Multi-Objective Planting Structure Optimisation in an Irrigation Area Using a Grey Wolf Optimisation Algorithm

Abstract

To improve agricultural production efficiency, increase farmers’ income, and promote sustainable development, we established a multi-objective optimisation model for crop planting structure in an irrigation area using the grey wolf optimisation (GWO) algorithm to comprehensively consider the resource, economic, and social objectives associated with agriculture. This model was subsequently applied to obtain the optimal planting structure in the southern bank of the Xiaolangdi Reservoir irrigation area in Henan Province, China. The planting areas of wheat, corn, autumn miscellaneous, and economic crops are 30,417; 25,050; 7157; and 1789 hm², respectively. The irrigation water is 8292.66 × 10⁴ m³, output value of crops is 105,721.37 × 10⁴ CNY, and crop yield is 34,280.31 × 10⁴ kg. Different solutions are used to solve the model to evaluate the results, and the order degree entropy method is used to evaluate and compare the results of multiple solutions. The optimisation scheme obtained with this model is consistent with the evaluation results of the cooperative game optimisation scheme, and the relative order degree entropy is 0.136, which is better than that in other schemes. Thus, the optimisation scheme of crop planting structure obtained via GWO comprehensively considers irrigation water consumption, economic benefits, and crop yield, which ensures coordinated development of resource, economic, and social systems and is conducive to promoting the benign development of the whole irrigation area system.

1. Introduction

Water resources are fundamental for maintaining ecological balance on Earth and promoting social development [1]. However, global climate change, population growth, and economic development have caused many regions to face severe challenges in balancing water supply against increasing demand [2]. In China, one of the world’s largest agricultural countries, agricultural water consumption accounts for a large proportion of total water consumption. Therefore, realising the efficient and intensive utilisation of water resources for agricultural irrigation is particularly urgent [3]. To do so, social and economic development, ecological environment protection, and other factors must be considered together; the crop planting structure must be optimised; and limited water and soil resources must be rationally allocated to different crops. In this manner, irrigated areas can be managed to realise comprehensive economic and social benefits and sustainable agricultural development [4].

International research on the optimisation of planting structures began decades ago with the work of Afzeal and Noble [5], who used linear programming to determine the optimal planting area and underground water intake for different crops during different periods. In research since then, Tasuku et al. [6] established a fuzzy optimisation-based model that considers uncertain factors in obtaining economically optimal agricultural planting structures. In addition to pursuing economic benefits, Khepar and Chaturvedi [7] argued that the soil environment should be considered when cultivating crops according to local conditions, and appropriate planting and groundwater modes should be selected to achieve the best net benefits. Furthermore, Montazar and Snyder [8] optimised crop planting structures in water-deficient areas based on a multi-attribute preference model. Birhanu et al. [9] utilised a chance-constrained linear programming model to maximise yield and benefits under limited land and water supply by optimising major crop planting patterns in Ethiopia’s Koga Irrigation District. Nguyen et al. [10] developed a new computationally efficient optimisation framework for crop and water allocation that employed domain knowledge and ant colony optimisation to reduce search space size. El Ghobashy et al. [11] determined the maximum net benefits and land use when intercropping peas and corn, changing corn varieties, and using other planting methods, reporting that intercropping peas and corn particularly improved net benefits and land productivity.

