ISCSA algorithm has the characteristics of Levitt's performance, which can effectively expand the exploring range and have strong global exploring ability, so that it has a higher optimization ability than the other algorithms. Since there are calculation errors in the Matlab software, the ISCSA algorithm can reach 0 in both the Rastrigin and Griewank multi-peak functions, which indicates that the ISCSA algorithm can effectively achieve the global optimal solution. Here, selecting the Rastrigin multi-peak function as a test function, the convergence process of the four algorithms were drawn in Figure 1.
Figure 1
Convergence process diagram of different algorithms.
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Convergence process diagram of different algorithms.

Figure 1
Convergence process diagram of different algorithms.
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Convergence process diagram of different algorithms.

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It can be seen from Figure 1 that the convergence speed of the ISCSA is the fastest of all the algorithms and the best results are realized. Therefore, it can be concluded that the convergence accuracy and global optimization ability of ISCSA are obviously better than that of SCSA, and the best optimization effect can be achieved by using the ISCSA algorithm in solving optimal models.

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Background

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The Xinxiang water supply region of China was selected as an example in this paper. The region is located on the north bank of the lower reaches of the Yellow River, with the geographical coordinates N34 °53′ ∼ N35 °50′, E113 °23′ ∼ E115 °01′. The region covers 12 water supply sub-regions with a total area of 8,249 km2. The Xinxiang water supply region has a total population of 6,130,000, of which 2,940,000 are rural and 3,190,000 are urban. The annual average precipitation is 580 mm. The annual average natural water resources in Xinxiang region are 1,697.0 million m3, of which surface water resources accounts for 744.0 million m3 and underground water resources accounts for 953.0 million m3. The shallow groundwater can be mainly classified as three groups, i.e. loose rock aquifer, carbonate aquifer and silt clay aquifer. In 2018, the effective agricultural irrigation area in the region reached to 335,400 hm2, and the urbanization rate was 52%.

Water resources optimal allocation scheme based on ISCSA

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Water resources system optimization allocation nodes design

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Based on the distribution of the water supply project system and water demand (domestic, industrial and agricultural production and eco-environment) in the region, and considering natural water resources components (such as surface and underground water), as well as the special water requirements of different sub-regions, the entire water supply region of Xinxiang can be divided into 12 sub-regions. The general nodes map of water supply in Xinxiang region is drawn as Figure 2.
Figure 2
Schematic of water supply nodes in Xinxiang water supply region.
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Schematic of water supply nodes in Xinxiang water supply region.

Figure 2
Schematic of water supply nodes in Xinxiang water supply region.
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Schematic of water supply nodes in Xinxiang water supply region.

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Construction of water resources optimal allocation models: objective functions

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(i) Economic objective

Maximizing economic benefit is one of the important goals for regional water resources allocation. The economic benefit can be calculated with Equation (6):
(6)
where indicates the economic benefits generated by per unit water consumption of j water users in the ith water supply sub-region, in Yuan/m3, indicates the allocated water amount for the jth water user in the ith water supply sub-region, in m3. I and J indicate the total number of water supply sub-regions and the total number of water users, respectively.

(ii) Social benefit objective

Here we use the minimum water shortage as the social economic objective, as Equation (7):
(7)
where is the weight coefficient of water shortage for the jth water user (where β1, β2 and β3 refer to the first, second and third production sectors), bij is water requirement of the jth water user in the ith water supply sub-region.

(iii) Ecological benefit objective

The objective is to realize minimum pollutants emissions; here, we select COD as an indicator:
(8)
where δij indicates the pollutants (COD) discharged by the jth water user per unit water consumption in the ith water supply sub-region, in t/m3, Pij indicates wastewater discharge coefficient of the jth water user in the ith water supply area, as a %, indicates the allocated water amount for the jth water user in the ith water supply sub-region, in m3.

Construction of water resources optimal allocation models: constraints

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(i) The water supply capacity
(9)
where W represents total amount of water supply, in m3; B represents the minimum ecological water demand in river channels, in m3, indicates the allocated water amount for the jth water user in the ith water supply sub-region, in m3.

