Multi-objective Water Resources Optimum Allocation Scheme Based on Improved Standard Cuckoo Searching Algorithm(ISCSA)

: The standard cuckoo searching algorithm （ SCSA ） is a population intelligent optimization algorithm, which is also a new heuristic searching algorithm. The advantages of SCSA (such as convenient operation, heuristic searching, etc.) make it easy to find the optimal solution and maintain wider searching range. However, SCSA also has some drawbacks, such as long searching time, easy to fall into local optimum. In order to solve the problems existed in SCSA, in this paper, the improved standard cuckoo searching algorithm (ISCSA) was studied, which includes chaotic initialization and Gaussian disturbed algorithm. As a case study, taking economic, social and ecological benefits as the objective function, the multi-objective water resources optimal allocation models were constructed in Xianxiang Region, China. The ISCSA was applied to solve the water allocation models and the multi-objective optimal water supply scheme for Xinxiang region was obtained. The water resources optimal allocation schemes in the planning level year (2025) for 12 water supply sub-regions were predicted. The desirable eco-environment and 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.

achieved, which can support the local water resources reasonable utilization and social economic development.

Standard cuckoo searching algorithm (SCSA)
The standard cuckoo searching algorithm (SCSA) is a population intelligent optimization algorithm, which is also a new heuristic searching algorithm (Salman A, et al. 2007, Solihin MI, Zanil MF 2016. It was proposed by Professor Yang in 2009(Tanyimboh TT, Czajkowska A 2018. SCSA algorithm has the advantages of convenient operation, easy to find the optimal solution and wider searching range path (Vasan A, Simonovic SP 2010). The main idea of SCSA is based on two strategies, i.e. the breeding and feeding behavior of cuckoo and the flight mechanism (Yang XS 2014).

Improved standard cuckoo searching algorithm (ISCSA) (1) Chaotic Initialization
Because of the bird's nest uneven distribution and influence of the initialization settings, there exists strong randomness in the SCSA. Therefore, the chaotic cubic mapping was put forward to initialize the bird's nest, in this way, the distribution of the SCSA in the specific solution space can be maintained with a certain rule so as to establish the foundation for the effective global searching.
·N-1 iterations are carried out for each dimension in each nest, then, N-1 bird's nests could be ·After all the bird nest iterations are finished, mapping the results to the solution space using Eq.2.
x id = L id + (1 + y id ) where, L id 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.
Firstly, the chaotic Logistic mapping is used to initialize the bird's nest position so that it could 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, the larger step size will be selected to search the optimum solution. The step size calculation is shown in Eq. 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 nest selection 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.
(2) Gaussian Disturbances In the standard cuckoo search algorithm (SCSA), it is easy to fall into local optimum, in fact, most of intelligent algorithms are difficult to jump out of the local optimal solution, which results in undesirable conclusion (Yasar M 2016). Therefore, in this paper, Gaussian micro-disturbance are performed to the former nest position for each iteration, as follows, G best = G × (1 + Gaussian(μ, σ)) (4) Where, Gaussian(.) is the function of Gaussian, 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 out of the local optimum domain to make systematic analysis, the efficiency and accuracy of the entire algorithm can be improved desirably, which could be helpful to increase the diversity of bird nest and improve the accuracy of solution (Razmkh et al. 2010, Ladon et al. 1999).

Algorithm validation
In order to test the superiority of ISCSA algorithm, six standard functions (see Table 1) were selected as the benchmark function (Wang et al. 2015, Zheng et al. 2013, OMID ＲG et al. 2014, in which f1~f3 are single peak functions, f4~f6 are multi-peak functions. According to Table2, 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. It is because that the optimization ability of GA algorithm is very limited, which could only be improved through mutation operation to create new solution space. In multi-peak function f4~f6, The optimization ability of the algorithm is ,in turn, ISCSA>SCSA>GA>PSO. It is because that the PSO algorithm 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 the highest optimization ability than other algorithms. Since there are calculation errors in the matlab software, the ISCSA algorithm can reach to 0 both in the multi-peak function Rastrigin and Griewank, which indicates that the ISCSA algorithm can effectively achieve the global optimal solution. Here, selecting the multi-peak function Rastrigin as a test function, the convergence process of four algorithms were drawn in Fig.1.