In the 21st century, Chinese scholars have attached considerable importance to coordinating regional, social, economic, and ecological benefits when studying planting structure optimisation and adjustment models. Furthermore, they have treated the optimisation of crop planting structure as a multi-objective problem to promote the sustainable development of agriculture. At present, a variety of optimisation algorithms have been applied to the optimisation of crop planting structures. There are three main methods to solve multi-objective decision-making problems, namely transforming multi objectives into single objectives, the rearrangement order method, and the hierarchical sequence method. In 2003, Chen and Ma [12] used the fuzzy weighting method to determine nonlinear comprehensive benefit coefficients in a multi-objective fuzzy optimisation model established to adjust the crop planting structure in an irrigation area by comprehensively considering economic, social, and ecological benefits; this method effectively solved the problem of highly nonlinear structural adjustments. Zhang and Xu [13] found that as the overall planting area increased in Shenyang, China, grain planting area decreased, and oil planting area increased the fastest. Notably, the innovation in agricultural facilities, rapid increases in cultivation efficiency, and cultivation of agricultural products with local characteristics have presented opportunities for local planting structure adjustment. Hou et al. [14] studied the local planting structure in the arid region of southern Ningxia by analysing the planting area and output value of crops such as grains, vegetables, and fruits, reporting that the benefits of the planting structure significantly improved when it was continuously optimised and adjusted. Guo [15] analysed the “grain-to-feed” process in the primary grain-producing areas of Northeast China based on the “three highs” problem associated with corn cultivation and proposed the optimisation and adjustment of the planting structure from the perspective of ecologically sustainable development by suggesting the interaction and integration of agriculture and animal husbandry. Finally, Qiu et al. [16] constructed a multi-objective structural optimisation model that was constrained by the resource, energy, and agricultural policies in Jilin Province and prioritised the development of low-water, low-energy, and high-yield crops; this approach provided a scientific basis for adjusting the local crop planting structure and corresponding resource allocation. Luo et al. [17] estimated the water-saving potential. They quantified the trade-off between water resources and agricultural production based on a model for planting structure optimisation and the elitist nondominated sorting genetic algorithm. Liu et al. [18] proposed a theoretical framework for comprehensively considering regional crop planting structure optimisation and a full-scale associated-benefits evaluation. The optimisation framework is verified in the upper-middle reaches of the Heihe River basin. Guangyao et al. [19] considered the minimum water shortage of agricultural irrigation and maximum economic benefit of crops as the objective function and ecological security constraints of the irrigation area to construct a multi-objective and multi-water source optimal allocation model based on planting structure optimisation. They further used the self-improved elite strategy and co-evolutionary genetic algorithm to solve the model and obtained the optimal allocation of water resources under different scenarios. Chunning et al. [20] selected the maximum economic benefit, comprehensive water productivity, and ecological benefit of crops as the goal. They used the multi-objective optimisation model to adjust the planting structure of crops in different years in Lijin County.

With in-depth research, various optimisation algorithms are constantly being improved to improve optimisation performance and reduce time costs. The grey wolf optimisation (GWO) algorithm is a swarm intelligence optimisation algorithm proposed by Mirjalili et al. [21]. It has strong convergence performance, a simple structure, fewer parameters to be adjusted, and is easy to implement. Convergence factors and information feedback mechanisms can be adaptively adjusted in the algorithm, achieving a balance between local optimisation and global search. Therefore, it performs well in terms of solution accuracy and convergence speed.

Given the advantages of the GWO, it is widely used in engineering optimisation, energy management [22], traffic system optimisation [23], scheduling system optimisation [24], and other fields. These applications demonstrate the flexibility and effectiveness of the GWO in solving practical problems. It suits various scenarios requiring complex decision-making and optimisation, especially when dealing with multivariate and multi-objective problems. However, existing studies have shown that the algorithm is less applied when optimising crop planting structures. Therefore, this paper attempts to apply this method to the multi-objective optimisation of crop planting structures. Based on the GWO, this paper optimises and adjusts the crop planting structure and solves the best decision variables and the best demand response scheme [25].

The south bank irrigation area of Xiaolangdi is one of the 172 national major water-saving and water supply projects determined by the State Council of the People’s Republic of China. Its agricultural planting aims to improve food production capacity and ensure food security. The irrigation area is in the Beimangling area of Luoyang City, Henan Province, China, bounded by the Yellow River to the north and the Luohe River to the south; this is a critical commodity grain production area in the province, but natural conditions restrict the water supply here and are susceptible to severe drought and shortages. Indeed, precipitation is scarce, and the available surface water is limited. Consequently, industrial, agricultural, and domestic water is primarily drawn from groundwater and the Xiaolangdi Reservoir on the Yellow River. These conditions indicate that the use of water resources for agriculture in this area must be minimised while maximising the social and economic benefits.

Agricultural products grown in the target irrigation area primarily comprise wheat, corn, miscellaneous autumn crops, and cash crops with a designed planting ratio of 0.8:0.6:0.3:0.1. The controlled cultivated land area is 50,672 hm², and the designed irrigation area is 35,804 hm². The multiple cropping index is 1.8, and the total annual water supply capacity to the irrigation area is 8293 × 10⁴ m³.

