According to the textbook of Sovetov and Yakovlev: "a model (Latin modulus - measure) is an object-substitute of the original object, providing the study of some properties of the original." (p. 6) "Replacing one object with another in order to obtain information about the most important properties of the original object with the help of a model object is called modeling." (p. 6) “Under mathematical modeling we will understand the process of establishing correspondence to a given real object of some mathematical object, called a mathematical model, and the study of this model, which allows obtaining the characteristics of the considered real object. The type of mathematical model depends on both the nature of the real object and the tasks of studying the object and the required reliability and accuracy of solving this problem.

Finally, the most concise definition of a mathematical model: "An equation expressing the idea».

Model classification

Formal classification of models

The formal classification of models is based on the classification of the mathematical tools used. Often built in the form of dichotomies. For example, one of the popular sets of dichotomies is:

etc. Each constructed model is linear or non-linear, deterministic or stochastic, ... Naturally, mixed types are also possible: concentrated in one respect (in terms of parameters), distributed models in another, etc.

Classification by the way the object is represented

Along with the formal classification, the models differ in the way they represent the object:

• Structural or functional models

Structural Models represent an object as a system with its own device and mechanism of functioning. functional models do not use such representations and reflect only the externally perceived behavior (functioning) of the object. In their extreme expression, they are also called "black box" models. Combined types of models are also possible, which are sometimes referred to as "models" gray box».

Content and formal models

Almost all authors describing the process of mathematical modeling indicate that first a special ideal construction is built, content model. There is no established terminology here, and other authors call this ideal object conceptual model , speculative model or premodel. In this case, the final mathematical construction is called formal model or just a mathematical model obtained as a result of the formalization of this content model (pre-model). The construction of a meaningful model can be carried out using a set of ready-made idealizations, as in mechanics, where ideal springs, rigid bodies, ideal pendulums, elastic media, etc. give ready-made structural elements for meaningful modeling. However, in areas of knowledge where there are no fully completed formalized theories (the cutting edge of physics, biology, economics, sociology, psychology, and most other fields), the creation of meaningful models is dramatically more complicated.

Meaningful classification of models

No hypothesis in science can be proven once and for all. Richard Feynman put it very clearly:

“We always have the ability to disprove a theory, but note that we can never prove that it is correct. Let's suppose that you put forward a successful hypothesis, calculated where it leads, and found that all its consequences are confirmed experimentally. Does this mean that your theory is correct? No, it simply means that you failed to refute it.

If a model of the first type is built, then this means that it is temporarily recognized as true and one can concentrate on other problems. However, this cannot be a point in research, but only a temporary pause: the status of the model of the first type can only be temporary.

Type 2: Phenomenological model (behave as if…)

The phenomenological model contains a mechanism for describing the phenomenon. However, this mechanism is not convincing enough, cannot be sufficiently confirmed by the available data, or does not agree well with the available theories and accumulated knowledge about the object. Therefore, phenomenological models have the status of temporary solutions. It is believed that the answer is still unknown and it is necessary to continue the search for "true mechanisms". Peierls refers, for example, the caloric model and the quark model of elementary particles to the second type.

The role of the model in research may change over time, it may happen that new data and theories confirm phenomenological models and they are promoted to the status of a hypothesis. Likewise, new knowledge may gradually come into conflict with models-hypotheses of the first type, and they may be transferred to the second. Thus, the quark model is gradually moving into the category of hypotheses; atomism in physics arose as a temporary solution, but with the course of history it passed into the first type. But the ether models have gone from type 1 to type 2, and now they are outside of science.

The idea of ​​simplification is very popular when building models. But simplification is different. Peierls distinguishes three types of simplifications in modeling.

Type 3: Approximation (something is considered very large or very small)

If it is possible to construct equations describing the system under study, this does not mean that they can be solved even with the help of a computer. A common technique in this case is the use of approximations (models of type 3). Among them linear response models. The equations are replaced by linear ones. The standard example is Ohm's law.

And here is type 8, which is widely used in mathematical models of biological systems.

Type 8: Possibility demonstration (the main thing is to show the internal consistency of the possibility)

These are also thought experiments. with imaginary entities demonstrating that supposed phenomenon consistent with basic principles and internally consistent. This is the main difference from models of type 7, which reveal hidden contradictions.

One of the most famous of these experiments is Lobachevsky's geometry (Lobachevsky called it "imaginary geometry"). Another example is the mass production of formally kinetic models of chemical and biological oscillations, autowaves, etc. The Einstein-Podolsky-Rosen paradox was conceived as a type 7 model to demonstrate inconsistency quantum mechanics. In a completely unplanned way, it eventually turned into a type 8 model - a demonstration of the possibility of quantum teleportation of information.

Example

Let us consider a mechanical system consisting of a spring fixed at one end and a load of mass , attached to the free end of the spring. We will assume that the load can move only in the direction of the spring axis (for example, the movement occurs along the rod). Let us construct a mathematical model of this system. We will describe the state of the system by the distance from the center of the load to its equilibrium position. Let us describe the interaction of a spring and a load using Hooke's law() after which we use Newton's second lawto express it in the form of a differential equation:

where means the second derivative of with respect to time: .

The resulting equation describes the mathematical model of the considered physical system. This pattern is called the "harmonic oscillator".

According to the formal classification, this model is linear, deterministic, dynamic, concentrated, continuous. In the process of its construction, we made many assumptions (about the absence of external forces, the absence of friction, the smallness of deviations, etc.), which in reality may not be fulfilled.

In relation to reality, this is most often a type 4 model. simplification(“we omit some details for clarity”), since some essential universal features (for example, dissipation) are omitted. In some approximation (say, as long as the deviation of the load from equilibrium is small, with little friction, for a not too long time and subject to certain other conditions), such a model describes a real mechanical system quite well, since the discarded factors have a negligible effect on its behavior . However, the model can be refined by taking into account some of these factors. This will lead to a new model, with a wider (though again limited) scope.

However, when the model is refined, the complexity of its mathematical study can increase significantly and make the model virtually useless. Often, a simpler model allows you to better and deeper explore the real system than a more complex (and, formally, “more correct”) one.

If we apply the harmonic oscillator model to objects that are far from physics, its meaningful status may be different. For example, when applying this model to biological populations, it should most likely be attributed to type 6 analogy(“Let’s take into account only some features”).

Hard and soft models

The harmonic oscillator is an example of a so-called "hard" model. It is obtained as a result of a strong idealization of a real physical system. To resolve the issue of its applicability, it is necessary to understand how significant are the factors that we have neglected. In other words, it is necessary to investigate the "soft" model, which is obtained by a small perturbation of the "hard" one. It can be given, for example, by the following equation:

Here - some function, which can take into account the friction force or the dependence of the coefficient of stiffness of the spring on the degree of its stretching - some small parameter. The explicit form of the function does not interest us at the moment. If we prove that the behavior of a soft model does not fundamentally differ from that of a hard model (regardless of the explicit form of the perturbing factors, if they are small enough), the problem will be reduced to studying the hard model. Otherwise, the application of the results obtained in the study of the rigid model will require additional research. For example, the solution to the equation of a harmonic oscillator are functions of the form , that is, oscillations with a constant amplitude. Does it follow from this that a real oscillator will oscillate indefinitely with a constant amplitude? No, because considering a system with an arbitrarily small friction (always present in a real system), we get damped oscillations. The behavior of the system has changed qualitatively.

