1. Introduction
In the context of global agricultural development, arable land is a crucial element of agricultural production [
1] and is critical to agricultural systems [
2,
3]. In China, arable land is the foundation of agricultural production and a vital source of income for farmers. Moreover, arable land acts as a key link in the implementation of rural revitalization strategies.
Since the onset of global modernization, particularly after China’s reform and opening-up policy, rapid urbanization and industrialization have led to substantial economic growth. This phenomenon has changed consumer behavior in China, shifting the demand for agricultural products from a singular focus to a diversified and refined spectrum [
4]. This evolution has driven the continuous transformation of crop-planting structures.
Given this context, and rooted in China’s national conditions and agricultural development strategy, optimizing crop planting structures is a critical task for rural agricultural advancement. This endeavor has profound implications for achieving agricultural modernization, ensuring food security, and promoting sustainable rural development.
Thus, multidimensional studies have investigated the structure of crop planting at various levels (e.g., country [
5], state [
6], county [
7]). Furthermore, remote sensing technology has been widely used for data acquisition because of its efficiency and cost-effectiveness [
8], enabling researchers to conduct in-depth analyses related to the spatiotemporal evolution and patterns of planting areas and structural types. Various analytical methods [
9,
10] have emerged to extract and analyze crop planting structure-related data from multiple perspectives.
Agricultural land use allocation is a core component of crop cultivation. Many agricultural regions have diverse topographies, geomorphologies [
11], and varying soil characteristics. Additionally, there is a wide array of crop types [
12] including grains, cash crops, fruits, and vegetables. This diversity is further complicated by various planting patterns (e.g., crop rotation [
13] and intercropping [
14]). In this complex context, the rational selection and planning of crop planting layouts present substantial challenges. This key problem in agricultural management science has both important theoretical implications and practical difficulties [
15].
Kar et al. [
16] used remote sensing and geographic information system technologies and found that rice, mung beans, and other crops were suitable for cultivation in irrigated areas and that low-irrigation zones were appropriate for initially planting rice and short-cycle vegetables, followed by legumes. Borah et al. [
17] found that Karbi farmers preferred to cultivate rice, maize, and vegetables in highly water-retentive black clay soil. Other studies [
18,
19] found that cover crop mixtures can provide nitrogen nutrients and that diverse crop rotations contribute to soil health, promoting sustainable agricultural development.
These studies on crop cultivation and land use offer valuable insights but have limitations. First, the focus has been on specific aspects (e.g., terrain morphology, soil characteristics, and cropping structures), without adequate investigation of the diversity of crop types. Second, most of the research has concentrated on macro-regional scales, neglecting the multiple interwoven constraints of individual plots in actual agricultural production contexts.
Currently, uncertainty modeling methods have demonstrated a broad applicability in interdisciplinary research. For example, Yang et al. [
20] proposed a modeling framework based on dynamic fault thresholds in mechanical system maintenance optimization, which constructs a life distribution model by incorporating multiple uncertainties. Chen et al. [
21] conducted a systematic review of uncertainty analysis in supply chain management and pointed out that the combination of abstract modeling and optimization techniques was a common approach to address complex dynamic environments. Han et al. [
22] developed a spatial uncertainty-aware method that combined deep learning and graph representation in geotechnical engineering, quantifying the spatial distribution and topological structure uncertainty of fault networks. Muñoz et al. [
23] proposed a cascaded uncertainty quantification method coupled with process models and machine learning for compound flood disasters, effectively analyzing the cascading effects of multiple risk factors. These interdisciplinary contributions provide methodological references for uncertainty research in agricultural systems.
