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Mathematical models of linear programming problems. Formulation of the main types of LP problems, construction of their mathematical models Construction of a mathematical model example

MATHEMATICAL MODEL - a representation of a phenomenon or process studied in concrete scientific knowledge in the language of mathematical concepts. In this case, a number of properties of the phenomenon under study are expected to be obtained through the study of the actual mathematical characteristics of the model. Construction of M.m. is most often dictated by the need to have a quantitative analysis of the phenomena and processes being studied, without which, in turn, it is impossible to make experimentally verifiable predictions about their course.

The process of mathematical modeling, as a rule, goes through the following stages. At the first stage, connections between the main parameters of the future M.m. are identified. We are talking primarily about a qualitative analysis of the phenomena under study and the formulation of patterns connecting the main objects of research. On this basis, objects that can be described quantitatively are identified. The stage ends with the construction of a hypothetical model, in other words, recording in the language of mathematical concepts qualitative ideas about the relationships between the main objects of the model, which can be characterized quantitatively.

At the second stage, the actual mathematical problems to which the constructed hypothetical model leads are studied. The main thing at this stage is to obtain empirically verifiable theoretical consequences (solution of the direct problem) as a result of mathematical analysis of the model. At the same time, there are often cases when, in order to construct and study M.m. in different areas of concrete scientific knowledge, the same mathematical apparatus is used (for example, differential equations) and mathematical problems of the same type arise, although very non-trivial in each specific case. In addition, at this stage, the use of high-speed computers (computers) becomes of great importance, which makes it possible to obtain approximate solutions to problems, often impossible within the framework of pure mathematics, with a degree of accuracy previously inaccessible (without the use of a computer).

The third stage is characterized by activities to identify the degree of adequacy of the constructed hypothetical M.M. those phenomena and processes for which it was intended to study. Namely, if all the parameters of the model have been specified, researchers try to find out to what extent, within the limits of observational accuracy, their results are consistent with the theoretical consequences of the model. Deviations beyond the limits of observational accuracy indicate the inadequacy of the model. However, there are often cases when, when constructing a model, a number of its parameters remain

uncertain. Problems in which the parametric characteristics of the model are established in such a way that the theoretical consequences are comparable, within the limits of observational accuracy, with the results of empirical tests are called inverse problems.

At the fourth stage, taking into account the identification of the degree of adequacy of the constructed hypothetical model and the emergence of new experimental data on the phenomena under study, subsequent analysis and modification of the model occurs. Here the decision made varies from the unconditional rejection of the applied mathematical tools to the acceptance of the constructed model as the foundation for the construction of a fundamentally new scientific theory.

First M.m. appeared in ancient science. Thus, to model the solar system, the Greek mathematician and astronomer Eudoxus gave each planet four spheres, the combination of the movements of which created a hippopedus - a mathematical curve similar to the observed movement of the planet. Since, however, this model could not explain all the observed anomalies in the motion of the planets, it was later replaced by the epicyclic model of Apollonius of Perga. The last model was used in his studies by Hipparchus, and then, having subjected it to some modification, by Ptolemy. This model, like its predecessors, was based on the belief that the planets undergo uniform circular motions, the overlap of which explained the apparent irregularities. It should be noted that the Copernican model was fundamentally new only in a qualitative sense (but not as a M.M.). And only Kepler, based on the observations of Tycho Brahe, built a new M.M. Solar system, proving that the planets move not in circular, but in elliptical orbits.

Currently, the most adequate ones are considered to be those constructed to describe mechanical and physical phenomena. On the adequacy of M.m. outside of physics one can, with some exceptions, speak with a fair amount of caution. Nevertheless, fixing the hypothetical nature, and often simply inadequacy of M.m. in various fields of knowledge, their role in the development of science should not be underestimated. There are often cases when even models that are far from adequate have significantly organized and stimulated further research, along with erroneous conclusions that also contained grains of truth that fully justified the efforts spent on developing these models.

Literature:

Math modeling. M., 1979;

Ruzavin G.I. Mathematization of scientific knowledge. M., 1984;

Tutubalin V.N., Barabasheva Yu.M., Grigoryan A.A., Devyatkova G.N., Uger E.G. Differential equations in ecology: historical and methodological reflection // Questions of the history of natural science and technology. 1997. No. 3.

Dictionary of philosophical terms. Scientific edition of Professor V.G. Kuznetsova. M., INFRA-M, 2007, p. 310-311.

According to the textbook by Sovetov and Yakovlev: “a model (Latin modulus - measure) is a substitute object for the original object, which ensures 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 using a model object is called modeling.” (p. 6) “By mathematical modeling we understand the process of establishing a correspondence to a given real object with a certain mathematical object, called a mathematical model, and the study of this model, which allows us to obtain the characteristics of the real object under consideration. The type of mathematical model depends both on 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 an idea».

