Research on Resource Allocation and Water Saving Strategies in Deep Bayesian Network Driven Farmland Irrigation Systems

Since entering the 21st century, how to build a moderately prosperous society in all aspects has become the top priority of China’s social and economic development. Based on the goal of building a moderately prosperous society in all aspects, the implementation of the strategy of rural revitalization points out the direction for solving the current problem of unbalanced and insufficient urban and rural development [1–2]. The strategy of rural revitalization is the general grasp of the work of the “three rural areas” in the new era, and it puts forward new requirements for the construction of a governance system that combines self-government, the rule of law, and the rule of ethics, and the modernization of rural governance capacity. Based on the macro framework of rural revitalization strategy, how to promote the effective governance of rural public affairs will become an important direction for the current agricultural economic management research [3–5]. Farmland irrigation system is an important foundation for strengthening the national agricultural production capacity, however, with the development of Chinese agriculture and rural areas, the contradiction of the irrigation capacity of farmland irrigation system lagging behind the needs of agricultural production caused by the shortage of water resources, the insufficient supply of farmland water conservancy facilities, and the lack of management and care has become an important problem facing Chinese agricultural development [6–9]. Government “suspension” and the market “predicament” coexist is the current Chinese farmland irrigation system governance predicament of the true picture, especially in rural areas, “two workers” system canceled, farmers can not manage, Collective management is not good, the state can not manage to become the norm. In the process of farmland irrigation system governance, the main responsibility avoidance and “free-riding” phenomenon, which is presented in the process of some people use, no one manages, has been the outstanding short board of China’s agricultural and rural development [10–13]. Deep Bayesian network can effectively complete the small sample learning task, so that the advantages of deep learning technology can still be applied in small data sets, the use of which to drive the resource allocation and water saving in the farmland irrigation system, you can fully mobilize the limited resources, improve the utilization rate of water resources, and actively play the role of the main stakeholders of farmland irrigation. The implementation of this strategy is expected to realize the self-organized governance of farmland irrigation affairs and improve the governance level of farmland irrigation system [14–18].

Taking farmland irrigation as the research object, this paper first introduces the characteristics and schemes of water resources optimal allocation. After that, we construct a multi-objective water resource allocation model based on deep Bayesian network to solve the uncertainty of the multi-objective water resource allocation model and the problem of selecting the optimal solution, so as to reasonably allocate the water resources in the irrigation area. Finally, based on the model, the optimal allocation of water resources in farmland irrigation system is investigated and analyzed, and the optimal scheme of water resources allocation in irrigation area is selected to realize the water-saving irrigation strategy in farmland.

2

Deep Bayesian-driven multi-objective optimization model of farmland irrigation system

2.1

Optimized allocation of water resources

2.1.1

Connotation and characteristics of optimal allocation of water resources

Optimal allocation of water resources is based on the balance of supply and demand, using engineering or non-engineering measures to rationally allocate limited water resources within a given area, thus ensuring coordinated economic, social and environmental development. Optimal allocation of water resources is characterized by rational planning, where supply is determined by demand and priority is given to meeting the water needs of domestic production, without reducing agricultural irrigation area.

The optimal allocation of water resources can effectively improve the efficiency of water resources distribution, reasonably allocate the proportion of water for each sector, and alleviate the tension between supply and demand of various types of water users. In addition, it can promote the improvement of water conservation mechanisms, improve the efficiency of water use in different sectors, and promote the efficient use of water resources in all industries. The optimal allocation of water resources belongs to the overall strategic planning, not only to take into account the innate conditions of water resources in different regions and basins, but also to achieve the coordinated development of different aspects of the region as a whole, in order to maximize the overall benefits.

2.1.2

Methods of water allocation

1)

Linear Programming

Linear programming is a mathematical method to rationally allocate and effectively use resources under the condition of limited resources to maximize the goal. First, construct a model with objective functions and constraints, and then use linear programming to find the optimal solution. Rational allocation of water resources is to meet the needs of water supply under the premise of maximizing the benefits of water supply, usually combined with linear programming and other methods used in the model solution to obtain the optimal solution.

2)

Dynamic planning

Dynamic planning can simplify the multidimensional decision-making problem into multiple one-dimensional decision-making problems. Dynamic programming models are used to deal with multilevel decision making by dividing the problem into different spatial or temporal features and making decisions continuously at each stage. At the same time, when using the principles and methods of dynamic planning, it is important to analyze the problem specifically based on determining the specific problem and finalizing the solution.

3)

Multi-objective planning method

Multi-objective planning, also known as multi-objective optimization, is mainly used to study the situation of multiple objective functions, in order to achieve overall objective optimization. There are two common multi-objective planning methods: the first is the linear weighting method, the ideal point method and other complex multi-objective problem into a simple objective problem method; the second is the hierarchical sequence method, which is in accordance with the degree of importance of the objectives of a method of categorization.

2.2

Multi-objective planning theory

2.2.1

Concept of multi-objective planning solutions

Pareto proposed the multi-objective optimization problem, which is the transformation of a multi-objective problem that is not easy to solve into a single-objective optimization problem that is easy to solve. After that, the first book on multi-objective planning was published, which then attracted many scholars to start researching this field, resulting in more theories and methods. 1)

Non-inferior solution

In general, there is no absolute optimal solution to a multi-objective planning problem, which requires the introduction of a new concept of “solution” - non-inferior solution. Let f(x) be the objective function of the multi-objective decision-making problem, and its components f_j(x), j = 1,2,⋯⋯,n are all larger and better. For x* ∈ X, if there is no x in X such that f_j(x) ≥ f_j(x*), j = 1,2,⋯⋯,n, and at least one j strict inequality holds, then x* is called a non-inferior solution to the optimization problem.