In the field of agricultural planning, researchers have begun to integrate uncertainty analysis to improve the robustness of decision-making. Among them, Alemany et al. [
24] planned tomato planting and harvesting within a multifarmer supply chain from a sustainability perspective, considering uncertain conditions. Borah et al. [
25] used four-dimensional single-valued neutrosophic sets to study the risks associated with mustard and rice cultivation in Assam. Liu et al. [
26] used imprecise fuzzy chance-constrained programming to plan water–food relationship systems in Jinan. However, existing studies have primarily focused on traditional uncertainty sources such as climate and production, while paying insufficient attention to market dimension uncertainty. As a critical disturbance factor in agricultural systems, market uncertainty manifests itself through fluctuations in key parameters such as sales volume [
27], selling price [
28], planting cost [
29], and yield per mu [
30]. Given the time-varying nature of market variables, constructing their exact probability distributions has often proved challenging.
To address the uncertainty quantification problem, existing studies have generally adopted three methods: stochastic, fuzzy, and interval [
31]. Stochastic uncertainty relies on probability distribution assumptions but is prone to modeling bias under limited data conditions. Fuzzy uncertainty effectively handles subjective perception differences but struggles to precisely characterize objective variable value fluctuations. In contrast, interval uncertainty can intuitively describe the market variable change boundaries by defining the parameter threshold ranges, requiring less information [
32]. However, the current literature on the interval uncertainty modeling of core market parameters remains limited, and further exploration of the related methods’ practical applications in agricultural decision-making is necessary.
In agriculture, the investigation of a wide range of uncertainty factors has led to the construction of effective objective planning models combined with intelligent algorithms. The integration of intelligent algorithms (e.g., genetic algorithms [
33,
34], simulated annealing [
35,
36], particle swarm optimization [
37,
38], and ant colony optimization [
39,
40]) with objective planning techniques has provided a new means of solving crop planting optimization problems. For instance, Wu et al. [
41] used game-theory algorithms to balance the objective function requirements to achieve an optimal crop-planting structure. Liu et al. [
42] used wavelet optimization algorithms to develop a crop cultivation planning model to maximize the expected total returns. Zheng et al. [
43] constructed a hybrid multiobjective firework optimization algorithm that targeted fertilization problems related to oilseed crops.
However, when random factors are considered simultaneously, the complexity of these problems increases significantly because of the unpredictability and dynamic nature of these random elements. Existing algorithms may have difficulty producing ideal solutions because of their limitations (e.g., inaccurate prediction and optimization results), deteriorating convergence properties, and increased data uncertainty.
In summary, despite progress in crop planting optimization, research on the planning of crop cultivation that has comprehensively considered specific plot constraints and uncertainty factors remains scarce. Notably, these two categories of factors are closely related: limiting factors (e.g., topography and soil characteristics) define suitable areas for crop cultivation, and uncertainty factors (e.g., market fluctuations) have a profound impact on the growth cycle of crops and their sales conditions.
In this study, we aimed to maximize the agricultural profits. We comprehensively considered multiple uncertainties, constructed the corresponding objective function, and established constraints based on both land constraints and uncertainty variables. To effectively solve this complex problem, we designed a decomposition-coordination algorithm architecture. This architecture decomposes the overall planning model into multiple submodels through the divide-and-conquer method and uses a greedy algorithm to solve each submodel, ultimately constructing a decomposition-based stochastic multilevel binary optimization model. To verify the performance of the model, we set up two benchmark models for comparative analysis: benchmark model 1 [
44] only considered land constraints, and benchmark model 2 [
45] only dealt with uncertainty factors. Empirical studies of different years and income scenarios have shown that our model is significantly superior to these two single-factor benchmark models in terms of optimization performance. This finding supports the necessity of the joint modeling of land constraints and uncertainty.
The rest of this paper is structured as follows.
Section 2 introduces the mathematical model and the solution method.
Section 3 validates the model through an application in the mountainous area of North China.
Section 4 reports the results, analyzes the profitability and planning of crop cultivation in rural areas, and evaluates the optimization of the cultivation model.
Section 5 of the system conducts a sensitivity analysis to explore the stability of the model by quantifying fluctuations in multiple key variables.
