vector of output variables, Y= t,

Z - vector of external influences, Z= t,

t - time coordinate.

Building mathematical model consists in determining the links between certain processes and phenomena, creating a mathematical apparatus that allows to express quantitatively and qualitatively the relationship between certain processes and phenomena, between physical quantities of interest to a specialist, and factors affecting the final result.

Usually there are so many of them that it is not possible to introduce their entire set into the model. When building mathematical model before the study, the task arises to identify and exclude from consideration factors that do not significantly affect the final result ( mathematical model usually includes a much smaller number of factors than in reality). Based on the experimental data, hypotheses are put forward about the relationship between the quantities expressing the final result and the factors introduced in the mathematical model. Such a relationship is often expressed by systems of differential partial differential equations(for example, in problems of mechanics solid body, liquid and gas, the theory of filtration, thermal conductivity, the theory of electrostatic and electrodynamic fields).

The ultimate goal of this stage is the formulation of a mathematical problem, the solution of which, with the necessary accuracy, expresses the results that are of interest to a specialist.

Form and principles of presentation mathematical model depends on many factors.

According to the principles of construction mathematical models divided into:

1. analytical;
2. imitation.

In analytical models, the processes of functioning of real objects, processes or systems are written in the form of explicit functional dependencies.

The analytical model is divided into types depending on the mathematical problem:

1. equations (algebraic, transcendental, differential, integral),
2. approximation problems (interpolation, extrapolation, numerical integration And differentiation),
3. optimization problems,
4. stochastic problems.

However, as the modeling object becomes more complex, the construction of an analytical model becomes an intractable problem. Then the researcher is forced to use simulation modeling .

IN simulation modeling the functioning of objects, processes or systems is described by a set of algorithms. Algorithms imitate real elementary phenomena that make up a process or system while maintaining their logical structure and sequencing over time. Simulation allows you to get information about the source data process states or systems at certain points in time, however, predicting the behavior of objects, processes or systems is difficult here. It can be said that simulation models- these are computer-based computational experiments With mathematical models, imitating the behavior of real objects, processes or systems.

Depending on the nature of the studied real processes and systems mathematical models can be:

1. deterministic,
2. stochastic.

In deterministic models, it is assumed that there are no random influences, the elements of the model (variables, mathematical relationships) are fairly well established, and the behavior of the system can be accurately determined. When constructing deterministic models, algebraic equations, integral equations, matrix algebra are most often used.

Stochastic model takes into account the random nature of the processes in the objects and systems under study, which is described by the methods of probability theory and mathematical statistics.

According to the type of input information, the models are divided into:

1. continuous,
2. discrete.

If the information and parameters are continuous, and the mathematical relationships are stable, then the model is continuous. And vice versa, if the information and parameters are discrete, and the connections are unstable, then mathematical model- discrete.

According to the behavior of models in time, they are divided into:

1. static,
2. dynamic.

Static models describe the behavior of an object, process or system at any point in time. Dynamic models reflect the behavior of an object, process or system over time.

According to the degree of correspondence between

A number of general properties of models follow directly from the structure of the adopted definition, which are usually taken into account in modeling practice.

• 1. The model is a "quadruple structure" whose components are:
  • - subject;
  • - the task solved by the subject;
  • - the original object and the language of description or the method of reproducing the model.

The problem solved by the subject plays a special role in the structure of the generalized model. Outside the context of a task or a class of tasks, the concept of a model is meaningless.

