Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 13, No. 2, 2024 

 

146 

Research	on	the	Optimization	Strategy	of	Crop	Planting	
Combining	Dynamic	Programming	and	Monte	Carlo	
Simulation	
Hongda Chen1, *, Wenbo Li1 and Qikai Chen1 

1School of Electrical Engineering, Xi 'an University of Technology, Xi’an, China 
*Corresponding author 

 

Abstract: This paper explores the optimization of crop planting strategies by combining dynamic programming with Monte 
Carlo simulation. In the multi-stage decision-making for planting optimization, we decomposed the optimal planting plans for 
crops from 2024 to 2030 into annual best planting plans. We established an objective function aimed at maximizing total profit, 
taking into account factors such as planting area, crop yield, sales price, and planting cost. A series of constraints were also 
introduced, including that the planting area should not exceed the cultivated land area and that the total crop production should 
not surpass the expected sales volume. By employing dynamic programming and greedy algorithm models, we utilized the PuLP 
linear programming library to define problems, add constraints, and invoke solvers to find the optimal solution. The model 
considered not only the planting costs and sales prices but also treated excess sales volume as waste. In the result analysis phase, 
we analyzed the cyclical changes in data across different years and observed that the total profit exhibited a certain fluctuation 
trend. We also conducted error analysis on the potential errors introduced by the greedy algorithm, running the model multiple 
times to verify the stability of the results and ensure the reliability of the model. Finally, we discussed the complementarity and 
substitutability between crops, which significantly impact the total income in the actual planting process. 

Keywords: Crop planting, dynamic programming, monte carlo simulation, optimization strategy. 
 

1. Introduction 
The traditional approach to crop planting has often been 

based on empirical methods and historical data, which may 
not fully account for the complexities and uncertainties 
inherent in agricultural production [1, 2]. To address these 
limitations, we propose a novel approach that combines 
advanced mathematical programming techniques with 
stochastic simulation.  Dynamic programming allows us to 
break down the complex, multi-year planting problem into a 
series of simpler [3], sequential decisions, each informed by 
the outcomes of the previous steps. Monte Carlo simulation 
[4], on the other hand, enables us to model and evaluate the 
impact of various uncertainties, such as weather fluctuations 
and market volatility, on the crop planting strategies. 

Our research begins with a thorough data preprocessing 
phase, utilizing Excel to visualize and analyze the existing 
arable land data in the North China mountain region.  This 
phase is crucial for understanding the current state of land use 
and crop distribution, which forms the foundation for our 
subsequent optimization efforts. We then delve into the multi-
stage decision-making process for planting optimization, 
establishing an objective function that maximizes total profit 
while adhering to a set of constraints that reflect real-world 
limitations and requirements. 

The constraints include not exceeding the cultivated area 
with crop planting, ensuring that total crop production does 
not surpass expected sales, and maintaining an appropriate 
balance in the planting area for different crops to avoid over-
saturation or under-utilization of land resources.  We also 
incorporate the concept of crop complementarity and 
substitutability, recognizing that certain crops can either 
compete for resources or benefit each other's growth, which 
in turn affects the overall agricultural output and profitability. 

We implement our dynamic programming and greedy 
algorithm models [5], using the PuLP linear programming 
library to define and solve the complex problem of crop 
planting optimization [6]. Our model evaluates planting 
strategies under different market conditions, considering both 
the costs of cultivation and the potential revenues from sales, 
including the treatment of excess sales volume as waste or 
discounted goods. 

2. Data Preprocessing 
In the process of our research on crop planting strategies, 

data preprocessing has become an essential part of our work. 
We first utilized Excel to visualize the existing arable land 
data of the rural areas in the North China mountain region, 
clearly presenting the distribution of planting areas for 
different land types through pie charts. Subsequently, based 
on the crop types and planting land conditions, we created 
corresponding charts, which helped us gain a more intuitive 
understanding of the distribution of crops and the use of 
arable land. To gain a deeper understanding of the planting 
area of crops in different quarters, we used the 2023 crop 
planting data to display this information. 

