MULTI-OBJECTIVE OPTIMIZATION: A MATHEMATICAL MODEL TO HELP MANAGE THE CROP-LIVESTOCK-FORESTRY INTEGRATION SYSTEM

Objective: The objective of this study is to develop a mathematical model based on multi-objective optimization to assist farmers who have adopted the Crop-Livestock-Forest Integration (ILPF) system, aiming to maximize production with resource limitations. Theoretical Framework: In this topic, sustainable agriculture, rural management, and mathematical optimization tools are discussed, providing a solid basis for understanding the research context. Method: It comprises an applied approach, considering the main inputs, costs and production revenues necessary to power the model. Microsoft Corporation's Excel Solver computational tool, version 7.3.5.2(x64)/LibreOffice Community, was used due to its accessibility and practicality for rural producers. Results and Discussion: The proposed mathematical model provides assistance in decision making, considering the distribution of areas for different productions with a view to achieving efficient results. In the discussion, these results are contextualized in light of the theoretical framework, highlighting the implications and relationships identified. Discrepancies and limitations of the study are also considered. Research Implications: The results can be applied to rural management and sustainable agriculture. These implications may include optimizing resources and improving the productive efficiency of farmers who use the ILPF system. Originality/Value: Considers the balance of economic, social and environmental aspects in the use of the ILPF system. The relevance and value of this research are evidenced by its potential to improve decision-making and maximize the efficient use of resources in sustainable agriculture.


INTRODUCTION
Farmers and ranchers face a series of difficulties when managing a rural enterprise due to price and market fluctuations, adverse weather conditions, variation in input costs, qualified labor, animal and plant health risks, and adaptation to new technologies, among several others.These difficulties increase when the producer is part of the Crop-Livestock-Forest Integration Systems (ILPF), as it requires integrated and complex management of different agricultural, livestock and forestry activities, demanding specific knowledge and management skills.On the other hand, Balbino et al. (2011) and Reis (2021) state that this system represents a promising strategy for sustainable development, offering benefits such as preservation of soil quality, water conservation, increased animal and plant productivity, recovery of degraded areas, production diversification, reducing environmental impacts and improving animal welfare.
These advantages make the ILPF system an attractive option for farmers looking to increase the sustainability and profitability of their operations, while promoting environmental conservation.However, the complexity of these systems requires an integrated approach and appropriate tools to assist rural producers in decision-making.One of the alternatives is multiobjective optimization, which, according to Arroyo (2002), serves to find solutions that maximize the efficiency of the ILPF system considering several factors, such as social, economic and environmental aspects.Coello (2006) defines a multi-objective optimization problem as one that involves the optimization of two or more objective functions, generally conflicting and subject to constraints.The solution to these problems, known as Pareto-optimal, consists of a set of solutions that represent a compromise between the objectives, where no other viable solution is capable of improving one criterion without worsening the other.Dealing with this category of problems can be challenging, but it is necessary.To overcome this challenge, Pereira (2004) suggests finding solutions that consider priorities or weights associated with objectives, transforming the multi-objective problem into a mono-objective problem.Some classic methods used are: weighted sum method, ε-restricted, goal programming, hierarchical optimization, negotiation, commitment optimization, exponential weighted criterion, weighted product, min-max, among others.Thus, the use of mathematics to solve rural production problems can optimize resources, increase efficiency and maximize results, offering tools and techniques for making strategic decisions.This article aims to develop a mathematical model based on multi-objective optimization to assist farmers who have adopted the ILPF system, seeking to provide rural 4 producers with a tool that considers the various aspects discussed above, by determining the best distribution of areas for crops.agriculture, livestock and forestry.Deb (2001) highlights that, although managers mainly seek the growth and profitability of the enterprise, the pursuit of these objectives results in conflicts due to the presence of several variables and restrictions in real problems, such as seasonality of demand, climatic influences and high costs. of raw material.In this context, the model was created to allow the development of effective management strategies for rural people, considering the balance of economic, social and environmental aspects, in addition to maximizing the use of the ILPF system.
The research adopts an applied approach, taking into account the main inputs required for agricultural production, as well as the costs and revenues associated with this production.
The proposed mathematical model was designed to result in a tool that assists in strategic decision making.It allows a detailed analysis of the distribution of areas for different types of production, always with the aim of achieving maximum efficiency in results.To implement this tool, Excel Solver, from Microsoft Corporation, and version 7.3.5.2 (x64) of LibreOffice Community were used.The choice of these computational tools is due to their accessibility and practicality, essential characteristics for rural producers who seek practical and efficient solutions.When considering the accessibility of the tool, the result allows even producers with less familiarity with advanced technologies to benefit from the proposed model.