2)

Pareto Optimality

Pareto optimality [19] refers to an ideal state of resource allocation, i.e., the situation that everyone is in is not getting worse, or at least someone is getting better. In the process of finding the optimal solution, multiple sub-objectives cannot achieve the optimized state at the same time, using Pareto optimality can allocate individual fitness and can obtain a non-inferior solution that satisfies the conditions and is acceptable.

2.2.2

Multi-objective planning solution methods

The following methods are commonly used to solve multi-objective planning problems: 1)

Comprehensive benefit optimization method

Comprehensive benefit optimization method needs to quantify multiple objectives, through certain rules to construct a utility function. And then the sum of multiple objectives is calculated, using the relationship between the objective function and the utility function to coordinate the conflict between the objectives. The method of utility function is based on the rationality of higher requirements, the need to accurately measure the degree of importance of objectives, and determine the size of the weights between objectives.

2)

Penalty function method

Penalty function method [20] can transform the constrained optimization problem into unconstrained optimization problem. For the multi-objective planning problem, the decision maker will explicitly give the ideal state of each objective value or the desired value, and determine the ideal solution by comparing the degree of deviation between the objective values of the variables in the actual model and the desired value. It is also necessary to determine the degree of importance and the size of the weights between the objectives. This method requires the decision maker to provide preference information and is highly subjective.

3)

Constraint modeling method

Considering that the decision maker may not be able to give an accurate ideal value of the objective, the constraint modeling method changes this fixed value into an interval value. At this point, the target value becomes a set of constraint intervals, which can be transformed into model constraints, and when multiple target values can be transformed into constraints, the multi-objective model can be transformed into a single-objective model, which reduces the difficulty of solving.

4)

Goal Reaching Method

This method solves the problem by setting relaxation variables. The model is transformed into a standard type, i.e., the minimum value of each objective is sought, and the ideal expectation of each objective is given. Next, the slack variables corresponding to the objective function are added and the corresponding weights are attached to each slack variable.

5)

Objective planning method

The goal programming method involves providing an expectation value for each objective beforehand, and then finding the closest solution to that expectation value while meeting certain constraints.

6)

Intelligent optimization algorithm

Intelligent optimization algorithms are a class of optimization methods established by simulating a natural phenomenon or process. Some commonly used algorithms are as follows:

Genetic algorithms are methods of searching for optimal solutions by converting the problem-solving process into a process similar to the crossover and mutation of chromosome genes in biological evolution. However, the algorithm has two drawbacks: first, the algorithm is greatly influenced by the initial solution, and the final optimization result depends to a greater extent on the merits of the initial solution. The second issue is that the computational efficiency is low due to the lack of effective and timely utilization of feedback information during network calculations.

Simulated annealing algorithm [21] is a stochastic optimization algorithm that simulates the annealing process of solid matter in physics. Substances are gradually transformed from a high-temperature liquid state to a low-temperature solid state, the internal molecules gradually find a suitable ordered position, the overall energy gradually from high to low, and the change of energy corresponds to the target value of the function. The algorithm introduces probabilistic jumps to avoid falling into local optimization.

In contrast, the integrated benefit optimization method is simple, but requires more rationality in the utility function. The penalty function method is more subjective and requires explicitly given the ideal objective value and weights. The constraint modeling method is not effective in transforming objectives into constraints. Goal reaching and goal planning methods are more suitable for linear programming problems. Intelligent optimization algorithms are capable of adapting to the problem, but require further improvement in their solution-search strategy.

2.3

Deep Bayesian-driven modeling of multi-objective farm irrigation systems

2.3.1

Basic concepts of multi-objective optimization problems

1)

Concept of multi-objective optimization

Franklin raised the problem about coordinating multi-objective conflicts, and French economist Pareto proposed the multi-objective optimization problem (MOP) [22].

With the formulation of this problem, people are paying more attention to research in this area. MOP as a branch of mathematics has been studied by a large number of researchers, in essence, it is to study how to optimize the values of multiple objectives at the same time under the special conditions. MOP has been widely used in daily life in the allocation of water resources for agricultural irrigation systems, reservoir operation and management, logistics and transportation, and Internet communication, among other things.

Generally, the defining equation of MOP is as follows: 1 $\begin{array}{l} n i n F (x) = f_{m} (x_{1}, x_{2}, ..., x_{n}) m; m = 1, 2, ..., M; \\ s . t . (1) g_{i} (x) = 0; i = 1, 2, ..., l; \\ (2) h_{j} (x) \geq 0; j = 1, 2, ..., J; \\ (3) x_{t}^{(L)} \leq x_{i} \leq x_{t}^{(U)} t = 1, 2, ..., T; \\ m i n F (x) = f_{m} (x_{1}, x_{2}, \dots, x_{n}) m; m = 1, 2, \dots, M; \\ s . t . (1) g_{i} (x) = 0; i = 1, 2, \dots, l; \\ (2) h_{j} (x) \geq 0; j = 1, 2, \dots, J; \\ (3) x_{t}^{(L)} \leq x_{i} \leq x_{t}^{(U)} t = 1, 2, ..., T; \end{array}$

2)

Pareto dominance relation

In single-objective, there is only one optimal solution to the solution problem, but in multi-objective problems, the unique global optimal solution does not exist, and there exists only a set of optimal solutions characterized by the existence of at least one objective that outperforms all the others, generally known as a non-inferiority solution, also called a Pareto solution.