Furthermore, we summarized the planting area and yield 
per acre of the same crops and visually presented them in the 
form of bar charts, which helped us analyze the relationship 
between crop yield and planting area. We also paid special 
attention to the correlation between crop planting area and 
yield per acre, analyzing the impact of the planting area of 
legumes and non-legumes on yield per acre through scatter 
plots. We found that after the planting area increased to a 
certain extent, the yield per acre reached its peak, and after 
exceeding this point, the yield per acre would actually 
decrease. 

In terms of total sales volume, we calculated the total sales 



 

147 

volume of each crop and compared them through bar charts, 
which helped us identify crops with high and low sales 
volumes. In addition, we conducted an in-depth analysis of 
the relationship between the planting cost and profit margin 
of each crop. By sorting and calculating the data from 2023, 
we obtained the average sales price and selling price for each 
crop. Combining the planting cost and sales price, we 
calculated the total cost and total profit for each crop, thereby 
deriving the profit margin. 

3. Multi-stage Decision-Making in 
Planting Optimization 

First, the optimal planting scheme for crops in 2024-2030 
was decomposed into the optimal planting scheme for each 
year. The planting plan of crops each year is relatively 
independent, but the planting plan of the previous year will 
have an impact on the planting plan of the following year, so 
dynamic planning is chosen. Then, according to the 
cultivation conditions, the crops are decomposed into food 
and non-food categories, and the greedy algorithm is used to 
find the optimal solution one by one. Finally, the optimal 
planting scheme of crops in the seven years from 2024 to 
2030 is obtained. 

3.1. Establishment of the Objective Function 
We set up the objective function as the maximum value of 

the total profit. 
The total production of the 𝑖 crop in that year was: 
 

𝑒 ∑ 𝑠 𝑦 𝑙                    (1) 

 

Where, 𝑀 is the amount of cultivated land, and 𝑙  is the 
planting yield per mu of the 𝑖 crop on the 𝑗 cultivated land, 
in units of jin. 

Let 𝑓  be the sales generated by the 𝑖 crop, 
 

𝑓 𝑒 𝑏                      (2) 

 

Where, 𝑏   is the unit price of the i crop, and the unit is 
yuan/jin. 

Planting cost is: 
 

cost ∑ ∑ 𝑠 𝑦 𝑎                (3) 

 

Where 𝑎   is the planting cost of the i crop on the j 
cultivated land, and the unit is jin/mu. 

In summary, the total profit of the objective function Z is 
 

𝑚𝑎𝑥 𝑍 ∑ 𝑓 cost ∑ ∑ 𝑠 𝑦 𝑙 ⋅ 𝑏
∑ ∑ 𝑠 𝑦 𝑎             (4) 

 

3.2. Constraints 
Constraint 1: The planting area of crops should not exceed 

the cultivated area. 
 

∑ 𝑠 𝑦 𝑑 , 𝑗 1,2,⋅⋅⋅ ,26             (5) 

 

Wherein, 𝑑   is the largest planting area of the 𝑗  -type 
cultivated land. 

Constraint 2: Total crop production per season cannot 
exceed expected sales. 

If the total output 𝑒   of the crop per season exceeds the 
expected sales 𝑐  , the excess part cannot be sold normally, 
which will cause a greater degree of loss, so the total output 
of the crop per season cannot exceed the expected sales: 

 

𝑒 ∑ 𝑠 𝑦 𝑙 𝑐 , 𝑖 1,2,⋅⋅⋅ ,15         (6) 

 

Constraint 3: The area of each crop in a single cultivated 
land should not be too small. 

Combined with Annex I and Annex II, it is obtained that 
the minimum area (A3) of the flat early field is 35 mu, and 35 
mu of corn crops are planted in the flat early field of A3, and 
there is no vacancy of arable land in all other cultivated fields. 
Similarly, the number of acres of crops in the flat terrace is 
not less than 20 acres, and the number of acres of crops in the 
hillside is not less than 13 acres. 

 
𝑠 35, 𝑗 1,2,⋅⋅⋅ ,6
𝑠 20, 𝑗 7,8,⋅⋅⋅ ,20
𝑠 13, 𝑗 21,22,⋅⋅⋅ ,26

            (7) 

 
Constraint 4: Each crop should not be planted too widely 

each season. 

