Agriculture is the main occupation in many countries around the world and with a growing population, which the UN projects will increase from 7.5 billion to 9.7 billion in 2050 [1]. This means that farmers will have to do more with less. According to the same survey, food production will have to increase by 60% to feed two billion more people. However, traditional methods are not enough to handle this huge demand. This drives farmers and agricultural businesses to nd new ways to increase production while conserving soil resources. The challenge is to increase global food production by 50% by 2050 [2] to feed two billion more people by practicing sustainable agriculture.
Crop rotation is a fundamental feature of all organic farming systems. Crop rotation means changing the type of crop grown in a particular piece of land from year to year. There are cyclical rotations that repeat the same sequence inde nitely, and noncyclical rotations that allow for changes in crop sequence, adapting to management decisions and evolving as market opportunities arise [3]. Greater soil fertility, fewer pests and crop diseases, and higher yields are just a few of the positive outcomes of rotation. Compared to continuous monoculture practices, according to [4] , rotation increased crop yields by 20 % on average and the bene ts are highly context-dependent.
The nancial stability of the agricultural sector could be improved by including a variety of cropping strategies. The pro ts of the farm would not be based solely on a single primary cash crop; rather, they would be based on a diverse collection of commercial crops that were evenly distributed over a number of periods. This could eventually lead to an improvement in cash ow by introducing regular incomes into the agribusiness [5].
In order to assist the farmer in choosing the appropriate crops for the rotation cycle, we proposed a Mixed-Integer Linear Programming (MILP) model to solve the crop rotation problem by exploring the bene t of including green manure. We studied the importance of including fertilizer plants in crop rotation schedule by using exact methods such as linear programming to determine the best solution of the proposed agricultural model and to test it in a real context. The experiment was conducted for a real planting area of average size with 9 plots, considering 8 crops from di erent botanical families and a two-year, three year and four year planting rotation.
Maximizing the overall net return is the aim of the crop rotation planning problem that they address in [6]. The branch and bound approach is used to optimize the crop rotation plans in an integer linear programming model. In addition to the agronomic, water supply, and seasonal demand constraints, the suggested model includes a new temporal preference constraint.
In order to optimize the e ciency of organic farming in the Philippines' second and third largest agricultural land areas, the authors suggested using a Mixed Integer Programming model as a decisionmaking tool. This tool considers a number of organic farming-related factors. With the matching pro t organic farms may make from planting them, the best-selling and most popular produce in the area is used [7].
Using a mixed integer linear programming model, their work [8] examines the crop rotation problem with water supply/demand and net return uncertainties that uctuate within the permitted rotation cycle. It renders the developed model numerically feasible, mainly when applied to intricate agricultural issues. Determining the best cropping plans, providing a fair income for the farmer, and strategically accounting for water uncertainties are the primary goals of this endeavor.
In order to optimize crop rotation in conventional organic farms with plot adjacency constraints and nutrient amendments, this work [9] makes use of real-world data and the CBC solver. The goal of the created rotations is to maximize farmers' income, and they have set durations for each plot. The suggested agricultural model's solution is found using a linear programming technique.
In this study [10], the authors examine the Crop Rotation Problem and its applicability to the combination of farm management and Precision Agriculture. They increased the problem's appeal for sustainability by presenting a novel mathematical method for the CRP based on crop requirements and nutrient balance. To optimize the CRP, a real-encoded genetic algorithm was created. The ndings show that mid-and long-term crop scheduling performed well.
A binary nonlinear bi-objective optimization model is presented in [11] to address the issue of agricultural cultivation planning that is sustainable by using a meta-heuristic technique based on a genetic algorithm and constructive heuristics. A planting schedule for crops to be grown in designated plots to limit the likelihood of pests proliferating and increase the process's pro tability.