If a system retains its qualitative behavior under a small perturbation, it is said to be structurally stable. The harmonic oscillator is an example of a structurally unstable (non-rough) system. However, this model can be used to study processes over limited time intervals.

Universality of models

The most important mathematical models usually have important property universality: fundamentally different real phenomena can be described by the same mathematical model. Let's say that the harmonic oscillator describes not only the behavior of the load on the spring, but also other oscillatory processes, often having a completely different nature: small oscillations of a pendulum, fluctuations in the liquid level in a -shaped vessel, or a change in the current strength in an oscillatory circuit. Thus, studying one mathematical model, we study at once a whole class of phenomena described by it. It is this isomorphism of laws expressed by mathematical models in various segments scientific knowledge, Ludwig von Bertalanffy's feat of creating "General Systems Theory".

Direct and inverse problems of mathematical modeling

There are many problems associated with mathematical modeling. First, it is necessary to come up with the basic scheme of the object being modeled, to reproduce it within the framework of the idealizations of this science. So, a train car turns into a system of plates and more complex bodies made of different materials, each material is given as its standard mechanical idealization (density, elastic moduli, standard strength characteristics), after which equations are drawn up, along the way some details are discarded as insignificant , calculations are made, compared with measurements, the model is refined, and so on. However, for the development of mathematical modeling technologies, it is useful to disassemble this process into its main constituent elements.

Traditionally, there are two main classes of problems associated with mathematical models: direct and inverse.

Direct problem: the structure of the model and all its parameters are considered known, the main task is to study the model to extract useful knowledge about the object. What static load can the bridge withstand? How it will react to a dynamic load (for example, to the march of a company of soldiers, or to the passage of a train at different speeds), how the plane will overcome the sound barrier, whether it will fall apart from flutter - these are typical examples of a direct task. Setting the correct direct problem (asking the correct question) requires special skill. If the right questions are not asked, the bridge can collapse, even if a good model has been built for its behavior. So, in 1879, a metal bridge across the River Tey collapsed in Great Britain, the designers of which built a bridge model, calculated it for a 20-fold margin of safety for the payload, but forgot about the winds constantly blowing in those places. And after a year and a half it collapsed.

In the simplest case (one oscillator equation, for example), the direct problem is very simple and reduces to an explicit solution of this equation.

Inverse problem: many possible models are known, it is necessary to choose a specific model based on additional data about the object. Most often, the structure of the model is known and some unknown parameters need to be determined. Additional information may consist in additional empirical data, or in the requirements for the object ( design task). Additional data can come regardless of the process of solving the inverse problem ( passive observation) or be the result of an experiment specially planned during the solution ( active surveillance).

One of the first examples of a virtuoso solution of an inverse problem with the fullest possible use of available data was the method constructed by I. Newton for reconstructing friction forces from observed damped oscillations.

Another example is mathematical statistics. The task of this science is the development of methods for recording, describing and analyzing observational and experimental data in order to build probabilistic models of mass random phenomena. Those. the set of possible models is limited by probabilistic models. In specific problems, the set of models is more limited.

Computer simulation systems

To support mathematical modeling, computer mathematics systems have been developed, for example, Maple, Mathematica, Mathcad, MATLAB, VisSim, etc. They allow you to create formal and block models of both simple and complex processes and devices and easily change model parameters during simulation. Block Models are represented by blocks (most often graphical), the set and connection of which are specified by the model diagram.

Additional examples

Malthus model

The growth rate is proportional to the current population size. It is described by the differential equation

where is a certain parameter determined by the difference between the birth rate and the death rate. The solution to this equation is exponential function. If the birth rate exceeds the death rate (), the population size increases indefinitely and very quickly. It is clear that in reality this cannot happen due to limited resources. When a certain critical population size is reached, the model ceases to be adequate, since it does not take into account the limited resources. A refinement of the Malthus model can be the logistic model , which is described by the Verhulst differential equation

where is the "equilibrium" population size, at which the birth rate is exactly compensated by the death rate. The population size in such a model tends to the equilibrium value , and this behavior is structurally stable.

predator-prey system

Let's say that two types of animals live in a certain area: rabbits (eating plants) and foxes (eating rabbits). Let the number of rabbits, the number of foxes. Using the Malthus model with the necessary corrections, taking into account the eating of rabbits by foxes, we arrive at the following system, which bears the name tray models - Volterra:

This system has an equilibrium state where the number of rabbits and foxes is constant. Deviation from this state leads to fluctuations in the number of rabbits and foxes, similar to fluctuations in the harmonic oscillator. As in the case of the harmonic oscillator, this behavior is not structurally stable: a small change in the model (for example, taking into account the limited resources needed by rabbits) can lead to a qualitative change in behavior. For example, the equilibrium state can become stable, and population fluctuations will fade. The opposite situation is also possible, when any small deviation from the equilibrium position will lead to catastrophic consequences, up to the complete extinction of one of the species. To the question of which of these scenarios is realized, the Volterra-Lotka model does not give an answer: additional research is required here.