Section 6 establishes two types of benchmark models as a comparative reference to comprehensively evaluate the model’s performance from different dimensions.
Section 7 summarizes the paper and outlines future research directions.
2. Model and Methods
This section comprises five components: the definition of variable symbols, construction of the objective function, construction of decision constraints, and the introduction of algorithms.
2.1. Definition of Variable Symbols
We introduced a series of symbols to denote the relevant elements and their characteristics. We let denote the index for the abstract land parcel set that characterizes different cultivable area units, denote the index for crop species, and denote the index for the planning periods.
The decision variable was defined as . It takes the value of 1 when crop is planted on plot during period , and 0 otherwise. The variable denotes the area allocated for planting crop on plot during period . denotes the total area of plot .
Given that random uncertainties are difficult to quantify precisely and fuzzy uncertainty depends on subjective perceptions, we used the interval uncertainty method to deal with market uncertainty. Specifically, we let the expected sales volume, yield per mu, planting cost, and selling price of agricultural products be the random variables , , , and , respectively. Their initial values were denoted , , , and . The annual growth rates of these variables were denoted as , , , and . In order to quantify the trend of the each variable, we used a uniform distribution to describe the annual rate of change of each variable and limit its fluctuation range, which was , , , and , where , , , and .
2.2. Construction of the Objective Function
In the construction of the objective function, we took maximizing the agricultural income as the core objective and comprehensively considered the impact of various uncertain factors. In the initial period, denoted by
, the yield per mu of crop
during period
was denoted as
. Expected sales volume was denoted by
. Actual sales volume was denoted by
. For a more concise representation, the actual sales volume was denoted by
. Selling price was denoted by
. Planting cost was referred to as
. Therefore, the formula for calculating the model return was as follows:
From the above formula, assuming a specific probabilistic scenario, , we let denote the realization of under this scenario (with analogous definitions for other variables). For land parcel and time period , the crop yield was influenced by factors (e.g., sales volume, yield per mu, planting cost, selling price). At time , an increase in expected sales volume contributed positively to profits. However, when both and are significantly large, even with rising prices, the overall profits may decline due to a substantial reduction in yield per mu. Additionally, if indicates an increase in planting cost, it negatively affects the profits. When is present and its magnitude is significant, the expected returns from cultivating crops tend to increase, influencing the decisions made during the optimization process .
2.3. Construction of Decision Constraints
Based on the land attributes and crop growth characteristics and taking into account the plot constraints and uncertain variables, we constructed the following six constraints.
The total area of the crops planted in a given plot should not exceed the overall area of the plot. The mathematical expression for this concept is as follows:
- 2.
Restrictions on repeated cropping
Monoculture, planting the same crop continuously on the same piece of land, can lead to imbalances in soil nutrients and outbreaks of pests and diseases, affecting the crop yield and quality. Mitigating these adverse effects requires restrictions on monoculture practices. The mathematical expression for this concept is as follows:
- 3.
Crop rotation association constraints
Crop rotation is a scientifically based agricultural method. Different crops have varying patterns of nutrient absorption and utilization in the soil, and crop rotation can help maintain balanced soil fertility. The mathematical expression for this concept is as follows:
In this context, denotes the minimum number of times is cultivated in the plot during the period .
- 4.
Constraint on planting dispersion
The degree of crop planting dispersion refers to the extent to which crops are distributed across plots. Excessive planting can significantly increase the ecological pressure on land, and overly dispersed planting may increase the management costs. Thus, to control this dispersion, regulating the number of plots designated for crop cultivation is essential. The mathematical expression for this concept is as follows:
In this context, denotes the maximum number of plots for the crop cultivation of crop , with a value range defined as .
- 5.
Constraints on the threshold of cultivated area for crops in the plot
The planting area of agricultural crops in a single plot is often too small to achieve economies of scale, and excessively large areas may exceed the management capabilities. These phenomena can lead to poor management practices, affecting the yield. The mathematical expression of this relationship is as follows:
where
and
denote the minimum and maximum values of the threshold coefficients for the crop
planting area in plot
, respectively.
