The concept of a mathematical model. Stages of mathematical modeling

Lecture #1

Introduction. The concept of mathematical models and methods

Section 1 Introduction

2. Methods for constructing mathematical models. The concept of a systematic approach. one

3. Basic concepts of mathematical modeling of economic systems.. 4

4. Methods of analytical, simulation and natural modeling. 5

Security questions.. 6

1. Content, goals and objectives of the discipline "Modeling methods"

This discipline is devoted to the study of modeling methods and practical application acquired knowledge. The purpose of the discipline is to teach students general issues of modeling theory, methods for constructing mathematical models and formal description of processes and objects, the use of mathematical models for conducting computational experiments and solving optimization problems using modern computing tools.

The tasks of the discipline include:

To acquaint students with the basic concepts of the theory of mathematical modeling, systems theory, similarity theory, theory of experiment planning and processing of experimental data used to build mathematical models,

To give students skills in the field of setting a modeling problem, mathematical description of objects / processes /, numerical methods for implementing mathematical models on a computer and solving optimization problems.

As a result of studying the discipline, the student must master the methods of mathematical modeling of processes and objects from the formulation of the problem to the implementation of mathematical models on a computer and the presentation of the results of the study of models.

The course of discipline is designed for 12 lectures and 12 practical works. As a result of studying the discipline, the student must master the methods of mathematical modeling from the formulation of the problem to the implementation of mathematical models on a computer

2. Methods for constructing mathematical models. The concept of a systematic approach

5. Solution of the problem.

The consistent use of operations research methods and their implementation on modern information and computer technology makes it possible to overcome subjectivism, to exclude the so-called volitional decisions based not on strict and accurate consideration of objective circumstances, but on random emotions and personal interest of managers at various levels, who, moreover, do not can agree on these volitional decisions.

System analysis makes it possible to take into account and use in management all the available information about the managed object, to coordinate the decisions made in terms of an objective, and not a subjective, criterion of effectiveness. Saving on calculations when driving is the same as saving on aiming when shooting. However, the computer not only makes it possible to take into account all the information, but also saves the manager from unnecessary information, and allows all the necessary information to bypass the person, presenting him only the most generalized information, the quintessence. The systems approach in economics is effective in itself, without the use of a computer, as a research method, while it does not change previously discovered economic laws, but only teaches how to use them better.

4. Methods of analytical, simulation and natural modeling

Simulation is a powerful technique scientific knowledge, which replaces the object under study with a simpler object called a model. The main varieties of the modeling process can be considered its two types - mathematical and physical modeling. In physical (natural) modeling, the system under study is replaced by another material system corresponding to it, which reproduces the properties of the system under study with the preservation of their physical nature. An example of this type of modeling is a pilot network, which is used to study the fundamental possibility of building a network based on certain computers, communication devices, operating systems and applications.

The possibilities of physical modeling are quite limited. It allows solving individual problems by specifying a small number of combinations of the studied parameters of the system. Indeed, in full-scale simulation of a computer network, it is almost impossible to check its operation for options using various types of communication devices - routers, switches, etc. Verification in practice of about a dozen different types of routers is associated not only with great effort and time, but also with considerable material cost.

But even in cases where network optimization does not change the types of devices and operating systems, but only their parameters, real-time experiments for huge amount all possible combinations of these parameters is practically impossible in the foreseeable future. Even a simple change in the maximum packet size in any protocol requires reconfiguring the operating system in hundreds of computers on the network, which requires a lot of work from the network administrator.

Therefore, when optimizing networks, in many cases it is preferable to use mathematical modeling. A mathematical model is a set of relationships (formulas, equations, inequalities, logical conditions) that determine the process of changing the state of the system depending on its parameters, input signals, initial conditions and time.

Simulation models are a special class of mathematical models. Such models are a computer program that, step by step, reproduces the events that occur in a real system. With regard to computer networks, their simulation models reproduce the processes of generating messages by applications, splitting messages into packets and frames of certain protocols, delays associated with the processing of messages, packets and frames within the operating system, the process of obtaining access by a computer to a shared network environment, the process of processing incoming packets by a router etc. When simulating a network, it is not necessary to purchase expensive equipment - its work is simulated by programs that accurately reproduce all the main features and parameters of such equipment.

The advantage of simulation models is the ability to replace the process of changing events in the system under study in real time with an accelerated process of changing events at the pace of the program. As a result, in a few minutes, you can reproduce the operation of the network for several days, which makes it possible to evaluate the performance of the network in a wide range of variable parameters.

The result of the simulation model is statistical data collected during the monitoring of ongoing events on the most important characteristics of the network: response times, utilization rates of channels and nodes, the probability of packet loss, etc.

There are special simulation languages ​​that facilitate the process of creating a software model compared to using universal programming languages. Examples of simulation languages ​​are languages ​​such as SIMULA, GPSS, SIMDIS.

There are also simulation modeling systems that focus on a narrow class of systems under study and allow you to build models without programming.

test questions

LECTURE 4

Definition and purpose of mathematical modeling

Under model(from the Latin modulus - measure, sample, norm) we will understand such a materially or mentally represented object that, in the process of cognition (study), replaces the original object, retaining some of its typical features that are important for this study. The process of building and using a model is called modeling.

essence mathematical modeling (MM) consists in replacing the object (process) under study with an adequate mathematical model and the subsequent study of the properties of this model using either analytical methods or computational experiments.

Sometimes it is more useful, instead of giving strict definitions, to describe a particular concept with a specific example. Therefore, we illustrate the above definitions of MM using the example of the problem of calculating the specific impulse. In the early 1960s, scientists were faced with the task of developing rocket fuel with the highest specific impulse. The principle of rocket movement is as follows: liquid fuel and oxidizer from the rocket tanks are fed into the engine, where they are burned, and the combustion products are released into the atmosphere. From the law of conservation of momentum, it follows that in this case the rocket will move with speed.

The specific impulse of a fuel is the resulting impulse divided by the mass of the fuel. The experiments were very expensive and led to systematic damage to the equipment. It turned out that it is easier and cheaper to calculate the thermodynamic functions of ideal gases, to calculate with their help the composition of the emitted gases and the plasma temperature, and then the specific impulse. That is, to carry out the MM of the fuel combustion process.

The concept of mathematical modeling (MM) today is one of the most common in the scientific literature. The vast majority of modern theses and dissertations are associated with the development and use of appropriate mathematical models. Computer MM today is an integral part of many areas of human activity (science, technology, economics, sociology, etc.). This is one of the reasons for today's shortage of specialists in the field of information technology.

The rapid growth of mathematical modeling is due to the rapid improvement of computer technology. If even 20 years ago only a small number of programmers were engaged in numerical calculations, now the amount of memory and speed of modern computers, which make it possible to solve problems of mathematical modeling, are available to all specialists, including university students.

In any discipline, a qualitative description of the phenomena is first given. And then - quantitative, formulated in the form of laws that establish relationships between various quantities (field strength, scattering intensity, electron charge, ...) in the form of mathematical equations. Therefore, we can say that in each discipline there is as much science as there are mathematicians in it, and this fact allows us to successfully solve many problems using mathematical modeling methods.