Notes

1. "A mathematical representation of reality" (Encyclopaedia Britanica)
2. Novik I. B., On philosophical questions of cybernetic modeling. M., Knowledge, 1964.
3. Sovetov B. Ya., Yakovlev S. A., Systems Modeling: Proc. for universities - 3rd ed., revised. and additional - M.: Higher. school, 2001. - 343 p. ISBN 5-06-003860-2
4. Samarsky A. A., Mikhailov A. P. Mathematical modeling. Ideas. Methods. Examples. - 2nd ed., corrected. - M .: Fizmatlit, 2001. - ISBN 5-9221-0120-X
5. Myshkis A. D., Elements of the theory of mathematical models. - 3rd ed., Rev. - M.: KomKniga, 2007. - 192 with ISBN 978-5-484-00953-4
6. Sevostyanov, A.G. Modeling of technological processes: textbook / A.G. Sevostyanov, P.A. Sevostyanov. - M.: Light and food industry, 1984. - 344 p.
7. Wiktionary: mathematical models
8. CliffsNotes.com. Earth Science Glossary. 20 Sep 2010
9. Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena, Springer, Complexity series, Berlin-Heidelberg-New York, 2006. XII+562 pp. ISBN 3-540-35885-4
10. “A theory is considered linear or non-linear, depending on what - linear or non-linear - mathematical apparatus, what - linear or non-linear - mathematical models it uses. ... without denying the latter. A modern physicist, if he happened to redefine such an important entity as non-linearity, would most likely act differently, and, preferring non-linearity as the more important and common of the two opposites, would define linearity as “non-non-linearity”. Danilov Yu. A., Lectures on nonlinear dynamics. Elementary introduction. Synergetics: from the past to the future series. Ed.2. - M.: URSS, 2006. - 208 p. ISBN 5-484-00183-8
11. “Dynamical systems modeled by a finite number of ordinary differential equations are called lumped or point systems. They are described using a finite-dimensional phase space and are characterized by a finite number of degrees of freedom. The same system in various conditions can be considered either concentrated or distributed. Mathematical models of distributed systems are partial differential equations, integral equations, or ordinary delay equations. The number of degrees of freedom of a distributed system is infinite, and an infinite number of data are required to determine its state. Anishchenko V.S., Dynamic Systems, Soros Educational Journal, 1997, No. 11, p. 77-84.
12. “Depending on the nature of the studied processes in the system S, all types of modeling can be divided into deterministic and stochastic, static and dynamic, discrete, continuous and discrete-continuous. Deterministic modeling displays deterministic processes, that is, processes in which the absence of any random influences is assumed; stochastic modeling displays probabilistic processes and events. … Static modeling is used to describe the behavior of an object at any point in time, while dynamic modeling reflects the behavior of an object over time. Discrete modeling serves to describe processes that are assumed to be discrete, respectively, continuous modeling allows you to reflect continuous processes in systems, and discrete-continuous modeling is used for cases where you want to highlight the presence of both discrete and continuous processes. Sovetov B. Ya., Yakovlev S. A. ISBN 5-06-003860-2
13. Usually, the mathematical model reflects the structure (arrangement) of the object being modeled, the properties and interconnections of the components of this object that are essential for the purposes of the study; such a model is called structural. If the model reflects only how the object functions - for example, how it reacts to external influences - then it is called a functional or, figuratively, a black box. Combined models are also possible. Myshkis A. D. ISBN 978-5-484-00953-4
14. “Obvious, but the most important initial stage of constructing or choosing a mathematical model is to obtain the clearest possible idea of ​​the object being modeled and to refine its content model based on informal discussions. Time and efforts should not be spared at this stage; the success of the entire study largely depends on it. More than once it happened that a significant amount of work expended on solving mathematical problem, turned out to be ineffective or even wasted due to insufficient attention to this side of the matter. Myshkis A. D., Elements of theory mathematical models. - 3rd ed., Rev. - M.: KomKniga, 2007. - 192 with ISBN 978-5-484-00953-4, p. 35.
15. « Description of the conceptual model of the system. At this sub-stage of building a system model: a) the conceptual model M is described in abstract terms and concepts; b) a description of the model is given using typical mathematical schemes; c) hypotheses and assumptions are finally accepted; d) the choice of a procedure for approximating real processes when building a model is substantiated. Sovetov B. Ya., Yakovlev S. A., Systems Modeling: Proc. for universities - 3rd ed., revised. and additional - M.: Higher. school, 2001. - 343 p. ISBN 5-06-003860-2, p. 93.
16. Blekhman I. I., Myshkis A. D.,

In the article brought to your attention, we offer examples of mathematical models. In addition, we will pay attention to the stages of creating models and analyze some of the problems associated with mathematical modeling.

Another issue of ours is mathematical models in economics, examples of which we will consider a definition a little later. We propose to start our conversation with the very concept of “model”, briefly consider their classification and move on to our main questions.

The concept of "model"

We often hear the word "model". What is it? This term has many definitions, here are just three of them:

• a specific object that is created to receive and store information, reflecting some properties or characteristics, and so on, of the original of this object (this specific object can be expressed in different forms: mental, description using signs, and so on);
• a model also means a display of any specific situation, life or management;
• a small copy of an object can serve as a model (they are created for a more detailed study and analysis, since the model reflects the structure and relationships).

Based on everything that was said earlier, we can draw a small conclusion: the model allows you to study in detail a complex system or object.

All models can be classified according to a number of criteria:

• by area of ​​use (educational, experimental, scientific and technical, gaming, simulation);
• by dynamics (static and dynamic);
• by branch of knowledge (physical, chemical, geographical, historical, sociological, economic, mathematical);
• according to the method of presentation (material and informational).

Information models, in turn, are divided into sign and verbal. And iconic - on computer and non-computer. Now let's move on to a detailed consideration of examples of a mathematical model.

Mathematical model

As it is not difficult to guess, a mathematical model reflects some features of an object or phenomenon with the help of special mathematical symbols. Mathematics is needed in order to model the laws of the world in its own specific language.

The method of mathematical modeling originated quite a long time ago, thousands of years ago, along with the advent of this science. However, the impetus for the development of this modeling method was given by the appearance of computers (electronic computers).

Now let's move on to classification. It can also be carried out according to some signs. They are presented in the table below.

We propose to stop and take a closer look at the last classification, since it reflects the general patterns of modeling and the goals of the models being created.

Descriptive Models

In this chapter, we propose to dwell in more detail on descriptive mathematical models. In order to make everything very clear, an example will be given.

To begin with, this view can be called descriptive. This is due to the fact that we simply make calculations and forecasts, but we cannot influence the outcome of the event in any way.

A striking example of a descriptive mathematical model is the calculation of the flight path, speed, distance from the Earth of a comet that invaded the expanses of our solar system. This model is descriptive, since all the results obtained can only warn us of some kind of danger. Unfortunately, we cannot influence the outcome of the event. However, based on the calculations obtained, it is possible to take any measures to preserve life on Earth.

Optimization Models

Now we will talk a little about economic and mathematical models, examples of which can be various situations. In this case we are talking about models that help to find the right answer in certain conditions. They must have some parameters. To make it very clear, consider an example from the agrarian part.

We have a granary, but the grain spoils very quickly. In this case, we need to choose the right temperature regime and optimize the storage process.

Thus, we can define the concept of "optimization model". In a mathematical sense, this is a system of equations (both linear and not), the solution of which helps to find the optimal solution in a particular economic situation. We have considered an example of a mathematical model (optimization), but I would like to add one more thing: this type belongs to the class of extreme problems, they help to describe the functioning of the economic system.

We note one more nuance: models can wear different character(see table below).

Multicriteria models

Now we invite you to talk a little about the mathematical model of multiobjective optimization. Before that, we gave an example of a mathematical model for optimizing a process according to any one criterion, but what if there are a lot of them?

A striking example of a multicriteria task is the organization of proper, healthy and at the same time economical nutrition of large groups of people. Such tasks are often encountered in the army, school canteens, summer camps, hospitals and so on.

What criteria are given to us in this task?

1. Food should be healthy.
2. Food expenses should be kept to a minimum.

As you can see, these goals do not coincide at all. This means that when solving a problem, it is necessary to look for the optimal solution, a balance between the two criteria.