(ii) Domestic water demand

According to the relevant policies, the domestic water demand should be firstly guaranteed, then,
(10)
where XD is the total amount of domestic water supply, in m3; indicates the total amount of domestic water demand of the ith users.

(iii) Pollutants emission control

According to the relevant regulations, water pollutants discharge should meet national and local standards.
(11)
where indicates the national water pollutants control standard, in kg/m3. is water pollutants volume of the ith water user, in kg/m3.

(iv) Ecological water use

Ecological water demand should be guaranteed first.
(12)
where is the ecological water use for the ith water supply sub-region, is the total ecological water supply, in m3.
(v) Maximum and minimum water demand
(13)
where Xijmin and Xijmax are the minimum and maximum water demand of jth water users in the ith water supply sub-region, and Dij is the water demand of the jth water user in the ith water supply sub-region.
(vi) Water balance constraint
(14)
is the total amount of water supply in Xinxiang region, in m3.
(vii) Non negative
(15)

Water demand forecast

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According to the investigation on the experiment sub-regions, in 2025, average irrigation quota will reach 3,700.5 m3/hm2, and the average irrigation water utilization coefficient will be 0.601. Domestic water use for urban residents will be 124.2 L/d.person. Domestic water use for rural residents will be 68.9 L/d.person. The irrigation quota for forest and fruit trees will be 7,230 m3/hm2. The irrigation area for forest and fruit trees will reach 2,530 hm2. The fish pond area will reach 3,700 hm2, which requires 11,290.95 m3/hm2 for water replenishment. In the animal husbandry sector, the large livestock water demand quota will be 29 L/d·head, and small livestock water demand quota will be 15 L/d.head. In 2025, the urbanization rate of Xinxiang region will reach 58%, and public urban water consumption will reach 58.67 million m3. Annual eco-environmental water demand will be increased by 10%, thus the eco-environmental water demand will be 33.80 million m3 in 2025. Table 3 shows the total water demand for different water supply sub-regions in 2025.

Table 3

Water demand of Xinxiang Region in 2025 (million m3)

Sub-regionAgricultureIndustryForestryHusbandryFisheryUrban publicdomesticEco-environmentTotal
Urban 68.8 89.2 1.3 0.4 2.2 22.6 60.4 15.7 260.6 
Xinxiang 96.4 77.3 0.0 0.9 2.70 3.0 19.0 2.7 202.0 
Weihui 96.2 20.8 0.0 4.3 3.9 5.2 16.0 2.6 149.0 
Huixian 152.9 90.2 3.2 5.4 1.6 9.2 24.8 2.2 289.5 
Huojia 122.6 20.5 8.2 1.8 4.7 3.4 14.7 2.3 178.2 
Yuanyang 216.0 23.0 0.0 2.9 8.2 3.3 19.5 1.8 274.7 
Yanjin 119.1 23.9 1.7 1.6 11.5 2.6 15.5 1.8 177.7 
Fengqiu 201.4 13.7 0.0 3.6 3.0 4.6 22.2 2.3 250.8 
Changyuan 167.8 41.4 3.7 1.9 3.5 4.9 26.4 2.4 252.0 
Total 1,241.2 400 18.1 22.8 41.4 58.7 218.5 33.8 2,034.5 
Sub-regionAgricultureIndustryForestryHusbandryFisheryUrban publicdomesticEco-environmentTotal
Urban 68.8 89.2 1.3 0.4 2.2 22.6 60.4 15.7 260.6 
Xinxiang 96.4 77.3 0.0 0.9 2.70 3.0 19.0 2.7 202.0 
Weihui 96.2 20.8 0.0 4.3 3.9 5.2 16.0 2.6 149.0 
Huixian 152.9 90.2 3.2 5.4 1.6 9.2 24.8 2.2 289.5 
Huojia 122.6 20.5 8.2 1.8 4.7 3.4 14.7 2.3 178.2 
Yuanyang 216.0 23.0 0.0 2.9 8.2 3.3 19.5 1.8 274.7 
Yanjin 119.1 23.9 1.7 1.6 11.5 2.6 15.5 1.8 177.7 
Fengqiu 201.4 13.7 0.0 3.6 3.0 4.6 22.2 2.3 250.8 
Changyuan 167.8 41.4 3.7 1.9 3.5 4.9 26.4 2.4 252.0 
Total 1,241.2 400 18.1 22.8 41.4 58.7 218.5 33.8 2,034.5 
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Model operation results analysis