Pareto Objective Dominance [23] Conditional Relationship: in a minimization multi-objective optimization problem, for n objective component f_i(x), i = 1,2,…,n, any given two decision variables X_a, X_b are said to X_a dominate X_b if the following two conditions hold.

(1)

For ∀i ∈ {1,…,n}, both f_i(X_a) ≤ f_i(X_b);

(2)

For ∃_iò{1,…,n}, such that f_i(X_a) ≤ f_i(X_b).

A decision variable is said to be an undominated solution if, for that decision variable, there exists no other decision variable that can dominate it.

2.3.2

Model for optimal allocation of water resources in agricultural irrigation systems

1)

Modeling ideas

Multi-objective optimal allocation of water resources in farmland irrigation system is a complex large system optimization problem. First of all, it is necessary to determine the objective of the optimal allocation of water resources in farmland irrigation system, and then construct a reasonable objective function and constraints, choose the solution algorithm, and finally get the water resource allocation scheme of farmland irrigation system. The model for optimal allocation of water resources in a farmland irrigation system is shown in Figure 1.

2)

Decision variables

When optimizing the allocation of water resources in farmland irrigation system, considering the different water consumption per unit area of each crop in farmland irrigation system, it is also necessary to optimize its planting and area together, so the decision variables include the water consumption and planting area of each crop in the fertility stage of farmland irrigation system.

3)

Objective function

The sustainable development of farmland irrigation system is the premise of water allocation in farmland irrigation system, this study takes the maximum economic net benefit of farmland irrigation system, the minimum total agricultural water consumption and the maximum total carbon absorption as the objective function.

(1)

Net economic benefits of farmland irrigation system

The economic net benefit of farmland irrigation system is an important criterion to measure the quality of life of people in farmland irrigation system, which is determined by the planting area and yield of crops in farmland irrigation system, and the rational allocation of water resources in farmland irrigation system is the main factor affecting the yield of crops. In order to maximize the economic net benefit of farmland irrigation system, under the condition of non-sufficient irrigation, by adjusting the planting structure of each crop and optimizing its irrigation system, so as to make full use of land resources and water resources of farmland irrigation system, the optimization objective function can be expressed as follows: 2 $\max F_{1} = \sum_{i = 1}^{n} (A_{i} C_{i} Y_{m i} - \frac{W_{i}}{ρ} B - S_{i} A_{i})$

Where, F_i is the net economic benefit of the farm irrigation system, yuan; A_i is the acreage of the i crop, acres: C_i is the current year price of the crop i, yuan; Y_mi is the current year’s yield of the crop i, kg/mu; W_i is the total water consumption of the crop i, m³; ρ is the integrated irrigation utilization coefficient; B is the current year’s price of water, yuan/m³; and S_i is the agricultural cost of the crop i, for: 3 $Y_{m i} = Y_{i} \sum_{j = 1}^{m} {(\frac{E T_{i, j}}{E T_{m i, j}})}^{λ_{i, j}}$

Where, Y_t is the yield of the crop under fully irrigated conditions, kg/mu: ET_i,j is the water consumption of the crop i under non-fully irrigated conditions in the j th time period, mm; ET_mi,j is the water consumption of the crop i under fully irrigated conditions in the j th time period, mm; and λ_i,j is the water deficit index of the crop i in the j th time period, which has the value of: 4 $W_{i} = A_{i} \sum_{j = 1}^{m} (W_{i, j} + G W_{i, j})$

Where W_i,j is the surface water consumption of crop i in j time, mm; GW_i,j is the groundwater consumption of crop i in j time, mm.

(2)

Total agricultural water consumption of farmland irrigation system

For the arid areas where water resources are already in short supply, excessive agricultural water consumption will squeeze the ecological and domestic water of the farmland irrigation system, which will have different degrees of impact on the sustainable development of the local area, this study adopts the objective function of minimizing the total water consumption of the farmland irrigation system, and its optimization objective function is as follows: 5 $\min F_{2} = \sum_{i = 1}^{n} A_{i} \sum_{j = 1}^{m} (W_{i, j} + G W_{i, j}) - W_{t o t a l}$

Where F₂ is the total water deficit of farmland irrigation system, m³; W_total is the total water supply of surface water and groundwater under the constraint of “three red lines” of farmland irrigation system, m³.

(3)

Total Carbon Sequestration in Farmland Irrigation System

With the goal of maximizing the total amount of carbon absorbed by the farmland irrigation system, the carbon sequestration potential of the crops themselves plays an important role in regional soil and water conservation, and mitigating soil desertification. According to the economic coefficients and carbon absorption rates of different kinds of crops, the carbon absorption during the reproductive period of the crops is measured, and the total carbon absorption of each crop is calculated, and the optimization objective function is as follows: 6 $\max F_{3} = \sum_{i = 1}^{n} \frac{A_{i} Y_{m i} E_{i} (1 - w c_{i})}{H_{i}}$

Where, F₃ is the total crop carbon uptake in the irrigated farmland system, kg: E_i is the crop i carbon uptake rate, %; wc_i is the water content of the crop i economic product fraction, %; and H_i is the economic coefficient of the crop i.

(4)

Constraints

According to the characteristics of water use in farmland irrigation system and water resource conditions, the “three red lines” are used as the constraints on the amount of water that can be supplied; the cultivated land in the farmland irrigation system is seriously damaged due to improper management, and the total irrigated area of the farmland irrigation system is used as the constraints on the land area in combination with the local planting conditions.