Game models

Speaking about game models, it is necessary to understand the concept of "game theory". Simply put, these models reflect mathematical models of real conflicts. It is only worth understanding that, unlike a real conflict, a game mathematical model has its own specific rules.

Now I will give a minimum of information from game theory, which will help you understand what a game model is. And so, in the model there are necessarily parties (two or more), which are usually called players.

All models have certain characteristics.

The game model can be paired or multiple. If we have two subjects, then the conflict is paired, if more - multiple. An antagonistic game can also be distinguished, it is also called a zero-sum game. This is a model in which the gain of one of the participants is equal to the loss of the other.

simulation models

In this section, we will focus on simulation mathematical models. Examples of tasks are:

• model of the dynamics of the number of microorganisms;
• model of molecular motion, and so on.

In this case, we are talking about models that are as close as possible to real processes. By and large, they imitate any manifestation in nature. In the first case, for example, we can model the dynamics of the number of ants in one colony. In this case, you can observe the fate of each individual. In this case, the mathematical description is rarely used, more often there are written conditions:

• after five days, the female lays eggs;
• after twenty days the ant dies, and so on.

Thus, are used to describe a large system. Mathematical conclusion is the processing of the received statistical data.

Requirements

It is very important to know that there are some requirements for this type of model, among which are those given in the table below.

Versatility	This property allows you to use the same model when describing groups of objects of the same type. It is important to note that universal mathematical models are completely independent of the physical nature of the object under study.
Adequacy	Here it is important to understand that this property allows the most correct reproduction of real processes. In operation problems, this property of mathematical modeling is very important. An example of a model is the process of optimizing the use of a gas system. In this case, calculated and actual indicators are compared, as a result, the correctness of the compiled model is checked.
Accuracy	This requirement implies the coincidence of the values ​​that we obtain when calculating the mathematical model and the input parameters of our real object
Economy	The requirement of economy for any mathematical model is characterized by implementation costs. If the work with the model is carried out manually, then it is necessary to calculate how much time it will take to solve one problem using this mathematical model. If we are talking about computer-aided design, then indicators of time and computer memory are calculated

Modeling steps

In total, it is customary to distinguish four stages in mathematical modeling.

1. Formulation of laws linking parts of the model.
2. Study of mathematical problems.
3. Finding out the coincidence of practical and theoretical results.
4. Analysis and modernization of the model.

Economic and mathematical model

In this section, we will briefly highlight the issue. Examples of tasks can be:

• formation of a production program for the production of meat products, ensuring the maximum profit of production;
• maximizing the profit of the organization by calculating the optimal number of tables and chairs to be produced in a furniture factory, and so on.

The economic-mathematical model displays an economic abstraction, which is expressed using mathematical terms and signs.

Computer mathematical model

Examples of a computer mathematical model are:

• hydraulics tasks using flowcharts, diagrams, tables, and so on;
• tasks for mechanics solid body, etc.

A computer model is an image of an object or system, presented as:

• tables;
• block diagrams;
• diagrams;
• graphics, and so on.

At the same time, this model reflects the structure and interconnections of the system.

Building an economic and mathematical model

We have already talked about what an economic-mathematical model is. An example of solving the problem will be considered right now. We need to analyze the production program to identify the reserve for increasing profits with a shift in the assortment.

We will not fully consider the problem, but only build an economic and mathematical model. The criterion of our task is profit maximization. Then the function has the form: Л=р1*х1+р2*х2… tending to the maximum. In this model, p is the profit per unit, x is the number of units produced. Further, based on the constructed model, it is necessary to make calculations and summarize.

An example of building a simple mathematical model

Task. The fisherman returned with the following catch:

• 8 fish - inhabitants of the northern seas;
• 20% of the catch - the inhabitants of the southern seas;
• not a single fish was found from the local river.

How many fish did he buy at the store?

So, an example of constructing a mathematical model of this problem is as follows. We denote the total number of fish as x. Following the condition, 0.2x is the number of fish living in southern latitudes. Now we combine all the available information and get a mathematical model of the problem: x=0.2x+8. We solve the equation and get the answer to main question: 10 fish he bought in the store.

Lecture 1

METHODOLOGICAL BASES OF MODELING

The current state of the problem of system modeling

Concepts of Modeling and Simulation

Modeling can be considered as a replacement of the investigated object (original) by its conditional image, description or another object, called model and providing behavior close to the original within certain assumptions and acceptable errors. Modeling is usually performed with the aim of knowing the properties of the original by examining its model, and not the object itself. Of course, modeling is justified in the case when it is simpler than creating the original itself, or when the latter, for some reason, is better not to create at all.

Under model a physical or abstract object is understood, the properties of which are in a certain sense similar to the properties of the object under study. In this case, the requirements for the model are determined by the problem being solved and the available means. There are a number of general requirements for models:

2) completeness - providing the recipient with all the necessary information

about the object;

3) flexibility - the ability to reproduce different situations in everything

range of changing conditions and parameters;

4) the complexity of development should be acceptable for the existing

time and software.

Modeling is the process of building a model of an object and studying its properties by examining the model.

Thus, modeling involves 2 main stages:

1) model development;

2) study of the model and drawing conclusions.

At the same time, at each stage, different tasks are solved and

essentially different methods and means.

In practice, various modeling methods are used. Depending on the method of implementation, all models can be divided into two large classes: physical and mathematical.

Mathematical modeling It is customary to consider it as a means of studying processes or phenomena with the help of their mathematical models.

Under physical modeling is understood as the study of objects and phenomena on physical models, when the process under study is reproduced with the preservation of its physical nature or another physical phenomenon similar to the one under study is used. Wherein physical models As a rule, they assume the real embodiment of those physical properties of the original that are essential in a particular situation. For example, when designing a new aircraft, its model is created that has the same aerodynamic properties; when planning a building, architects make a layout that reflects the spatial arrangement of its elements. In this regard, physical modeling is also called prototyping.

HIL Modeling is a study of controlled systems on simulation complexes with the inclusion of real equipment in the model. Along with real equipment, the closed model includes impact and interference simulators, mathematical models of the external environment and processes, for which a sufficiently accurate mathematical description is not known. The inclusion of real equipment or real systems in the circuit for modeling complex processes makes it possible to reduce a priori uncertainty and explore processes for which there is no exact mathematical description. With the help of semi-natural simulation, studies are performed taking into account small time constants and non-linearities inherent in real equipment. In the study of models with the inclusion of real equipment, the concept is used dynamic simulation, in the study complex systems and phenomena - evolutionary, imitation and cybernetic simulation.

Obviously, the real benefit of modeling can only be obtained if two conditions are met:

1) the model provides a correct (adequate) display of properties

the original, significant from the point of view of the operation under study;

2) the model makes it possible to eliminate the problems listed above, which are inherent

conducting research on real objects.

