Utilization threshold of surface water and groundwater based on the system optimization of crop planting structure

Based on the diversity of the agricultural system, this research calculates the planting structures of rice, maize and soybean considering the optimal eco-nomic-social-ecological aspects. Then, based on the uncertainty and randomness of the water resources system, the interval two-stage stochastic programming method, which introduces the uncertainty of the interval number, is used to calculate the groundwater exploitation and the use ef ﬁ ciency of surface water. The method considers the minimum cost of water as the objective of the uncertainty model for surface water and groundwater joint scheduling optimization for different planting structures. Finally, by calculating harmonious entropy, the optimal exploitation utilization interval of surface water and groundwater is determined for optimal cultivation in the Sanjiang Plain. The optimal matching of the planting structure under the economic system is suitable when the mining ratio of the surface is in 44.13 % – 45.45 % and the exploitation utilization of groundwater is in 54.82 % – 66.86 % , the optimal planting structure under the social system is suitable when surface water mining ratio is in 47.84 % – 48.04 % and the groundwater exploitation threshold is in 67.07 % – 72.00 % . This article optimizes the economic-social-ecological-water system, which is important for the development of a water- and food-conserving society and providing a more accurate management environment.


Introduction
As an agricultural country, in China, more than 70% of the total water used is for agriculture [1] . However, the distribution of water resources over time and space is uneven in China, particularly in the northern region. This region includes 59.2% of the country's arable land but only 14.7% of the country's water resources [2] . Water is scarce in China and the tens of billions of kilogram decrease in grain production caused by water shortages each year threatens food security. Thus, water has become the most important factor affecting crop yield [3] . From the perspective of sustainable development, when a shortage of water resources occurs in the region, it can only be overcome through the rational allocation of limited water resources, in order to achieve the optimum comprehensive benefit and achieve sustainable development in agriculture [4] . Along with the rapid development of the social economy, the water use in industry, in cities and towns, and in the environment have also generated increasing competition with water use agriculture, caused the shortfall between the supply and demand for agricultural water to increase. Therefore, scientific planning of crop planting structures, establishing models of optimal allocation of water for the efficient use of the limited water resources for agriculture, and promoting the sustainable development of the agricultural ecological environment and economy are of great significance [5] .
As the difference between the supply and demand of agricultural water resources increases, the development of agricultural water resources simulation and optimal allocation have been widely used. Grundmann et al. [6] presented a new simulation-based integrated water management tool for sustainable water resources management in arid coastal environments. Huo et al. [7] distributed irrigation water based on a soil moisture dynamic simulation optimization model of irrigation systems and the crop water production function. Environmental problems are triggered by the over-exploitation of groundwater; thus, joint groundwater and surface water scheduling problems are gaining increasing attention. The main methods used to solve such problems are the simulation optimization method [8] , linear method [9,10] , dynamic programming model [11] , and numerical simulation analysis method [12] .
Because the level of agricultural water use and unit cost are uncertain, and the water resources management system is an uncertain complex system, an uncertainty optimization method is suitable for solving the uncertainty factors for such a system. In recent years, this method has been widely used in many areas by, for instance, Maqsood [13] . An accurate rough interval, fuzzy linear programming method is applied to the water resources allocation of an agricultural irrigation system. Li et al. [14,15] used an interval fuzzy multistage method for water resources management and established a random two-stage interval quadratic programming model for water quality management. Xie and Fu [16][17][18] used two-stage stochastic programming to optimize the allocation of water resources.
However, few studies have obtained a multi-objective cultivation solution by calculating the utilization threshold of the surface water and groundwater. Therefore, this study considers the economic-social-ecological and water resources system using a multi-objective two-stage stochastic programming model with intervals and agricultural water system entropy considerations to analyses the surface water and groundwater exploitation ratio for the Sanjiang Plain to overcome the complexity and uncertainty of the agricultural water resources system and to obtain an optimization solution and use ratio threshold of surface water and groundwater in agriculture. Multi-objective optimization and two-stage interval programing model are combined to establish the objective solution function under different solutions. Moreover, it can be employed for quantitatively analyzing a variety of alternative solutions with different stream flow levels and allocation proportions. Then, the optimal scheme is obtained by the comparison of the different solutions through the harmonious entropy. Using this approach, a new method for optimizing the allocation of water resources is proposed.