This course is designed for students majoring in applied mathematics who are completing their thesis under the supervision of leading scientists working in various fields. Therefore, this course is necessary not only as a teaching material, but also as a preparation for a thesis. To study this course, we will need the following sections of mathematics:

1. Equations of mathematical physics (Kantian mechanics, gas and hydrodynamics)

2. Linear algebra (the theory of elasticity)

3. Scalar and vector fields (field theory)

4. Probability theory ( quantum mechanics, statistical physics, physical kinetics)

5. Special features.

6. Tensor analysis (theory of elasticity)

7. Mathematical analysis

MM in natural science, engineering, and economics

Let us first consider the various branches of natural science, technology, economics, in which mathematical models are used.

natural science

Physics, which establishes the basic laws of natural science, has long been divided into theoretical and experimental. The derivation of equations describing physical phenomena, engaged in theoretical physics. Thus, theoretical physics can also be considered one of the areas of mathematical modeling. (Recall that the title of the first book on physics - "The Mathematical Principles of Natural Philosophy" by I. Newton can be translated into modern language as "Mathematical Models of Natural Science".) Based on the laws obtained, engineering calculations are carried out, which are carried out in various institutes, firms, design bureaus. These organizations develop technologies for the manufacture of modern products that are science-intensive. Thus, the concept of science-intensive technologies includes calculations using appropriate mathematical models.

One of the most extensive branches of physics - classical mechanics(sometimes this section is called theoretical or analytical mechanics). This section of theoretical physics studies the motion and interaction of bodies. Calculations using the formulas of theoretical mechanics are necessary when studying the rotation of bodies (calculating the moments of inertia, gyrostats - devices that keep the axes of rotation stationary), analyzing the movement of a body in a vacuum, etc. One of the sections of theoretical mechanics is called the theory of stability and underlies many mathematical models describing the movement of aircraft, ships, rockets. Sections of practical mechanics - courses "Theory of machines and mechanisms", "Machine parts", are studied by students of almost all technical universities (including MGIU).

Theory of elasticity- part of a section continuum mechanics, which assumes that the material of an elastic body is homogeneous and continuously distributed throughout the volume of the body, so that the smallest element cut out of the body has the same physical properties as the whole body. The application of the theory of elasticity - the course "strength of materials", is studied by students of all technical universities (including MGIU). This section is required for all strength calculations. Here is the calculation of the strength of the hulls of ships, aircraft, missiles, the calculation of the strength of steel and reinforced concrete structures of buildings, and much more.

Gas and hydrodynamics, as well as the theory of elasticity - part of the section continuum mechanics, considers the laws of motion of liquid and gas. The equations of gas and hydrodynamics are necessary when analyzing the movement of bodies in a liquid and gaseous medium (satellites, submarines, rockets, shells, cars), when calculating the outflow of gas from the nozzles of rocket and aircraft engines. Practical Application of Fluid Dynamics – Hydraulics (Brake, Rudder,…)

The previous sections of mechanics considered the movement of bodies in the macrocosm, and the physical laws of the macrocosm are not applicable in the microcosm, in which particles of matter move - protons, neutrons, electrons. Here, completely different principles operate, and to describe the microworld, it is necessary to quantum mechanics. The basic equation describing the behavior of microparticles is the Schrödinger equation: . Here, is the Hamiltonian operator (Hamiltonian). For a one-dimensional particle motion equation https://pandia.ru/text/78/009/images/image005_136.gif" width="35" height="21 src=">-potential energy. The solution of this equation is a set of energy eigenvalues and eigenfunctions..gif" width="55" height="24 src=">– probability density. Quantum mechanical calculations are needed for the development of new materials (microcircuits), the creation of lasers, the development of spectral analysis methods, etc.

A large number of tasks are solved kinetics describing the motion and interaction of particles. Here and diffusion, heat transfer, the theory of plasma - the fourth state of matter.

statistical physics considers ensembles of particles, allows you to say about the parameters of the ensemble, based on the properties of individual particles. If the ensemble consists of gas molecules, then the properties of the ensemble derived by the methods of statistical physics are the equations of the gas state well known from high school: https://pandia.ru/text/78/009/images/image009_85.gif" width="16" height="17 src=">.gif" width="16" height="17">-molecular weight of the gas. K is the Rydberg constant. Statistical methods are also used to calculate the properties of solutions, crystals, and electrons in metals. The MM of statistical physics is the theoretical basis of thermodynamics, which underlies the calculation of engines, heat networks and stations.

Field theory describes by MM methods one of the main forms of matter - the field. In this case, electromagnetic fields are of primary interest. Equations electromagnetic field(electrodynamics) were derived by Maxwell:, , , . Here and https://pandia.ru/text/78/009/images/image018_44.gif" width="16" height="17"> - charge density, - current density. Electrodynamic equations underlie propagation calculations electromagnetic waves necessary to describe the propagation of radio waves (radio, television, cellular communications), explain the operation of radar stations.

Chemistry can be represented in two aspects, highlighting descriptive chemistry - the discovery of chemical factors and their description - and theoretical chemistry - the development of theories that allow generalizing the established factors and presenting them in the form of a specific system (L. Pauling). Theoretical chemistry is also called physical chemistry and is, in essence, a branch of physics that studies substances and their interactions. Therefore, everything that has been said about physics fully applies to chemistry. Sections of physical chemistry will be thermochemistry, which studies the thermal effects of reactions, chemical kinetics (reaction rates), quantum chemistry (the structure of molecules). At the same time, the problems of chemistry are extremely complex. So, for example, to solve the problems of quantum chemistry - the science of the structure of atoms and molecules, programs are used that are comparable in volume to the air defense programs of the country. For example, in order to describe a UCl4 molecule, consisting of 5 atomic nuclei and +17 * 4) electrons, you need to write down the equation of motion - equations in partial derivatives.

Biology

Mathematics really came into biology only in the second half of the 20th century. The first attempts to mathematically describe biological processes are related to models of population dynamics. A population is a community of individuals of the same species occupying a certain area of ​​space on Earth. This area of ​​mathematical biology, which studies the change in population size under various conditions (the presence of competing species, predators, diseases, etc.), continued to serve as a mathematical testing ground on which mathematical models were "performed" in various fields of biology. Including models of evolution, microbiology, immunology and other areas related to cell populations.
The very first famous model, formulated in a biological setting, is the famous Fibonacci series (each subsequent number is the sum of the previous two), which Leonardo from Pisa cites in his work in the 13th century. This is a series of numbers describing the number of pairs of rabbits that are born each month, if the rabbits start breeding from the second month and produce a pair of rabbits each month. The row represents a sequence of numbers: 1, 1, 2, 3, 5, 8, 13, 21, ...

8, 13, …

Another example is the study of ionic transmembrane transport processes on an artificial bilayer membrane. Here, in order to study the laws of formation of a pore through which an ion passes through the membrane into the cell, it is necessary to create a model system that can be studied experimentally, and for which a well-developed physical description can be used.