An integrated strategic-tactical planning model for the supply chain issue involving sugar beets is presented by the authors this study. In order to minimize the overall operational costs, including transportation and inventory of processed and unprocessed beets, a binary integer programming model is developed . To enable crop rotation planning across many cropping seasons, a special temporal dimension was introduced to the planning horizon [12] 3. Methodology
According to FAO's annual report in 2006, nearly 69% of the 56,600 km2 of the Togolese territory is agricultural land, and 38% of this land is exploited. Rain-fed agriculture is practiced in Togo, and nearly 75% of the country's working population works as small farmers using traditional farming techniques. The two principal crops are maize in the south and focus of the nation and sorghum/millet in the north. 70% of the country's population, or 5.75 million people in 2010, is supported by agriculture, which also makes up nearly 40% of the country's GDP. From the south to the north, Togo is divided into ve regions: the Maritime, Plateaux, Central, Kara, and Savanes. From one region to the next, pedoclimatic conditions and the availability of agricultural production factors, particularly land, vary signi cantly [13]. Togo's climate is tropical, and it varies a lot from south to north. The climate is sub-Sahelian (hot and dry) in the north, sub-Sahelian (rainy) in the center, and Guinea-Sudanian (hot and humid) in the south. The regions' intra-annual rainfall distribution is also unique. There are two rainy seasons in the Maritime and Plateaux regions each year March/April to July and September to mid-November and two dry seasons August and mid-November to March/April with 900 and 1500 mm of precipitation per year. There is a dry season from November to April in the Central, Kara, and Savanes regions, with annual rainfall ranging from 1200 to 1500 mm. As a result, there are two agricultural seasons in the southern parts of Togo, while there is only one in the northern parts. Despite the signi cance of agriculture to the Togolese economy, it has remained traditional and lacked access to organic and mineral fertilizers. Agricultural production includes:
• Cereals: Maize, Millet, and Sorghum • Tubers: Yams and Cassava • Legumes: Beans, Groundnuts, and Soya Our research focuses primarily on the subtropical climate of the region, taking into account the potential crops.
Several factors, including market demands, soil characteristics, and crop nutrient requirements from a climatic perspective, are taken into consideration when choosing a cropping sequence. Plots are used to divide an agricultural area. Di erent kinds of crops can be grown on each plot [14]. Planning crops on agricultural land while taking into account the primary factors that a ect crop yields (economic, environmental, and ecological) presents a challenge when developing a crop rotation system [9]. Utilizing the principles of crop rotation to make farmers as much money as possible while taking into consideration restrictions based on demand, crop characteristics, production times, and plot conditions is the primary objective of this work. The proposed model takes into account the following constraints:
1. Sowing Period : It is essential to observe each crop's sowing and production times. 2. Cultural continuity among families: Di erent botanical families contain the cultures. It is not recommended to grow cultures from the same botanical family in succession on the same plot [5]. This issue is primarily brought about by the fact that crops in the same family have similar de ciencies (the risk of acquiring the same diseases or weeds) and nutrient requirements. The cropping system's ability to last is jeopardized as a result of this. 3. Crops belonging to the same family's neighbors: On two adjacent plots, two crops belonging to the same botanical family cannot be sown simultaneously [5]. Characteristics are shared by cultures of the same family. Therefore, staking them simultaneously on two plots that are adjacent to one another is the same as staking the same crop on these two plots. This makes more resources available to pests, which in turn increases their population and the damage they can cause. 4. Needs for nutrients: The quantity of soil nitrogen, phosphorus, and potassium required to start a crop at any given time is referred to as its nutrient requirements. We de ne the nutrient of a crop as the quantity of nitrogen, phosphorus, and potassium that this crop requires, and the nutrient of a plot as the amount of nitrogen, phosphorus, and potassium applied to this plot over a speci c time period.
a positive e ect on the soil and, as a result, increases yield. 6. Fallow Period: To allow the soil to regain its moisture and fertility, each cycle ought to include one or more periods of fallowness. 7. Market demand: The distribution of cash crops is signi cantly restricted as a result of this. Each culture has a preexisting market; the demand needs to be met.
The crop rotation problem is a complicated combinatorial optimization problem that changes depending on the model's scope, considering the aforementioned constraints. Along the rotation cycle, it is desirable to determine the best crop combinations to plant in each plot at each time. The problem is solved using the constructed mathematical model, and the following decisions are made:
• Determine the crop-speci c area needed to satisfy demand.
• During the rotation cycle, determine the crop sequence in each plot.
• Obtain the various agricultural operations' calendar.
The crop rotation problem is a complicated combinatorial optimization problem that changes depending on the model's scope, considering the aforementioned constraints. Along the rotation cycle, it is desirable to determine the best crop combinations to plant in each plot at each time. The problem is solved using the constructed mathematical model, and the following decisions are made:
• Determine the crop-speci c area needed to satisfy demand.
• During the rotation cycle, determine the crop sequence in each plot.
• Obtain the various agricultural operations' calendar.
The planting area is thought to be divided into plots by us. If two parcels share a boundary that is not reduced to a discrete set of points, they are neighbors. We will consider the opposite plots to be adjacent, for instance, if the planting area is divided into four plots.