- 6.
Uncertain variable range of change
Since the annual growth rates of the uncertain variables, namely the expected sales volume, yield per mu, planting cost, and selling price, follow a uniform distribution within certain ranges, these relationships can be set as constraint conditions. The specific expressions are as follows:
2.4. Divide-and-Conquer Decomposition with Coordinated Land—Crop Dimensionality
2.4.1. Divide-and-Conquer Approach
The divide-and-conquer approach is an effective granularity computation approach centered on repartitioning the attribute space into multiple subsystem decision tables. This technique enables the decomposition of large datasets into smaller subsets, a characteristic that facilitates parallel processing, thereby significantly improving the computational efficiency. In large-scale optimization problems, the divide-and-conquer method has significant advantages [
46]. For instance, Zhang et al. [
47] applied it to solve a large-scale capacitated arc-routing problem via a route-cutting-off operator, boosting the decomposition efficiency. Ren et al. [
48] formulated a bilevel cooperative coevolution algorithm. This algorithm initializes populations using elite subspace solutions, thereby improving the optimization results. Ji et al. [
49] proposed the AutoLifter synthesis tool to streamline divide-and-conquer automation, resolving scalability challenges in program synthesis.
These case studies demonstrate that the divide-and-conquer approach holds significant application value in solving large-scale problems across various domains. By decomposing complex problems into multiple subsystem problems, it enhances the overall problem-solving efficiency and effectiveness, enabling subproblems to be addressed more efficiently.
2.4.2. Criteria for Divide-and-Conquer Approach
In agricultural production, due to the need to consider many complex conditions, we chose two criteria for model division: land type and crop type, in order to optimize more effectively. The reasons are as follows:
In terms of land, agricultural land varies significantly in soil quality, irrigation conditions, and geographical location. Fertile soil is suitable for high-yield crops such as corn and soybeans, while poor soil is more suitable for hardy crops such as sweet potatoes and buckwheat. Irrigation conditions are also crucial. Well-irrigated lands are suitable for crops that require a lot of water, such as rice and lotus root, while land with limited water sources is more suitable for drought-resistant crops such as sorghum and sunflowers. Location is also a key factor: plots close to markets can prioritize perishable produce such as strawberries and grapes to reduce transportation losses, while remote plots are better suited for storage crops such as wheat and corn. In addition, the complementarity of crops can also bring benefits; for example, legumes can improve the soil fertility through nitrogen fixation. By implementing a reasonable crop rotation strategy, not only can fertilizer use be reduced, but sustainable land use can also be promoted. Therefore, the characteristics of the land are an important dimension of the agricultural system that cannot be ignored, directly affecting crop planting planning and resource allocation efficiency.
In terms of the crop, the characteristics of different crops vary significantly, and they have distinct requirements for conditions such as land type, light, temperature, and humidity. For example, rice, which thrives in water, is suitable for cultivation in plots with good irrigation conditions, while edible fungi rely on the stable environment of greenhouses. In addition, the growth cycles of crops vary; for instance, wheat is a single-season crop, whereas cucumbers are multiseason crops. These differences significantly impact planting planning and resource allocation. There are also notable differences in the economic value and market demand of crops. High-value crops such as blueberries and avocados, despite their higher planting cost, can achieve a better balance between costs and benefits through separate planning. Therefore, crops represent a crucial dimension of the agricultural system. By integrating these characteristics, scientific agriculture can be achieved, thereby enhancing the overall economic efficiency.
2.4.3. Planning Steps for Optimizing Model Subdivision Subsystems
We defined subsets of parcels that satisfied and . This represented a partitioning of the entire parcel set into multiple non-overlapping parcel subsets. This partitioning enabled us to analyze and optimize each subset individually based on its unique characteristics, ensuring that the planting scheme was optimally adapted to the specific conditions of each parcel subset.