Surface water constraint: 7 $\sum_{i = 1}^{n} \frac{W_{i}}{ρ} \leq W$

Groundwater constraints: 8 $\sum_{i = I}^{n} \sum_{j = I}^{m} \frac{G W_{i, j}}{ρ} \leq G W$

In Eq., W is the amount of surface water available under the constraints of the “three red lines” of the farmland irrigation system, and in Eq. m³, GW is the amount of groundwater available under the constraints of the “three red lines” of the farmland irrigation system, and m³.

The land area constraint: 9 $\sum_{i = 1}^{n} \frac{A_{j}}{ε} \leq A$ 10 $A_{L} \leq \sum_{i = 1}^{r} \frac{A_{i}}{ε}$

Where, A is the total irrigated area of the farmland irrigation system, mu; ε is the replanting coefficient, which is required to be not lower than the lower limit value set by the local national economic development, and due to the serious salinization of the land in the farmland irrigation system, the replanting of the local crops is not taken into account in this study: A_i is the planting area of the food crops t in the farmland irrigation system, hm² ; A_L is the minimum planting area of the food crops in the farmland irrigation system, hm².

Water requirement constraints for each crop at each fertility stage: 11 $0 \leq E T_{i j} \leq E T_{m i, j}$

Non-negative constraints: 12 $0 \leq W_{i, j}, G W_{i, j}, A_{i}$

2.3.3

Deep Bayesian-driven modeling of on-farm irrigation systems

DNNs can fit nonlinear mapping relations from inputs to outputs, and thus, based on the historical dataset computed by multi-objective planning, nonlinear mapping relation Ω₁ in the set of nonlinear equations formed by the KKT condition and nonlinear mapping relation Ω₂ in the set of multi-objective equations formed by the multi-objective equations. 1)

Deep Neural Network

Based on the three aspects of network framework, cost function and stochastic gradient descent (SGD) method, the following content will elaborate the DNNs model:

(1)

Network framework

DNNs are structurally composed of input, hidden and output layers. Among them, each layer consists of several neurons, and each neuron represents an element in a vector. With x denoting the feature vector of the input, h denoting the intermediate vector of the hidden layer, y denoting the resultant vector of the output, b denoting the bias vector, and w denoting the weight matrix, the relationship between the layers of DNNs is as follows: 13 $\begin{matrix} h^{[1]} = g^{[1]} (W^{[1] T} x + b^{[1]}) \\ h^{[2]} = g^{[2]} (W^{[2] T} h^{[1]} + b^{[2]}) \\ ...... \\ h^{[L]} = g^{[L]} (W^{[L] T} h^{[L - 1]} + b^{[L]}) \\ y = g^{[L + 1]} (W^{[L + 1] T} h^{[L]} + b^{[L + 1]}) \end{matrix}$ where L is the number of hidden layer layers; g(·) is the activation function acting on each neuron.

(2)

Cost function

Most DNNs are trained based on the Maximum Likelihood Estimation (MLE) method in point estimation, aiming to learn a parameter ϖ* = {W[t]r*, …, W[t]r*, b^[1]*, …, b^[t]*} that maximizes the likelihood function with the training objective: 14 $ϖ^{*} = \arg \max \prod_{i = 1}^{N} p (y^{(i)} | x^{(i)}; ω) = \arg \max \sum_{i = 1}^{N} \log p (y^{(i)} | x^{(i)}; ϖ)$ where argmax represents the value of the variable underneath when the right-hand side function obtains the maximum value: p(y | x;ϖ) is the model distribution of the DNNs, which determines the specific form of the cost function.

Since the variables in multi-objective planning are continuous, the output layer of the DNNs used to learn the nonlinear mapping relationships therein is usually chosen to be a linear unit, whereby, the linear output unit layer can output a predicted value $\hat{y}$ as the mean of the conditional Gaussian distribution p(y | x;ϖ). Therefore, the cost function of DNNs is to minimize the mean square error between the true value y and the predicted value $\hat{y}$ , which is expressed as: 15 $J (ϖ) = \frac{1}{N} {\sum_{i = 1}^{N} ‖ {\hat{y}}^{(i)} - y^{(i)} ‖}^{2} = \frac{1}{N} {\sum_{i = 1}^{N} ‖ f^{α} (x^{(i)}) - y^{(i)} ‖}^{2}$

Style: 16 $f^{x} (x^{(t)}) = W^{[L + 1] Y} g^{[L]} (W^{[L] T} (... g^{[2]} (W^{[2] T} (g^{[l]} (W^{[l] Y} x + b^{[l]})) + b^{[2]})) + b^{[L]}) + b^{[L + 1]}$

f^σ represents the feedforward computation of DNNs with implied number of layers L.

(3)

Stochastic Gradient Descent

SGD is one of the parameter learning methods commonly used in deep learning, and its specific steps are as follows: first, the initial values of the network parameters ϖ = {W^[T], …, W^[L]T, b^[L], …, b^[L]} are given: second, the gradient is computed: 17 $\nabla_{ω} J (ϖ) = \frac{1}{N} \sum_{i = 1}^{N} \nabla_{ω} | | f^{\infty} (x^{(i)}) - y^{(i)} | |^{2}$

Finally, update the gradient: ϖ ← ϖ – α∇_aJ(ϖ), where α is the learning rate.