Modeling
2.1 Optimization of the crop planting structure under each system The agricultural system is a complex system. A crop planting structure optimization model is established based on the latest research results of crop planting structure adjustment based on the aspects of production, life and ecology to determine the areas of economic benefit, social benefit and ecological benefit target function.
One of the largest economic benefits is to make crops produce more, which involves maximizing the use of water resources to achieve the efficient utilization of water resources [19] .
Social benefits arise from making the output of grain meet the demands of human society, achieving fair social distribution, meeting the social demands of people's daily consumption and increasing farmers' income to maintain social stability and raise the quality of life of individuals [19] .
Ecological sustainability involves guaranteeing ecological water use to ensure the natural condition of water resource natural ecological service function and regulating human activities and allowing water to better serve humans [19] .
According to the above analysis and considering the limitations of water and soil resources, a multi-objective economic, social and ecological optimization model is proposed as follows: where decision vector X refers to the crop planting area and 3 are the economic, social and ecological benefits of the objective function, respectively. The constraint conditions include water and soil resource constraints.

Joint surface water and groundwater configuration under uncertainty
Considering the various benefits of crop planting area, due to the importance of the water resources system and considering the joint use of surface water and groundwater to meet a variety of crop water requirements, the goal is to reduce the cost of water. Using the two-stage uncertainty interval stochastic programming method with the minimum agricultural water comprehensive cost as the objective function and the water supply in study area as the constraint conditions, the first step to determining the three solutions for crop planting area for known conditions is the introduction of the total water cost and water dynamic resources and evaluating water resources and different water costs to calculate the optimal configuration of water.
Because the water resources system contains many sources of uncertainty, the prediction of the supply of water is challenging and the early crop water supply target is uncertain. The difference between the water resources and the water supply route leads to water transfer cost and crop price changes [20] . The output of water makes the presence of a penalty coefficient uncertain. To account for this uncertainty, this research introduced the interval parameter and used the upper and lower limit based on scientific considerations and rationality. The basic two-stage interval stochastic programming model is as follows: where f AE is the cost of agricultural water (CNY), T AE i is the water conveyance cost (CNY), W AE i is the prediction of water under different solutions (ten thousand m 3 ), C AE i is lack of punishment coefficient, indicating that the actual water supply cannot meet the forecast (CNY), and S AE ik is the water deficit forecast for the year (ten thousand m 3 ). These values are significantly influenced by the amount of rainfall, which is difficult to forecast. This study predicts different years by analyzing the water deficit cases according to discrete function processing [21] . p k is the probability of the water level. k = 1, 2, 3…… k, where k = 1 predicts the year of minimal water deficit and high flow. When k = 2, water flow is moderate, and the water deficit is small. When k = k there is little water and low flow, resulting in a large water deficit. Different water resources are represented (surface water and groundwater). The constraints are as follows: (1) Minimum water demand constraints: where W AE imin ensures the normal growth of minimum water requirement (hundred million m 3 ).
(2) Water supply capacity constraints: where Q AE i is the amount of available water in different regions in early water planning, and q AE ik is the water in different regions with different inflows during the planning period; only rainfall is considered here.
(3) Maximum water consumption constraint: (4) Variable nonnegative constraints: Due to the linear programming solution of constraints, when W AE i , the uncertain input parameters cannot calculate the model solution. Therefore, this article introduces another decision variable z i , among z i 2 ½0,1, When z i ¼ 1, W AE i reaches the upper limit value, and the corresponding water consumption is at a maximum. When z i ¼ 0, W AE i reaches the lower limit value and the corresponding water consumption is expected to reach the minimum.