- 2.
Definition of a subset of parcels
Crop subsets were defined so that and . This partitioning divided all crops into distinct non-overlapping groups. Through this categorization, targeted planting and marketing strategies could be formulated based on attributes like growth cycles, market demands, and cultivation requirements specific to each crop group.
- 3.
Construction of subsystem model objective function
For each combination of crop subset
and plot subset
, we constructed a subsystem model
. The objective function of the subsystem model was designed to maximize the total profit generated by planting a subset of crops
on a subset of plots
during different periods. This can be expressed as:
This objective function incorporates factors such as the expected sales volume, selling price, and planting cost over different periods to achieve the maximum profit.
- 4.
Constraint construction for subsystem modeling
The constraints for subsystem-model
were
These constraints were derived from the original model and were optimized and adjusted for different subsets of plots and crops after model decomposition. On the one hand, these constraints inherited the uncertainty-related constraints from the original model; on the other hand, they incorporated practical planting rules based on specific plot–crop combination scenarios. By restricting the subset of crops on the plot , the planting arrangements of each subsystem were ensured to align closely with the actual production constraints, thereby providing support for scientific and feasible agricultural planning.
2.5. Greedy Coordination Algorithm for Multisubsystem Crop Optimization
2.5.1. Greedy Algorithm
In the field of algorithm design, the greedy algorithm has been widely recognized as a classical and pivotal strategic approach. Its core tenet lies in making locally optimal decisions at each step based on the current state, thereby progressively approximating a globally satisfactory solution. Wang [
50] systematically expounded on the fundamental concepts, characteristics, and application domains of the greedy algorithm, further demonstrating its extensive applicability and efficacy across diverse problem contexts.
From a macroscopic perspective of the research field, the greedy algorithm demonstrates an extremely wide range of applications. It exhibits unique value in problem scenarios such as path planning [
51,
52] and resource allocation [
53,
54]. Particularly in domains requiring high real-time performance and rapid decision-making, greedy algorithms have become a prevalent choice for researchers due to their simplicity and efficiency. For instance, Liu et al. [
55] integrated telescope characteristics with space debris dynamics, designed an objective function, employed the pixelated sphere method to partition the sky area, and generated an optimal sky observation strategy using the greedy algorithm. Basile et al. [
56] introduced the PRO-ACTION greedy algorithm to address the issue of high lone-block risk in traditional mining strategies. This algorithm comprehensively evaluates transaction fees, block propagation speeds, and other factors at each step to select optimal transactions, ultimately balancing the benefits and risks to achieve global optimization. Švarcmajer et al. [
57] utilized a greedy graph coloring algorithm to systematically achieve rational network resource allocation based on node connectivity and coloring information, thereby enhancing system stability and load adaptability.
It is evident that the greedy algorithm demonstrates remarkable adaptability and efficacy across diverse domains. Whether addressing complex spatial observation challenges or optimizing the equilibrium between economic gains and risks in mining strategies, this algorithm can deliver efficient and viable solutions through locally optimal decisions. Although it does not inherently guarantee a globally optimal outcome, its operational efficiency and practical utility in real-world implementations have established it as a pivotal tool in algorithmic design with broad application prospects.
The core advantage of the greedy algorithm in resource allocation problems—step-by-step decision-making and rapid response—is aligned with the requirements of the crop allocation model in this study. In the model, the dynamic matching of land resources and crop demands essentially constitutes an immediate resource optimization problem under multiple constraints. By iteratively selecting the current optimal “land–crop” match (e.g., prioritizing high-value crops for high-quality plots), the algorithm can generate feasible solutions within a limited time frame. This approach not only inherits the efficiency of path planning and blockchain optimization, but also avoids the computational bottlenecks of traditional planning methods in complex agricultural scenarios, thereby helping to meet the real-time and actionable requirements of agricultural decision-making.