• 2. Generally speaking, each material object corresponds to an uncountable set of equally adequate, but essentially different models associated with different tasks.
• 3. A pair of task-object also corresponds to a set of models containing, in principle, the same information, but differing in the forms of its presentation or reproduction.
• 4. By definition, a model is always only a relative, approximate similarity of the original object and, in terms of information, is fundamentally poorer than the latter. This is its fundamental property.
• 5. The arbitrary nature of the original object, which appears in the accepted definition, means that this object can be material-material, can be purely informational in nature, and, finally, can be a complex of heterogeneous material and informational components. However, regardless of the nature of the object, the nature of the problem being solved, and the method of implementation, the model is an information entity.
• 6. Particular, but very important for theoretically developed scientific and technical disciplines, is the case when the role of the modeling object in a research or applied task is played not by a fragment of the real world, considered directly, but by some ideal construct, i.e., in fact, another model created earlier and practically reliable. Such secondary, and in the general case, n-fold modeling can be carried out by theoretical methods with subsequent verification of the results obtained against experimental data, which is typical for fundamental natural sciences.

In theoretically less developed areas of knowledge (biology, some technical disciplines), the secondary model usually includes empirical information that is not covered by existing theories.

The properties of any model are as follows:

• - finiteness: the model reflects the original only in a finite number of its relations and, in addition, the modeling resources are finite;
• - simplicity: the model displays only the essential aspects of the object;
• - Approximation: the reality is displayed roughly or approximately by the model;
• - adequacy: the model successfully describes the modeled system;
• - informativeness: the model should contain sufficient information about the system - within the framework of the hypotheses adopted in the construction of the model.

Classification of mathematical models. When designing technical objects, many types of mathematical models are used. In this regard, there are mathematical models of elements and systems. When moving to a higher hierarchical level of block structuring, a lower-level system becomes an element of a new level system, and vice versa, when moving to a lower level, an element becomes a system. Consequently, the most complex mathematical models are used at the lower levels.

At higher levels, simpler models can be successfully applied. They can be obtained by approximating models of lower hierarchical levels.

In the general case, the equations of the mathematical model relate physical quantities, which characterize the state of the object and are not related to the output, internal and external parameters listed above. These quantities are: speeds and forces - in mechanical systems. The quantities characterizing the state of a technical object in the process of its operation are called phase variables (phase coordinates).

The vector of phase variables defines a point in a space called phase space. Mathematical models are subject to the requirements of adequacy, economy, universality. These requirements are contradictory, therefore, usually for the design of each object, their original model is used. The model is considered adequate if it reflects the studied properties with acceptable accuracy.

The accuracy is estimated by the degree of coincidence of the values ​​of the output parameters predicted during the computational experiment on the model with their true values. At the same time, the mathematical model should be as simple as possible, but at the same time provide an adequate description of the analyzed process.

Classification of mathematical models used in design technical systems, shown in the figure.

Figure 1. - Classification of mathematical models:

According to the form of representation of mathematical models, invariant, algorithmic, analytical and graphical models of the design object are distinguished.

In an invariant form, a mathematical model is represented by a system of equations (differential, algebraic), without regard to the method for solving these equations.

In the algorithmic form, the relations of the model are associated with the chosen numerical solution method and are written as an algorithm for the sequence of calculations.

The analytical model is an explicit dependence of the desired variables on the given values ​​(usually, the dependence of the object's output parameters on internal and external parameters).

A graphical (circuit) model is presented in the form of graphs, equivalent circuits, dynamic models, diagrams, etc.

Among the algorithmic models, simulation models are distinguished, designed to simulate the physical and information processes that occur in an object when it functions under the influence of various environmental factors.

Structural models display only the structure of objects and are used in solving problems of structural synthesis. The parameters of structural models are called morphological variables.

Functional models describe the processes of functioning of technical objects and have the form of systems of equations. They take into account the structural and functional properties of both the object and allow solving problems of both parametric and structural synthesis.

According to the methods of obtaining functional mathematical models are divided into theoretical and experimental.

Theoretical models are obtained on the basis of a description of the physical processes of the object's functioning, and experimental models - on the basis of studying the object's behavior in the external environment, considering it as a cybernetic black box. Experiments can be physical (on a technical object or its physical model) or computational (on a theoretical mathematical model).

When constructing theoretical models, physical and formal approaches are used.