is used to determine the value. With the introduction of decision variables, the optimal value z iopt can be solved. The water resources optimal allocation solution system cost is determined to obtain the optimal value W AE . The model is transformed into two certain temperament models representing the upper and lower limits. Such a deformation process based on an interactive algorithm is different from the commonly used interval analysis and case analysis [22] because the model uses the minimum cost of water. Therefore, the first solution fand its corresponding model are For the model, z i and si are the decision variables. The definitions of z iopt and siopt are the solutions for the model.
the optimal water diversion goals can be calculated by In the same manner, z iopt is changed to the upper limit of the model: Subject to After solving the calculated s þ iopt and f þ opt merging two sub models, the solution of the two-stage stochastic programming model is obtained as follows: One of the most optimal allocations of water is OPT -

Computing the harmonious entropy
There are contradictions and competition between economic, societal, and ecological resources. The question of how to achieve harmoniously balanced development requires quantitative evaluation. This research established a calculation method for the entropy of the harmonious degree and determined the harmonious entropy of each subsystem. The evolution of the system direction is often determined by entropy change theory based on the maximum entropy principle, defining the system of relative entropy of harmonious entropy and using the relative harmony entropy of the agricultural system efficiency under different water utilization thresholds for comparative analysis. The modeling process is as follows: (1) Confirm the economic, social, ecological and water resources system parameters. For the dimensionless processing of various parameters, the optimal methods adopt the following formulas to eliminate the dimension influence on the evaluation results: where r ij is the relative membership degree of the i system parameters under the j plan structure; x ij is the system parameter values of i under thej plan structure; and x ijmin , x ijmax and x ijmin are the respective thresholds of the i system parameters under the j plan structure.
(2) Confirm the relative membership degree of each subsystem of the harmonious degree, hereinafter referred to as the subsystem relative harmony degree u AE i : where u i is the relative degree of the subsystem of i,u i 2 ½0,1; m is the total parameter number of i subsystem; and w ij is the weight of the i system parameters under the j plan structure.
(3) Confirm the relative harmony entropy of the agricultural water system EðtÞ AE : where EðtÞ AE is the relative harmony entropy of the agricultural water system under the j plan structure; n is the total number of the system; and k is the ratio coefficient, which typically takes k ¼ -1.
According to the relationship between entropy change, determine the evolution of the water resource system [23] to evaluate water resources allocation rationality: when EðtÞ<Eðt þ 1Þ, the system entropy increases, there is disharmonious degree enhancement, the system is in an unstable state, the system is in the process of evolution in a vicious cycle, and the allocation of water resources is unreasonable; when EðtÞ > Eðt þ 1Þ, the system entropy decreases, the disharmony degree is reduced, the system is in a state of virtuous cycle, the allocation of water resources is reasonable; when EðtÞ<Eðt þ 1Þ, for a certain time interval, the system entropy has no change, the system is in a stable state, and the water resources allocation is more reasonable. By comparing various solutions of harmony entropy, where lower values indicate the solution is relatively optimal, the utilization threshold of the groundwater and surface water conditions under the solution of crop planting structure are improved.