A classic example of MM is also the Drosophila population. An even more convenient model is viruses, which can be propagated in a test tube. The methods of modeling in biology are the methods of the dynamic theory of systems, and the means are differential and difference equations, methods of the qualitative theory of differential equations, simulation modeling.
Goals of modeling in biology:
3. Elucidation of the mechanisms of interaction between the elements of the system
4. Identification and verification of model parameters using experimental data.
5. Assessment of the stability of the system (model).

6. Prediction of system behavior under various external influences, various ways management and so on.
7. Optimal control of the system in accordance with the chosen optimality criterion.

Technique

A large number of specialists are engaged in the improvement of technology, who in their work rely on the results of scientific research. Therefore, the MM in technology are the same as the MM in natural science, which were discussed above.

Economy and social processes

It is generally accepted that mathematical modeling as a method of analyzing macroeconomic processes was first used by the physician of King Louis XV, Dr. François Quesnay, who in 1758 published the work "Economic Table". In this work, the first attempt was made to quantitatively describe the national economy. And in 1838 in the book O. Cournot"Investigation of the mathematical principles of the theory of wealth" quantitative methods were first used to analyze competition in the market for goods in various market situations.

Malthus's theory of population is also widely known, in which he proposed the idea that population growth is far from always desirable, and this growth is faster than the growing possibilities of providing the population with food. The mathematical model of such a process is quite simple: Let - population growth over time https://pandia.ru/text/78/009/images/image027_26.gif" width="15" height="24"> the number was equal to . and are the coefficients taking into account the birth and death rates (persons/year).

https://pandia.ru/text/78/009/images/image032_23.gif" width="151" height="41 src=">Instrumental and mathematical methods" href="/text/category/instrumentalmznie_i_matematicheskie_metodi/" rel ="bookmark">mathematical methods of analysis (for example, in recent decades in the humanities, mathematical theories of the development of culture appeared, mathematical models of mobilization, the cyclical development of sociocultural processes, a model of interaction between the people and the government, a model of the arms race, etc., were built and studied).

In the most general terms, the MM process of socio-economic processes can be conditionally divided into four stages:

formulating a system of hypotheses and developing a conceptual model; development mathematical model; analysis of the results of model calculations, which includes their comparison with practice; formulation of new hypotheses and refinement of the model in case of discrepancy between the results of calculations and practical data.

Note that, as a rule, the process of mathematical modeling is cyclical, since even in the study of relatively simple processes it is rarely possible to build an adequate mathematical model from the first step and select its exact parameters.

At present, the economy is considered as a complex developing system, for the quantitative description of which dynamic mathematical models of varying degrees of complexity are used. One of the areas of research of macroeconomic dynamics is associated with the construction and analysis of relatively simple nonlinear simulation models that reflect the interaction of various subsystems - the labor market, the goods market, the financial system, the natural environment, etc.

The theory of catastrophes is successfully developing. This theory considers the question of the conditions under which a change in the parameters of a nonlinear system causes a point in the phase space that characterizes the state of the system to move from the region of attraction to the initial equilibrium position to the region of attraction to another equilibrium position. The latter is very important not only for the analysis of technical systems, but also for understanding the sustainability of socio-economic processes. In this regard, the findings about the significance of the study of nonlinear models for management. In the book "The Theory of Catastrophes", published in 1990, he, in particular, writes: "... the current restructuring is largely due to the fact that at least some feedback mechanisms (fear of personal destruction) have begun to operate."

(model parameters)

When building models of real objects and phenomena, one often encounters a lack of information. For the object under study, the distribution of properties, the parameters of the impact and the initial state are known with varying degrees of uncertainty. When building a model, the following options for describing uncertain parameters are possible:

Classification of mathematical models

(implementation methods)

MM implementation methods can be classified according to the table below.

Under mathematical modeling, in the narrow sense of the word, they understand the description in the form of equations and inequalities of real physical, chemical, technological, biological, economic and other processes. In order to use mathematical methods for the analysis and synthesis of various processes, it is necessary to be able to describe these processes in the language of mathematics, that is, describe them in the form of a system of equations and inequalities.

Mathematical methods act as a way of obtaining new knowledge about an object. This doesn't just apply to systems. Looking back, turning to the history of science, the researcher sees that the entire dynamics of science can be viewed as a continuous process of building new, more advanced and powerful models. The notion that “all knowledge is a simulation” (N.Amosov) has taken root. Under the influence of the general theory of systems, there was a rethinking, reassessment and classical ideas. The concept of mathematical modeling began to be interpreted so broadly that it included all the formalization and mathematization of knowledge. " A mathematical model is only a special method of description that allows using the formal logical apparatus of mathematics for analysis."(Moiseev N.N., 1973).

But models of complex and large systems are something else fundamentally and qualitatively. The analytical, formal-logical apparatus is no longer enough here. Within the framework of this work, a mathematical model is understood as any mathematical construction that is a large and/or complex dynamic system and has the property of structural-functional isomorphism with respect to the system under study (original system).

There is a profound difference between modeling and obtaining a quantitative or qualitative result by mathematical methods. The use of mathematics becomes possible when it becomes clear what and for what purpose to determine, evaluate, measure, what and how to process by mathematical methods. The model does not serve these purposes. Mathematical modeling is not the application of a mathematical tool to an object, it is not the solution of specific problems by mathematical means. This is the construction by formal methods and means of an abstract object isofunctional to the object under study for the subsequent application of mathematical methods of quantitative and qualitative analysis. At the same time, the use of mathematics as a language (metatheory) in modeling gives the obtained conclusions an evidentiary force. The activity of building models does not belong to mathematics and is performed (should be performed) not by mathematicians, but by specialists in a particular field of knowledge.

To build a system model, those meaningful empirical ideas, those descriptive sciences that precede the emergence of formalized sciences, are needed. These descriptions are not included in the form of components in a formalized science, but only facilitate the process of formalization, enrich the heuristic possibilities of formalization. The model does not require a preliminary description of the modeled object, because it is itself a form of description.

The relation of model and reality is different from the relation of reality and mathematical formula. The formula is a hieroglyph, a sign of reality. The model is reality itself. It can be objected that a physicist or mathematician perfectly feels the dynamics, the real relationships that are hidden behind the formula, does not perceive it as a hieroglyph, and, moreover, modern mathematics is far from simple and not only a formula. And yet, a scientist cannot think with formulas. Another thing is the model. She has dynamics, she lives (not only figuratively, sometimes in literally the words). The researcher can think in models, he gets the opportunity of figurative thinking. In the world of models, the artistic and logical perception of reality merges.

Mathematical modeling does not exclude the use of classical mathematics, moreover, as part of the model, mathematics receives the power and universality of penetration, which it was deprived of in the classical era.

If we consider some object as a whole, given by its external properties, we can effectively use analytical methods of description for processes occurring outside this whole. But as soon as we set the task of describing the internal description of a large and/or complex system, describing the interactions between its parts, elements and subsystems using the methods of classical mathematics, we immediately encounter insurmountable difficulties.