The physical approach is reduced to the direct application of physical laws to describe objects, for example, the laws of Newton, Hooke, Kirchhoff, Fourier, etc.

The formal approach uses general mathematical principles and is used in the construction of both theoretical and experimental models.

Functional mathematical models can be linear and non-linear.

Linear models contain only linear functions phase variables and their derivatives. Mathematical models of such objects include non-linear functions of phase variables and (or) their derivatives and are non-linear.

If the simulation takes into account the inertial properties of a technical object and (or) the change in time of the parameters of the object or the environment, then the model is called dynamic. Otherwise, the model is static.

Most design procedures are performed on deterministic models. A deterministic mathematical model is characterized by a one-to-one correspondence between an external influence on a dynamic system and its response to this influence. In a computational experiment, when designing, some standard typical effects on an object are usually set: stepwise, impulse, harmonic, piecewise linear, exponential, etc.

They are called test influences.

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 is based on the idea of ​​modeling scientific research, at the same time, in theoretical methods, various kinds of sign, abstract models are used, in experimental ones - subject models.

In the study, a complex real phenomenon is replaced by some simplified copy or scheme, sometimes such a copy serves only to remember and to recognize the desired phenomenon at the next meeting. 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 implies 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 found to be 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 distinguished, 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 an object can be associated with appropriate 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 details and constituent parts can be written using mathematical relations: 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 . IN 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.

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

Models are discrete And continuous, and mixed type. IN 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, there are no general rules.

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 an acceptable 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 different type(differential, algebraic, etc.), establishing 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 outside 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 for educational purposes are unacceptable; here one cannot do without a model.

Summarizing what has been said, we can answer the question: what are models for? In order 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 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 corresponding to this goal are highlighted.
• 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 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 the 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

1.2.1 Classification taking into account the time factor and the area of ​​\u200b\u200buse (Makarova N.A.) Static model - it is like a one-time slice of information on the object (the result of one survey) Dynamic model-allows see changes in the object over time (Card in the clinic) Models can be classified according to what field of knowledge do they belong to(biological, historical, ecological, etc.) Return to start

1.2.2 Classification by area of ​​​​use (Makarova N.A.) Training- visual aids, trainers , oh thrashing programs Experienced models-reduced copies (car in a wind tunnel) Scientific and technical synchrophasotron, stand for testing electronic equipment Game- economic, sports, business games simulation- Not they simply reflect reality, but imitate it (drugs are tested on mice, experiments are carried out in schools, etc.. This modeling method is called trial and error Return to start

1.2.3 Classification according to the method of presentation Makarova N.A.) material models- otherwise can be called subject. They perceive geometric and physical properties original and always have a real embodiment Informational models-not allowed touch or see. They are based on information. .Information model is a set of information that characterizes the properties and states of an object, process, phenomenon, as well as the relationship with the outside world. Verbal model - information model in a mental or conversational form. Iconic model-informational model expressed by signs , i.e.. by means of any formal language. Computer model - m A model implemented by means of a software environment.

1.2.4 Classification of models given in the book "Land of Informatics" (Gein A.G.)) "...here is a seemingly simple task: how long will it take to cross the Karakum desert? Answer, of course depends on the mode of travel. If travel on camels, then one term will be required, another if you go by car, a third if you fly by plane. And most importantly, different models are required for travel planning. For the first case, the required model can be found in the memoirs of famous desert explorers: after all, one cannot do without information about oases and camel trails. In the second case, irreplaceable information contained in the atlas of roads. In the third - you can use the flight schedule. These three models differ - memoirs, atlas and timetable and the nature of the presentation of information. In the first case, the model is represented by a verbal description of the information (descriptive model), in the second - like a photograph from nature (natural model), in the third - a table containing symbols: time of departure and arrival, day of the week, ticket price (the so-called sign model) However, this division is very conditional - maps and diagrams (elements of a full-scale model) can be found in memoirs, there are symbols on the maps (elements of a sign model), a decoding of symbols (elements of a descriptive model) is given in the schedule. So this classification of models ... in our opinion is unproductive" In my opinion, this fragment demonstrates the descriptive (wonderful language and style of presentation) common to all Gein's books and, as it were, the Socratic style of teaching (Everyone thinks that this is so. I completely agree with you, but if you look closely, then ...). In such books it is quite difficult to find a clear system of definitions (it is not intended by the author). In the textbook edited by N.A. Makarova demonstrates a different approach - the definitions of concepts are clearly distinguished and somewhat static.