Model classification

Formal classification of models

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

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

Classification according to the way the object is represented

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

Structural or functional models

Structural models represent an object as a system with its own structure and functioning mechanism. 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 called “ gray box».

Content and formal models

Almost all authors describing the process of mathematical modeling indicate that first a special ideal structure 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 simply a mathematical model obtained as a result of the formalization of a given meaningful model (pre-model). The construction of a meaningful model can be done using a set of ready-made idealizations, as in mechanics, where ideal springs, rigid bodies, ideal pendulums, elastic media, etc. provide 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 becomes dramatically more difficult.

Content classification of models

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

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

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

Type 2: Phenomenological model (we behave as if…)

A phenomenological model contains a mechanism for describing a phenomenon. However, this mechanism is not convincing enough, cannot be sufficiently confirmed by the available data, or does not fit well with existing 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 the search for the “true mechanisms” must continue. Peierls includes, for example, the caloric model and the quark model of elementary particles as the second type.

The role of the model in research may change over time, and it may happen that new data and theories confirm phenomenological models and they are promoted to the status of a hypothesis. Likewise, new knowledge can gradually come into conflict with models-hypotheses of the first type, and they can be translated into 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 became the first type. But the ether models have made their way from type 1 to type 2, and are now outside science.

The idea of simplification is very popular when building models. But simplification comes in different forms. Peierls identifies three types of simplifications in modeling.

Type 3: Approximation (we consider something very big or very small)

If it is possible to construct equations that describe 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 (type 3 models). Among them linear response models. The equations are replaced by linear ones. A standard example is Ohm's law.

Here comes Type 8, which is widespread in mathematical models of biological systems.

Type 8: Feature 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 vibrations, autowaves, etc. The Einstein-Podolsky-Rosen paradox was conceived as a type 7 model to demonstrate the inconsistency of 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

Consider a mechanical system consisting of a spring, fixed at one end, and a mass of mass , attached to the free end of the spring. We will assume that the load can only move in the direction of the spring axis (for example, movement occurs along the rod). Let's build 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 the spring and the load using Hooke's law() and then use Newton's second law to 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 model is called a "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 met.

In relation to reality, this is most often a type 4 model simplification(“we will omit some details for clarity”), since some essential universal features (for example, dissipation) are omitted. To some approximation (say, while the deviation of the load from equilibrium is small, with low friction, for not too much 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 of applicability.

However, when refining the model, the complexity of its mathematical research can increase significantly and make the model virtually useless. Often, a simpler model allows for better and deeper exploration of a real system than a more complex one (and, formally, “more correct”).

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

Hard and soft models

The harmonic oscillator is an example of the 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 the factors that we have neglected are. In other words, it is necessary to study 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 is some function that can take into account the friction force or the dependence of the spring stiffness coefficient on the degree of its stretching - some small parameter. We are not interested in the explicit form of the function at the moment. If we prove that the behavior of the soft model is not fundamentally different from the behavior of the hard one (regardless of the explicit type of perturbing factors, if they are small enough), the problem will be reduced to studying the hard model. Otherwise, the application of the results obtained from studying the rigid model will require additional research. For example, the solution to the equation of a harmonic oscillator is 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 arbitrarily small friction (always present in a real system), we get damped oscillations. The behavior of the system has changed qualitatively.

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

Versatility of models

The most important mathematical models usually have the important property versatility: Fundamentally different real phenomena can be described by the same mathematical model. For example, a harmonic oscillator describes not only the behavior of a load on a spring, but also other oscillatory processes, often of a completely different nature: small oscillations of a pendulum, fluctuations in the level of a liquid in an A-shaped vessel, or a change in current strength in an oscillatory circuit. Thus, by studying one mathematical model, we immediately study a whole class of phenomena described by it. It is this isomorphism of laws expressed by mathematical models in various segments of scientific knowledge that inspired Ludwig von Bertalanffy to create the “General Theory of Systems”.

Direct and inverse problems of mathematical modeling

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

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

Direct task: the structure of the model and all its parameters are considered known, the main task is to conduct a study of the model to extract useful knowledge about the object. What static load will 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 problem. Setting the right direct problem (asking the right question) requires special skill. If the right questions are not asked, a bridge may collapse, even if a good model for its behavior has been built. So, in 1879, a metal bridge across the River Tay collapsed in Great Britain, the designers of which built a model of the bridge, calculated it to have a 20-fold safety factor for the action of 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, a specific model must be selected 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 of additional empirical data, or requirements for the object ( design problem). Additional data can arrive 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 masterly solution to an inverse problem with the fullest 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 to develop 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 to probabilistic models. In specific tasks, 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 modeling. Block models are represented by blocks (most often graphic), 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 death rate. The solution to this equation is an 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 limited resources. A refinement of the Malthus model can be a 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 an 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 amendments to take into account the eating of rabbits by foxes, we arrive at the following system, named models Trays - Volterra:

This system has an equilibrium state when the number of rabbits and foxes is constant. Deviation from this state results in fluctuations in the numbers of rabbits and foxes, similar to the fluctuations of a harmonic oscillator. As with the harmonic oscillator, this behavior is not structurally stable: a small change in the model (for example, taking into account the limited resources required by rabbits) can lead to a qualitative change in behavior. For example, the equilibrium state may become stable, and fluctuations in numbers will die out. 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. The Volterra-Lotka model does not answer the question of which of these scenarios is being realized: additional research is required here.