To realise high-quality development of irrigated agricultural areas, preserve water resources, and realise the sustainable development of agriculture, this study comprehensively considered the resource, economic, and social objectives in the study area to establish a multi-objective crop planting structure optimisation model using an improved GWO to efficiently obtain a solution. The efficacy of the proposed model was subsequently demonstrated through application to optimise a crop planting structure, and the relative order degree entropies of the results were compared with those obtained using other optimisation algorithms. The model integrates the needs of society, the economy, and resources in the irrigation area, and the grey wolf optimisation algorithm has good global search ability. It can quickly search for the optimal solution. The obtained crop planting structure optimisation scheme can be the basis for decision-makers.

2. Materials and Methods

2.1. Multi-Objective Crop Planting Structure Optimisation Model

The optimisation model was constructed using the crop planting area as the decision variable, considering the irrigation resource planning and economic and social objectives of the planting industry. Furthermore, each crop’s irrigation area, water supply capacity, and the minimum planting area proportion were used as constraints.

2.1.1. Objective Functions

(1)

Resource Objective

The objective functions employed in the proposed model comprised resource, economic, and social objectives. The resource objective was to minimise water consumption in the irrigated area and was defined as follows:

$$\min f_1={\displaystyle {\sum }_{j=1}^n{X}_{Iaj}}\times W_{Iqj},$$

(1)

$X_{Iaj}$ is the irrigated area of crop j, and $W_{Iqj}$ is the irrigation quota of crop j.

(2)

Economic Objective

The economic objective was to maximise the net output value of crops, defined as the difference between the input and output costs, and reflects the degree of resource utilisation, production efficiency, and farmer income; it was defined as follows:

$$\max f_2={\displaystyle {\sum }_{j=1}^n{X}_{Iaj}}\times Y_j\times N_j,$$

(2)

$Y_j$ is the yield per unit area of crop j, and $N_j$ is the net income of crop j.

(3)

Social Objective

The social objective was to maximise the crop yield, which reflects the production of food for society, and was defined as follows:

$$\max f_3={\displaystyle {\sum }_{j=1}^n{X}_{Iaj}}\times Y_j.$$

(3)

2.1.2. Constraint Functions

The constraints employed in the proposed model comprised restrictions on the crop planting area, irrigation area, and water resources, as well as a non-negative constraint. The available crop planting area is limited by regional land planning. This constraint was defined to adhere to the most stringent farmland protection system to ensure the implementation of basic farmland protection objectives and was defined as follows:

$$p_{j\max }\times X_{IA}\le {\displaystyle {\sum }_{j=1}^n{X}_{Iaj}\le p_{j\min }}\times X_{IA},$$

(4)

where $X_{IA}$ is the total crop planting area, $P_{jmin}$ is the minimum crop planting ratio, and $P_{jmax}$ is the maximum crop planting ratio.

The irrigation area constraint requires that the total irrigated area for all crops be less than the available irrigated area as follows:

$${\displaystyle {\sum }_{j=1}^A{X}_{Iaj}\le I_M\times A_I},$$

(5)

where $I_M$ is the multiple crop index, and $A_I$ is the total irrigated area.

The water resources management constraint requires that the irrigation water used be less than the total planned water supply of the irrigation area as follows:

$${\displaystyle {\sum }_{j=1}^n{X}_{Iaj}}\times W_{Iqj}\le W_{ST},$$

(6)

where $W_{ST}$ is the total planned water supply for irrigation.

Finally, the non-negative constraint was expressed as follows:

$$X_{Iaj}\ge 0.$$

(7)

2.2. Application of Grey Wolf Optimisation Algorithm

The GWO algorithm is an optimisation algorithm inspired by the behaviour of grey wolves in nature. It simulates the social hierarchy and hunting strategy of grey wolves. Grey wolves are divided into four roles: wolf leader (Alpha), wolf deputy (Beta), counsellor (Delta), and working wolf (Omega) [26].

This hierarchical structure helps grey wolves efficiently organise hunting and resource allocation. In the algorithm, the Alpha represents the optimal solution, while Beta and Delta represent the suboptimal and the third optimal solutions. Omega grey wolves follow the guidance of these leaders to explore the search space. The operation process of the algorithm is like the grey wolf group’s cooperative hunting, which gradually approaches the optimal solution by tracking, encircling, and attacking the prey [21].