Fig.1 Convergence process diagram of different algorithms
It can be seen from Fig.1 that convergence speed of the ISCSA is the fastest in all 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 ISCSA algorithm in solving optimal models.
3 Case study

Background
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 o 50', E113°23'~E115°01'. The region covers 12 water supply sub-regions with the total area of 8249 km 2 . The Xinxiang water supply region has a total population of 6130000, of which 2940000 are rural and 3190000 are urban population. The annual average precipitation is 580 mm. The annual average natural water resources in Xinxiang region is 1697.0 million m 3 , of which surface water resources accounts for 744.0 million m 3 and underground water resources accounts for 953.0 million m 3 . In 2018, agricultural effective irrigation area in the region reached to 335400 hm 2 , urbanization rate was 52%.

Water resources system optimization allocation nodes design
Based on the distribution of 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  (1) Economic objective Maximizing economic benefit is one of the important goals for regional water resources allocation.
The economic benefit can be calculated with Eq.6.
where α ij indicates the economic benefits generated by per unit water consumption of j water users in the ith water supply sub-region, Yuan/m 3 , x ij indicates the allocated water amount for the jth water user in the ith water supply sub-region, m 3 . I,J indicate the total number of water supply sub-regions and the total number of water users, respectively.
(2) Social benefit objective Here we use the minimum water shortage as the social economic objective, as Eq.7.
where β j is weight coefficient of water shortage for the jth water user, 1 is for the first production sector; 2 is for the second production sector; 3 is for others, b ij is water requirement of the jth water user in the ith water supply sub-region. (

3) Ecological benefit objective
The objective is to realize minimum pollutants emission, here, we select COD as an indicator.
where δij indicates the pollutants (COD) discharged by the jth water user per unit water consumption in the ith water supply sub-region, t/m 3 , Pij indicates wastewater discharge coefficient of the jth water user in the ith water supply area,%, x ij indicates the allocated water amount for the jth water user in the ith water supply sub-region, m 3 .

Constraints
(1) The water supply capacity where W represents total amount of water supply, m 3 ；B represents the minimum ecological water demand in river channels，m 3 , x ij indicates the allocated water amount for the jth water user in the ith water supply sub-region, m 3 .
(2) Domestic water demand According to the relevant policies, the domestic water demand should be firstly guaranteed, then, where XD is the total amount of domestic water supply,m 3 ; x id indicates the total amount of domestic water demand of the ith users. (

3) Pollutants emission control
According to the relevant regulations, water pollutants discharge should meet national and local standards.
where (COD) N indicates the national water pollutants control standard, kg/m 3 . (COD) i is water pollutants volume of the ith water user, kg/m 3 .
(4) Ecological water use Ecological water demand should be guaranteed firstly.
where x ie is the ecological water use for the ith water supply sub-region, X E is the total ecological water supply, m 3 .
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.
(6) Water balance constraint X ij is the total amount of water supply in Xinxiang region,m 3 .

Water demand forecast
According to the investigation on the experiment sub-regions, in 2025, average irrigation quota will reach to 3700.5 m 3 /hm 2 , average irrigation water utilization coefficient will be 0.601. Domestic water use for urban residents will be 124.2L/d.person. Domestic water use for rural residents will be 68.9L/d.person. the irrigation quota for forest and fruit trees will be 7230 m 3 /hm 2 . The irrigation area for forest and fruit trees will reach to 2530 hm 2 . Fish pond area will reach to 3700 hm 2 , which requires 11290.95 m 3 /hm 2 for water replenishment. In animal husbandry sector, large livestock water demand quota will be 29L/d·head, and small livestock water demand quota will be 15 L/d.head. In 2025, urbanization rate of Xinxiang region will reach to 58%, public urban water consumption will reach to 58.67 million m 3 . Annual eco-environmental water demand will be increased by 10 %, thus, eco-environmental water demand will be 33.80 million m 3 in 2025. Table 1 shows the total water demand for different water supply sub-regions in 2025.