1.2.5 Classification of models given in the manual of A.I. Bochkin There are many ways to classify .We present just a few of the more well-known foundations and signs: discreteness And continuity, matrix and scalar models, static and dynamic models, analytical and information models, subject and figurative-sign models, large-scale and non-scale... Every sign gives a certain knowledge about the properties of both the model and the modeled reality. The sign can serve as a hint about the way the simulation has been performed or is to be done. Discreteness and continuity discreteness - a characteristic feature of computer models .After all a computer can be in a finite, albeit very large, number of states. Therefore, even if the object is continuous (time), in the model it will change in jumps. It could be considered continuity a sign of non-computer type models. Randomness and determinism . Uncertainty, accident initially opposed to the computer world: The algorithm launched again must repeat itself and give the same results. But for imitation random processes using pseudo-random number generators. Introducing randomness into deterministic problems leads to powerful and interesting models (Random Tossing Area Calculation). Matrix - scalar. Availability of parameters matrix model indicates its greater complexity and, possibly, accuracy compared to scalar. For example, if we do not single out all age groups in the country's population, considering its change as a whole, we get a scalar model (for example, the Malthus model), if we single out, a matrix (gender and age) model. It was the matrix model that made it possible to explain the fluctuations in the birth rate after the war. static dynamism. These properties of the model are usually predetermined by the properties of the real object. There is no freedom of choice here. Just static model can be a step towards dynamic, or some of the model variables can be considered unchanged for the time being. For example, a satellite moves around the Earth, its movement is influenced by the Moon. If we consider the Moon to be stationary during the satellite's revolution, we obtain a simpler model. Analytical Models. Description of processes analytically, formulas and equations. But when trying to build a graph, it is more convenient to have tables of function values ​​​​and arguments. simulation models. simulation models appeared a long time ago in the form of large-scale copies of ships, bridges, etc. appeared a long time ago, but in connection with computers they are considered recently. Knowing how connected model elements analytically and logically, it is easier not to solve a system of certain relationships and equations, but to map the real system into computer memory, taking into account the links between memory elements. Information Models. Informational It is customary to oppose models to mathematical ones, more precisely algorithmic ones. The data/algorithm ratio is important here. If there is more data or they are more important, we have an information model, otherwise - mathematical. Subject Models. This is primarily a children's model - a toy. Figurative-sign models. It is primarily a model in the human mind: figurative, if graphic images predominate, and iconic, if there are more than words and/or numbers. Figurative-sign models are built on a computer. scale models. TO large-scale models are those of the subject or figurative models that repeat the shape of the object (map).

Computers have firmly entered our lives, and there is practically no such area of ​​human activity where computers would not be used. Computers are now widely used in the process of creating and researching new machines, new technological processes and looking for them best options; when solving economic problems, when solving problems of planning and managing production at various levels. The creation of large objects in rocketry, aircraft construction, shipbuilding, as well as the design of dams, bridges, etc., is generally impossible without the use of computers.

For the use of computers in solving applied problems, first of all applied task must be "translated" into a formal mathematical language, i.e. for a real object, process or system, its mathematical model.

The word "Model" comes from the Latin modus (copy, image, outline). Modeling is the replacement of some object A with another object B. The replaced object A is called the original or the modeling object, and the replacement B is called the model. In other words, a model is an object-replacement of the original object, providing the study of some properties of the original.