On the other hand, an attempt to describe a certain system by procedural methods, in general, without penetrating into its internal structure, into its structure and functions of elements, as a rule, will not lead to a significant result. Each method has its place.

In the mathematics of analytic structures, we must first understand and then describe. In modeling, in the mathematics of algorithmic processes, the very process of describing what is not yet understood often becomes a means of understanding.

As a methodology for scientific research, mathematical modeling combines the experience of various branches of science about nature and society, applied mathematics, computer science and system programming to solve fundamental problems. Mathematical modeling of objects of complex nature - a single end-to-end development cycle from fundamental research problems to specific numerical calculations of the object's performance indicators. The result of the development is a system of mathematical models that describe qualitatively heterogeneous patterns of the functioning of an object and its evolution as a whole as a complex system under various conditions. Computational experiments with mathematical models provide initial data for evaluating the performance indicators of an object. Therefore, mathematical modeling as a methodology for organizing scientific expertise of major problems is indispensable in the development of national economic solutions. (First of all, this applies to the modeling of economic systems). At its core, mathematical modeling is a method for solving new complex problems, so research on mathematical modeling should be ahead of the curve. It is necessary to develop new methods in advance, to train personnel who know how to apply these methods with knowledge to solve new practical problems. A mathematical model can arise in three ways:1. As a result of direct study of the real process. Such models are called phenomenological.2. As a result of the process of deduction. New model is a special case of some general model. Such models are called asymptotic.3. As a result of the process of induction. The new model is a generalization of elementary models. Such models are called ensemble models. The modeling process begins with the modeling of a simplified process, which, on the one hand, reflects the main qualitative phenomena, and on the other hand, allows for a fairly simple mathematical description. As research deepens, new models are built that describe the phenomenon in more detail. Factors that are considered secondary at this stage are discarded. However, at the next stages of the study, as the model becomes more complex, they can be included in the consideration. Depending on the purpose of the study, the same factor can be considered the main or secondary factor. The mathematical model and the real process are not identical. As a rule, a mathematical model is built with some simplification and with some idealization. It only approximately reflects the real object of study, and the results of the study of a real object by mathematical methods are approximate. The accuracy of the study depends on the degree of adequacy of the model and the object and on the accuracy of the applied methods of computational mathematics. The scheme for constructing mathematical models is as follows:1. Selection of the parameter or function to be investigated.2. The choice of the law to which this value obeys.3. The choice of the area in which you want to study this phenomenon.

The theoretical discipline becomes exact science when it operates with quantitative characteristics. The qualitative description of the model is followed by the second phase of abstraction - the quantitative description of the model. Even Galileo Galilei said that the book of nature is written in the language of mathematics. Immanuel Kant proclaimed that "in any science there is as much truth as there is mathematics in it." And David Hilbert owns the words: “Mathematics − basis of all exact natural science.

Mathematical modeling is a theoretical and experimental method of cognitive and creative activity, it is a method of research and explanation of phenomena, processes and systems (original objects) based on the creation of new objects - mathematical models.

A mathematical model is commonly understood as a set of relationships (equations, inequalities, logical conditions, operators, etc.) that determine the characteristics of the states of the modeling object, and through them the output values ​​- reactions, depending on the parameters of the original object, input actions , initial and boundary conditions, and time.

A mathematical model, as a rule, takes into account only those properties (attributes) of the original object that reflect, determine and are of interest from the point of view of the goals and objectives of a particular study. Therefore, depending on the goals of modeling, when considering the same original object from different points of view and in different aspects, the latter may have different mathematical descriptions and, as a result, be represented by different mathematical models.

Taking into account the above, we will give the most general, but at the same time rigorous constructive definition of a mathematical model, formulated by P.J. Cohen.

Definition 4.1. A mathematical model is a formal system that is a finite collection of symbols and is completely strict rules operating with these symbols in conjunction with the interpretation of the properties of a particular object by some relations, symbols or constants.

As follows from the above definition, the final collection of symbols (alphabet) and completely strict rules for operating these symbols (the "grammar" and "syntax" of mathematical expressions) lead to the formation of abstract mathematical objects (AMO). Only interpretation makes this abstract object a mathematical model.

A mathematical model is a quantitative formalization of abstract ideas about the phenomenon or object being studied.

Mathematical models can be represented by various mathematical means:

· real or complex quantities;

· vectors, matrices;

· geometric images;

· inequalities;

· functions and functionalities;

· sets, various equations;

· probability distribution functions, statistics, etc.

"In physical science − Thompson wrote, − in the study of any object, the first and most essential step is to find the principles of numerical evaluation and practical methods for measuring some quantity inherent in this object.

The transition from the first to the second phase of abstraction, i.e. from a physical model to a mathematical one often frees the model from the specific features inherent in a given phenomenon or object under study. Many mathematical models, having lost their physical or technical shell, acquire universality, i.e. the ability to quantitatively describe processes that are different in their physical nature or according to the technical purpose of objects. This manifests one of the most important properties of the mathematical formalization of the subject of research, due to which, when setting and solving applied problems, in most cases it is not necessary to create a new mathematical apparatus, but you can use the existing one, with the improvement and interpretation necessary for a particular situation. Thus, one mathematical model can be used to solve a large number of particular, specific problems, and in this sense it expresses one of the main practical purposes of the theory.

Of course, the construction of a physical model is often inextricably linked with the construction of a mathematical model, and both of these processes represent two sides of a single process of abstraction.

We are surrounded by complex technical objects (technical systems) created by man. In the process of designing a new or upgrading an existing technical system, the tasks of calculating the parameters and studying the processes in this system are solved. When carrying out multivariate calculations, the real system is replaced by a model. In a broad sense, a model is defined as a reflection of the most essential properties of an object.

Definition 4.2 . A mathematical model of a technical object is a set of mathematical objects and relations between them that adequately reflects the properties of the object under study that are of interest to the researcher (engineer).

The model can be represented in various ways.

Model View Forms

· invariant - recording the model relations using the traditional mathematical language, regardless of the method of solving the model equations;

· analytical - record of the model in the form of the result of the analytical solution of the initial equations of the model;

· algorithmic - recording the relations of the model and the selected numerical method of solution in the form of an algorithm;

· schematic (graphic) - representation of the model in some graphic language (for example, the language of graphs, equivalent circuits, diagrams, etc.);

· physical;

· analog;

Mathematical modeling is the most universal description of processes.

The concept of mathematical modeling sometimes includes the process of solving a problem on a computer (which, in principle, is not entirely true, since solving a problem on a computer involves, among other things, the creation of an algorithmic and software model that implements the calculation in accordance with the mathematical model).

Definition 4.3.MM is the image of the object under study, created in the mind of the subject-researcher with the help of certain formal (mathematical) systems in order to study (evaluate) certain properties of this object.

Let some object Q has some property of interest to us C 0 .

To obtain a mathematical model describing this property, it is necessary:

1. Determine the indicator of this property(those. determine the measure of a property in some system of measurement).

2. Set property list C 1 , ..., ~ m, with which the propertyFROM 0 connected by some relationships (these can be internal properties of the object and properties external environment affecting the object).

