## 4 A Mathematical Model
### Remarks for Restricting a Possible Application of the Mathematical Model  
We next provide the following remarks on the model. **Remark 1**. : _The annual costs for the purchase of a tractor (or combine) of grade \(j\) are calculated according to the following formula:_  
\[a_{j}=\frac{A_{j}}{T_{j}}, \tag{3}\]  
_where \(A_{j}\) determines the initial cost of purchasing a tractor (or combine) of grade \(j\); and \(T_{j}\) determines the standard useful life of a tractor (or combine) of grade \(j\)._  
**Remark 2**. : _The annual costs for the purchase of agricultural machinery (tools) of grade \(k\) are calculated according to the following formula:_  
\[b_{k}=\frac{B_{k}}{t_{k}}, \tag{4}\]  
_where \(B_{k}\) determines the initial cost of acquiring an agricultural machine (tool) of grade \(k\); and \(t_{k}\) determines the normative useful life of an agricultural machine (tool) of grade \(k\)._  
_The Formulas (3) and (4) are based on the linear method of calculating depreciation. One should take into account that there are more approaches which may be more precise and can be used if required._  
**Remark 3**. : _If for some reasons the purchase of certain brands of tractors or other agricultural machines is not possible or their parameters (price and service life) are not known, then the value of the \(a_{i}\) (\(b_{k}\)) for these brands is taken to be equal to some large number (for example, \(10^{10}\)). Then, due to the fact that the target function tends to a minimum, these brands will not be among those recommended for purchase._  
**Remark 4**. : _If the agricultural enterprise does not have the opportunity to conclude a leasing contract for the supply of a specific brand of a tractor or other agricultural machinery, then the value of \(a_{i}^{0}\) (\(b_{k}^{0}\)) for these brands is taken to be equal to some large number (for example, \(10^{10}\)). Then, due to the fact that the target function tends to a minimum, these brands are guaranteed not to fall into the optimal plan._  
**Remark 5**. : _The question of choosing a monetary unit in which all the cost parameters of the model will be expressed is quite important. Given that the planning period in this task is one year, it is possible either to make calculations in Belarusian rubles (BYR) and take into account the inflation factor by multiplying all cost parameters by the project price index, or to use in the calculations a nominally more stable monetary unit, for example, USD or EUR. Note that this issue is important only for determining the costs associated with the formation and use of the MT unit, but does not affect the structure of the fleet and the schedule of mechanized work._  
**Remark 6**. : _An important role is played by the correct determination of the values of costs \(a_{i}\), \(b_{k}\), \(a_{i}^{0}\) and \(b_{k}^{0}\). To select their correct values, the marketing service (a specialist or a subdivision replacing it) must conduct a study of the market for agricultural machinery. For this purpose, specialized catalogs, websites of manufacturers and distributors of tractors and agricultural machines, as well as information from exhibitions, forums, etc. can be used._  
It should be noted that at the current stage of development of economic science, most seriously applied tasks in the field of economics are characterized by a large quantity of input data and numerical parameters. Dealing with thousands of variables and hundreds of constraints manually is extremely difficult. In this regard, there is a need to employ specialized software products to solve optimization problems. However, even this does not guarantee the desired results, since the vast majority of optimization packages are based on the use of the method of generalized reduced gradient or its analogues. To apply this approach correctly, both the objective function and the functions involved in writing constraints must be smooth.