Observing Annex II, it can be found that only one crop is 
planted in the same plot (for example, A1 flat early field, B1 
terrace and C5 only grow wheat). Therefore, it is stipulated 
that no more than three crops can be grown on the same 
cultivated land. The decision variable 𝑦  is applied. 

 

∑ 𝑦 3, 𝑗 1,2,3,⋅⋅⋅ ,26            (8) 

 

Constraint 5: Each crop can not be planted in the same plot 
of land. 

First, build the indicator variables 𝑚 𝑡   and 𝑛 𝑡  . 
Since the same crop cannot be grown on the same plot for two 
consecutive years, the relationship between 𝑚 𝑡   and 
𝑛 𝑡  is: 

 

𝑛 𝑡 1 0，𝑚 𝑡 1 1
𝑛 𝑡 1 1，𝑚 𝑡 1 0           (9) 

 

After the feasibility condition 𝑛 𝑡   is introduced, the 
above objective functions and constraints need to be modified. 

The output is modified to: 
 

𝑒 ∑ 𝑠 𝑦 𝑛 𝑡 ⋅ 𝑙 , 𝑖 1,2,⋅⋅⋅ 15       (10) 

 

The planting cost is revised as: 
 

cost ∑ ∑ 𝑠 𝑦 𝑛 𝑡 ⋅ 𝑎          (11) 

 

The objective function is modified to: 
 



 

148 

𝑚𝑎𝑥 𝑍 ∑ 𝑒 𝑏 cost ∑ ∑ 𝑠 𝑦 𝑛 𝑡 ⋅ 𝑙 ⋅
𝑏 ∑ ∑ 𝑠 𝑦 𝑛 𝑡 ⋅ 𝑎        (12) 

 

Constraint 6: Plant pulses at least once in three years from 
2023. 

Since the planting of soybeans produces legumes in the soil 
that are conducive to the growth of other crops, every plot 
(including greenhouses) is required to plant legumes at least 
once in three years from 2023. 

To sum up, this paper establishes a multi-stage 
mathematical programming model by determining objective 
function, constraint condition, decision variable and time 
variable t. Since pulses are planted at least once in three years, 
the model for 2024 does not need to consider constraint 6, but 
the model for 2025 and later years does. 

3.3. Establishment of the Model 
The mathematical planning planting model for 2024 crops 

is as follows: 
 

𝑚𝑎𝑥 𝑍 ∑ ∑ 𝑠 𝑦 𝑛 𝑡 ⋅ 𝑙 ⋅ 𝑏
∑ ∑ 𝑠 𝑦 𝑛 𝑡 ⋅ 𝑎            (13) 

 

𝑠. 𝑡. 1

⎩
⎪
⎨

⎪
⎧

∑ 𝑠 𝑦 𝑛 2024 𝑑 , 𝑗 1,2,⋅⋅⋅ ,26

𝑒 ∑ 𝑠 𝑦 𝑛 2024 ⋅ 𝑙 𝑐 , 𝑖 1,2,⋅⋅⋅ ,15
𝑛 2024 0, 𝑚 2023 1
𝑛 2024 1, 𝑚 2023 0

(14) 

 

𝑠. 𝑡. 2

⎩
⎪
⎨

⎪
⎧ 𝑠 35, 𝑗 1,2,⋅⋅⋅ ,6

𝑠 20, 𝑗 7,8,⋅⋅⋅ ,20
𝑠 13, 𝑗 21,22,⋅⋅⋅ ,26

∑ 𝑦 3, 𝑗 1,2,⋅⋅⋅ ,26

          (15) 

 

The mathematical planning planting model for crops in 
2025 and later years is as follows: 

 
𝑚𝑎𝑥 𝑍 𝑡 ∑ ∑ 𝑠 𝑦 𝑛 𝑡 ⋅ 𝑙 ⋅ 𝑏

∑ ∑ 𝑠 𝑦 𝑛 𝑡 ⋅ 𝑎             (16) 

 

𝑠. 𝑡.