The optimisation process of the GWO algorithm includes steps such as social hierarchy stratification, tracking, encircling, and attacking prey. Firstly, by constructing the social hierarchy model of grey wolves, the grey wolves of α, β, δ, and ω levels are determined. Then, the algorithm finds the optimal solution to the problem by simulating the predation behaviour of grey wolves, such as encircling prey, hunting prey, and attacking prey. In this process, the α wolf guides other levels of grey wolves to the optimal solution through its social hierarchy leadership to achieve global optimisation. In addition, the GWO algorithm also uses basic GWO to calculate the fitness value of each grey wolf. It uses the niche radius as a limit to compare the fitness values of the grey wolf individuals. The global search ability is improved by imposing a penalty function on the grey wolf individuals with poor fitness values. This algorithm not only simulates the predatory behaviour of nature but also integrates the idea of swarm intelligence, which makes it perform well in solving optimisation problems.

The crop planting structure was optimised in this study using the GWO algorithm to determine the planting structure variables according to the following process [27]. The calculation formula comes from the reference [21].

• Input the objective function parameter values for $W_{Iqj}$, $N_j$, and $Y_j$ as well as each constraint function value according to the water resource conditions in the irrigation area;
• Set the wolf population size N, search space dimension D, and maximum number of iterations T. Control the initial and termination values of the convergence factor a and the output $X_{\alpha }$;
• Initialise the wolf population $\left\{\overrightarrow{X_i, }i=1, 2, \dots , N\right\}$, and calculate the parameters A, C, and a as follows:

  $$A=2a\times r_1-a,$$

  (8)

  $$C=2r_2,$$

  (9)

  $$a=2\times (1-t/T)$$

  (10)

  where A and C are coefficient vectors, r₁ and r₂ are random vectors between 0 and 1, and a is defined using a linearly decreasing variation law and can be calculated as $a=2\times (1-t/T)$;
• Calculate the objective value for each individual in the population and use the multi-objective model to calculate the corresponding planting structure when each objective is optimal. Apply this objective value as the fitness of the individual wolf $\left\{f\left(X_i\right), i=1, 2, \dots , N\right\}$, sort the fitness values for all wolves, then record those of the top three individuals as $\alpha$, $\beta$, $\delta$ and their corresponding locations as $X_{\alpha }$, $X_{\beta }$, and $X_{\delta }$, which guide the wolves to move towards the grey.
• Update the parameters a, A, and C and recalculate the locations of the top three wolves according to

  $$D_{\alpha }=\left|C_1\times X_{\alpha }(t)-X_i(t)\right|,$$

  (11)

  $$D_{\beta }=\left|C_2\times X_{\beta }(t)-X_i(t)\right|,$$

  (12)

  $$D_{\delta }=\left|C_3\times X_{\delta }(t)-X_i(t)\right|,$$

  (13)
  And update the location vector of each wolf according to

  $$X_1(t)=X_{\alpha }(t)-A_1\times D_{\alpha },$$

  (14)

  $$X_2(t)=X_{\beta }(t)-A_2\times D_{\beta },$$

  (15)

  $$X_3(t)=X_{\delta }(t)-A_3\times D_{\delta }.$$

  (16)

  $$X(t+1)={\displaystyle \frac{X_1(t)+X_2(t)+X_3(t)}{3}},$$

  (17)

  where $X_i\left(t\right)$ is the position vector of the current grey wolf, and D is the distance between grey wolves and prey;
• When the maximum number of iterations T is reached, select the fittest individual as the optimal solution, ending the algorithm. The fittest individual is the optimal scheme of the crop planting structure model. Otherwise, update the parameters and location vector of the wolves, and return to step 4;
• Output $X(t+1)$.

2.3. Comparison of Optimal Crop Planting Structures Based on Relative Order Degree Entropy

The proposed GWO algorithm-based multi-objective optimisation model and the rationality of its results were evaluated in this study by calculating the change in relative order degree entropy of the obtained scheme and comparing it with those obtained using other optimisation algorithms.

The relative order degree entropy is an index used to measure and compare the degree of order in different systems or processes. It is a kind of relative entropy which can be used to compare the orderliness or randomness of the states of two systems to evaluate their intrinsic quality or the complexity of the system.