The overall concept of the model can be summarized as shown in Fig. 1.   1 Analytic framework of the model base and commodity grain production base. The allocation of water resources in the Sanjiang Plain is not optimal, which results in a decrease in the capacity of regional water resources. Therefore, for the protection of national food security and based on the premise of regional ecological security, adjustments to the agricultural planting structure and the sustainable utilization of water resources are used to ensure the food security and sustainable development of the social economy in our country. The regional geographical locations are shown in Fig. 2. 3.2 Target of the crop planting structure under each system The research data mainly originate from the Heilongjiang Province economic statistical yearbook, the water resources in Heilongjiang Province, the Heilongjiang Province statistical yearbook and the Heilongjiang Province yearbook. Considering the data sources and the cultivation characteristics of the Sanjiang Plain, three main cropsrice ðx 1 Þ, corn ðx 2 Þ and soybeanðx 3 Þare chosen for use in the surface water and groundwater joint scheduling considering the economic, social and ecological benefits. Three types of solutions are developed with the corresponding objective functions as follows: Solution 1: economic benefit maximum (Â10 8 CNY). According to the data analysis and the comprehensive consideration of many years of rice, maize and soybean sales prices, the sales of Sanjiang Plain food produce economic benefits according to the objective function: Solution 2: social benefit maximum (Â10 8 kg) According to the data analysis and the comprehensive consideration of many years of rice, maize and soybean yield per unit area, the goal of maximizing the Sanjiang Plain grain output function yields the following objective function: Solution 3: ecological benefit maximum (Â10 8 CNY) According to the data analysis, the comprehensive consideration of the size of the agricultural land area and the relationship between the ecological service values, the following Sanjiang Plain ecological service value function is maximized: The constraint conditions of the objective function are as follows: (1) The full irrigation of crops water requirement constraint (10 4 m 3 ) According to the data analysis and the comprehensive consideration of the Sanjiang Plain rice, maize and soybean supply situation, the amount of water available for the three crops does not exceed a maximum of 16.64Â10 9 m 3 .  x 1 þ x 2 þ x 3 £228:38 According to the data analysis and the comprehensive consideration of the Sanjiang Plain crops with a planting area for rice, maize and soybean of no more than 2.2838 Â 10 4 hm 2 over many years, the largest acreage was less than 95 Â 10 4 hm 2 , 128 Â 10 4 hm 2 and 50 Â 10 4 hm 2 for rice, maize, and soybean, respectively.
The benefit of each system for crop planting area and the corresponding benefit value are calculated [24] , and the results are shown in Table 1.

Optimal allocation of surface water and groundwater under uncertainty
With more than three solutions for planting area under known conditions, an interval, two-stage, stochastic programming model has been developed using the method of interval two-stage stochastic programming and having the minimum comprehensive cost of the Sanjiang Plain water as the goal. Groundwater and surface water are optimized; then, the water resources system status is analyzed according to the result of the optimal configuration and analysis of the surface water and groundwater exploitation utilization. Table 2 lists the allowable water usage in the irrigation of the three crops in Sanjiang Plain for different forecast years. The research of the future water levels is divided into three categories, namely, low, medium, and high, according to the historical statistical rainfall and runoff data of Sanjiang Plain. The probability of medium is higher than the high flow and low flow, and the high flow and low flow have similar probabilities. Therefore, this article assumes water level probabilities of 0.2, 0.6, and 0.2.
In water resources planning, if the amount of water available is expected to meet the water demand, there are only water costs; if the amount of water fails to meet the water demand, there are punishment costs. Table 3 shows the different cost of water diversion, the water shortage penalty coefficient and the amount of water available in early planning [25] .

Optimal allocation results for water resources
The solutions are obtained using Matlab and Excel. The optimal water supply of surface water and groundwater under different solutions can be obtained by, The water deficit is forecast according to the model calculation results, which are shown in Table 4. Then, the optimal allocation of water resources under different solutions is calculated for different levels of water. The value is the available water quantity minus the water deficit OPT AE ¼ W AE iopt -S AE iopt , as shown in Fig. 3. The results in Table 4 show that each solution in the optimal water resources allocation has a certain water shortage. The difference between the supply and demand of water resources is significant in the study area. Research   into water-saving irrigation and improving the efficiency of water use has a profound significance for the Sanjiang Plain. Reducing the total cost of water is also important to further increase income among regions and to reduce the losses caused by the exploitation of water; thus, the solutions of water resources allocation are smaller. Figure 3 shows that the changing trends of the optimal configuration of water under different water levels in Solutions 1 and 2 are small. The change in the third solution of the optimal allocation of water under different water levels is larger, which indicates that the considerations for the economic system and social system in the natural and outside factors of water are small, whereas the ecosystem response to the external system is large, which results in a larger change.