2)

Deep Bayesian statistical models

Deep Bayesian statistical models (BDNNs) are a combination of DNNs and Bayesian statistical models. By introducing stochastic factors in the model parameters and inferring the distribution of model parameters, BDNNs can provide a probabilistic interpretation of DNNs. Thus, Bayesian statistical models confer the following three advantages to the original DNNs: 1) low risk of overfitting: 2) ability to fit small data sets: 3) ability to model uncertain variables probabilistically.

Given datasets X = {x₁, …, x_i, …, x_N} and Y = {y₁, …, y_i,…, y_N} with input x and output y, BDNNs aim to learn the probability distributions of model parameters ϖ. Compared to DNNs, BDNNs are able to provide a range of model parameters ϖ with uncertainty modeling capabilities.

In multi-objective planning computations, the ultimate goal is: given a new input x^ncw, to obtain the probability distribution of the output y^ncw, i.e., p(y^ncw | x^ncw). Historical datasets X and Y are the basis for inference p(y^ncw | x^ncw). Thus, the Bayesian inference process for fitting BDNNs with nonlinear mapping relationships in multi-objective planning computations can be represented as: 18 $p (y^{n e w} | x^{n e w}; X, Y) = \int p (y^{n e w} | x^{n e w}; w) p (ϖ | X, Y) d ϖ$ where p(ϖ | X, Y) is the posterior distribution of model parameter ϖ, i.e., the probability distribution of the model parameter given the historical data set: p(y^zcw | x^zcw;ϖ) is the likelihood function with the following expression: 19 $p (y^{a c w} | x^{a c w}; ϖ) = N (y^{a c w} | f^{a c w} (x^{a c w}), β^{- 1} I)$ where N(·) represents the Gaussian distribution; β⁻¹I represents the accuracy matrix, reflecting the deviation of the predicted values f^α(x^mew).

How to find the posterior distribution p(ϖ | X, Y) of the model parameters ϖ based on the historical datasets x and Y is one of the key tasks in the training of BDNNs, i.e., the Bayesian inference process. Therefore, the posterior distribution p(ϖ | X, Y) of ϖ is specified according to the Bayesian formula: 20 $p (ϖ | X, Y) = \frac{p (Y | X, ϖ) p (ϖ)}{p (Y | X)}$ where p(ϖ) is the prior distribution of model parameters ϖ : p(Y | X, ϖ) is the likelihood function: p(Y | X) is the evidence for the model based on the historical dataset.

3)

Variational Inference

The actual posterior distributions p(ϖ | X, Y) of the model parameters are often difficult to compute directly and are usually solved with the help of variational inference methods. The goal of the variational inference method is to approximate the actual posterior distribution with a parameterized variational distribution q_ϕ(ϖ), where the distribution function of q_s(ϖ) is of a simpler form, usually chosen as an exponential family of distributions. To measure the effectiveness of the variational distribution q_s(ϖ) in approximating the actual posterior distribution p(ϖ | X, Y), the KL dispersion between the two is as follows: 21 $D_{K L .} (q_{ϕ} (ϖ) | | P (ϖ | X, Y)) = \int q_{ϕ} (ϖ) \log (\frac{q_{ϕ} (ϖ)}{p (ϖ | X, Y)}) d ϖ$ where D_KL (· | ·) denotes the KL scatter between the two distributions. Minimizing the KL scatter is equivalent to maximizing the lower bound on the evidence, so the goal of variational inference can be transformed into: 22 $L_{N 1} (ϕ) = \int q_{ϕ} (ϖ) \log p (ϖ | X, Y) d ϖ - D_{K L} (q_{ϕ} (ϖ) | | p (ϖ))$

The above equation is the theoretical basis for the modeling framework of BDNNs and the design of the training objective function (loss function).

3

Results and analysis

3.1

Example analysis of optimal scheduling of water resources in field irrigation systems

3.1.1

Water allocation in field irrigation systems

The Deep Bayesian Multi-objective Optimization Algorithm toolbox is used to perform the calculation. This calculation mainly uses DBNNs algorithm toolbox, and the running environment is win7. The specific calculation process is as follows:

The Deep Bayesian Multi-objective Optimization runs in Python and is built based on the open-source machine learning library PyTorch and the probability model library Pyro. The specific calculation process is as follows: 1)

Choose to adopt the water resources optimization scheduling model in chapter three.

2)

Select and code the optimized data units. The coding of water resource allocation optimization unit of field irrigation system is shown in Table 1.

3)

Program the objective function, constraints.

4)

Determine the parameters of DBNNs algorithm itself, n = 50, Pc = 0.25, Pm = 0.1, and the number of DBNNs generations is 1000.

5)

Firstly calculate the water resource scheduling in the case of objective 1 using DBNNs algorithm toolbox, secondly calculate the water resource scheduling in the case of objective 2 using DBNNs algorithm toolbox and get the results as follows, finally calculate the water resource scheduling in the case of integrated objective using DBNNs algorithm toolbox using the weighted processing method.

Table 1.

Optimization unit coding of water allocation in irrigation system

Date		Irrigation rate
Month	Time period	Barley barley	Spring wheat	Potato	Beans	Cole
4	Early days	A1	A2
	Middle days			A3	A4	A5
	Late days
5	Early days	A6	A7
	Middle days			A8	A9	A10
	Late days	A11	A12
11	Early days	A13	A14	A15	A16	A17
	Middle days
	Late days

3.1.2

Results of water scheduling under different scenarios

1)

Water resource scheduling results in the case of objective 1

The optimization process curve of the DBNN algorithm for objective 1 is shown in Fig. 2. It can be seen that it tends to stabilize after 1000 iterations.