The purpose of the simulation are the receipt, processing, presentation and use of information about objects that interact with each other and external environment; and the model here acts as a means of knowing the properties and patterns of the behavior of the object.

Modeling is widely used in various fields of human activity, especially in the areas of design and management, where the processes of making effective decisions based on the information received are special.

A model is always built with a specific goal in mind, which influences which properties of an objective phenomenon are significant and which are not. The model is, as it were, a projection of objective reality from a certain point of view. Sometimes, depending on the goals, you can get a number of projections of objective reality that come into conflict. This is typical, as a rule, for complex systems, in which each projection singles out what is essential for a specific purpose from a set of non-essential ones.

Modeling theory is a branch of science that studies ways to study the properties of original objects based on replacing them with other model objects. The theory of similarity underlies the theory of modeling. When modeling, absolute similarity does not take place and only strives to ensure that the model reflects the studied side of the object's functioning well enough. Absolute similarity can take place only when one object is replaced by another exactly the same.

All models can be divided into two classes:

1. real,
2. ideal.

In turn, real models can be divided into:

1. natural,
2. physical,
3. mathematical.

Ideal Models can be divided into:

1. visual,
2. iconic,
3. mathematical.

Real full-scale models are real objects, processes and systems on which scientific, technical and industrial experiments are performed.

Real physical models- these are models, models that reproduce the physical properties of the originals (kinematic, dynamic, hydraulic, thermal, electrical, light models).

Real mathematical are analog, structural, geometric, graphic, digital and cybernetic models.

Ideal visual models are diagrams, maps, drawings, graphs, graphs, analogues, structural and geometric patterns.

Ideal sign models are symbols, alphabet, programming languages, ordered notation, topological notation, network representation.

Ideal mathematical models- these are analytical, functional, simulation, combined models.

In the above classification, some models have a double interpretation (for example, analog). All models, except for full-scale ones, can be combined into one class of mental models, since they are the product of man's abstract thinking.

Let us dwell on one of the most universal types of modeling - mathematical, which puts in correspondence with the simulated physical process a system of mathematical relations, the solution of which allows you to get an answer to the question about the behavior of an object without creating a physical model, which often turns out to be expensive and inefficient.

Math modeling is a means of studying a real object, process or system by replacing them mathematical model, more convenient for experimental research with the help of a computer.

Mathematical model is an approximate representation of real objects, processes or systems, expressed in mathematical terms and retaining the essential features of the original. Mathematical models in a quantitative form, with the help of logical and mathematical constructions, they describe the main properties of an object, process or system, its parameters, internal and external relations.

In general, model is a reflection of a real object. Such a reflection of an object can be represented by a sketch, diagram, photograph, graph, table, etc.

We will consider only mathematical models of various economic processes that are described by mathematical symbols and solved using appropriate mathematical methods.

In economics, mainly mathematical models are used that describe the phenomenon under study with the help of mathematical apparatus (functions, equations, inequalities, their systems).

In the theory of optimal solutions the main role assigned to mathematical modeling. To build a mathematical model, it is necessary to have a strict understanding of the purpose of the functioning of the system under study and to have information about the restrictions that determine the range of acceptable values ​​of the controlled variables. Both the goal and the constraints must be represented as functions of the controlled variables. The analysis of the model should lead to the determination of the best control action on the control object when all the established restrictions are met.

The model of a managed object is built in order to apply some computing device to optimize the functioning of this object (maximum possible increase in the efficiency of its work). The development of a model is almost always associated with an attempt to achieve two conflicting goals: to reflect real processes as accurately as possible and to get the model as simple as possible so that it is easy to work with.

To apply quantitative methods for studying economic processes, it is required to build a mathematical model of the optimization object. When building a model, an object is usually simplified, schematized, and the object's scheme is described using one or another mathematical apparatus.