Notes

“A mathematical representation of reality” (Encyclopaedia Britanica)
Novik I. B., On philosophical issues of cybernetic modeling. M., Knowledge, 1964.
Sovetov B. Ya., Yakovlev S. A., Modeling of systems: Proc. for universities - 3rd ed., revised. and additional - M.: Higher. school, 2001. - 343 p. ISBN 5-06-003860-2
Samarsky A. A., Mikhailov A. P. Math modeling. Ideas. Methods. Examples. - 2nd ed., rev. - M.: Fizmatlit, 2001. - ISBN 5-9221-0120-X
Myshkis A. D., Elements of the theory of mathematical models. - 3rd ed., rev. - M.: KomKniga, 2007. - 192 with ISBN 978-5-484-00953-4
Sevostyanov, A.G. Modeling of technological processes: textbook / A.G. Sevostyanov, P.A. Sevostyanov. – M.: Light and food industry, 1984. - 344 p.
Wiktionary: mathematical model
CliffsNotes.com. Earth Science Glossary. 20 Sep 2010
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
“A theory is considered linear or nonlinear depending on what kind of mathematical apparatus - linear or nonlinear - and what kind of linear or nonlinear mathematical models it uses. ...without denying the latter. A modern physicist, if he had to re-create the definition of such an important entity as nonlinearity, would most likely act differently, and, giving preference to nonlinearity as the more important and widespread of the two opposites, would define linearity as “not nonlinearity.” Danilov Yu. A., Lectures on nonlinear dynamics. Elementary introduction. Series “Synergetics: from past to future.” Edition 2. - M.: URSS, 2006. - 208 p. ISBN 5-484-00183-8
“Dynamical systems modeled by a finite number of ordinary differential equations are called concentrated 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 under different 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.
“Depending on the nature of the processes being studied 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 reflects deterministic processes, that is, processes in which the absence of any random influences is assumed; stochastic modeling depicts probabilistic processes and events. ... Static modeling serves to describe the behavior of an object at any point in time, and dynamic modeling reflects the behavior of an object over time. Discrete modeling is used to describe processes that are assumed to be discrete, respectively, continuous modeling allows us to reflect continuous processes in systems, and discrete-continuous modeling is used for cases when they want to highlight the presence of both discrete and continuous processes.” Sovetov B. Ya., Yakovlev S. A. ISBN 5-06-003860-2
Typically, a mathematical model reflects the structure (device) of the modeled object, the properties and relationships of the components of this object that are essential for the purposes of research; 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 functional or, figuratively, a black box. Combined models are also possible. Myshkis A. D. ISBN 978-5-484-00953-4
“The obvious, but most important initial stage of constructing or selecting a mathematical model is obtaining as clear a picture as possible about the object being modeled and refining its meaningful model, based on informal discussions. You should not spare time and effort at this stage; the success of the entire study largely depends on it. It has happened more than once that significant work spent on solving a 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 the theory of mathematical models. - 3rd ed., rev. - M.: KomKniga, 2007. - 192 with ISBN 978-5-484-00953-4, p. 35.
« Description of the conceptual model of the system. At this substage 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 standard mathematical schemes; c) hypotheses and assumptions are finally accepted; d) the choice of procedure for approximating real processes when constructing a model is justified.” Sovetov B. Ya., Yakovlev S. A., Modeling of systems: Proc. for universities - 3rd ed., revised. and additional - M.: Higher. school, 2001. - 343 p. ISBN 5-06-003860-2, p. 93.
Blekhman I. I., Myshkis A. D.,

What is a mathematical model?

The concept of a mathematical model.

A mathematical model is a very simple concept. And very important. It is mathematical models that connect mathematics and real life.

In simple terms, a mathematical model is a mathematical description of any situation. That's all. The model can be primitive, or it can be super complex. Whatever the situation, such is the model.)

In any (I repeat - in any!) in a case where you need to count and calculate something - we are engaged in mathematical modeling. Even if we don’t suspect it.)

P = 2 CB + 3 CM

This entry will be a mathematical model of the costs of our purchases. The model does not take into account the color of the packaging, expiration date, politeness of cashiers, etc. That's why she model, not an actual purchase. But expenses, i.e. what we need- we will find out for sure. If the model is correct, of course.