The watering quota for the scenario 1 case is shown in Table 2. The experimental results show that under scenario 1 situation, this paper plans to water a total of 280, 180, 325, 175 and 230 cubic meters of irrigation per acre for barley, spring wheat, potatoes, beans and oilseed rape.

2)

Water resource scheduling results in the objective 2 case

The curve of the optimization calculation process of DBNNs algorithm in the objective 2 case is shown in Fig. 3.

The watering quota for the objective 2 case is shown in Table 3. The experimental results show that under scenario 2 situation, this paper plans to water a total of 305, 285, 367, 232 and 245 m³ per acre of irrigation for barley, spring wheat, potato, beans and oilseed rape. Compared to scenario 1, scenario 2 increased the amount of irrigation water by 25, 105, 42, 57 and 15 cubic meters per acre for the five types of crops, respectively.

3)

Water resource scheduling results with integrated objectives

The curve of the optimization calculation process of DBNN’s algorithm under the comprehensive objective is shown in Fig. 4.

The irrigation quotas for the integrated target scenario are shown in Table 4. The experimental results showed that a total of 335, 243, 384, 220 and 266 m³ per acre of irrigation was planned in this paper for barley, spring wheat, potato, beans and oilseed rape under the integrated target scenario. Compared to Scenario 1, the integrated target scenario increased the amount of water irrigated by 55, 63, 59, 45, and 36 cubic meters per acre for the five crop categories, respectively. Compared to Scenario 2, the integrated target scenario increased irrigation per acre of cropland by 30, 17 and 21 m³ per acre for barley, potato and oilseed rape, respectively; while for spring wheat and beans, irrigation decreased by 42 and 12 m³ per acre compared to Scenario 2.

Table 2.

The amount of irrigation in the case of solution 1

Date		Irrigation rate(m³/acre)
month	Time period	Barley barley	Spring wheat	Potato	Beans	Cole
3	Early days	0	0	0	0	0
	Middle days	0	0	0	0	0
	Late days	0	0	0	0	0
4	Early days	50	30	0	0	0
	Middle days	0	0	55	15	30
	Late days	0	0	0	0	0
5	Early days	50	20	0	0	0
	Middle days	0	0	55	15	30
	Late days	45	20	0	0	0
6	Early days	0	0	0	40	40
	Middle days	40	40	50	0	0
	Late days	0	0	0	45	45
7	Early days	0	0	0	0	0
	Middle days	45	45	0	45	45
	Late days	0	0	40	0	0
8	Early days	0	0	0	0	0
	Middle days	0	0	0	0	0
	Late days	0	0	40	0	0
9	Early days	0	0	0	0	0
	Middle days	0	0	35	0	0
	Late days	0	0	0	0	0
10	Early days	0	0	0	0	0
	Middle days	0	0	0	0	0
	Late days	0	0	0	0	0
11	Early days	50	25	50	15	40
	Middle days	0	0	0	0	0
	Late days	0	0	0	0	0
12	Early days	0	0	0	0	0
	Middle days	0	0	0	0	0
	Late days	0	0	0	0	0
Total		280	180	325	175	230

Table 3.

The amount of irrigation in the case of solution 2

Date		Irrigation rate(m³/acre)
Month	Time period	Barley barley	Spring wheat	Potato	Beans	Cole
3	Early days	0	0	0	0	0
	Middle days	0	0	0	0	0
	Late days	0	0	0	0	0
4	Early days	55	40	0	0	0
	Middle days	0	0	55	25	35
	Late days	0	0	0	0	0
5	Early days	55	50	0	0	0
	Middle days	0	0	55	55	55
	Late days	45	55	0	0	0
6	Early days	0	0	0	40	40
	Middle days	55	55	62	0	0
	Late days	0	0	0	45	45
7	Early days	0	0	0	0	0
	Middle days	45	45	0	45	45
	Late days	0	0	40	0	0
8	Early days	0	0	0	0	0
	Middle days	0	0	0	0	0
	Late days	0	0	55	0	0
9	Early days	0	0	0	0	0
	Middle days	0	0	40	0	0
	Late days	0	0	0	0	0
10	Early days	0	0	0	0	0
	Middle days	0	0	0	0	0
	Late days	0	0	0	0	0
11	Early days	50	40	60	22	25
	Middle days	0	0	0	0	0
	Late days	0	0	0	0	0
12	Early days	0	0	0	0	0
	Middle days	0	0	0	0	0
	Late days	0	0	0	0	0
Total		305	285	367	232	245

Table 4.

The amount of irrigation in the comprehensive target situation

Date		Irrigation rate(m³/acre)
Month	Time period	Barley barley	Spring wheat	Potato	Beans	Cole
3	Early days	0	0	0	0	0
	Middle days	0	0	0	0	0
	Late days	0	0	0	0	0
4	Early days	62	35	0	0	0
	Middle days	0	0	62	25	40
	Late days	0	0	0	0	0
5	Early days	62	36	0	0	0
	Middle days	0	0	52	29	42
	Late days	52	40	0	0	0
6	Early days	0	0	0	40	40
	Middle days	52	52	62	0	0
	Late days	0	0	0	52	52
7	Early days	0	0	0	0	0
	Middle days	45	45	0	45	45
	Late days	0	0	52	0	0
8	Early days	0	0	0	0	0
	Middle days	0	0	0	0	0
	Late days	0	0	52	0	0
9	Early days	0	0	0	0	0
	Middle days	0	0	42	0	0
	Late days	0	0	0	0	0
10	Early days	0	0	0	0	0
	Middle days	0	0	0	0	0
	Late days	0	0	0	0	0
11	Early days	62	35	62	29	47
	Middle days	0	0	0	0	0
	Late days	0	0	0	0	0
12	Early days	0	0	0	0	0
	Middle days	0	0	0	0	0
	Late days	0	0	0	0	0
Total		335	243	384	220	266

