## 4 A Mathematical Model
### Remarks for Restricting a Possible Application of the Mathematical Model
At the same time, in certain cases, in the process of optimizing the use of the MTF of an agricultural enterprise in agricultural production, there may be objects and processes in the description of which specialists have to use discontinuous functions, i.e., functions that have breaks (of the first or second type). The economic meaning of the gaps in the objective function can be interpreted as follows. If the total actual output of tractors (combines) of this type exceeds the normative output, then depreciation deduction should be adjusted upwards (by multiplying by the corrective coefficient \(\mathrm{g}_{0}\)). For example, if the regulatory annual fund of the combined operating time, according to technical requirements, is 900 h, and it has worked 1000 h, then depreciation should be written off from the cost of finished products at an increased rate, for example, 8% more than in a normal operation. It is especially important to use this approach in intense agro-technical periods. The considered problem belongs to the class of non-smooth optimization problems, since there are gaps of the first type in the objective function. In this regard, it is required to select such tools that are able to successfully solve problems of this type. At the present stage of development of specialized software designed to solve optimization problems, there are a number of problems associated with the processing of non-smooth functions. Algorithms and computer programs that are capable of solving this kind of problem are usually distributed on a commercial basis. Thus, not every agricultural enterprise can afford to purchase such a software product and periodically allocate considerable funds for its renewal. Moreover, modern software products, even the most powerful, have limitations on the dimensionality of the problem being exactly solved. To apply the above mathematical model to the optimization of the MTF of real organizations of the agro-industrial complex, the problem must be brought into a form that can be solved by standard means, in particular, by the Solver package included in the MS Excel environment or by a free version of software products such as GAMS (e.g., version 24.5). There are several ways to solve such problems. It is possible to divide a problem with gaps into a finite number of sub-problems, each of which is considered at a continuous scope of definition in the absence of gaps. Having solved each of the sub-problems, it is possible to choose the best solutions obtained and it may be practically sufficient for the original problem with gaps. In the Equation (1), \(\tau\) denotes a given constant whose value is greater than 1. Note that according to Equation (1), the coefficient of \(g_{j}\) is a discontinuous function that depends on the variables \(x_{ijkt}\) and \(y_{ijkt}\). It is required to select a continuously differentiated function \(\widetilde{g}_{j}\) of variables \(x_{ijkt}\) and \(y_{ijkt}\) that would accurately approximate the change in the coefficient of \(g_{j}\). To do this, we transform the objective function so that it becomes smooth, i.e., we eliminate the gaps of the first type. To construct such a function \(\widetilde{g}_{j}\) we propose to use the following logistic function:  
\[\varphi(z)=\frac{1}{1+\exp(-z)} \tag{5}\]  
The graph of the Function (5) is shown in Figure 1.  
Let \(f(X)\) denote a function that depends on the multidimensional array \(X\). The value of this function is denoted as \(y\). Let \(h(y)\) denote a function dependent on the scalar \(y\) (which describes the values of the function \(f(X)\)) given by the formula:  
\[h(y)= \begin{cases}c,\text{ }if\ y\leq a,\\ d,\text{ }if\ y>a.\end{cases}  \tag{6}\]Ultimately, it is necessary to obtain a functional dependence of the form \(h[f(x)]\). By means of elementary transformations (compression, stretching and parallel transfer), it is possible to select a function \(\stackrel{{\sim}}{{h}}(y)\) such that it can be used for the approximation of the function \(g_{j}\).