The primary steps of this process are as follows [28]:

• According to the planting characteristics, define the resource, economic, and social objective functions described in Equations (1)–(3) as subsystems and the total amount of irrigation water, total output value of the agricultural industry, and crop yield as the corresponding order parameters;
• Conduct dimensionless processing of the order parameters to eliminate the influence of their dimensions on the results as follows:
  $$\mathrm{When} \mathrm{the} \mathrm{larger} \mathrm{the} \mathrm{index} \mathrm{is}, \mathrm{the} \mathrm{better}, r_{cd}={\displaystyle \frac{y_{cd}-y_{cd\min }}{y_{cd\max }-y_{cd\min }}},$$

  (18)

  $$\mathrm{When} \mathrm{the} \mathrm{smaller} \mathrm{the} \mathrm{index} \mathrm{is}, \mathrm{the} \mathrm{better}, r_{cd}={\displaystyle \frac{y_{cd\max }-y_{cd}}{y_{cd\max }-y_{cd\min }}},$$

  (19)

  where $r_{cd}$ is the relative membership of the dth order parameter of subsystem c, $y_{cd}$ is the dth order parameter value of subsystem c, and $y_{cdmax}$ and $y_{cdmin}$ are the upper and lower limits of the dth order parameter of subsystem c, respectively.
• Determine the relative order of the subsystem $u_c$ as follows:

  $$u_c={\displaystyle \frac{1}{1+\frac{{\displaystyle {\sum }_{d=1}^m{\left[w_{cd}(1-r_{cd})\right]}^2}}{{\displaystyle {\sum }_{d=1}^m{(w_{cd}{r}_{cd})}^2}}}},$$

  (20)

  where $u_c\in \left[\mathrm{0,1}\right]$, m is the total number of order parameters in subsystem c, and $w_{cd}$ is the weight coefficient of the dth order parameter of subsystem c, reflecting the role of the jth order parameter in maintaining the orderly operation of subsystem c.
• Calculate the entropy $E\left(p\right)$ of the pth scheme as follows:

  $$E(p)=-k{\displaystyle {\sum }_{c=1}^n{u}_c\ln u_c},$$

  (21)

  where $k$ is a constant related to the sample size; generally, k = 1. If $E\left(p+1\right)>E\left(p\right)$, the system entropy increases, as does its disorder, and the crop planting structure is unstable, indicating that the agricultural system devolves in a vicious cycle and that scheme $p+1$ is unreasonable; if $E\left(p+1\right)<E\left(p\right)$, scheme $p+1$ is reasonable; if $E\left(p+1\right)=E\left(p\right)$, the system entropy does not change within the considered time interval and the crop planting structure is in a fixed state.

3. Results and Discussion

3.1. Case Study

We apply the proposed model to the Xiaolangdi Reservoir irrigation area in the South Bank in Henan Province, China. The contradiction between the supply and demand in the Xiaolangdi Reservoir irrigation area in the South Bank is prominent, and water resource allocation conflicts have become a serious problem.

3.2. Selection of Model Parameters

The multi-objective crop planting structure optimisation model was established by taking the area of wheat ($x_1$), corn ($x_2$), miscellaneous autumn crops ($x_3$), and cash crops ($x_4$) in the irrigated area as the decision variables and the model parameters from Wu et al. [29] as listed in

Table 1. Model parameters.

Metric	Wheat	Corn	Miscellaneous Autumn Crops	Cash Crops
Irrigation norm (m3/hm2)	1650	975	750	1650
Economic benefits (CNY/hm2)	18,200	16,000	10,200	16,700
Social benefits (kg/hm2)	6000	5500	2300	3400

.

Because wheat and corn are the primary crops in the irrigated area, the designed proportion of wheat, corn, miscellaneous autumn crops, and cash crops in the irrigated area must be between 0.7:0.4:0.2:0.05 and 0.9:0.7:0.4:0.2. Given a total designed area of 35,787 hm², considering the multiple cropping index of 1.8, the total irrigation area was 64,416 hm². Finally, the volume of irrigation water consumed must be less than 8293 × 10⁴ m³.

3.3. Model Solution

MATLAB R2016b was used to compile the GWO functions, read the model parameters, and optimise them to obtain a solution.

First, the parameters in the algorithm were initialised and the maximum number of iterations was set to T = 200. The wolf population size was set to N = 20, 30,..., 100 to determine the effects of wolf population on the results, and leading wolves α, β, δ and the initial location of the pack were determined accordingly.

Next, the crop-designed proportion constraint was defined as the search range, and the irrigation area and water supply constraints were considered penalty functions to construct the fitness function. This function was defined and employed to conduct a global search while updating the locations of the three lead wolves using the convergence factor as follows:

$$\max F(x)={\displaystyle \frac{f_{1\min }(x^{*})}{f_1(x)}}\times {\displaystyle \frac{f_2(x)}{f_{2\max }(x^{*})}}\times {\displaystyle \frac{f_3(x)}{f_{3\max }(x^{*})}},$$

(22)

where $f_{1min}\left(x^{*}\right)$, $f_{2max}\left(x^{*}\right)$, and $f_{3max}\left(x^{*}\right)$ are the single-objective optimal values shown in

Table 2. Single-objective optimisation values of crop planting structure optimisation model.