3.2

Optimized multi-objective water allocation for agricultural irrigation in flat water years

3.2.1

Decision analysis of water use for agricultural irrigation in level water years

The farmland irrigation system irrigates orchards and cropland wheat once during winter irrigation, cropland corn once during spring irrigation, and cropland corn once during summer irrigation under flat water year conditions. The statistical results of farmland irrigation water use in the flat water year are shown in Table 5. The ratio of canal and well water use in Scheme 1 is 0.357:0.361; the ratio of canal and well water use in Scheme 2 is 0.711:0.631; and the ratio of canal and well water use in the comprehensive scheme is 0.867:0.699.

Table 5.

Statistical results of irrigation water in pingshui year

Scheme	Winter irrigation(10⁸·m³)
Scheme	Natural irrigation	Well irrigation	Canal completion
Solution 1	0.714	0.722	1.436
Solution 2	0.711	0.631	1.342
Comprehensive plan	0.867	0.699	1.566

The statistical results of agricultural irrigation water use in the districts in the flat water year are shown in Table 6. The combined irrigation water for winter, spring, and summer canal wells was 0.252, 0.876, and 1.104 (10⁸·m³) in Scenario 1, and 0.630, 0.735, and 0.858 (10⁸·m³) in Scenario 2; the combined scenario was 0.882, 1.611, and 1.962 (10⁸·m³).

Table 6.

Statistical results of irrigation water in each area of the sea

Scheme		Solution 1	Solution 2	Comprehensive plan
Natural irrigation(10⁸·m³)	Natural irrigation	0.134	0.343	0.477
	Well irrigation	0.129	0.298	0.427
	Canal completion	0.252	0.630	0.882
Well irrigation(10⁸·m³)	Natural irrigation	0.453	0.399	0.852
	Well irrigation	0.434	0.347	0.781
	Canal completion	0.876	0.735	1.611
Canal completion(10⁸·m³)	Natural irrigation	0.569	0.465	1.034
	Well irrigation	0.546	0.404	0.95
	Canal completion	1.104	0.858	1.962

3.2.2

Water Program for Agricultural Irrigation in Peaceful Water Years

The results of irrigation water use statistics for cropland in each district in Pingshui year are shown in Table 7. The results show that the irrigation water consumption of canal irrigation and well irrigation in the five types of farmland of “barley, spring wheat, potato, beans and rape” in winter irrigation of Program 1 is 0.0160-0.0458 and 0.0142-0.1018, respectively; Water use in spring irrigation ranged from 0.0461-0.0732 and 0.0382-0.0629, respectively; and in summer irrigation from 0.0382-0.0811 and 0.0311-0.2304, respectively. In scheme 2, the irrigation water consumption of canal irrigation and well irrigation in five kinds of farmland of “barley, spring wheat, potato, bean and rape” was 0.0399-0.2021, 0.0307-0.0741, 0.0314-0.0655, 0.0513-0.0563, 0.0465-0.1051 and 0.0438-0.0523, respectively. In the comprehensive plan, the irrigation water consumption of canal irrigation and well irrigation in five kinds of farmland, namely “barley, spring wheat, potato, bean and rape”, was 0.0485-0.0546, 0.0365-0.0692, 0.0227-0.0735, 0.0395-0.0715, 0.0438-0.1043 and 0.0394-0.1583, respectively.

Table 7.

Statistical results of irrigation water in each area of the sea

Scheme	Irrigation season	Irrigation mode	Barley barley	Spring wheat	Potato	Beans	Cole
Solution 1	Winter irrigation	Natural irrigation	0.0275	0.0160	0.0458	0.0389	0.0295
	Winter irrigation	Well irrigation	0.0387	0.0142	0.0344	0.0245	0.1018
	Spring irrigation	Natural irrigation	0.0461	0.0597	0.0349	0.0732	0.0607
	Spring irrigation	Well irrigation	0.0524	0.0607	0.0579	0.0629	0.0382
	Summer irrigation	Natural irrigation	0.0433	0.0811	0.0456	0.0576	0.0648
	Summer irrigation	Well irrigation	0.0486	0.2304	0.0518	0.0311	0.0571
Solution 2	Winter irrigation	Natural irrigation	0.2021	0.0399	0.0597	0.0569	0.0552
	Winter irrigation	Well irrigation	0.0741	0.0415	0.0307	0.0561	0.0482
	Spring irrigation	Natural irrigation	0.0502	0.0368	0.0483	0.0314	0.0655
	Spring irrigation	Well irrigation	0.0529	0.0524	0.0546	0.0513	0.0563
	Summer irrigation	Natural irrigation	0.0489	0.1051	0.0488	0.0512	0.0465
	Summer irrigation	Well irrigation	0.0472	0.0509	0.0438	0.0493	0.0523
Comprehensive plan	Winter irrigation	Natural irrigation	0.0546	0.0541	0.0485	0.0518	0.0535
	Winter irrigation	Well irrigation	0.0365	0.0612	0.0692	0.0388	0.0404
	Spring irrigation	Natural irrigation	0.0319	0.0331	0.0463	0.0227	0.0735
	Spring irrigation	Well irrigation	0.0715	0.0528	0.0566	0.0395	0.0401
	Summer irrigation	Natural irrigation	0.0446	0.0439	0.0576	0.1043	0.0438
	Summer irrigation	Well irrigation	0.0689	0.0394	0.1583	0.1059	0.0635