2. Basic concepts of mathematical modeling

The solution of practical problems by mathematical methods is consistently carried out by formulating the problem (development of a mathematical model), choosing a method for studying the obtained mathematical model, and analyzing the obtained mathematical result. The mathematical formulation of the problem is usually presented in the form of geometric images, functions, systems of equations, etc. The description of an object (phenomenon) can be represented using continuous or discrete, deterministic or stochastic and other mathematical forms.

Theory of mathematical modeling ensures the identification of regularities in the flow of various phenomena of the surrounding world or the operation of systems and devices by their mathematical description and modeling without field tests. In this case, the provisions and laws of mathematics are used that describe the simulated phenomena, systems or devices at a certain level of their idealization.

Mathematical Model (MM) is a formalized description of a system (or operation) in some abstract language, for example, in the form of a set of mathematical relations or an algorithm scheme, i.e. e. such a mathematical description that provides an imitation of the operation of systems or devices at a level sufficiently close to their real behavior obtained during full-scale testing of systems or devices.

Any MM describes a real object, phenomenon or process with some degree of approximation to reality. The type of MM depends both on the nature of the real object and on the objectives of the study.

Mathematical modeling social, economic, biological and physical phenomena, objects, systems and various devices is one of the most important means of understanding nature and designing a wide variety of systems and devices. There are known examples of the effective use of modeling in the creation of nuclear technologies, aviation and aerospace systems, in the forecast of atmospheric and oceanic phenomena, weather, etc.

However, such serious areas of modeling often require supercomputers and years of work by large teams of scientists to prepare data for modeling and its debugging. Nevertheless, in this case, too, mathematical modeling of complex systems and devices not only saves money on research and testing, but can also eliminate environmental disasters - for example, it makes it possible to abandon the testing of nuclear and thermonuclear weapons in favor of their mathematical modeling or testing aerospace systems before their real flights. Meanwhile, mathematical modeling at the level of solving simpler problems, for example, from the field of mechanics, electrical engineering, electronics, radio engineering and many other areas of science and technology, has now become available to perform on modern PCs. And when using generalized models, it becomes possible to model quite complex systems, for example, telecommunication systems and networks, radar or radio navigation systems.

The purpose of mathematical modeling is the analysis of real processes (in nature or technology) by mathematical methods. In turn, this requires the formalization of the MM process to be investigated. The model can be a mathematical expression containing variables whose behavior is similar to the behavior of a real system. The model can include elements of randomness that take into account the probabilities of possible actions of two or more"players", as, for example, in game theory; or it may represent the real variables of the interconnected parts of the operating system.

Mathematical modeling for studying the characteristics of systems can be divided into analytical, simulation and combined. In turn, MM are divided into simulation and analytical.

Analytical Modeling

For analytical modeling it is characteristic that the processes of functioning of the system are written in the form of some functional relations (algebraic, differential, integral equations). The analytical model can be investigated by the following methods:

1) analytical, when they strive to get into general view explicit dependencies for system characteristics;

2) numerical, when it is not possible to find a solution to equations in general form and they are solved for specific initial data;

3) qualitative, when, in the absence of a solution, some of its properties are found.

Analytical models can be obtained only for relatively simple systems. For complex systems, large mathematical problems often arise. To apply the analytical method, one goes to a significant simplification of the original model. However, a study on a simplified model helps to obtain only indicative results. Analytical models mathematically correctly reflect the relationship between input and output variables and parameters. But their structure does not reflect the internal structure of the object.

In analytical modeling, its results are presented in the form of analytical expressions. For example, by connecting RC- circuit to a constant voltage source E(R, C and E are the components of this model), we can make an analytical expression for the time dependence of the voltage u(t) on the capacitor C:

This is a linear differential equation (DE) and is an analytical model of this simple linear circuit. Its analytical solution, under the initial condition u(0) = 0 , meaning a discharged capacitor C at the beginning of the simulation, allows you to find the required dependence - in the form of a formula:

u(t) = E(1− exp(- t/RC)). (2)

However, even in this simplest example, certain efforts are required to solve differential equation (1) or to apply computer mathematics systems(SCM) with symbolic calculations - computer algebra systems. For this quite trivial case, the solution of the problem of modeling a linear RC-circuit gives an analytical expression (2) of a rather general form - it is suitable for describing the operation of the circuit for any component ratings R, C and E, and describes the exponential charge of the capacitor C through a resistor R from a constant voltage source E.

Undoubtedly, finding analytical solutions in analytical modeling turns out to be extremely valuable for revealing the general theoretical laws of simple linear circuits, systems and devices. However, its complexity increases sharply as the impact on the model becomes more complex and the order and number of equations of state that describe the modeled object increase. You can get more or less visible results when modeling objects of the second or third order, but even with a higher order, analytical expressions become excessively cumbersome, complex and difficult to comprehend. For example, even a simple electronic amplifier often contains dozens of components. However, many modern SCMs, such as systems of symbolic mathematics Maple, Mathematica or Wednesday MATLAB are capable of automating to a large extent the solution of complex problems of analytical modeling.

One type of modeling is numerical simulation, which consists in obtaining the necessary quantitative data about the behavior of systems or devices by any suitable numerical method, such as the Euler or Runge-Kutta methods. In practice, the modeling of nonlinear systems and devices using numerical methods is much more efficient than the analytical modeling of individual private linear circuits, systems or devices. For example, to solve DE (1) or DE systems in more complex cases, the solution in an analytical form is not obtained, but numerical simulation data can provide sufficiently complete data on the behavior of the simulated systems and devices, as well as plot graphs describing this behavior of dependencies.

Simulation

At imitation In modeling, the algorithm that implements the model reproduces the process of the system functioning in time. The elementary phenomena that make up the process are imitated, with the preservation of their logical structure and the sequence of flow in time.

The main advantage of simulation models compared to analytical ones is the ability to solve more complex problems.

Simulation models make it easy to take into account the presence of discrete or continuous elements, non-linear characteristics, random effects, etc. Therefore, this method is widely used at the design stage of complex systems. The main tool for the implementation of simulation modeling is a computer that allows digital modeling of systems and signals.

In this regard, we define the phrase " computer modelling”, which is increasingly used in the literature. We will assume that computer modelling- this is mathematical modeling using computer technology. Accordingly, computer simulation technology involves the following actions:

1) definition of the purpose of modeling;

2) development of a conceptual model;

3) formalization of the model;

4) software implementation of the model;

5) planning of model experiments;

6) implementation of the experiment plan;

7) analysis and interpretation of simulation results.

At simulation modeling the used MM reproduces the algorithm (“logic”) of the functioning of the system under study in time for various combinations of values ​​of the parameters of the system and the environment.

An example of the simplest analytical model is the equation of uniform rectilinear motion. When studying such a process using a simulation model, observation of the change in the path traveled over time should be implemented. Obviously, in some cases, analytical modeling is more preferable, in others - simulation (or a combination of both). To make a good choice, two questions must be answered.

What is the purpose of modeling?

To what class can the simulated phenomenon be assigned?