A crop rotation schedule with a clearly de ned planning horizon (T) is established by a producer with the intention of maximizing pro ts following an agricultural season. It has a number of crops (N) that are part of various botanical families (p), where F(p) is the number of crops in the family (p) and surfaces that are divided into plots (K) and naturally contain certain amounts of nutrients (Nitrogen, Phosphorus, and Potassium). Pro tability (l), production cost (CP), production time (Z), average production (Q) per hectare, market demand (D), and nutrient requirements are all associated with each crop. Additionally, the farmer is restricted by the following restrictions: sowing time, continuity and neighbor for crops belonging to the same family, green fertilization, and fallow time.
The tables de ne all indices 1, parameters/data, and variables 2 for the proposed model. The index of fertilization (↵) is determined by the model parameters fertilization interval (✓) and planning horizon (T). For instance, if the planning horizon is 24 periods and the fertilization interval is 12 periods (✓ = 12), then the set ⌦ is 1, 2 because ↵ = 1 and ↵ = 2 satisfy the de nition of ↵ • ✓ T .
The entire model is displayed below. The bene ts of crop planning (calendar), the costs of fertilization, and other production costs (land preparation, transportation, labor) are evaluated by the objective function in Equation (1). Fertilization costs are also included in, with the goal of reducing reliance on external chemical fertilizers.
Subject to (1)- (15). Other production costs incurred on plot j at period t due to crop i. (Preparation of the land,purchase of seeds for sowing, transport, labor)
With t = 1, . . . , T ; j = 1, . . . , K.
It is not allowed to plant two crops in the same plot [15,16,10]. Indeed, a crop occupies the entire plot throughout its growth and harvest. The constraint de ned in equation (2) forbids scheduling more than one culture during the same period. There is a spatial constraint.
With p = 1, . . . , N f ; t = 1, . . . , T ; j = 1, . . . , K.
It is not recommended to plant two crops from the same botanical family on adjacent plots at the same time [15,9,16,10]. The fact that the cultures of the same family share characteristics is the source of this issue. Therefore, planting them simultaneously on two plots that are adjacent to one another amounts to planting the same crop on both plots, which does not maximize crop distribution on the various plots. If a crop i of the same botanical family p has already been sown on plot k, constraint (3) states that the number of crops on all plots S j adjacent to plot j during their production periods Z i must be equal to zero; otherwise, the number of crops is at most equal to the number of independent plots.
With p = 1, . . . , N f ; t = 1, . . . , T ; j = 1, . . . , K.
On the same plot, cultures belonging to the same botanical family cannot be grown immediately [15,16]. This issue is primarily brought about by the fact that crops in the same family have similar de ciencies (the risk of acquiring the same diseases or weeds) and nutrient requirements. The cropping system's agronomic viability is jeopardized as a result of this. Constraint in equation ( 4) sets a maximum of one crop as the sum of all crops i in the botanical family p over their production period Z i .
Additionally, green manure improves the structure and fertility of the soil while also enhancing its organic matter enrichment [16]. Green manure adds nitrogen to the rotation by planting legumes [17]. Constraint (5) guarantees that every plot gets at least one implementation of green manure (legumes).
With n = N + 1, j = 1, . . . , K.
The period of frost or fallow enables the plot to restock its production capacity, water reserves, and other resources. as well as to restrict excessive agricultural production. Constraint de ned in equation ( 6) ensures that each plot has at least one freezing period. During the frost period, there are no restrictions on neighborhood and consecutive planting.
With i = 1, . . . , N.
Following the recommended planting date is critical for allowing the crop to express its yield potential and lowering crop protection costs. By preventing allocation outside of this window, the constraint outlined in equation (7) ensures that crop scheduling takes place only during the appropriate sowing period.
When nutrients are present in su cient quantities and in mineral forms that can be absorbed by plants, a soil is more fertile [9]. They deplete the soil of the essential nutrients they require as they grow.
With t = 1, . . . , T ; ↵ 2 ⌦.
With t = 1, . . . , T ; ↵ 2 ⌦.
With t = 1, . . . , T ; ↵ 2 ⌦.
The quantity of nitrogen, phosphorus, and potassium that the soil requires to initiate a crop at any given time is referred to as the minimum nutrient requirement. A plot's nutrient amendment is de ned here as the amount of nitrogen, phosphorus, and potassium applied to it over a speci c time period. Let ↵ be the size of the interval that is appropriate for sowing crop i, the interval that is used to apply the amendments F N↵j ,F P ↵j and F K↵j , B Nj ,B P j and B Kj , which represent the initial composition of the Nitrogen, Phosphorus, and Potassium soil of plot j per ha [10,9] , R Ni ,R P i and R Ki , which represent the minimum amount of nutrients that crop i requires.