3.2.3

Analysis of Optimization Scenarios for Agricultural Irrigation in Flat Water Years

The groundwater level in each irrigation season of the level water year is shown in Fig. 5. As can be seen from the figure, under the condition of an optimized integrated scheme, the groundwater level decreased slightly after winter, spring, and summer irrigation and rebounded gradually. Since the ratio of winter, spring and summer irrigation is 0.441:0.8055:0.9810, the variation of groundwater level is larger after winter and summer irrigation, and the change of water level is not obvious after spring irrigation. Due to the lower proportion of canal and well water use in the integrated program, the more balanced winter, spring and summer irrigation ratio, the water resource recovery and consumption within the farm irrigation system roughly offset each other, and the groundwater level did not change much in the ending period compared to the initial one, and the fluctuation amplitude within the year was small.

3.3

Decision-making analysis of water use for agricultural irrigation in exceptionally dry years

The water level changes in the observation wells in the special dry water year are shown by Fig. 6. The results show that under the optimized comprehensive scheme, the groundwater level decreases slightly after winter-spring-summer irrigation and gradually rises. Since the ratio of winter, spring and summer irrigation is 0.630:0.735:0.858, the variation of groundwater level is larger after winter and summer irrigation, and the change of water level is not obvious after spring irrigation. Due to the higher proportion of canal and well water use and the larger proportion of summer irrigation in Scenario 2, the groundwater level within the farm irrigation system decreased slightly at the end of the period compared to the initial one, and the fluctuation within the year was small.

4

Conclusions and outlook

4.1

Conclusion

In this paper, a multi-objective deep Bayesian network-driven farm irrigation system based on multi-objective is constructed from the resource allocation and water saving in farm irrigation system. This model is used to optimize the allocation of farmland irrigation water resources. The primary conclusions are as follows:

In this paper, five kinds of farmland crops, namely barley, spring wheat, potato, beans and oilseed rape, were investigated, and the total annual watering amount of the integrated target program for the five crops was finally determined to be 335, 243, 384, 220 and 266 cubic meters through Program 1 and Program 2, respectively; and the total ratio of the canal-well water use in its flat water year was 0.867:0.699. In addition, the irrigation water of the integrated program for the winter, spring, and summer canal-wells combined was were 0.882, 1.611, and 1.962 (10⁸·m³), respectively. In the integrated plan, the irrigation water consumption of canal irrigation and well irrigation in five types of farmland of “highland barley, spring wheat, potato, bean and rape” was 0.0485-0.0546, 0.0365-0.0692, 0.0227-0.0735, 0.0395-0.0715, 0.0438-0.1043 and 0.0394-0.1583 (10⁸·m³), respectively. Under the condition of optimized comprehensive scheme, the irrigation ratio in winter, spring and summer flat water year is 0.441:0.8055:0.9810, the groundwater level varies greatly after winter and summer irrigation, and the change of water level is not obvious after spring irrigation. The ratio of irrigation in winter, spring and summer in the special dry water year was 0.630:0.735:0.858; at this time, the groundwater level in the farmland irrigation system decreased slightly at the end of the period compared with the initial one, and the fluctuation amplitude during the year was small.

4.2

Outlook

The optimal allocation of water resources in canal-well combination irrigation areas is a very complex multi-objective optimization problem. In this study, the optimal allocation of water resources in irrigation areas was carried out by constructing a deep Bayesian network-driven statistical model, and the different canal-well irrigation water use and water use ratios of different crops in canal-well combination irrigation areas in different irrigation seasons were determined, which provided a strong technical support for the sustainable use of water resources in the irrigation areas.

Subsequent studies need to further improve the water cycle simulation model of the combined canal and well irrigation area, such as considering the specific canal water irrigation area and well water irrigation area in each sub-district, and simulating the distribution of specific canals and their leakage more accurately in the irrigation area. As irrigation districts investigate water resource simulation and optimization models, the combined simulation models need to take into account more specific subsurface conditions.

As the amount of water available for agricultural irrigation gradually decreases, water-saving irrigation is more beneficial to the future development of agriculture, and water-saving irrigation should be taken into account in the future research on the optimal allocation of water resources in irrigation districts, and therefore the optimal allocation of water resources in irrigation districts under different water-saving scenarios should be further considered.

In the process of future agricultural development, due to the improvement of living standards and changes in dietary structure, the proportion of facility agriculture in the irrigation area will become larger and larger, so the proportion of facility agriculture should be considered in the study of crop planting structure in the irrigation area. In addition, the ecological environment of the irrigation area is also the focus of future research, in the optimization of water resources in the irrigation area should also be considered in the allocation of water quality-related issues, such as the extraction of groundwater with excessive mineralization and mixing with canal water for irrigation.

Lingua:: Inglese

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Research on Resource Allocation and Water Saving Strategies in Deep Bayesian Network Driven Farmland Irrigation Systems

Shuai Cui

Yiming Gao

Pubblicato online: 19 mar 2025

Ricevuto: 01 nov 2024

Accettato: 04 feb 2025

DOI: https://doi.org/10.2478/amns-2025-0462

Parole chiaveMulti-objective optimization, Agricultural irrigation, Optimal resource allocation, Deep Bayesian statistical modeling

© 2025 Shuai Cui et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Parole chiave
Multi-objective optimization, Agricultural irrigation, Optimal resource allocation, Deep Bayesian statistical modeling