Objective Function	Water Consumption (Single)	Economic Objective (Single)	Social Objective (Single)
f1(x)/104 m3	7770
f2(x)/104 CNY		30,694
f3(x)/104 kg			106,500

.

An optimal candidate solution $X_i(t+1)$ was subsequently calculated by combining the adaptive weight values of the top three wolves. Figure 1 shows the optimal planting area for each crop when optimised for each single objective; the optimal solution $f\left(x\right)$ for the entire model was applied to obtain the optimal solution for the high-efficiency, high-yield, and low-consumption objectives as shown in Table 2.

The location of each wolf was subsequently updated, another candidate solution was obtained through a global search, and the fitness values of the previous and current solutions were compared to select the optimal solution. The optimal planting area of crops in the irrigated area was subsequently obtained through iteration.

Finally, the maximum fitness values of the iteration curves for different wolf population sizes were plotted as shown in Figure 2, which indicates that when the wolf population was less than 50, the fitness of the local optimal solution was limited. By contrast, when the grey wolf population was greater than 70, there was little difference between the fitness values of the obtained optimal solutions, and the results are shown in Figure 3. The optimal solution for this model was determined to be 30,417; 25,050; 7157; and 1789 hm² for wheat, corn, miscellaneous autumn, and cash crops, respectively. The optimal results obtained using the proposed method are compared with those obtained using other optimisation algorithms in

Table 3. Optimal crop planting structures obtained using different optimisation algorithms.

Algorithm	Crop Planting Area				Objective Function
	${\mathit{x}}_1$ (104 m3)	${\mathit{x}}_2$ (104 m3)	${\mathit{x}}_3$ (104 m3)	${\mathit{x}}_4$ (104 m3)	${\mathit{f}}_1\left(\mathit{x}\right)$ (104 m3)	${\mathit{f}}_2\left(\mathit{x}\right)$ (104 CNY)	${\mathit{f}}_3\left(\mathit{x}\right)$ (104 kg)
GWO	30,414	25,050	7157	1789	8292.66	105,721.37	34,280.31
cooperative game	30,411	25,050	7157	1789	8292.16	105,715.89	34,278.50
Competition game	30,365	25,050	7157	1789	8284.60	105,632.49	34,251.01
Fuzzy multi-target	30,349	25,050	7157	1789	8281.92	105,602.95	34,241.27
Original design scheme	28,629	21,472	10,736	3579	8213.04	103,387.68	32,673.23
Resources (single)	25,050	23,261	14,315	1790	7770.17	100,399.20	31,724.60
Economy (single)	32,208	17,894	7157	7157	8276.67	106,501.29	33,245.99
Society (single)	26,984	15,960	14,315	7157	8262.99	101,200.37	34,694.23

.

To compare the results of the GWO algorithm with other methods, the cooperative game method [29], competitive game method [30], and fuzzy optimisation method [31] are used to solve the above multi-objective model, respectively. The calculation results and value of each objective function are shown in Table 3. To intuitively compare the rationality and effectiveness of the solution results of each method, the irrigation area design scheme and the schemes obtained with each single objective are also listed in Table 3.

The calculation results show that the difference between the planting area of each crop solved with the GWO method and the cooperative game method, the competitive game method, and the fuzzy optimisation method is small. Only the planting area of wheat is slightly different, indicating that the global optimisation performance of the above four methods is good. Still, the value of each objective function is different. To objectively evaluate the effectiveness of the GWO method, the order degree entropy values E(p) of the above schemes are evaluated.

3.4. Evaluation of Results

Selecting the optimal function value obtained with each algorithm model as the order parameter, the single- and multiple-objective optimal planting structures presented in Table 2 and Table 3, respectively, were compared by applying the maximum and minimum value of the dimensionless values in Equations (18) and (19) to determine the relative membership of the order parameter. Considering the weight of each objective in the model $\omega =\left[0.333, 0.333, 0.334\right]$, the relative order u_i of each subsystem for each scheme was calculated using Equation (20), and Equation (21) was applied to calculate the relative order degree entropy E(p) of each scheme with the results shown in

Table 4. Evaluation of optimal crop structure schemes.