⎩
⎪⎪
⎨

⎪⎪
⎧

∑ 𝑠 𝑦 𝑛 𝑡 𝑑 , 𝑗 1,2,⋅⋅⋅ ,26; 𝑡 2025,⋅⋅⋅ ,2030
𝑒 𝑡 ∑ 𝑠 𝑦 𝑛 𝑡 ⋅ 𝑙 𝑐 , 𝑖 1,2,⋅⋅⋅ ,15; 𝑡 2025,⋅⋅⋅ ,2030

𝑠 35, 𝑗 1,2,⋅⋅⋅ ,6
𝑠 20, 𝑗 7,8,⋅⋅⋅ ,20
𝑠 13, 𝑗 21,22,⋅⋅⋅ ,26

∑ 𝑦 3, 𝑗 1,2,⋅⋅⋅ ,26

(17) 

 

Then, we set up the mathematical model of rice, vegetable 
and edible fungus crop planning problem, which also includes 
the objective function and constraint conditions. 

3.4. Model Solving 
In the model solving stage, we use Python programming 

language, combined with Jupyter Notebook, a powerful 
interactive computing environment, to implement our 
dynamic programming and greedy algorithm model. We first 
define decision variables that represent the choice to grow a 
particular crop on a particular plot. Next, we construct the 
objective function to maximize the total profit, i.e. the total 
selling price minus the total cost. In addition, we have 

included a series of constraints to ensure the viability of the 
planting program, including that the crop area cannot exceed 
the cultivated area and the total crop production cannot 
exceed the expected sales volume. 

To solve the model, we use the linear programming library 
PuLP, which allows us to define linear problems, add 
constraints, and invoke solvers to find the optimal solution. 
Our model takes into account not only the cost of growing the 
crop and the selling price, but also the way in which the excess 
sales volume is treated as waste in case one and sold at half 
price in case two. In this way, we are able to evaluate planting 
strategies under different market conditions. 

After solving the model, we obtained the optimal planting 
scheme for each year from 2024 to 2030. The plans detail how 
each crop will be planted on each piece of land, including the 
area planted and the expected yield. We further analyzed these 
results to assess the impact of different planting strategies on 
total profitability. 

In the result analysis stage, we first analyzed the periodic 
changes of the data and found that the total profit showed a 
certain fluctuation trend in different years. This may be due 
to changes in planting costs, selling prices and market demand. 
We also perform error analysis to evaluate the errors that the 
greedy algorithm may introduce in the solution process. By 
running the model several times, we verify the stability of the 
results and ensure the reliability of the model. 

4. Crop Complementarity and 
Substitutability 

In the actual planting process, there are certain 
substitutability and complementarity between crops, which 
will affect the total income. Alternative crops compete with 
each other for land, and complementary crops promote the 
growth of another crop to increase yields. Therefore, it is 
necessary to weigh the planting area of different crops and 
consider the impact of crops of the same plot type. 

In addition, in the market, there is a relationship between 
the pre-sale of crops and the yield, sales and planting cost per 
mu, which can reflect the scientific rationality of resource 
utilization and the law of market fluctuation. 

4.1. Analysis of Crop Substitutability 
Substitutability is manifested as the increase of planting 

area of one crop will lead to the decrease of planting area of 
another crop, so the substitutability constraint model is 
proposed. Substitution constraint model: Substitution can be 
represented by setting the constraint conditions of land 
resources. 

Take corn and wheat for example (the numbers of corn and 
wheat are i=6,7 respectively), two alternative crops cannot be 
planted in the same cultivated land at the same time, and the 
constraint conditions are expressed as: 

 

𝑦 𝑡 𝑦 𝑡 1, 𝑗 1,2, ⋯ ,26
𝑦 𝑡 , 𝑦 𝑡 0 or 1, 𝑗 1,2, ⋯ ,26      (18) 

 

Among them, 𝑗 represents 26 cultivated land such as flat 
dry land and terraced fields. 

