Principles of constriction of mathematical models and their classification in economics

A mathematical model is a system of mathematical relationships, approximately, describing the process or system under study in abstract form.

Conducting an operational study, building and calculating a mathematical model allows us to analyze the situation and select optimal solutions for managing it or substantiate the proposed solutions. The use of mathematical models is necessary in cases where the problem is complex, depends on a large number of factors that affect its solution in different ways.

The use of mathematical models allows you to pre-select the optimal or close to them solutions for certain criteria. They are scientifically based, and the decision maker can be guided by them when choosing a final decision. It should be understood that there are no optimal solutions. Any solution obtained with the help of a mathematical model is optimal according to one or several criteria proposed by the task director and the researcher.

Currently, mathematical models are used for analysis, forecasting and selection of optimal solutions in various areas of the economy. These are planning and operational management of production, human resources management, inventory management, resource allocation, planning and placement of facilities, project management, investment distribution and many more.

The basic principles of the construction of a mathematical model:

• 1)    It is necessary to measure the accuracy and detail of the model of the first, with the accuracy of the initial data available to the

researcher, and second, with the results that one is afraid to obtain.

• 2)    The mathematical model should reflect the essential features of the phenomenon under study and should not overly simplify it.

• 3)    A mathematical model cannot be fully adequate to a real phenomenon; therefore, to study it, it is better to use several models, and the construction of which used various mathematical methods. If this results in similar results, the study ends. If the results are very different, then the problem should be revised.

• 4)    Any complex system is always subject to small external internal influences, therefore, the mathematical model must be stable (preserve the properties and structure under these effects).

By the number of efficiency criteria, mathematical models are divided into single criteria and many criteria (contain two or more criteria).

By taking into account unknown factors, mathematical models are divided into deterministic, stochastic, and models with uncertainty elements.

In stochastic models, unknown factors are random variables for which distribution functions and various statistical characteristics are known (expectation, variance, standard deviation, etc.). These models include:

• 1)    Stochastic programming models in which either the objective function or the constraints include random variables;

• 2)    Models of the theory of random processes, designed to study the processes whose state at each moment in time is a random variable;

• 3)    Models of queuing theory, in which multi-channel systems are studied, servicing requirements also for stochastic models;

• 4)    Models of utility theory, search and decision making can be used to simulate situations depending on factors for which it is impossible to collect statistical data and values of which are not determined models with el-

• ements of vagueness.

In models of game theory, the task is represented as a game in which several players participate, pursuing different goals, for example, organizing an enterprise under competitive conditions.

In simulation models, the real process takes place in machine time and the results of random effects on it are traced, for example, the organization of the production process.

In deterministic models unknown factors are not taken into account. Despite the seeming simplicity of these models, many practical tasks, including most economic tasks, are reduced to them.

By the form of the objective function and constraints, deterministic models are divided

In linear models, the objective function and constraints are linear in the control variables. Construction and calculation of linear models are the most developed section of mathematical modeling, therefore, other tasks are often reduced to them either at the stage of formulation or in the process of solution. For non-linear models are models in which either the objective function, the constraints (or all restrictions) are non-linear to variables. For non-linear models there is no single dependence depending on the type of nonlinearity, properties, it is possible to propose various ways of solving the problem and so that for the nonlinear set, there exists a calculation method. Dynamic models take into account time dependence. The criterion for dynamic models may be the most common function, but certain properties must be observed for it. The calculation of dynamic models of a specific problem requires the development of solutions. Graphic models are used when it is more convenient to solve a problem using a graphical structure.

into: linear, nonlinear, dynamic, and graphic.