MATERIAL IS METHOD
In this section, the process by which the mathematical model was developed is described.It does not apply to all problems faced by rural producers, therefore, it is essential to follow the steps below to avoid unrealistic expectations and optimize its effectiveness in the specific context for which this work is proposed.The construction process of the mathematical model used in this work is in accordance with Vanderplaats (1999), Lobato (2008) and Kagan et al. (2009), who state that an optimization problem is characterized by the presence of the following mathematical elements: 1. Mathematical Model Variables: These are variables that represent the choices or actions that can be taken within the context of the problem, allowing different scenarios to be explored and acting in conjunction with the objective functions; 2. Objective Function: It is the mathematical function that you want to optimize; 3. Constraints: These are the conditions or limitations that the variables must satisfy, defining the space of variables or feasible set of candidate solutions.
To understand the need to obtain the elements that compose it, a survey of the parameters, variables, restrictions and objective functions that will feed the model was carried out, organized in the following sections.

DEFINITION OF VARIABLES AND PARAMETERS
This section is crucial for properly constructing a mathematical model so that it faithfully reflects the reality of the problem.Parameters represent fixed values or constants, while variables are controllable elements of the problem that can be adjusted 3 to find the optimal solution.Correctly identifying these elements is essential to define the scope of the model and the possibilities for optimization.
To facilitate the survey of parameters, below is a list according to CONAB (2020) of the main components of the production cost and revenue calculation: Here, care must be taken with items IV and V as, being fixed, they can cause problems to the system as their values are not proportional, so it is recommended to leave them out of the model.To facilitate the organization, as an example, the relationship between variables and parameters, associated with costs and revenues, follows.In order not to overload the model, only costs related to inputs and labor will be considered, however, using similar reasoning, it can be done for all components of the cost calculation listed above.6 Let P Ij and QIj, respectively, be the price and unit quantity of input j (R$/kg) used in crop i (kg/ha).Thus, the cost of inputs will be: (1) Let PMOj and QMOj be, respectively, the price and unit quantity of labor j (R$/hour) used in crop i (hour/ha).Thus, the cost of services will be: (2) Therefore, the total variable and operational cost of production is given by: (3) Revenue: Production in the ILPF system, as it is diversified, comes from several sources.Thus, let Pj and Qj be, respectively, the price and unit quantity of product j (R$/kg) of crop i (kg/ha).Thus, the recipe will be: (4) In the case of other expenses and income, simply follow the reasoning above and add them to the equations given, respectively, in (1) and (2).7 The main reason for the relationship of variables and parameters is to avoid exchanges between these two elements, as they are essential for developing the model and their roles are distinct.With the parameters and variables defined, the objective functions and restrictions are defined.

FUNCTIONS OBJECTIVES AND RESTRICTIONS
In this section, the restrictions associated with investment resources, available area and non-negativity and the three objective functions will be defined, as follows: 1. Regarding the restrictions -let Ω be the set of points x = (x1, x2, n that satisfy the restrictions: Resource limitation: Let C be the investment capital, then Resource limitation: Let C be the investment capital, then: (5) Area limitation: Let A be the area available for crops, then: (6) Non-negativity constraints: (7) Other restrictions will be inserted according to the proposed problem.(10) Having the parameters, variables, restrictions and objective functions, the mathematical model follows.

MATHEMATICAL MODEL
The mathematical model that meets the problem proposed in this work, and whose solution will be configured using Excel Solver, is defined using the vector function and must be maximized: where: fi is defined as ( 8), ( 9) e (10).

Note:
The minus sign that appears with the second function serves to standardize.F must be maximized and as f2 is associated with the environmental issue (emissions), the problem is assumed to be mathematically equivalent to the problem of minimizing f2.
The model can be summarized by: (12) There are several tools for solving multi-objective problems.Since the purpose of this work is its accessibility not only for people familiar with sophisticated computational tools, therefore, to solve the multi-objective problem according to the model given in ( 7), it will be necessary to transform it into a mathematical model, inserted in the class of 6 scalar problems, whose solution methodology refers to techniques for classical problems with feasibility restrictions and a single objective function.Basic information about the methods is presented below: Programming by goals and weighted sum.
Goal programming method is an optimization technique that allows the simultaneous consideration of several goals and is formulated as follows based on Gomes & Chaves (2004).11 These elements are necessary to compose the model, however, the variables, parameters and restrictions are adjustable according to individual realities.