3. Describe the properties of the external environment in the chosen format system as external factors х 1 , ..., x n , affecting the desired indicatorY,internal properties of the object as z parameters 1 , ..., z r , and unaccounted properties are assigned to the group of unaccounted factors .

4. Find out, if possible, the relationship betweenYand all factors and parameters taken into account, and draw up a mathematical description(model).

A real object is characterized by the following functional relationship between the indicators of its properties:

However, the model displays only those factors and parameters of the original object that are essential for solving the problem under study. In addition, measurements of significant factors and parameters almost always contain errors caused by the inaccuracy of measuring instruments and ignorance of some factors. Because of this, MM is only an approximate description of the properties of the object under study.

The mathematical model can also be defined as abstraction studied real entities.

Models usually differ from the originals in the nature of their internal parameters. The similarity lies in the adequacy of the reaction Y model and original to change external factors. Therefore, in the general case, the mathematical model is a function

where are the internal parameters of the model, adequate to the parameters of the original.

Depending on the applied methods of mathematical description of the studied objects (phenomena, processes), MMs can be analytical, logical, graphical, automatic, etc.

The main issue of mathematical modeling is the question of how accurately compiled MM reflects the relationship between factors taken into account, parameters and indicator Y evaluated property of a real object, i.e. how exactly equation (4.2) corresponds to equation (4.1). Sometimes equation (4.2) can be obtained immediately in an explicit form, for example, in the form of a system of differential equations, or in the form of other explicit mathematical relations.

In more complex cases, the form of equation (4.2) is unknown, and the task of the researcher is, first of all, to find this equation. At the same time, among the variable parameters , are all considered external factors and parameters of the object under study, and among the desired parameters include the internal parameters of the model , linking factors , with the indicator Y"the most plausible relation. The theory of experiment deals with the solution of this problem. The essence of this theory is that, based on selective measurements of the values ​​of the parameters , and the indicator Y", find the parameters for which the function (4.2) most accurately reflects the real regularity (4.1).

The concept of model and simulation.

Model in a broad sense- this is any image, analogue of a mental or established image, description, diagram, drawing, map, etc. of any volume, process or phenomenon, used as its substitute or representative. The object, process or phenomenon itself is called the original of this model.

Modeling - this is the study of any object or system of objects by building and studying their models. This is the use of models to determine or refine the characteristics and rationalize the ways of constructing newly constructed objects.

Any method of scientific research is based on the idea of ​​modeling, while theoretical methods use various kinds of symbolic, abstract models, while experimental methods use subject models.

In the study of a complex real phenomenon, it is replaced by some simplified copy or scheme, sometimes such a copy serves only to remember and at the next meeting to find out the desired phenomenon. Sometimes the constructed scheme reflects some essential features, allows you to understand the mechanism of the phenomenon, makes it possible to predict its change. Different models can correspond to the same phenomenon.

The task of the researcher is to predict the nature of the phenomenon and the course of the process.

Sometimes, it happens that an object is available, but experiments with it are expensive or lead to serious environmental consequences. Knowledge about such processes is obtained with the help of models.

An important point is that the very nature of science involves the study of not one specific phenomenon, but a wide class of related phenomena. It suggests the need to formulate some general categorical statements, which are called laws. Naturally, with such a formulation, many details are neglected. In order to more clearly identify the pattern, they deliberately go for coarsening, idealization, schematicity, that is, they study not the phenomenon itself, but a more or less exact copy or model of it. All laws are laws about models, and therefore it is not surprising that, over time, some scientific theories are recognized as unusable. This does not lead to the collapse of science, since one model has been replaced by another. more modern.

A special role in science is played by mathematical models, the building material and tools of these models - mathematical concepts. They have accumulated and improved over thousands of years. Modern mathematics provides exceptionally powerful and universal means of research. Almost every concept in mathematics, every mathematical object, starting from the concept of a number, is a mathematical model. When constructing a mathematical model of an object or phenomenon under study, those of its features, features and details are singled out, which, on the one hand, contain more or less complete information about the object, and, on the other hand, allow mathematical formalization. Mathematical formalization means that the features and details of the object can be matched with suitable adequate mathematical concepts: numbers, functions, matrices and so on. Then the connections and relationships found and assumed in the object under study between its individual parts and components can be written using mathematical relationships: equalities, inequalities, equations. The result is a mathematical description of the process or phenomenon under study, that is, its mathematical model.

The study of a mathematical model is always associated with some rules of action on the objects under study. These rules reflect the relationships between causes and effects.

Building a mathematical model is a central stage in the study or design of any system. The whole subsequent analysis of the object depends on the quality of the model. Building a model is not a formal procedure. It strongly depends on the researcher, his experience and taste, always relies on certain experimental material. The model should be accurate enough, adequate and should be convenient for use.

Math modeling.

Classification of mathematical models.

Mathematical models can bedetermined and stochastic .

Deterministic model and - these are models in which a one-to-one correspondence is established between the variables describing an object or phenomenon.

This approach is based on knowledge of the mechanism of functioning of objects. The object being modeled is often complex and deciphering its mechanism can be very laborious and time-consuming. In this case, they proceed as follows: experiments are carried out on the original, the results are processed, and, without delving into the mechanism and theory of the modeled object, using the methods of mathematical statistics and probability theory, they establish relationships between the variables describing the object. In this case, getstochastic model . AT stochastic model, the relationship between variables is random, sometimes it happens fundamentally. The impact of a huge number of factors, their combination leads to a random set of variables describing an object or phenomenon. By the nature of the modes, the model isstatistical and dynamic.

Statisticalmodelincludes a description of the relationships between the main variables of the simulated object in the steady state without taking into account the change in parameters over time.

AT dynamicmodelsdescribes the relationship between the main variables of the simulated object in the transition from one mode to another.

Models are discrete and continuous, as well as mixed type. AT continuous variables take values ​​from a certain interval, indiscretevariables take isolated values.

Linear Models- all functions and relations that describe the model are linearly dependent on the variables andnot linearotherwise.

Math modeling.

Requirements , presented to the models.

1. Versatility- characterizes the completeness of the display by the model of the studied properties of the real object.

1. Adequacy - the ability to reflect the desired properties of the object with an error not higher than the specified one.
2. Accuracy - is estimated by the degree of coincidence of the values ​​of the characteristics of a real object and the values ​​of these characteristics obtained using models.
3. economy - is determined by the cost of computer memory resources and time for its implementation and operation.

Math modeling.

The main stages of modeling.

1. Statement of the problem.

Determining the purpose of the analysis and ways to achieve it and develop a common approach to the problem under study. At this stage, a deep understanding of the essence of the task is required. Sometimes, it is not less difficult to correctly set a task than to solve it. Staging is not a formal process, general rules no.

2. The study of the theoretical foundations and the collection of information about the object of the original.

At this stage, a suitable theory is selected or developed. If it is not present, causal relationships are established between the variables describing the object. Input and output data are determined, simplifying assumptions are made.