Answers to both of these questions can be obtained during the execution of the first two stages of modeling.

Simulation models not only in properties, but also in structure correspond to the object being modeled. In this case, there is an unambiguous and explicit correspondence between the processes obtained on the model and the processes occurring on the object. The disadvantage of simulation modeling is that it takes a long time to solve the problem in order to obtain good accuracy.

The results of simulation modeling of the operation of a stochastic system are realizations random variables or processes. Therefore, to find the characteristics of the system, multiple repetition and subsequent data processing are required. Most often, in this case, a type of simulation is used - statistical

modeling(or the Monte Carlo method), i.e. reproduction in models of random factors, events, quantities, processes, fields.

Based on the results of statistical modeling, estimates of probabilistic quality criteria, general and particular, characterizing the functioning and efficiency of the controlled system are determined. Statistical modeling is widely used to solve scientific and applied problems in various fields of science and technology. Methods of statistical modeling are widely used in the study of complex dynamic systems, evaluation of their functioning and efficiency.

The final stage of statistical modeling is based on the mathematical processing of the obtained results. Here, methods of mathematical statistics are used (parametric and non-parametric estimation, hypothesis testing). An example of a parametric assessment is the sample mean of a performance measure. Among the nonparametric methods, the most widely used histogram method.

The considered scheme is based on multiple statistical tests of the system and methods of statistics of independent random variables. This scheme is far from always natural in practice and optimal in terms of costs. Reduction of system testing time can be achieved by using more accurate estimation methods. As is known from mathematical statistics, effective estimates have the highest accuracy for a given sample size. Optimal filtering and the maximum likelihood method give general method obtaining such estimates. In the problems of statistical modeling, the processing of realizations of random processes is necessary not only for the analysis of output processes.

It is also very important to control the characteristics of input random effects. The control consists in checking whether the distributions of the generated processes correspond to the given distributions. This task is often formulated as hypothesis testing task.

The general trend in computer-assisted simulation of complex controlled systems is the desire to reduce the simulation time, as well as to conduct research in real time. Computational algorithms are conveniently represented in a recurrent form that allows their implementation at the pace of current information.

PRINCIPLES OF A SYSTEM APPROACH IN MODELING

Fundamentals of systems theory

The main provisions of the theory of systems arose in the course of the study of dynamic systems and their functional elements. A system is understood as a group of interrelated elements acting together to perform a predetermined task. Systems analysis allows you to determine the most real ways accomplishment of the set task, ensuring the maximum satisfaction of the set requirements.

The elements that form the basis of systems theory are not created with the help of hypotheses, but are discovered experimentally. In order to start building a system, it is necessary to have general characteristics of technological processes. The same is true for the principles of creating mathematically formulated criteria that a process or its theoretical description must satisfy. Modeling is one of the most important methods of scientific research and experimentation.

When building models of objects, a systematic approach is used, which is a methodology for solving complex problems, which is based on the consideration of an object as a system operating in a certain environment. The system approach involves the disclosure of the integrity of the object, the identification and study of its internal structure, as well as connections with the external environment. In this case, the object is presented as a part of the real world, which is identified and studied in connection with the problem of building a model being solved. Besides, systems approach involves a consistent transition from the general to the particular, when the consideration is based on the design goal, and the object is considered in relation to the environment.

A complex object can be divided into subsystems, which are parts of the object that meet the following requirements:

1) the subsystem is a functionally independent part of the object. It is connected with other subsystems, exchanges information and energy with them;

2) for each subsystem, functions or properties that do not coincide with the properties of the entire system can be defined;

3) each of the subsystems can be further subdivided to the level of elements.

In this case, an element is understood as a subsystem of the lower level, the further division of which is inexpedient from the standpoint of the problem being solved.

Thus, a system can be defined as a representation of an object in the form of a set of subsystems, elements, and relationships for the purpose of its creation, research, or improvement. At the same time, an enlarged representation of the system, which includes the main subsystems and connections between them, is called a macrostructure, and a detailed disclosure of the internal structure of the system to the level of elements is called a microstructure.

Along with the system, there is usually a supersystem - a system of a higher level, which includes the object under consideration, and the function of any system can be determined only through the supersystem.

It is necessary to single out the concept of the environment as a set of objects of the external world that significantly affect the efficiency of the system, but are not part of the system and its supersystem.

In connection with the systematic approach to building models, the concept of infrastructure is used, which describes the relationship of the system with its environment (environment). In this case, the selection, description and study of the properties of an object that are essential within a specific task is called the stratification of an object, and any model of an object is its stratified description.

For a systematic approach, it is important to determine the structure of the system, i.e. set of links between the elements of the system, reflecting their interaction. To do this, we first consider the structural and functional approaches to modeling.

With a structural approach, the composition of the selected elements of the system and the links between them are revealed. The totality of elements and relationships makes it possible to judge the structure of the system. The most general description of a structure is a topological description. It allows you to define the components of the system and their relationships using graphs. Less general is the functional description when individual functions are considered, i.e., algorithms for the behavior of the system. At the same time, a functional approach is implemented that determines the functions that the system performs.

On the basis of a systematic approach, a sequence of model development can be proposed, when two main stages of design are distinguished: macro-design and micro-design.

At the macro-design stage, a model of the external environment is built, resources and constraints are identified, a system model and criteria for assessing adequacy are selected.

The stage of microdesign largely depends on the specific type of model chosen. In the general case, it involves the creation of information, mathematical, technical and software support for the modeling system. At this stage, the main technical characteristics of the created model are established, the time of working with it and the cost of resources to obtain the specified quality of the model are estimated.

Regardless of the type of model, when building it, it is necessary to be guided by a number of principles of a systematic approach:

1) consistent progress through the stages of creating a model;

2) coordination of information, resource, reliability and other characteristics;

3) the correct ratio of different levels of model building;

4) the integrity of the individual stages of model design.

Mathematical modeling

1. What is mathematical modeling?

Since the middle of the XX century. in various fields of human activity, mathematical methods and computers began to be widely used. New disciplines such as "mathematical economics", "mathematical chemistry", "mathematical linguistics", etc., have emerged that study mathematical models of relevant objects and phenomena, as well as methods for studying these models.

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.

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.

3. Classification of models

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.

4. Examples of mathematical models

1) Problems about the movement of the projectile.

Consider the following problem in mechanics.

The projectile is launched from the Earth with an initial velocity v 0 = 30 m/s at an angle a = 45° to its surface; it is required to find the trajectory of its movement and the distance S between the start and end points of this trajectory.

Then, as it is known from the school physics course, the motion of the projectile is described by the formulas:

where t - time, g = 10 m / s 2 - free fall acceleration. These formulas give the mathematical model of the task. Expressing t in terms of x from the first equation and substituting it into the second, we get the equation for the trajectory of the projectile:

This curve (parabola) intersects the x-axis at two points: x 1 \u003d 0 (the beginning of the trajectory) and (the place where the projectile fell). Substituting the given values ​​v0 and a into the obtained formulas, we obtain

answer: y \u003d x - 90x 2, S \u003d 90 m.