To further understand the exploitation of surface water and groundwater, this research used the optimal allocation of water resources determined above as the foundation and the medium water level was selected to calculate the proportion of surface water and groundwater available for  each situation and to calculate the surface water and groundwater proportions of the total water use of each situation. The reasonable proportion of surface water and groundwater usage is determined. The calculation result is shown in Fig. 4. As seen from Fig. 4., the overall surface mining ratio of Sanjiang Plain is [40%, 55%], and the groundwater exploitation threshold is [50%, 80%], which indicates a higher percentage of groundwater exploitation but reasonable control (within 80%). Reasonable development of groundwater can greatly improve the efficiency of groundwater; in each region, the surface water use ratio is approximately 70%, and the groundwater use ratio is approximately 30%. For the Sanjiang Plain agricultural irrigation to achieve economic-social-ecological optimization and for the sound development of the water resources system, the manager should give priority to surface water, with groundwater being complementary. 3.5 Using the entropy of the harmony degree to compare solutions According to the characteristics of the Sanjiang Plain and the related system principle of determining the parameters, the output of the three crops is chosen as the order parameter of the economic system, per capita gain is chosen as the parameter for the social system, the per capita ecological service value is chosen as the ecological order parameter, and the extraction threshold of groundwater and surface water utilization are chosen for the water resources system parameters. The weight of each of the economic-social-ecological-water systems is 0.25, which embodies the principle of balanced and harmonious development. Finally, the harmonious entropy of each solution and corresponding exploitation of groundwater and surface water utilization are calculated, as shown in Fig. 5.  Figure 5 shows that the entropy of harmony degree of the third solution is the largest, the corresponding surface water and groundwater exploitation utilization ratio is the largest, the agricultural system is relatively unstable, and the planting structure is not suitable. For the first and second solutions, due to the mining of groundwater and the surface water use ratio of a certain interval, with the surface water utilization within 44.13%-45.45% and the threshold of groundwater exploitation within 54.82%-66.86%, the minimum entropy value of harmony degree and the corresponding crop planting structure of the first solution are more suitable. For surface water utilization within 47.84%-48.04% and a threshold of groundwater exploitation within 67.07%-72.00%, the entropy of the harmonious degree of the second solution is smaller, and the corresponding crop planting structure is more reasonable. Figure 5 shows that the three solutions of surface water and groundwater exploitation are better than the current usage. Solution 1 considers the economic benefit most and the mining quantity and costs less. The optimal solution 2 maximizes the social benefits, and the exploitation of water and cost are greater. Solution 3 maximizes the ecological benefits. The mining quantity is greater, the cost is the largest, and the entropy of harmonious degree is the maximum, which indicates the system is the least stable. The solution cannot blindly pursue economic-socialecological benefits, causing the water resources system to face excessive risk. The solution also cannot only consider the existence of risk and ignore the development of the society. Therefore, the benefits and water resources system must be balanced to select the appropriate solution.

Conclusions
This research establishes an objective optimization model for the benefit of the comprehensive agricultural planting structure and the corresponding water cost minimum target optimization model considers the sustainable development of agriculture. The objective functions of the model represent the four aspects of economy, society, ecology and water resources.
After a variety of crop planting structures were determined, a reasonable allocation of water resources was studied, with the objective function of minimizing the cost of crop water. A two-stage uncertainty interval stochastic programming model was applied for the optimal allocation of agricultural water resources in the Sanjiang Plain. After the introduction of water cost and shortage cost to the cost of water, the optimal allocation of surface water and groundwater under various solutions can be calculated under different water levels, which helps policy makers save water and improve the use efficiency of water.
The greatest advantage of this model is using the entropy of harmonious degree to evaluate the social-economicecological-water resources system comprehensively for each solution. Through the harmonious entropy, this research provides the optimal allocation of surface water and groundwater in the rational mining range. When the utilization threshold of surface water is within 44.13%-45.45% and the exploitation threshold of groundwater is within 54.82%-66.86%, the structure of crop planting is based on achieving the maximum benefits of the economic system. When surface water utilization is within 47.84%-48.04% and the groundwater exploitation threshold is within 67.07%-72.00%, the structure of crop planting is based on achieving the maximum benefits of the social system. This reasonable exploitation is an important reference value and theoretical basis for agricultural policy, which is useful for the sustainable utilization of water resources in the Sanjiang Plain.