3. Formalization.

It consists in choosing a system of symbols and using them to write down the relationship between the components of the object in the form of mathematical expressions. A class of tasks is established, to which the resulting mathematical model of the object can be attributed. The values ​​of some parameters at this stage may not yet be specified.

4. Choice of solution method.

At this stage, the final parameters of the models are set, taking into account the conditions for the operation of the object. For the obtained mathematical problem, a solution method is selected or a special method is developed. When choosing a method, the knowledge of the user, his preferences, as well as the preferences of the developer are taken into account.

5. Implementation of the model.

Having developed an algorithm, a program is written that is debugged, tested, and a solution to the desired problem is obtained.

6. Analysis of the received information.

The received and expected solution is compared, the modeling error is controlled.

7. Checking the adequacy of a real object.

The results obtained by the model are comparedeither with the information available about the object, or an experiment is carried out and its results are compared with the calculated ones.

The modeling process is iterative. In case of unsatisfactory results of the stages 6. or 7. a return to one of the early stages, which could lead to the development of an unsuccessful model, is carried out. This stage and all subsequent stages are refined, and such refinement of the model occurs until acceptable results are obtained.

A mathematical model is an approximate description of any class of phenomena or objects of the real world in the language of mathematics. The main purpose of modeling is to explore these objects and predict the results of future observations. However, modeling is also a method of cognition of the surrounding world, which makes it possible to control it.

Mathematical modeling and the associated computer experiment are indispensable in cases where a full-scale experiment is impossible or difficult for one reason or another. For example, it is impossible to set up a full-scale experiment in history to check “what would happen if...” It is impossible to check the correctness of this or that cosmological theory. In principle, it is possible, but hardly reasonable, to set up an experiment on the spread of some disease, such as the plague, or to carry out nuclear explosion to study its implications. However, all this can be done on a computer, having previously built mathematical models of the phenomena under study.

1.1.2 2. Main stages of mathematical modeling

1) Model building. At this stage, some "non-mathematical" object is specified - a natural phenomenon, construction, economic plan, production process, etc. In this case, as a rule, a clear description of the situation is difficult. First, the main features of the phenomenon and the relationship between them at a qualitative level are identified. Then the found qualitative dependencies are formulated in the language of mathematics, that is, a mathematical model is built. This is the most difficult part of the modeling.

2) Solving the mathematical problem that the model leads to. At this stage, much attention is paid to the development of algorithms and numerical methods for solving the problem on a computer, with the help of which the result can be found with the required accuracy and within the allowable time.

3) Interpretation of the obtained consequences from the mathematical model.The consequences derived from the model in the language of mathematics are interpreted in the language accepted in this field.

4) Checking the adequacy of the model.At this stage, it is found out whether the results of the experiment agree with the theoretical consequences from the model within a certain accuracy.

5) Model modification.At this stage, either the model becomes more complex so that it is more adequate to reality, or it is simplified in order to achieve a practically acceptable solution.

1.1.3 3. Model classification

Models can be classified according to different criteria. For example, according to the nature of the problems being solved, models can be divided into functional and structural ones. In the first case, all quantities characterizing a phenomenon or object are expressed quantitatively. At the same time, some of them are considered as independent variables, while others are considered as functions of these quantities. A mathematical model is usually a system of equations of various types (differential, algebraic, etc.) that establish quantitative relationships between the quantities under consideration. In the second case, the model characterizes the structure of a complex object, consisting of separate parts, between which there are certain connections. Typically, these relationships are not quantifiable. To build such models, it is convenient to use graph theory. A graph is a mathematical object, which is a set of points (vertices) on a plane or in space, some of which are connected by lines (edges).

According to the nature of the initial data and prediction results, the models can be divided into deterministic and probabilistic-statistical. Models of the first type give definite, unambiguous predictions. Models of the second type are based on statistical information, and the predictions obtained with their help are of a probabilistic nature.

MATHEMATICAL MODELING AND GENERAL COMPUTERIZATION OR SIMULATION MODELS

Now, when almost universal computerization is taking place in the country, one can hear statements from specialists of various professions: "Let's introduce a computer in our country, then all tasks will be solved immediately." This point of view is completely wrong, computers themselves cannot do anything without mathematical models of certain processes, and one can only dream of universal computerization.

In support of the foregoing, we will try to justify the need for modeling, including mathematical modeling, reveal its advantages in the knowledge and transformation of the external world by a person, identify existing shortcomings and go ... to simulation modeling, i.e. modeling using computers. But everything is in order.

First of all, let's answer the question: what is a model?

A model is a material or mentally represented object that, in the process of cognition (study), replaces the original, retaining some typical properties that are important for this study.

A well-built model is more accessible for research than a real object. For example, experiments with the country's economy in educational purposes, a model is indispensable here.

Summarizing what has been said, we can answer the question: what are models for? To

• understand how an object works (its structure, properties, laws of development, interaction with the outside world).
• learn to manage an object (process) and determine the best strategies
• predict the consequences of the impact on the object.

What is positive in any model? It allows you to get new knowledge about the object, but, unfortunately, it is not complete to one degree or another.

Modelformulated in the language of mathematics using mathematical methods is called a mathematical model.

The starting point for its construction is usually some task, for example, an economic one. Widespread, both descriptive and optimization mathematical, characterizing various economic processes and events such as:

• resource allocation
• rational cutting
• transportation
• consolidation of enterprises
• network planning.

How is a mathematical model built?

• First, the purpose and subject of the study are formulated.
• Secondly, the most important characteristics appropriate for this purpose.
• Thirdly, the relationships between the elements of the model are verbally described.
• Further, the relationship is formalized.
• And the calculation is carried out according to the mathematical model and the analysis of the obtained solution.

Using this algorithm, you can solve any optimization problem, including a multicriteria one, i.e. one in which not one, but several goals, including contradictory ones, are pursued.

Let's take an example. Queuing theory - the problem of queuing. You need to balance two factors - the cost of maintaining service devices and the cost of staying in line. Having built a formal description of the model, calculations are made using analytical and computational methods. If the model is good, then the answers found with its help are adequate to the modeling system; if it is bad, then it must be improved and replaced. The criterion of adequacy is practice.

Optimization models, including multicriteria ones, have a common property - a goal (or several goals) is known to achieve which one often has to deal with complex systems, where it is not so much about solving optimization problems, but about researching and predicting states depending on chosen control strategies. And here we are faced with difficulties in implementing the previous plan. They are as follows:

• a complex system contains many connections between elements
• the real system is influenced by random factors, it is impossible to take them into account analytically
• the possibility of comparing the original with the model exists only at the beginning and after the application of the mathematical apparatus, because intermediate results may not have analogues in a real system.

In connection with the listed difficulties that arise when studying complex systems, the practice required a more flexible method, and it appeared - simulation modeling " Simujation modeling".

Usually, a simulation model is understood as a set of computer programs that describes the functioning of individual blocks of systems and the rules of interaction between them. Usage random variables makes it necessary to conduct repeated experiments with a simulation system (on a computer) and subsequent statistical analysis of the results obtained. A very common example of the use of simulation models is the solution of a queuing problem by the MONTE CARLO method.

