Optimality Conditions of Agricultural Production With Fixed Input Costs

We present a computational procedure to maximize the production of a given agricultural crop with limited inputs (water-nitrogen), and where a fixed cost (or expense) of the inputs (general problem of agricultural production) is imposed. Theoretically the procedure is based on the Karush-Kuhn-Tucker optimality conditions and numerically was tested with three different scenarios defined in the literature, for the cultures: Lettuce, Oats, Onions and Melons. In each agricultural scenario considered, it was possible to verify that the procedure is a reliable alternative in making agribusiness economic decisions.


Introduction
Optimizing the production of agricultural crops, with limited inputs and spending on fixed inputs, has as a central tool in its mathematical modeling, the analytical quantification of productivity in response to the total inputs applied. Water and nitrogen are essential for the development of agricultural crops, and when they are correlated with the production obtained, there is the production function or water-nitrogen-culture response function.
The use of analytical production functions in the analysis of the results of agricultural experiments is widespread (Mousinho et al., 2003;Frizzone et al., 2005;Monteiro et al., 2006;Silva et al., 2008;Carvalho et al., 2009;Delgado et al., 2010 andTeodoro et al., 2013). Climatic variations, physical attributes related to the soil, the plant, and many other factors, make it difficult to accurately predict crop yields. In practice, linear and/or quadratic regressions are generated to represent "good approximations" of the response or agricultural production functions.
We will assume that the inputs water and nitrogen are limited above and below. In addition, a limit on ceiling is imposed on the inputs used for the development of each culture. The resulting model presented is a nonlinear programming problem with linear constraints, considering that the objective is a quadratic function in two variables: water and nitrogen.
This work presents a computational model that directly maximizes the production of agricultural crops with limited inputs (water and nitrogen) and fixed spending on inputs. Some numerical tests with data known in the literature are performed, with the purpose of testing the effectiveness of the proposed procedure. We consider four annual agricultural crops (oat, onion and melon lettuce), representatives of the kingdom vegetables, fruits and cereals.

Method
Let be the nonlinear analytical function of production or response of a given culture in relation to the water depth and nitrogen dose ; , are the lower and upper limits of and respectively; the cost of a water depth and the cost of a dose of nitrogen .
Suppose that represents a fixed cost intended exclusively to cover expenditure on water-nitrogen inputs. The problem that maximizes the production of a given agricultural crop with limited inputs and fixed spending on inputs, can be written mathematically as the problem of nonlinear programming with linear constraints: In what follows, (quadratic form in variables and ); and where with . Thus is a strictly concave function and therefore reaches its maximum at the intersection of the two-dimensional box and the plane . Note that we impose a limit on the spending on inputs e , of reais per hectare.
The problem (P) can be written as: In matrix form the previous problem can be written as: . To solve problem (P) using the primal-dual interior points method, we apply the Newton method to the disturbed KKT (barrier) conditions. It is given by: , , and , where is fixed and tends to zero.
The conceptual procedure for solving (1)-(3) works as follows: given a strictly positive initial solution and an initial parameter , the Newton direction associated with the system (1)- (3) and a step length are calculated step in such a way that it allows iteratively to generate new strictly positive points. Then we make a decrease of and repeat the process until a stop criterion is satisfied. The Appendix contains the implemented computational procedure in To computationally test the procedure, several numerical tests were performed using information known in the literature. Table 1 presents in an analytical way the response or production functions of the cultures: Lettuce (Silva et al., 2008), Oats (Frizzone et al., 1995), Onion (Baptestini, JCM, 1982) and Melon (Monteiro et al., 2006).

Results and Discussion
Tables 3, 4, and 5 show the results obtained for each numerical test implemented. Table 3 informs that in the two-dimensional scenario and for a fixed cost of inputs, water-nitrogen of R $ 500, Lettuce reaches its maximum production at the point , Oats for a fixed cost of inputs of R$ 100, at the point , Onions for a fixed cost of inputs of R$ 200, at point and Melon for a fixed input cost of R$ 500, at the point . ISSN 2166-0379 2021  It is possible to show graphically the trajectory of points generated by the implemented procedure, converging to the optimal solution of the problem, for each culture considered. Figure 1, for example, shows for the Oats culture, the sequence of interior points in the two-dimensional box generated by the implemented procedure and converging to the optimal solution . Note that this optimal solution satisfies the plane equation: .     (Table 1) with those obtained in the second scenario (Table 2), we can observe that the lettuce culture, for the same fixed cost , presented equal results in relation to water depth, nitrogen dose and productivity. The same happened with the culture of Oats, for a fixed cost of .

Journal of Agricultural Studies
As for the onion crop, for the same fixed cost , it can be noted that in the second scenario (Table 2), there was a reduction of in relation to the water depth, a slight increase of in relation to the nitrogen dose, and a drop in productivity of . In the case of Melon, for the same fixed cost , we obtained a drop of in relation to the water depth, an increase of in relation to the nitrogen dose, and a significant drop in productivity of . ISSN 2166-0379 2021 Finally, Table 5 presents the results obtained for the third scenario performed in the two-dimensional box . Note that the Lettuce and Oats crops again remain invariant in relation to the first two scenarios.

Journal of Agricultural Studies
For Onion there was an increase in relation to the water depth of around , a reduction in the nitrogen dose of and a decrease in productivity of .
Finally, in relation to Melon, and for a fixed cost of inputs of R$ 500, we achieved the highest productivity and the lowest dose of nitrogen among the three numerical scenarios considered.  It is important to emphasize that the proper management of the water layer is fundamental, considering that the agricultural sector is the largest consumer of water, and that water resources are essential and strategic in the development of agriculture. Also, considering that currently the costs of nitrogen fertilization, specifically nitrogen , are increasingly variable, and that the demand in Brazil grows every day, it is necessary to respect the environmental and soil preservation issues, as a fundamental part for sustainable agriculture. In relation to fixed costs , we will have a fixed amount in R$, for each crop and each scenario considered.

Conclusions
• We present a computational procedure based on the conditions of optimality Karush-Kuhn-Tucker, to maximize the production of a certain agricultural culture with limited inputs (water-nitrogen), and a fixed cost (or expense) of the inputs.
• For each agricultural scenario considered, we were able to confirm that all the optimal solutions of (P) generated by the procedure, satisfy the imposed restrictions. ISSN 2166-0379 2021 • In the three numerical scenarios presented, Lettuce and Oats were the only agricultural crops considered to remain invariant in relation to the water depth, nitrogen dose and productivity.

Journal of Agricultural Studies
• Finally, when making agricultural production decisions (agribusiness), it is important to deal with problems where it is desired to maximize the production of a given agricultural crop with limited inputs (water-nitrogen), and where there must be a fixed cost for the input expenses. .
The previous system of equations is nonlinear with 17 equations and 17 variables, and can be solved using Newton's method.

NEWTON DIRECTION:
We search for a direction: such that: . So, the linear system for determining Newton's direction is given by: (4) (8) . (20)