Note that a number of assumptions were used in the construction of this model: for example, it is assumed that the Earth is flat, and the air and rotation of the Earth do not affect the movement of the projectile.

2) The problem of a tank with the smallest surface area.

It is required to find the height h 0 and radius r 0 of a tin tank with a volume V = 30 m 3, having the shape of a closed circular cylinder, at which its surface area S is minimal (in this case, the smallest amount of tin will be used to manufacture it).

Let's write down the following formulas for the volume and surface area of ​​a cylinder of height h and radius r:

V = p r 2 h, S = 2p r(r + h).

Expressing h in terms of r and V from the first formula and substituting the resulting expression into the second, we get:

Thus, from a mathematical point of view, the problem is reduced to determining the value of r at which the function S(r) reaches its minimum. Let us find those values ​​of r 0 for which the derivative

goes to zero: You can check that the second derivative of the function S(r) changes sign from minus to plus when the argument r passes through the point r 0 . Therefore, the function S(r) has a minimum at the point r0. The corresponding value h 0 = 2r 0 . Substituting the given value V into the expression for r 0 and h 0, we obtain the desired radius and height

3) Transport task.

There are two flour warehouses and two bakeries in the city. Every day, 50 tons of flour are exported from the first warehouse, and 70 tons from the second to the factories, with 40 tons to the first and 80 tons to the second.

Denote by a ij cost of transporting 1 ton of flour from the i-th warehouse to j-th plant(i, j = 1.2). Let be

a 11 \u003d 1.2 p., a 12 \u003d 1.6 p., a 21 \u003d 0.8 p., a 22 = 1 p.

How should transportation be planned so that their cost is minimal?

Let's give the problem a mathematical formulation. We denote by x 1 and x 2 the amount of flour that must be transported from the first warehouse to the first and second factories, and through x 3 and x 4 - from the second warehouse to the first and second factories, respectively. Then:

x 1 + x 2 = 50, x 3 + x 4 = 70, x 1 + x 3 = 40, x 2 + x 4 = 80. (1)

The total cost of all transportation is determined by the formula

f = 1.2x1 + 1.6x2 + 0.8x3 + x4.

From a mathematical point of view, the task is to find four numbers x 1 , x 2 , x 3 and x 4 that satisfy all given conditions and give the minimum of the function f. Let us solve the system of equations (1) with respect to xi (i = 1, 2, 3, 4) by the method of elimination of unknowns. We get that

x 1 \u003d x 4 - 30, x 2 \u003d 80 - x 4, x 3 \u003d 70 - x 4, (2)

and x 4 cannot be uniquely determined. Since x i i 0 (i = 1, 2, 3, 4), it follows from equations (2) that 30J x 4 J 70. Substituting the expression for x 1 , x 2 , x 3 into the formula for f, we obtain

f \u003d 148 - 0.2x 4.

It is easy to see that the minimum of this function is achieved at the maximum possible value of x 4, that is, at x 4 = 70. The corresponding values ​​of other unknowns are determined by formulas (2): x 1 = 40, x 2 = 10, x 3 = 0.

4) The problem of radioactive decay.

Let N(0) be the initial number of atoms of the radioactive substance, and N(t) be the number of undecayed atoms at time t. It has been experimentally established that the rate of change in the number of these atoms N "(t) is proportional to N (t), that is, N" (t) \u003d -l N (t), l > 0 is the radioactivity constant of a given substance. In the school course of mathematical analysis, it is shown that the solution to this differential equation has the form N(t) = N(0)e –l t . The time T, during which the number of initial atoms has halved, is called the half-life, and is an important characteristic of the radioactivity of a substance. To determine T, it is necessary to put in the formula Then For example, for radon l = 2.084 10–6, and hence T = 3.15 days.

5) The traveling salesman problem.

A traveling salesman living in city A 1 needs to visit cities A 2 , A 3 and A 4 , each city exactly once, and then return back to A 1 . It is known that all cities are connected in pairs by roads, and the lengths of roads b ij between cities A i and A j (i, j = 1, 2, 3, 4) are as follows:

b 12 = 30, b 14 = 20, b 23 = 50, b 24 = 40, b 13 = 70, b 34 = 60.

It is necessary to determine the order of visiting cities, in which the length of the corresponding path is minimal.

Let's depict each city as a point on the plane and mark it with the corresponding label Ai (i = 1, 2, 3, 4). Let's connect these points with line segments: they will depict roads between cities. For each “road”, we indicate its length in kilometers (Fig. 2). The result is a graph - a mathematical object consisting of a certain set of points on the plane (called vertices) and a certain set of lines connecting these points (called edges). Moreover, this graph is labeled, since some labels are assigned to its vertices and edges - numbers (edges) or symbols (vertices). A cycle on a graph is a sequence of vertices V 1 , V 2 , ..., V k , V 1 such that the vertices V 1 , ..., V k are different, and any pair of vertices V i , V i+1 (i = 1, ..., k – 1) and the pair V 1 , V k are connected by an edge. Thus, the problem under consideration is to find such a cycle on the graph passing through all four vertices for which the sum of all edge weights is minimal. Let's search through all the different cycles passing through four vertices and starting at A 1:

1) A 1, A 4, A 3, A 2, A 1;
2) A 1, A 3, A 2, A 4, A 1;
3) A 1 , A 3 , A 4 , A 2 , A 1 .

Now let's find the lengths of these cycles (in km): L 1 = 160, L 2 = 180, L 3 = 200. So, the route of the smallest length is the first one.

Note that if there are n vertices in a graph and all vertices are connected in pairs by edges (such a graph is called complete), then the number of cycles passing through all vertices is equal. Therefore, in our case there are exactly three cycles.

6) The problem of finding a connection between the structure and properties of substances.

Consider a few chemical compounds called normal alkanes. They consist of n carbon atoms and n + 2 hydrogen atoms (n = 1, 2 ...), interconnected as shown in Figure 3 for n = 3. Let the experimental values ​​of the boiling points of these compounds be known:

y e (3) = - 42°, y e (4) = 0°, y e (5) = 28°, y e (6) = 69°.

It is required to find an approximate relationship between the boiling point and the number n for these compounds. We assume that this dependence has the form

y » a n+b

where a, b - constants to be determined. For finding a and b we substitute into this formula successively n = 3, 4, 5, 6 and the corresponding values ​​of the boiling points. We have:

– 42 » 3 a+ b, 0 » 4 a+ b, 28 » 5 a+ b, 69 » 6 a+b.

To determine the best a and b there are many different methods. Let's use the simplest of them. We express b in terms of a from these equations:

b" - 42 - 3 a, b » – 4 a, b » 28 – 5 a, b » 69 – 6 a.