2.5.2. Steps for Applying the Algorithm
The decision-making mechanism of the greedy algorithm has significant advantages in our model. Its ability to iteratively select the local optima and dynamically adapt can efficiently handle the matching of land resources and crop needs, providing practical and actionable solutions for agricultural decision-making. The specific process flowchart is shown in
Figure 1.
For the subsystem-programming model
, under the conditions of plot
and period
, we calculated the expected profit
for each type of crop
. We then selected the crop
that maximized
and denoted
as the decision to plant
. Thus,
was formulated as follows:
- 2.
Examination and adjustment of constraints
After completing greedy selection, it was necessary to examine and adjust the set of constraints
in the subsystem model. For
, we let
denote the decision domain
. Specifically, we introduced a constraint violation flag
. If
(indicating
), we defined
as the set of crop types
from the original set
that caused the constraint violation. That is, when considering the decision-making variables related to
, the constraint
was not satisfied. Then, we recalculated the relevant
for the violated part, re-selected a new
from the valid crop set
, set
, and repeated the constraint check. This process was continued until all
were satisfied. Thus,
was formulated as follows:
- 3.
Iterative computation
For the parcel with and period , we repeatedly performed greedy selections and constraint adjustments. Specifically, in each iteration, we first executed the greedy selection step to determine the optimal crop. Then, we checked whether all constraints were satisfied. If violations exist, adjust as per the rules. The above steps were repeated until the entire agricultural planting planning cycle was completed. This process generates an optimal decision set for maximizing agricultural planting profit.
3. Case Study
3.1. Study Area
The study area is located in the mountainous region of North China (
Figure 2), characterized by a combination of temperate continental climate and mountain microclimate. The average annual temperature is relatively low, with significant diurnal temperature variations, and precipitation is primarily concentrated in the summer. However, the region enjoys sufficient sunlight throughout the year. The complex topography has fostered a diverse agricultural system: in flat dry lands, sandy loam soil with moderate water and nutrient retention capacity is suitable for drought-resistant crops such as corn and legumes; in terraced fields, the distinct soil layers and high organic matter content make them ideal for root crops such as potatoes; in slope areas, where soil erosion risk is higher, improved plots can support nitrogen-fixing pioneer crops like alfalfa, which also contribute to ecological restoration.
The region is currently undergoing a critical transition from traditional dry farming (primarily wheat-based) to diversified and modern agricultural production models. In open-field cultivation, the ratio of traditional staple crops (e.g., wheat) to cash crops (e.g., broccoli) is dynamically adjusted based on market demand. In facility agriculture, ordinary greenhouses extend the production cycle of spring and autumn vegetables, while intelligent greenhouses enable precise environmental control for the efficient cultivation of off-season vegetables and edible fungi. Since 2023, the region has implemented a three-year legume crop rotation system, prohibiting continuous cropping on all plots (including greenhouses) alongside the promotion of large-scale planting layouts to enhance mechanized operation efficiency.
As a typical representative of agricultural modernization in the mountainous areas of North China, the unique topographic and climatic diversity of this region, combined with the integration of traditional and modern technologies, provide an experimental environment for exploring crop adaptation strategies. This study adopted a decomposition-based stochastic multilevel binary optimization model to reveal planting rules through data-driven methods and formulate adaptation plans to help achieve sustainable agricultural development in similar mountainous areas.
3.2. Data
3.2.1. Data Sources
The data for this study were collected by the research team in collaboration with the local agricultural department in the mountainous area of North China. The main sources included historical records of land use, crop planting structure, and yields provided by the county-level agricultural department, detailed information on land use patterns, crop rotation practices, and greenhouse management obtained through field research in multiple representative villages. For detailed data, please refer to the
Supplementary Materials.