Mathematical model- this is an approximate description of any object or class of phenomena of the external world, expressed with the help of mathematical apparatus and mathematical symbols.

Mathematical models have a number of advantages over other types of models. The most important of them include the following:

a wide range of applications,

low cost of creating a model compared to other types,

the speed of obtaining research results when using electronic computers,

the possibility of experimenting with the studied economic process,

· the possibility of checking the correctness of the put forward prerequisites and conditions of the set economic task.

The mathematical model of any economic problem includes an objective function, a system of constraints, and an optimality criterion.

The objective function relates the various values ​​of the model to each other. Typically, the target is economic indicator(profit, cost, profitability, etc.). Therefore, the objective function is sometimes called economic, criterion.

objective function- a characteristic of the object from the condition of further search for the optimality criterion, mathematically connecting one or another factor of the object of study.

When solving optimization problems, it is necessary to determine the optimality criterion, i.e. a sign by which a comparative evaluation of alternatives is carried out and the best among them is chosen from the point of view of the optimization goal.

Optimality criterion- This is an indicator that, as a rule, has an economic meaning, which serves to formalize the specific goal of managing the object of study and is expressed using the objective function.

The operation optimality criterion performs such an important function as a comparative evaluation of the chosen strategies (solutions) before their implementation and at the final stage of the operation. It allows you to analyze the results and draw a conclusion about which of the strategies would be optimal.

The values ​​that are changed during optimization and included in the mathematical model of the optimization object are called optimization parameters, and the ratios that set the limits for the possible change in these parameters are restrictions.

Restrictions- these are ratios that narrow the area of ​​feasible, acceptable or admissible solutions, and fix the main external and internal properties of the object. These constraints can be given in the form of equalities or inequalities (or their systems).

Decision mathematical model of the economic problem, or acceptable plan, is a set of unknown values ​​that satisfies its system of constraints. The model can have many solutions, or feasible plans, among which it is necessary to find the only one that satisfies the system of constraints and the objective function.

A feasible plan that satisfies the objective function is called optimal .

If the problem model has a set of optimal plans, then for each of them the value of the objective function is the same.

Thus, to make an optimal solution to any economic problem, it is necessary to build its mathematical model, which in structure includes a system of constraints, an objective function, an optimality criterion and a solution.

The process of building a mathematical model is called mathematical modeling .

Drawing up a model of an object requires understanding the essence of the described phenomenon and knowledge of the mathematical apparatus.

Modeling and construction of a mathematical model of an economic object make it possible to reduce the economic analysis of production processes to mathematical analysis and the adoption of effective (optimal) decisions.

When constructing a mathematical model, it is important to avoid, on the one hand, excessive simplification of an economic phenomenon or process (since excessive simplification does not reflect reality), on the other hand, its excessive detailing and complication (since this leads to a large number variables and complicates the construction of the model).

The main elements of the model:

1) Initial data:

determined,

random.

2) Required variables:

continuous,

discrete.

3) Dependencies:

linear (variables are included in the first degree and there is no product),

non-linear (variables are included in degrees higher than the first or there is a product of variables).

The combination of various elements of the model leads to different classes of optimization problems (topic 2) that require different methods of solution.

When solving a specific economic problem, the use of methods of optimal solutions involves:

construction of mathematical models for decision-making problems in complex situations or under conditions of uncertainty,

Studying the relationships that subsequently determine decision-making, and establishing optimality criteria that allow evaluating the advantage of one or another variant of action.

To the main methods making the best decisions include:

1) Methods of mathematical programming:

linear programming,

non-linear programming

integer programming,

dynamic programming,

convex programming,

geometric programming,

parametric programming

stochastic programming,

heuristic programming.

2) Methods of the theory of queuing.

3) Methods of game theory.

4) Classical optimization methods (Lagrange method, gradient method).

5) Network planning and management methods.