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According to the above models and water demand in the region, the ISCSA algorithm was used to search the water resources optimal allocation scheme; 200 seed groups were selected and 100 iterations were calculated. The water resources allocation schemes of Xinxiang water supply region in 2025 were optimized. The comparison of optimal results among the different schemes can be seen in Figures 3 and 4.
Figure 3
Optimal solution set of the SCSA algorithm.
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Optimal solution set of the SCSA algorithm.

Figure 3
Optimal solution set of the SCSA algorithm.
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Optimal solution set of the SCSA algorithm.

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Figure 4
Optimal solution set of the ISCSA algorithm.
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Optimal solution set of the ISCSA algorithm.

Figure 4
Optimal solution set of the ISCSA algorithm.
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Optimal solution set of the ISCSA algorithm.

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Comparing with the SCSA algorithm in Figure 3, the optimal solution set generated by the ISCSA algorithm in Figure 4 is more uniform and close to a smooth curve, which shows that the results of the ISCSA are better than those of the SCSA.

The optimal allocation scheme of water resources in Xinxiang region

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Based on the water resources optimal allocation models (Equations (6)–(15)), and using the ISCSA algorithm, the optimal allocation scheme of water resources in Xinxiang water supply region were able to be achieved as shown in Table 4.

Table 4

The water resources optimal allocation scheme in Xinxiang Region (million m3)

Sub-regionWater supply
Increased water supply ability
Total water supply abilityWater demandBenefit (108 Yuan)
Surface waterGround waterRain harvestRural waterUrban waterSewage reuseS-N water transfer
Urban 38.3 0.0 0 0.8 0 0.87 2.30 277.9 260.6 522.4 
Xinxiang 30.3 113.7 0 0.4 0 0 57.6 202.0 202.0 379.9 
Weihui 45.2 56.5 2.6 0.4 0 0 45.0 149.7 149.0 281.5 
Huixian 60.6 163.9 10.3 1.9 0 0 57.5 294.2 289.5 553.3 
Huojia 124.3 33.9 0 0.8 0 0 19.8 178.8 178.2 336.3 
Yuanyang 250.0 11.5 0 1.6 15.0 0 0 278.1 274.7 523.0 
Yanjin 57.8 106.4 0 0.8 1.0 0 0 166.0 177.7 312.2 
Fengqiu 83.2 164.3 0 1.2 0 0 0 248.7 250.8 467.7 
Changyuan 146.5 85.1 0 3.7 0 10.0 0 245.3 252.0 461.3 
Total 836.2 735.2 12.9 11.6 16.0 18.7 410.0 2,040.7 2,034.5 3,837.6 
Sub-regionWater supply
Increased water supply ability
Total water supply abilityWater demandBenefit (108 Yuan)
Surface waterGround waterRain harvestRural waterUrban waterSewage reuseS-N water transfer
Urban 38.3 0.0 0 0.8 0 0.87 2.30 277.9 260.6 522.4 
Xinxiang 30.3 113.7 0 0.4 0 0 57.6 202.0 202.0 379.9 
Weihui 45.2 56.5 2.6 0.4 0 0 45.0 149.7 149.0 281.5 
Huixian 60.6 163.9 10.3 1.9 0 0 57.5 294.2 289.5 553.3 
Huojia 124.3 33.9 0 0.8 0 0 19.8 178.8 178.2 336.3 
Yuanyang 250.0 11.5 0 1.6 15.0 0 0 278.1 274.7 523.0 
Yanjin 57.8 106.4 0 0.8 1.0 0 0 166.0 177.7 312.2 
Fengqiu 83.2 164.3 0 1.2 0 0 0 248.7 250.8 467.7 
Changyuan 146.5 85.1 0 3.7 0 10.0 0 245.3 252.0 461.3 
Total 836.2 735.2 12.9 11.6 16.0 18.7 410.0 2,040.7 2,034.5 3,837.6 
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Table 4 shows that the total water supply can be raised to 2,040.7 million m3 in 2025. The increased water supply includes rural water supply, urban water supply, water transfer from south to north, etc. Sewage water reuse will be increased to 18.7 million m3. Rain water collection and utilization will be increased to 12.9 million m3. Considering investigation into social and economic development in the region, the water resources utilization and benefits were calculated for the optimal allocation scheme. It can be seen from Table 4 that the benefits (GDP) of Xinxiang region will reach 383.76 billion (383.76 × 108) Yuan in 2025. The benefits generated from the optimal water resources allocation scheme will be 153.5 billion Yuan in 2025.