3.2.2. Data Description and Data Processing
We focused on collecting data related to rural cultivated land and crops in 2023 (
Table 1). Cultivated land covers six types, namely flat dry lands, terraced fields, sloping lands, irrigated lands, ordinary greenhouses, and smart greenhouses. Different types of cultivated land are suitable for corresponding planting modes due to differences in soil characteristics and environmental conditions. At the crop level, there were a total of 41 types, broken down into 16 types of food crop, 21 types of vegetable, and 4 types of edible fungi. According to the planting cycle, there were 16 single-season records and 25 double-season records: single-season crops are mainly distributed in arid land and some irrigated areas, relying on the natural climate to complete single-season growth; double-season crops, on the other hand, achieve annual production through irrigation systems and environmental control technology, thereby improving land utilization.
Additionally, the 2023 crop planting dataset included detailed information such as sales volume, yield per mu, planting cost, and selling price. For example, in plot A1 (flat dry lands), 80 mu of wheat was planted, with a yield of 800 jin per mu, a planting cost of 450 yuan per mu, and a selling price ranging from CNY 3.00 to 4.00 per jin. These data reflect the economic performance and production characteristics of the local agriculture, providing a solid foundation for subsequent research and decision-making.
In the data preprocessing stage, we implemented a series of measures to address the challenges posed by multisource data and inconsistent formats. First, we conducted data cleaning to eliminate records with abnormal formats, ensuring data integrity. Second, rule-based logic was applied to exclude records missing key information. Finally, a “crop–plot–season” ternary composite label was constructed to accurately capture the dynamic planting patterns. These steps ensured data accuracy and usability.
3.3. Application in Rural Crop Planting Planning
Crop cultivation in rural areas is characterized by many uncertainties, with various interrelated factors influencing the outcomes. These factors include the crop yield per mu, selling prices, expected sales volume, and planting cost. Notably, the crop yield per mu and the selling price of edible fungi followed a uniform distribution. Specifically, the range for crop yield varied between −10% and 10%, and the selling price of edible fungi ranged between −5% and −1%. Additionally, the planting cost, expected sales volume, and vegetable selling prices were projected to increase annually by 5%, while the crop selling prices were projected to remain stable.
Based on the empirical data of the villages in the mountainous areas of North China, the model proposed in this study was divided into four subplanning models in the two dimensions of land resource management and crop planting planning, as follows:
Given the unique high water consumption characteristics of rice, whose growth conditions and management modes differ from other crops, we classified rice cultivation on irrigated lands as a subsystem.
We let
denote the set of irrigated lands,
the set of rice,
as the time, and
as a very small number. As no constraints were related to restrictions on repeated cropping, crop rotation association constraints, or constraints on planting dispersion for rice cultivation in irrigated lands, we defined
and
. The model was formulated as follows:
- 2.
Subsystem-planning model II: Planting planning model for non-rice seasonal food crops in flat dry lands, terraced fields, and sloping lands
Non-rice food crops have strong adaptability to water, can be grown in a wide range of land, and are all single-season crops. Therefore, their cultivation in flat drylands, terraced fields, and sloping lands was classified into a subsystem.
We let
denote the collection of flat dry lands, terraced fields, and sloping lands,
denote the set of non-rice seasonal food crops, and
denote the time set. Let
be a very small number. To ensure that the cultivation of non-rice seasonal food crops was not overly dispersed and to adhere to the constraints regarding the threshold area for crop planting on these parcels, we defined
,
,
, and
. The model was formulated as follows:
- 3.
Subsystem-planning model III: Planting planning model for double-crop vegetables in irrigated lands, ordinary greenhouses, and smart greenhouses
Double-cropping vegetables adopt double-cropping planting. Irrigated lands secure water sources, and greenhouses control the basic temperature and humidity. Due to these characteristics, the planting model in water-irrigated lands and greenhouses formed a subsystem.
We let
denote the collection of irrigated lands, ordinary greenhouses, and smart greenhouses,
the collection of double-crop vegetables, and
denotes the time set. To ensure that the planting of double-crop vegetables did not become overly dispersed and to adhere to constraints on the threshold area for crop cultivation within these plots, we defined
,
,
, and
. Thus, the model was as follows:
- 4.