Thus, work with the simulation system is an experiment carried out on a computer. What are the benefits?

– Greater proximity to the real system than mathematical models;

– The block principle makes it possible to verify each block before it is included in the overall system;

– The use of dependencies of a more complex nature, not described by simple mathematical relationships.

The listed advantages determine the disadvantages

– to build a simulation model is longer, more difficult and more expensive;

– to work with the simulation system, you must have a computer that is suitable for the class;

– interaction between the user and the simulation model (interface) should not be too complicated, convenient and well known;

- the construction of a simulation model requires a deeper study of the real process than mathematical modeling.

The question arises: can simulation modeling replace optimization methods? No, but conveniently complements them. A simulation model is a program that implements some algorithm, to optimize the control of which an optimization problem is first solved.

So, neither a computer, nor a mathematical model, nor an algorithm for studying it separately can solve a rather complicated problem. But together they represent the power that allows you to know the world, manage it in the interests of man.

1.2 Model classification

A mathematical model of a technical object is a set of mathematical objects and relations between them that adequately reflects the properties of the object under study that are of interest to the researcher (engineer).

The model can be represented in various ways.

Model representation forms:

invariant - recording model relations using a traditional mathematical language, regardless of the method for solving model equations;

analytical - recording the model in the form of the result of an analytical solution of the initial equations of the model;

algorithmic - recording the relations of the model and the selected numerical method of solution in the form of an algorithm.

schematic (graphic) - representation of the model in some graphic language (for example, the language of graphs, equivalent circuits, diagrams, etc.);

physical

analog

The most universal is the mathematical description of processes - mathematical modeling.

The concept of mathematical modeling also includes the process of solving a problem on a computer.

Generalized mathematical model

The mathematical model describes the relationship between the initial data and the desired values.

The elements of the generalized mathematical model are (Fig. 1): a set of input data (variables) X,Y;

X - set of variable variables; Y - independent variables (constant);

mathematical operator L that defines operations on these data; which is understood as a complete system of mathematical operations that describe numerical or logical relationships between sets of input and output data (variables);

set of output data (variables) G(X,Y); is a set of criterion functions, including (if necessary) the objective function.

The mathematical model is a mathematical analogue of the designed object. The degree of adequacy of its object is determined by the formulation and correctness of solutions to the design problem.

The set of variable parameters (variables) X forms the space of variable parameters Rx (search space), which is metric with dimension n equal to the number of variable parameters.

The set of independent variables Y form the metric space of input data Ry. In the case when each component of the space Ry is given by a range of possible values, the set of independent variables is mapped to some limited subspace of the space Ry.

The set of independent variables Y determines the environment for the operation of the object, i.e. external conditions in which the designed object will operate

It can be:

• - technical specifications an object that is not subject to change during the design process;
• - physical perturbations of the environment with which the design object interacts;
• - tactical parameters that the design object should achieve.

The output data of the considered generalized model form a metric space of criterial indicators RG.

The scheme of using a mathematical model in a computer-aided design system is shown in Fig.2.

Requirements for the mathematical model

The main requirements for mathematical models are the requirements of adequacy, universality and economy.

Adequacy. The model is considered adequate if it reflects the given properties with acceptable accuracy. Accuracy is defined as the degree of agreement between the values ​​of the output parameters of the model and the object.

The accuracy of the model is different in different conditions of the functioning of the object. These conditions are characterized by external parameters. In the space of external parameters, select the region of model adequacy, where the error is less than the specified maximum permissible error. Determination of the model adequacy region is a complex procedure that requires large computational costs, which grow rapidly with an increase in the dimension of the space of external parameters. This task can significantly exceed the task of parametric optimization of the model itself in volume, therefore, it may not be solved for newly designed objects.

Universality - is determined mainly by the number and composition of external and output parameters taken into account in the model.

The economy of the model is characterized by the cost of computing resources for its implementation - the cost of computer time and memory.

Inconsistency of requirements to the model wide area adequacy, a high degree of universality and high cost-effectiveness determines the use of a number of models for objects of the same type.

Model Retrieval Methods

Obtaining models in the general case is an unformalized procedure. The main decisions regarding the choice of the type of mathematical relationships, the nature of the variables and parameters used, are made by the designer. At the same time, such operations as the calculation of the numerical values ​​of the model parameters, the determination of adequacy areas, and others are algorithmized and solved on a computer. Therefore, the modeling of the elements of the designed system is usually performed by specialists in specific technical fields using traditional experimental studies.

Methods for obtaining functional models of elements are divided into theoretical and experimental.

Theoretical methods are based on the study of the physical regularities of the processes occurring in the object, determining the mathematical description corresponding to these regularities, substantiating and accepting simplifying assumptions, performing the necessary calculations and bringing the result to the accepted form of model representation.

Experimental methods are based on the use of external manifestations of the properties of an object, fixed during the operation of objects of the same type or during targeted experiments.

Despite the heuristic nature of many operations, modeling has a number of provisions and techniques common to obtaining models of various objects. Enough general character have

macro modeling technique,

mathematical methods for planning experiments,

algorithms for formalized operations for calculating the numerical values ​​of parameters and determining the areas of adequacy.

Using Mathematical Models

The computing power of modern computers, combined with the provision of all system resources to the user, the possibility of an interactive mode when solving a problem and analyzing the results, make it possible to minimize the time for solving a problem.

When compiling a mathematical model, the researcher is required to:

study the properties of the object under study;

the ability to separate the main properties of the object from the secondary ones;

evaluate the assumptions made.

The model describes the relationship between the input data and the desired values. The sequence of actions that must be performed in order to move from the initial data to the desired values ​​is called an algorithm.

The algorithm for solving the problem on a computer is associated with the choice of a numerical method. Depending on the form of representation of the mathematical model (algebraic or differential form), various numerical methods are used.

The essence of economic and mathematical modeling lies in the description of socio-economic systems and processes in the form of economic and mathematical models.

Let's consider questions of classification of economic and mathematical methods. These methods, as noted above, are a complex of economic and mathematical disciplines that are an alloy of economics, mathematics and cybernetics.

Therefore, the classification of economic and mathematical methods is reduced to the classification of the scientific disciplines included in their composition. Although the generally accepted classification of these disciplines has not yet been developed, with a certain degree of approximation, the following sections can be distinguished in the composition of economic and mathematical methods:

• * economic cybernetics: system analysis of economics, theory of economic information and theory of control systems;
• * mathematical statistics: economic applications of this discipline - sampling method, analysis of variance, correlation analysis, regression analysis, multivariate statistical analysis, factor analysis, index theory, etc.;
• * Mathematical economics and econometrics that studies the same issues from a quantitative point of view: the theory of economic growth, the theory of production functions, intersectoral balances, national accounts, analysis of demand and consumption, regional and spatial analysis, global modeling, etc.;
• * methods for making optimal decisions, including the study of operations in the economy. This is the most voluminous section, which includes the following disciplines and methods: optimal (mathematical) programming, including branch and bound methods, network planning and control methods, program-targeted planning and control methods, inventory management theory and methods, queuing theory , game theory, decision theory and methods, scheduling theory. Optimal (mathematical) programming includes, in turn, linear programming, non-linear programming, dynamic programming, discrete (integer) programming, fractional linear programming, parametric programming, separable programming, stochastic programming, geometric programming;
• * Methods and disciplines that are specific to both a centrally planned economy and a market (competitive) economy. The former include the theory of optimal functioning of the economy, optimal planning, the theory of optimal pricing, models of logistics, etc. The latter include methods that allow developing models of free competition, models of the capitalist cycle, models of monopoly, models of indicative planning, models of the theory of the firm etc.