It can be seen from Table 4 that according to the water resources optimal allocation scheme studied in this paper, the amount of groundwater exploitation will be reduced to 735.2 million m3 in 2025, which means that 217.8 million m3 underground water resources were saved compared with that in 2009. Under the condition of strengthening water saving measures, the balance between water supply and demand in Xinxiang region can be realized. The water use for eco-environment will be 33.8 million m3 in 2025, thus, the eco-environment will be improved greatly.

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In this paper, the ISCSA algorithm was studied and applied to multi-objective water resources allocation models to obtain a water resources optimal allocation scheme in the studied region. The deficiencies in the SCSA algorithm (such as the ease of falling onto a local optimimum in the searching process, its long searching time, slow convergence and low optimization precision, etc.) have been overcome and the convergence speed of the SCSA algorithm is accelerated.

The performance of the ISCSA algorithm was tested using test functions (Sphere, Schwefel's 2.22, Schwefel's 1.2, Rastrigin, Ackley, Griewank). Intelligent algorithms such as the genetic algorithm (GA) and particle swarm optimization (PSO) were used to compare with ISCSA algorithm.

It can be concluded that the convergence speed of the ISCSA should be faster than other methods. The desirable results were realized by using the ISCSA algorithm. It can also be concluded that the accuracy and global optimization ability of the ISCSA algorithm are obviously better than that of the SCSA algorithm, and the best optimization effect was achieved by using the ISCSA algorithm.

Through the case study, water resources optimum allocation models were constructed and solved by using the ISCSA algorithm, and an optimal allocation scheme of regional water resources was obtained. The results show that the ISCSA algorithm can achieve high convergence speed, which can meets the operational requirement of multi-objective function, and the results of water resources allocation are more reasonable.

Table 4 shows that water supply ability can be greatly improved by using a water resources allocation optimization scheme and the ISCSA algorithm. The total water supply could reach 2,040.7 million m3 in 2025, in which the total increased water supply in Xinxiang region will reach 410.0 million m3 compared to 1630.7 million m3 in 2019. Sewage water reuse will be 18.7 million m3. Rain water collection and utilization will be increased to 12.9 million m3. Considering social and economic development in the region, the benefits of water resources utilization will be raised under application of the water resources optimal allocation scheme. The benefits generated from the water resources optimal allocation scheme will reach 153.5 billion Yuan.

It can be forecast that according to the water resources optimal allocation scheme studied in this paper, the amount of groundwater exploitation will be reduced to 735.2 million m3 in 2025, which means that 217.8 million m3 underground water resources will be saved compared to 2009. The water use for the eco-environment will be 33.8 million m3. Thus, the eco-environment will be improved. Under the condition of strengthening water saving measures, the balance between water supply and demand in Xinxiang region will be realized.

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The study was supported and funded by the Natural Science Fund of China (No.50579020).

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All the data and materials in the current study are available from the corresponding author on reasonable request.

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The author declares that there are no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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

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The authors declare there is no conflict.

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