Scheme Set	Subsystem Relative Membership			${\mathit{u}}_{\mathit{i}}$	$\mathit{E}($$\mathit{p})$
	Resource Subsystem	Economic Subsystem	Social Subsystem
Grey wolf algorithm	0.872	0.481	0.861	0.853	0.136
Cooperative game	0.871	0.481	0.860	0.853	0.136
Competition game	0.858	0.489	0.851	0.851	0.138
Fuzzy multi-target	0.853	0.492	0.847	0.850	0.138
Original design scheme	0.490	0.560	0.319	0.436	0.362
Resources (single)	0.000	1.000	0.000	0.329	0.366
Economy (single)	1.000	0.000	0.512	0.545	0.331
Society (single)	0.131	0.510	1.000	0.489	0.350

.

Comparing the evaluation results in Table 4, the entropy values for the single-optimisation models are relatively large, indicating that these planting structures are unreasonable. In contrast, the GWO and competitive game algorithms exhibit the same low relative order degree entropy; this indicates that the planting structures determined using these algorithms provide reasonable crop yield, economic benefits, and irrigation water consumption. According to the calculation results in Table 3, when the planting area, agricultural planting planning, and water supply conditions of the irrigation area are limited, the minimum irrigation water consumption, the maximum net income of crops, and the maximum grain yield of crops are taken as the objectives. The optimal planting area of wheat, corn, autumn miscellaneous, and economic crops in the irrigation area is 30,414; 25,050; 7157; and 1789 hm², and the optimal planting ratio is 0.85:0.7:0.2:0.05. Indeed, the proposed algorithm provided a water consumption of 82,926,600 m³, a net income of CNY 1,057,213,700, and a total crop output of 342,803,100 kg. Compared with the original design scheme of the irrigation area, the irrigation water amount of the crop planting structure increased by 0.97%, the net income of the crop increased by 2.26%, and the total grain yield increased by 4.92%. Although the water consumption of this planting ratio is slightly higher than that of the original design scheme, the planting ratio meets the requirements of the upper and lower limits of the crop planting ratio in the development plan of the irrigation area, and the water consumption and planting area are within the constraints.

The evaluation results E(p) show that the E(p) of the GWO is the same as that of the cooperative game method, which is smaller than that of other schemes and far less than the E(p) of the original design scheme of the irrigation area. Therefore, the proposed GWO algorithm-based multi-objective optimisation model provided the coordinated consideration of resource, economic, and social systems to promote the sustainable development of the entire irrigation area.

4. Conclusions

The optimisation of crop planting structure is a complex systematic project that requires the comprehensive consideration of factors such as the regional resources, economy, and society. Therefore, a multi-objective model and optimisation method based on the GWO algorithm was established in this study to reasonably plan the planting structure for an irrigation area. The proposed method was subsequently applied to a target irrigation area to demonstrate its efficacy compared to other optimisation methods. The following conclusions were drawn from the results of this study:

• The proposed model integrates resource, economic, and social objectives to optimise the planting structure in a target irrigation area. The resulting structure not only meets the needs of crop production in the irrigation area but also meets those of farmers’ income and water conservation;
• The GWO algorithm exhibited an excellent global search ability when solving the multi-objective model. The solution’s fitness reached its maximum value when the population of wolves in the algorithm was greater than 70, and an optimal solution was obtained. The results indicated a water consumption of 82,926,600 m³, a net income of 1,057,213,700 CNY, and a total crop output of 342,803,100 kg;
• The relative order degree entropies of the results obtained using different optimisation algorithms were compared to evaluate the solution provided via the proposed method. The results indicate that solving the multi-objective optimisation model using the GWO can effectively optimise the crop structure in an irrigation area by coordinating the requirements of resource, economic, and social systems to promote the sustainable development of the entire irrigation area;
• The parameters in the multi-objective model will change with different irrigation areas. Therefore, it is necessary to determine the model parameters according to the specific conditions of the irrigation area to solve the optimal planting structure for decision-makers to reference;
• In this study, the model is only applied to the south bank irrigation area of the Yellow River Xiaolangdi, and it needs to be tested in different irrigation areas in the future to verify its effectiveness and applicability;
• With the development of the irrigation area, the requirements of the ecological environment will be further improved. The ecological environment and water demand in the irrigation area can be considered as the research goal in the future.