Subsystem-planning model IV: Cultivation planning model for edible fungi in ordinary greenhouses during the second season
Edible fungi demand a stable temperature and humidity. Open-air cultivation may reduce the yield and quality, so greenhouse facilities are needed for precise control. Given its environmental sensitivity, its greenhouse cultivation mode was set as a separate submodel.
Ordinary greenhouses were denoted as
, with
representing the collection of edible fungi and
denoting the time set. When cultivating edible fungi in ordinary greenhouses, there were no constraints related to restrictions on repeated cropping, crop rotation association constraints, or constraint on planting dispersion. We defined
and
. Thus, the model was as follows:
7. Conclusions
In the study of crop planting planning, we focused on specific land and uncertainty constraints and constructed a decomposition-based stochastic multilevel binary optimization model. The model aims to maximize agricultural profits by taking into account the uncertainties in sales volume, selling price, yield per mu, and planting cost. In addition, we established six types of constraints, including thresholds for the crop area planted per plot, restrictions on repeated cropping, associations with crop rotation, degree of planting dispersion, management of the threshold for the crop area within plots, and the range of variation of uncertain variables, to ensure the comprehensiveness and practicality of the model.
Faced with the complexity of the planting planning problem, we adopted a divide-and-conquer approach, decomposing the problem from the perspectives of crops and plots, and transforming it into multiple manageable subproblems. In this way, we reduced the difficulty and complexity of solving the problem. Subsequently, we used a greedy algorithm to gradually solve these subproblems, which further improved the model solving process.
To verify the effectiveness of the model, we conducted an empirical study with a rural village in the mountainous area of North China as an example, predicted the village’s total agricultural revenue for the three years from 2024 to 2026, and formulated specific crop planting plans for different plots. To evaluate the optimization performance of the model, we used the Herfindahl–Hirschman Index (HHI) and the frequency of legume crop rotation indicators. The results show that from 2024 to 2026, the planting dispersion and rotation frequency of legume crops improved significantly compared with those in 2023. In addition, we conducted a sensitivity analysis, dividing the uncertain variables into endogenous dynamic variables and exogenous benchmark variables, and observed the impact of their coordinated changes on the model’s benefits. In particular, the model output showed good stability when the endogenous dynamic variables (such as the annual rate of change in yield per mu and the selling price of edible fungi) and exogenous benchmark variables (such as planting cost, selling prices, and expected sales volumes) changed.
In order to fully verify the superiority of the proposed model, we set up two benchmark models for comparative analysis: benchmark model 1 only considered the land resource constraints, and benchmark model 2 only considered the uncertainty factors. Through comparative experiments in different years and multiple income scenarios, the results showed that the performance of our model was significantly better than these two benchmark models. This comparison not only verified the effectiveness of the model, but also highlighted the combined advantages of considering both the land constraints and uncertainty factors.
Although the model performed well in both theoretical and empirical studies, it still had some limitations. First, we used a greedy algorithm in the solution process. Although this method can efficiently find a local optimal solution, it may not guarantee a global optimal solution. This means that in some cases, the optimization results of the model may not be the most ideal planting plan. Second, the model’s assumptions about uncertainty may not fully reflect the complexity and dynamic changes in reality. For example, extreme weather events (such as droughts and floods) or sudden market fluctuations (such as price crashes or supply chain disruptions) can have a significant impact on agricultural production.
In summary, our model provides a reference for crop planting planning by comprehensively considering the land constraints and uncertainty factors. It aims to help agricultural managers optimize resource allocation and promote sustainable agricultural development. In order to further improve the practicality and stability of the model, future research should focus on optimizing the model in complex and uncertain environments, incorporate more uncertainty factors, and further improve the algorithm to overcome the limitations of the greedy algorithm to better support agricultural production practices.