Let us take as the desired b the arithmetic mean of these values, that is, we put b » 16 - 4.5 a. Let us substitute this value b into the original system of equations and, calculating a, we get for a the following values: a» 37, a» 28, a» 28, a» 36 a the average value of these numbers, that is, we put a» 34. So, the desired equation has the form

y » 34n – 139.

Let's check the accuracy of the model on the initial four compounds, for which we calculate the boiling points using the obtained formula:

y r (3) = – 37°, y r (4) = – 3°, y r (5) = 31°, y r (6) = 65°.

Thus, the calculation error of this property for these compounds does not exceed 5°. We use the resulting equation to calculate the boiling point of a compound with n = 7, which is not included in the initial set, for which we substitute n = 7 into this equation: y р (7) = 99°. The result turned out to be quite accurate: it is known that the experimental value of the boiling point y e (7) = 98°.

7) The problem of determining the reliability of the electrical circuit.

Here we consider an example of a probabilistic model. First, let's give some information from the theory of probability - a mathematical discipline that studies the patterns of random phenomena observed during repeated repetition of an experiment. Let's call a random event A a possible outcome of some experience. Events A 1 , ..., A k form a complete group if one of them necessarily occurs as a result of the experiment. Events are called incompatible if they cannot occur simultaneously in the same experience. Let the event A occur m times during the n-fold repetition of the experiment. The frequency of the event A is the number W = . Obviously, the value of W cannot be predicted exactly until a series of n experiments has been carried out. However, the nature of random events is such that in practice the following effect is sometimes observed: with an increase in the number of experiments, the value practically ceases to be random and stabilizes around some non-random number P(A), called the probability of the event A. For an impossible event (which never occurs in the experiment) P(A)=0, and for a certain event (which always occurs in the experiment) P(A)=1. If events A 1 , ..., A k form a complete group of incompatible events, then P(A 1)+...+P(A k)=1.

Let, for example, the experience consists in throwing a dice and observing the number of dropped points X. Then we can introduce the following random events A i =(X = i), i = 1, ..., 6. They form a complete group of incompatible equally probable events, therefore P(A i) = (i = 1, ..., 6).

The sum of events A and B is the event A + B, which consists in the fact that at least one of them occurs in the experiment. The product of events A and B is the event AB, which consists in the simultaneous occurrence of these events. For independent events A and B, the formulas are true

P(AB) = P(A) P(B), P(A + B) = P(A) + P(B).

8) Consider now the following task. Suppose that three elements are connected in series in an electric circuit, working independently of each other. The failure probabilities of the 1st, 2nd and 3rd elements are respectively P 1 = 0.1, P 2 = 0.15, P 3 = 0.2. We will consider the circuit reliable if the probability that there will be no current in the circuit is not more than 0.4. It is required to determine whether the given chain is reliable.

Since the elements are connected in series, there will be no current in the circuit (event A) if at least one of the elements fails. Let A i be the event that i-th element works (i = 1, 2, 3). Then P(A1) = 0.9, P(A2) = 0.85, P(A3) = 0.8. Obviously, A 1 A 2 A 3 is the event that all three elements work simultaneously, and

P(A 1 A 2 A 3) = P(A 1) P(A 2) P(A 3) = 0.612.

Then P(A) + P(A 1 A 2 A 3) = 1, so P(A) = 0.388< 0,4. Следовательно, цепь является надежной.

In conclusion, we note that the above examples of mathematical models (among which there are functional and structural, deterministic and probabilistic) are illustrative and, obviously, do not exhaust the whole variety of mathematical models that arise in the natural and human sciences.

The tasks solved by LP methods are very diverse in content. But their mathematical models are similar and are conditionally combined into three large groups of problems:

• transport tasks;
• planning tasks;

Let us consider examples of specific economic problems of each type, and dwell in detail on building a model for each problem.

Transport task

On two trading bases BUT and AT There are 30 sets of furniture, 15 for each. All furniture needs to be delivered to two furniture stores, With and D and in With you need to deliver 10 headsets, and in D- 20. It is known that the delivery of one headset from the base BUT to the store With costs one monetary unit, to the store D- in three monetary units. According to the base AT to shops With and D: two and five monetary units. Make a transportation plan so that the cost of all transportation is the least.
For convenience, we mark these tasks in a table. At the intersection of rows and columns are numbers characterizing the cost of the respective transportation (Table 3.1).

Table 3.1

Let's make a mathematical model of the problem.
Variables must be entered. The wording of the question says that it is necessary to draw up a transportation plan. Denote by X 1 , X 2 number of headsets transported from the base BUT to shops With and D respectively, and through at 1 , at 2 - the number of headsets transported from the base AT to shops With and D respectively. Then the amount of furniture removed from the warehouse BUT, equals ( X 1 + X 2) well from stock AT - (at 1 + at 2). Store need With is equal to 10 headsets, and they brought it ( X 1 + at 1) pieces, i.e. X 1 + at 1 = 10. Similarly, for the store D we have X 2 + at 2 = 20. Note that the needs of stores are exactly equal to the number of headsets in stock, so X 1 + at 2 = 15 and at 1 + at 2 = 15. If you took away less than 15 sets from the warehouses, then the stores would not have enough furniture to meet their needs.
So the variables X 1 , X 2 , at 1 , at 2 are non-negative in the meaning of the problem and satisfy the system of constraints:
(3.1)
Denoting through F shipping costs, let's count them. for the transportation of one set of furniture from BUT in With spend one day. units, for transportation x 1 sets - x 1 day units Likewise, for transportation x 2 sets of BUT in D cost 3 x 2 days units; from AT in WITH - 2y 1 day units, from AT in D - 5y 2 days units
So,
F = 1x 1 + 3x 2 + 2y 1 + 5y 2 → min (3.2)
(we want the total cost of shipping to be as low as possible).
Let's formulate the problem mathematically.
On the set of solutions of the constraint system (3.1), find a solution that minimizes the objective function F(3.2), or find the optimal plan ( x 1 , x 2, y 1 , y 2) determined by the system of constraints (3.1) and the objective function (3.2).
The problem that we have considered can be represented in a more general form, with any number of suppliers and consumers.
In the problem we have considered, the availability of cargo from suppliers (15 + 15) is equal to the total need of consumers (10 + 20). Such a model is called closed, and the corresponding task is balanced transport task.
In economic calculations, the so-called open models, in which the indicated equality is not observed, also play a significant role. Either the supply of suppliers is greater than the demand of consumers, or demand exceeds the availability of goods. note that then the system of constraints of the unbalanced transport problem, along with the equations, will also include inequalities.

Questions for self-control
1. Statement of the transport problem. describe the construction of a mathematical model.
2. What is a balanced and unbalanced transport problem?
3. What is calculated in the objective function of the transport task?
4. What does each inequality of the system of constraints of the plan problem reflect?
5. What does each inequality of the system of constraints of the mixture problem reflect?
6. What do the variables mean in the plan problem and the mixture problem?