Model operation Results analysis
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 different schemes can be seen in Fig.3 and Fig.4.  Comparing with the SCSA algorithm in Fig .3, the optimal solution set generated by the ISCSA algorithm in Fig.4. is more uniform and close to a smooth curve, which shows that the results of ISCSA is better than that of SCSA.
3.2.5 The optimal allocation scheme of water resources in Xinxiang region Based on the water resources optimal allocation models (Eq.6 to Eq.15), and using the ISCSA algorithm, the optimal allocation scheme of water resources in Xinxiang water supply region can be achieved as Table2.
Table2 The water resources optimal allocation scheme in Xinxiang Region (million m 3 ) It shows in Table 2 that the total water supply ability can be raised to 2040.7 million m 3 in 2025.
The increased water supply ability includes rural water supply, urban water supply, water transfer from south to north, etc. Sewage water reuse will be increased to 18.7 million m 3 . Rain water collection and utilization will be increased to 12.9 million m 3 . Considering investigation on social and economic development in the region, the water resources utilization and benefits were calculated for the optimal allocation scheme. It can be seen form Table2 that the benefits (GDP) of Xinxiang region will reach to 383.76 billion Yuan in 2025. The benefits generated from the optimal water resources allocation scheme will be 153.5 billion Yuan RMB in 2025.
It can be seen from Table2 that according to water resources optimal allocation scheme studied in this paper, the amount of groundwater exploitation will be reduced to 735.2 million m 3 in 2025, which means that 217.8 million m 3 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 m 3 in 2025, thus, the eco-environment will be improved greatly.

Conclusions
In this paper, the ISCSA algorithm was studied and applied to the multi-objective water resources allocation models to obtain water resources optimal allocation scheme in the studied region. The Deficiencies in the SCSA algorithm (such as easy falling into a local optimization in the searching process, long searching time, slow convergence and low optimization precision, etc.) have been overcome and the convergence speed of SCSA algorithm is accelerated.
The performance of ISCSA algorithm was tested using test functions(Sphere, Schwefel's 2.22, Schwefel's 1.2, Rastrigin, Ackley, Griewank). The intelligent algorithms of genetic algorithm (GA) and particle swarm optimization (PSO) were used to compare with ISCSA algorithm.
It can be concluded that convergence speed of the ISCSA should be faster than other methods. The desirable results were realized by using ISCSA algorithm. It can also be concluded that the accuracy and global optimization ability of ISCSA algorithm are obviously better than that of SCSA algorithm, and the best optimization effect was achieved by using ISCSA algorithm.
Through case study, water resources optimum allocation models were constructed and solved by using the ISCSA algorithm, and the optimal allocation scheme of regional water resources was obtained. It shows from the results 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.
It shows from Table 2 that water supply ability can be greatly improved by using water resources allocation optimization scheme and the ISCSA algorithm. The total water supply ability can reach to 2040.7million m 3 in 2025, in which the total increased water supply in Xinxiang region will reach to 410.0 million m 3 compared with that in 2019. Sewage water reuse will be 18.7million m 3 .
Rain water collection and utilization will be increased to 12.9 million m 3 . 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 be reached to 153.5 billion Yuan.
It can be forecasted that according to water resources optimal allocation scheme studied in this paper, the amount of groundwater exploitation will be reduced to 735.2 million m 3 in 2025, which means that 217.8 million m 3 underground water resources will be saved compared with that in 2009.
The water use for eco-environment will be 33.8 million m 3 .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.

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

Conflicts of interest/Competing interests
There is no conflict of interest in this manuscript.

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

Authors' contributions
The work has not been published before and it is not under consideration for publication anywhere else. The work has been completed by Ke ZHOU.

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

Consent for publication.
All the data in the paper can be published without any competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.