SOLUTION OF THE PROBLEM
A linear problem can be solved using various commercially and non-commercially available computational tools.
The weighted sum method will be used to solve the problem because, in addition to converting a multi-objective problem into a single-objective problem, it considers weights that can be adjusted to reflect the decision maker's preferences, allowing great flexibility in modeling the objectives.As a computational tool, Microsoft Corporation's Excel Solver, version 7.3.5.2(x64)/LibreOfficeCommunity, was considered due to its practicality and easy access.
The Solver is configured to find the optimal solution that maximizes the merit functions aiming at the best viable combination of values for the decision variables in the format proposed by this work.To use it, follow the steps below and whenever the function fi appears, consider i ∈ {1, 2, 3}: Step 1. Build the table with the main variables and parameters according to Table 1, which should be seen as an Excel table, where the numbering corresponds to the number of lines constructed to better understand the Solver configuration.
Table 1 Model of how the spreadsheet should be filled out in Excel, with suggested organization of the elements that make up the model.12 Table 2 Configuration of fi functions in the Excel spreadsheet.
Source: The author.
Step 2. Insert the objective functions in the same Excel spreadsheet (see Table 1) as defined in the previous section.Note that each fi function defined in Table 2 is associated with one of the factors: economic, environmental and social.
Step 3. Find a solution that optimizes the objective functions, and will be done using Excel Solver and a weighted linear combination of the weights w1, w2 and w3 with w1 + w2 + w3 = 1.Thus, we have Table 3 with the overall function to be optimized.
Note that the function F defined in ( 6) is a vector function, while the function in Table 3 is scalar (not vector) and was called function G.

Table 3
Function to be increased and defined through linear combination.
Source: The author.
Step 4. Define the restrictions, which are in Table 4.Here C is the total cost available for investment and A is the total area available for crops.
Step 5. Configure the Solver: For better understanding, follow Figure 1, following the sequence: 1.In "Objective cell" there must be the G function to be optimized and configured according to Table 3; 2. As the objective is to maximize, check if it is already checked in "Optimize for"; 3.In "Variable cells" are the variables that are in column A. Note in Table 1 that the space is blank and will be filled with the solution found by Solver; *Hi is the amount produced in xi and Ii is the amount used in supplementing animals with i ∈ {1, 2, ..., n}.Source: The author.
4. In "Set of restrictions" there is the information from Table 4 and, in each line of Figure 1, there will be a line from Table 4; 5.It is important that the variables are non-negative, so click on the "Options" button and check "Assume variables as non-negative"; 6.Finally, by clicking on the "Solve" button located in the last line on the right, a "Solver result" rectangle will appear as shown in Figure 1.
Source: The author.
In "Solver result" the increased value will appear.When clicking on "Keep the result", the values of the decision variables, the value of the objective function and the values in the restrictions will appear in Table 1.Now, it's time to evaluate the results and how they can be interpreted by rural producers.
Always observe the variation of the f ′ i s functions, remembering that they, in isolation, reflect the following aspects: economic, social and environmental.

RESULTS AND DISCUSSIONS
In the previous section, the optimization model was presented, highlighting the elements that compose it.To illustrate its practical relevance, an explanatory example will be provided, as shown in Figure 2.This way, a producer will be able to find the optimized solution for their agricultural production, as they will be familiar with the components of the model and will know how to solve it.
To resolve this issue in Excel Solver, follow these steps: * Define variable cells: Define the cells in green colors (cells A1 and A2 in Figure 2) as variables.They will be filled in by Solver and will result in the values that increase the function.Suppose that cell A2 is the value of the area to be occupied by cattle and A3 by grains and that the other information in lines 2 and 3 are associated with these crops, with their appropriate measurements and that these values are per hectare.Units were not added to the table to avoid overloading it.* Configure the function to be optimized: The function F is given by the linear combination as shown in line 5 of Figure 2).Thus, the first part of the model is configured in cell D5 as follows: (17) To obtain the second part of the model, consider the constraints.
* Configure restrictions: Suppose the owner has 300 hectares to be used, has R$200,000.00to invest and needs to plant at least 10 hectares of grains.Like this: • The area used must be less than or equal to 300, therefore, in cell C13, enter = A2 + A3 and we have line 13 in Figure 2; • The resource limitation must be less than or equal to R$ 200,000.00,so, in cell C15, enter = SOMARPRODUTO (A2 : A3; C2 : C3) and we have line 15 of Figure 2; • Considering that you need to have at least 10 hectares of grains, in cell C17 put = A2, you have line 19 in Figure 2.
The spreadsheet is ready to run the solver.
• Run Solver: Open Solver in Excel (usually Tools > Solver).Follow as Figure 3: • The function to be optimized must be in "Objective cell" and it is configured in cell D5; As the objective is to maximize, it is already marked in "Optimize for"; • In "Variable cells" there will be the variables that are in cells A2 and A3; • In "Set of restrictions", the configuration will be in "Cell reference", respectively, in cells C13, C15 and C17; • Click on the "Options" button and check "Assume variables as non-negative"; • Run Solver and you will have Figure 3.
• The Excel Solver revealed, through Figure 3, that the optimal distribution occurs when 168 hectares are allocated to the livestock area and 10 hectares to grains.
Source: The author.
The values of the functions  have been filled in and reflect the economic, social and environmental aspects of the problem.It is important to highlight that, if desired, it is possible to weight one of the functions .This action is equivalent to setting the remaining weights  = 0 for  ≠ .
It is important to note that it was not necessary to use the entire area or all available resources.
It is possible to add more restrictions and/or adjust function weights according to specific needs.If the results are satisfactory, click "Keep the result".Now, it is crucial to evaluate the results obtained and consider how they can be interpreted by rural producers to make the best decision.This process involves a careful analysis of the economic, social and environmental aspects involved.
If the Solver extension is not available for access via the toolbar, you can find the correct guidance to activate the Excel extension at: https://support.microsoft.com/pt-br/office/Carregar-o -add-in-solver-in-excel-612926fc-d53b-46b4-872c-e24772f078ca.
In an ILPF agricultural model, parameters such as the amount of rainfall, temperature or prices of agricultural products can change due to climatic, economic conditions or other external influences.Therefore, for future work, it will be interesting to investigate new proposals with the presence of uncertainties about the parameters.It is very important to evaluate the sensitivity of the model and understand whether small perturbations in the parameters cause small perturbations in the solution initially found.