Many of the methods developed for a centrally planned economy may also be useful in economic and mathematical modeling in a market economy;

* methods of experimental study of economic phenomena. These include, as a rule, mathematical methods of analysis and planning of economic experiments, methods of machine simulation (simulation), business games. This also includes methods of expert assessments developed to evaluate phenomena that cannot be directly measured.

Let us now turn to the questions of classifying economic and mathematical models, in other words, mathematical models of socio-economic systems and processes.

A unified classification system for such models currently does not exist either, however, more than ten main features of their classification, or classification headings, are usually distinguished. Let's take a look at some of these sections.

According to the general purpose, economic and mathematical models are divided into theoretical and analytical, used in the study of general properties and patterns of economic processes, and applied, used in solving specific economic problems of analysis, forecasting and management. Various types of applied economic and mathematical models are considered in this tutorial.

According to the degree of aggregation of modeling objects, the models are divided into macroeconomic and microeconomic. Although there is no clear distinction between them, the first of them include models that reflect the functioning of the economy as a whole, while microeconomic models are associated, as a rule, with such parts of the economy as enterprises and firms.

According to a specific purpose, i.e., according to the purpose of creation and application, balance models are distinguished, expressing the requirement that the availability of resources correspond to their use; trend models, in which the development of the modeled economic system is reflected through the trend (long-term trend) of its main indicators; optimization models designed to select the best option from a certain number of production, distribution or consumption options; simulation models intended for use in the process of machine simulation of the systems or processes under study, etc.

According to the type of information used in the model, economic-mathematical models are divided into analytical, built on a priori information, and identifiable, built on a posteriori information.

By taking into account the time factor, the models are divided into static, in which all dependencies are related to one point in time, and dynamic, which describe economic systems in development.

By taking into account the uncertainty factor, the models are divided into deterministic ones, if the output results in them are uniquely determined by control actions, and stochastic (probabilistic), if when a certain set of values ​​is specified at the model input, different results can be obtained depending on the action of a random factor.

Economic and mathematical models can also be classified according to the characteristics of the mathematical objects included in the model, in other words, according to the type of mathematical apparatus used in the model. On this basis, matrix models, linear and non-linear programming models, correlation-regression models,

Basic concepts of mathematical modeling of the queuing theory model, network planning and control model, game theory model, etc.

Finally, according to the type of approach to the studied socio-economic systems, descriptive and normative models are distinguished. With a descriptive (descriptive) approach, models are obtained that are designed to describe and explain actually observed phenomena or to predict these phenomena; As an example of descriptive models, we can cite the previously named balance and trend models. In the normative approach, one is interested not in how the economic system is organized and develops, but in how it should be arranged and how it should operate in the sense of certain criteria. In particular, all optimization models are of the normative type; normative models of standard of living can serve as another example.

Let us consider as an example the economic-mathematical model of the input-output balance (EMM IOB). Taking into account the above classification headings, this is an applied, macroeconomic, analytical, descriptive, deterministic, balance, matrix model; There are both static methods and dynamic methods.

Linear programming is a particular branch of optimal programming. In turn, optimal (mathematical) programming is a branch of applied mathematics that studies problems of conditional optimization. In economics, such problems arise in the practical implementation of the principle of optimality in planning and management.

A necessary condition for using the optimal approach to planning and management (the principle of optimality) is flexibility, alternativeness of production and economic situations in which planning and management decisions have to be made. It is these situations that, as a rule, constitute the daily practice of an economic entity (the choice production program, attaching to suppliers, routing, cutting materials, preparing mixtures, etc.).

The essence of the principle of optimality is the desire to choose such a planning and management decision X = (xi, X2 xn), where Xu, (y = 1. x) - its components, which would best take into account the internal capabilities and external conditions of the production activity of an economic entity .

The words "in the best way" here mean the choice of some criterion of optimality, i.e. some economic indicator that allows you to compare the effectiveness of certain planning and management decisions. Traditional optimality criteria: “maximum profit”, “minimum costs”, “maximum profitability”, etc. The words “would take into account the internal capabilities and external conditions of production activity” mean that a number of conditions are imposed on the choice of a planning and management decision (behavior), t .e. the choice of X is carried out from a certain region of possible (admissible) solutions D; this area is also called the problem definition area. a general problem of optimal (mathematical) programming, otherwise, a mathematical model of an optimal programming problem, the construction (development) of which is based on the principles of optimality and consistency.

A vector X (a set of control variables Xj, j = 1, n) is called a feasible solution, or an optimal programming problem plan, if it satisfies the system of constraints. And the plan X (admissible solution) that delivers the maximum or minimum of the objective function f(xi, *2, ..., xn) is called the optimal plan (optimal behavior, or simply solution) of the optimal programming problem.

Thus, the choice of optimal managerial behavior in a specific production situation is associated with conducting economic and mathematical modeling from the standpoint of consistency and optimality and solving the problem of optimal programming. Optimal programming problems in the most general view classified according to the following criteria.

• 1. By the nature of the relationship between variables -
• a) linear
• b) non-linear.

In case a) all functional connections in the system of restrictions and the goal function are linear functions; the presence of a nonlinearity in at least one of the mentioned elements leads to case b).

• 2. By the nature of the change in variables --
• a) continuous
• b) discrete.

In case a) the values ​​of each of the control variables can completely fill a certain area of ​​real numbers; in case b) all or at least one variable can take only integer values.

• 3. By taking into account the time factor -
• a) static
• b) dynamic.

In tasks a), modeling and decision-making are carried out under the assumption that the elements of the model are independent of time during the period of time for which a planning and management decision is made. In case b), such an assumption cannot be accepted with sufficient reason and the time factor must be taken into account.

• 4. According to the availability of information about variables --
• a) tasks under conditions of complete certainty (deterministic),
• b) tasks in conditions of incomplete information,
• c) tasks under conditions of uncertainty.

In problems b), individual elements are probabilistic quantities, but their distribution laws are known or additional statistical studies can be established. In case c), one can make an assumption about the possible outcomes of random elements, but it is not possible to draw a conclusion about the probabilities of the outcomes.

• 5. According to the number of criteria for evaluating alternatives -
• a) simple, single-criteria tasks,
• b) complex, multicriteria tasks.

In tasks a) it is economically acceptable to use one optimality criterion or it is possible by special procedures (for example, “priority weighting”)