4.2. Analysis of Crop Complementarity 
Complementarity: Complementarity shows that the 

nutrient elements of one crop can promote the growth of 



 

149 

another crop or enhance the resistance to diseases and pests, 
thereby improving crop yields and increasing earnings. 
Legumes, for example, have the ability to fix nitrogen, which 
can increase soil effort and facilitate the cultivation of other 
crops. 

Complementarity model: To achieve complementarity, 
rewards can be added to the objective function. That is, if two 
complementary crops are planted on the same arable land, 
they can bring benefits. This synergistic effect can be realized 
by setting the correlation coefficient h, which is concretely 
expressed as an additional reward term in the objective 
function. 

There are two crops 𝑖  and 𝑖  in cultivated land j, and the 
two crops are complementary, and the correlation coefficient 
between them is ℎ , then the income of the two crops will 

be a part more than the original income, let the original 
income is Z_0, and the income after planting complementary 
crops is Z, Z can be expressed as: 

 

𝑍 𝑍 ∑ ∑ ∑ ℎ 𝑦 𝑠            (19) 

 

4.3. Construction of Mutually Exclusive 
Complementary Matrix 

According to the crop planting situation in 2023, crops can 
be divided into legumes, cereals, vegetable crops and fungus 
crops. In order to construct a scientific mutually exclusive 
complementary matrix, the characteristics of the above crop 
types were analyzed separately. 

Soil containing rhizobacteria of legumes is good for crop 
growth, and legumes have the ability to fix nitrogen, which 
can improve soil nutrients, which is good for the growth of 
other crop types. There are complementarities between 
legumes and other types of crops that can increase crop yields 
and yields. However, there is substitution between legumes 
and legumes, and they will compete for land and fertility 
resources, and there is a competitive relationship and mutual 
exclusion. 

Grain crops have similar growth environment. If two kinds 
of grain are planted in the same arable land at the same time, 
their similar demand for soil and nutrients will lead to 
competition for limited resources, which will reduce soil 
fertility, which is not conducive to crop growth and 
development, and reduce yield and income. Therefore, cereal 
crops should not be planted in the same arable land at the 
same time, and crop rotation can be adopted to reduce the 
consumption of soil fertility. 

Vegetable crops can maintain soil fertility by rotating with 
other crops, so vegetable crops and other crops are 
complementary. The quality and yield of crops can be 
improved through reasonable and scientific crop rotation 
methods. However, vegetable crops need to compete with 
each other for light, water and land resources, and there is a 
competitive relationship, and the complementarity between 
them is weak. 

Fungus crops grow in a special environment, they grow on 
different cultivation media, so there is no competition with 
plant crops, with good complementarity. Fungal crops can use 
agricultural by-products (such as straw, straw) to achieve 
adaptation with most crops, suitable for rotation. 

The Seaborn model in python was used to make a display 
diagram of complementarity and substitution between crops, 
as shown in Fig. 1. 

 
Figure 1. Complementary and substitutability display of 

crops 

4.4. Results Analysis 
python is used to obtain the total profit per year from 2024 

to 2030 and the optimal planting strategy from 2024 to 2030. 
Taking food crops as an example, the optimal annual planting 
plan of food crops is shown in Fig. 2. By observing the 
optimal planting plan each year, it can be found that wheat 
and barley have strong substitution. When the planting area 
of wheat is large, the planting area of barley is almost not 
changed. The substitution of corn and sorghum is weak, and 
the planting area of the two is similar in many years. By 
observing soybeans from 2023 to 2030, it can be seen that 
soybean planting meets every three years, although some land 
types of soybeans are the same as food crops, but they do not 
compete with food crops, so they are complementary to food 
crops. In line with the natural growth law of crops, the data is 
reliable. 

 

 
Figure 2. Soybean planting area in 2023-2030 

 

5. Conclusion 
Rotary kiln is also used in the production of saloon with 

production capacity by burning specific clay soil that 
possesses adequate quantity of silica, alumina, and iron 
oxides. The external diameter of the kiln is. 

The main purpose of a rotary kiln hydrolyser is to convert 
olive pits into char fated to the production of activated char. 
The capacity of plant is about of wet olive pit, distribution of 
pyrolysis products as function of the process temperature 

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