CONCLUSION
This study proposed a mathematical model based on multi-objective optimization as a tool to assist farmers in managing the ILPF system.Through the use of Excel Solver, it was possible to propose an instrument that allows the distribution of areas for different activities, considering different aspects, such as profit, social impact and environmental sustainability, adjusting the objective functions.The result offers rural producers another alternative for decision-making, contributing to improving the efficiency and viability of the ILPF system.
Furthermore, the mathematical model was developed to be flexible and adaptable to the different conditions and needs of rural producers, allowing adjustments according to specific variables of each region and type of cultivation.This level of customization is essential to maximize the applicability and positive impact of the model in daily agricultural practice.This study highlights the relevance of applying mathematical tools to topics related to agriculture to promote sustainable and efficient practices, aiming to make better use of available resources and the development of more effective agricultural systems.
1. Costing Expenses: Operation with animals; airplane operation; operation with own machines; rental of machines and animals; labor; seeds and seedlings; fertilizers; pesticides; revenue; others; 2. Other Expenses: External transport; administrative costs; storage expenses; processing; production and credit insurance; technical assistance; taxes and fees; 3. Financial Expenses: Interest on financing; 4. Depreciation: Depreciation of improvements and installations; depreciation of machines, implements and irrigation sets; crop depreciation or crop depletion; 5. Other Fixed Costs: Periodic maintenance of improvements and facilities; social charges; fixed capital insurance; lease; 6. Revenue: Revenue in an ILPF system comes from different sources, depending on the combination of crops associated with livestock, farming and forestry.
___________________________________________________________________________ Rev. Gest.Soc.Ambient.| Miami | v.18.n.9 | p.1-18 | e07628 | Optimization: a Mathematical Model to Help Manage the Crop-Livestock-Forestry Integration System ___________________________________________________________________________ Rev. Gest.Soc.Ambient.| Miami | v.18.n.9 | p.1-18 | e07628 | 2024.x1, x2, . . ., xn) ∈ Ω ⊂ ℝ n are the decision variables; fi denote the objective functions; gi are the goals defined by the manager; d + i e d − i are the auxiliary deviation variables for each goal gi associated with fi and reveal how far we will be from the goal (beyond or short) when considering some solution; w + i e w − i are the weights associated with the deviations d + i e d − i. Weighted sum method transforms a multi-objective optimization problem into a singlecriterion optimization problem, by assigning weights to different objectives and is defined based on Zadech (1963).The weighted sum function F, given in (7), is defined as: number of objective functions; fi is the objective function i; wi is the weight assigned to objective function i, with A greater weight assigned to an objective indicates that it is relatively more important in the production process.Multi-Objective Optimization: a Mathematical Model to Help Manage the Crop-Livestock-Forestry Integration System ___________________________________________________________________________ Rev. Gest.Soc.Ambient.| Miami | v.18.n.9 | p.1-18 | e07628 | 2024.

Figure 2
Figure 2Spreadsheet with information necessary to feed the model.

Table 4 Table with
basic restrictions.