Equations ( 8), ( 9),( 10) are used to calculate fertilization balances based on the surface nutrient budget.
With i = 1, . . . , N.
Demand from the market is another signi cant constraint. The farm's crop yields are constrained by this constraint to meet the anticipated demand for the crop i. To avoid issues like prolonged product storage or conservation, which can result in additional costs and increase the risk of income variability, we believe that the farmer should not exceed the estimated demand. The constraint outlined in the equation ( 11) is used to evaluate the crop's production requirements.
Each of the Boolean decision variables x itj represents the schedule (planning) of crop i during period j on plot t. When and where crops are planted are tracked by them. The variables of fertilization F N↵j ,F P ↵j and F K↵j are actual variables.
With i = 1, . . . , N; t = 1, . . . , T ; j = 1, . . . , K.
With j = 1, . . . , K; ↵ 2 ⌦.
With j = 1, . . . , K; ↵ 2 ⌦.
With j = 1, . . . , K; ↵ 2 ⌦.
Python-MIP, a collection of Python tools for modeling and solving Mixed-Integer Linear Programs (MIPs), was used for the model's implementation and evaluation and CBC Solver as Solver since it does not require a license unlike Gurobi. The solver uses the branch and bound algorithm to nd the optimal solution for the crop rotation problem. The study will focus on eight crops from ve botanical families. Crop data, as well as production parameters like planting, harvesting dates and nutrient requirements, are listed in Table 4. The planning horizon is divided into month. At each rotation schedule, each area adopted at least one fallow period and one green manure crop. The proposed model can be solved in less than a minute and includes 1998 variables and 23049 linear constraints.
The proposed model will be evaluated in two scenarios. Firstly, we present the impact of including nutrient amendment in the crop rotation schedule on the net pro t (Instance 1). We evaluate the income and crop rotation schedule in this experiment, taking into consideration whether or not soil amendments are present. Besides, we present the alteration of the design of the plots on the establishing region which restricts the crop allocation. Then, we evaluate our model using a variety of two, three, and four years planning horizons (Instance 2).
Table 4 presents details about the crop, planting and harvest times and Nitrogen, Phosphorus and Potassium requirement for each crop . In this study, there are a total of eight crops (N = 8) and ve families of crops (Nf = 5). Our planning horizon is two years, divided into 24 periods (T = 24). Table 5 describes the adjacency among plots and the cultivable area of each plot. We do not take into consideration the main diagonal, but the other positions in the table lled in Green represent the plots that are adjacent to each other. Details such as number of variables and linear constraint about the two models are displayed in Table 3. In the rst instance of our problem, the solver reached the value of the objective function in 9,66 seconds for the model without nutrient amendments which is 30,639,998 XOF and in 61,21 seconds for the model with nutrient amendments which is 22,087,660 XOF. Planning is presented di erently in each model. The representation of the CBC solver-derived solutions for the two models is depicted in Figures 1 and 2. Figures 1 and 2 both demonstrate compliance with the adjacency constraint. 4 . The Cbc solver respectively a value of 31.366.080 XOF and 88.327.840 XOF after 30 seconds and 28 seconds of running time, and the associated rotation for each plot is given in Figure 3.
As can be seen in Figure 4, the net pro t increases with the planning horizon. This necessitates consideration of the horizon selection and crop selection for the rotation. But planning crops ahead of time over several years can be risky because there are many things that can a ect market demand, like the weather, and market prices can change a lot from season to season.
This article highlights the signi cant role of legumes in modeling a crop rotation system, considering constraints such as contiguity and the use of mineral fertilizers for soil amendment. To address the problem of increasing the net income of farmers in a crop rotation system, We used a mixed integer linear programming model (MILP). Our model proposes the best possible solution by maximizing the objective, in this case farmers' income, while respecting a set of constraints, such as crop rotation rules. The best solution of the proposed model was tested in an experiment conducted for a medium-sized real plantation area with nine plots, considering eight crops from ve botanical families and a two-year plantation rotation. The results of the study indicate that farmer's incomes improve when a longer planning period is considered.
However, the solutions proposed by MILP may lack the exibility to adapt to rapid changes in the eld, such as a sudden drought or market price uctuations. This aspect will be taken into consideration in our future work by introducing a model based on stochastic linear programming model. Additionally, integrating a dynamic subdivision could further boost farmers' incomes.
Either: The author(s) have not employed any Generative AI tools.
For instance 2, we want to compare the net returns from the crop rotation planning so we consider for experiment two-year, three-year and four-year period for the horizon of planning with the same crop