It is useful to imagine what a mathematical model is, but it is not enough. The most important thing is to be able to build these models.

Drawing up (construction) of a mathematical model of the problem.

To create a mathematical model means to translate the conditions of the problem into mathematical form. Those. turn words into an equation, formula, inequality, etc. Moreover, transform it so that this mathematics strictly corresponds to the source text. Otherwise, we will end up with a mathematical model of some other problem unknown to us.)

More specifically, you need

There are an endless number of tasks in the world. Therefore, offer clear step-by-step instructions for drawing up a mathematical model any tasks are impossible.

But there are three main points that you need to pay attention to.

1. Any problem contains text, oddly enough.) This text, as a rule, contains explicit, open information. Numbers, values, etc.

2. Any problem has hidden information. This is a text that assumes additional knowledge in your head. There is no way without them. In addition, mathematical information is often hidden behind simple words and... slips past attention.

3. Any task must be given connection of data with each other. This connection can be given in plain text (something equals something), or it can be hidden behind simple words. But simple and clear facts are often overlooked. And the model is not compiled in any way.

I’ll say right away: in order to apply these three points, you have to read the problem (and carefully!) several times. The usual thing.

And now - examples.

Let's start with a simple problem:

Petrovich returned from fishing and proudly presented his catch to the family. Upon closer examination, it turned out that 8 fish came from the northern seas, 20% of all fish came from the southern seas, and not a single one came from the local river where Petrovich was fishing. How many fish did Petrovich buy in the Seafood store?

All these words need to be turned into some kind of equation. To do this you need, I repeat, establish a mathematical connection between all the data in the problem.

Where to start? First, let's extract all the data from the task. Let's start in order:

Let's pay attention to the first point.

Which one is here? explicit mathematical information? 8 fish and 20%. Not a lot, but we don’t need a lot.)

Let us pay attention to the second point.

Are looking for hidden information. It's here. These are the words: "20% of all fish"Here you need to understand what percentages are and how they are calculated. Otherwise, the problem cannot be solved. This is exactly the additional information that should be in your head.

There is also mathematical information that is completely invisible. This task question: "How many fish did I buy..." This is also a number. And without it, no model will be formed. Therefore, let's denote this number by the letter "X". We don’t yet know what x is equal to, but this designation will be very useful to us. More details on what to take for X and how to handle it are written in the lesson How to solve problems in mathematics? Let’s write it down right away:

x pieces - total number of fish.

In our problem, southern fish are given as percentages. We need to convert them into pieces. For what? Then what in any the problem of the model must be drawn up in the same type of quantities. Pieces - so everything is in pieces. If given, say, hours and minutes, we translate everything into one thing - either only hours, or only minutes. It doesn't matter what it is. It is important that all values were of the same type.

Let's return to information disclosure. Whoever doesn’t know what a percentage is will never reveal it, yes... But whoever knows will immediately say that the percentages here are based on the total number of fish. And we don’t know this number. Nothing will work!

It’s not for nothing that we letter the total number of fish (in pieces!) "X" designated. It won't be possible to count the number of southern fish, but we can write them down? Like this:

0.2 x pieces - the number of fish from the southern seas.

Now we have downloaded all the information from the task. Both obvious and hidden.

Let us pay attention to the third point.

Are looking for mathematical connection between task data. This connection is so simple that many do not notice it... This often happens. Here it’s useful to simply write down the collected data in a pile and see what’s what.

What do we have? Eat 8 pieces northern fish, 0.2 x pieces- southern fish and x fish- total amount. Is it possible to link this data together somehow? Yes Easy! Total number of fish equals the sum of southern and northern! Well, who would have thought...) So we write it down:

x = 8 + 0.2x

This is the equation mathematical model of our problem.

Please note that in this problem We are not asked to fold anything! It was we ourselves, out of our heads, who realized that the sum of the southern and northern fish would give us the total number. The thing is so obvious that it goes unnoticed. But without this evidence, a mathematical model cannot be created. Like this.

Now you can use the full power of mathematics to solve this equation). This is precisely why the mathematical model was compiled. We solve this linear equation and get the answer.

Answer: x=10

Let's create a mathematical model of another problem:

They asked Petrovich: “Do you have a lot of money?” Petrovich began to cry and answered: “Yes, just a little. If I spend half of all the money, and half of the rest, then I’ll only have one bag of money left...” How much money does Petrovich have?

Again we work point by point.

1. We are looking for explicit information. You won’t find it right away! Explicit information is one money bag. There are some other halves... Well, we’ll look into that in the second point.

2. We are looking for hidden information. These are halves. What? Not very clear. We are looking further. There is one more question: "How much money does Petrovich have?" Let us denote the amount of money by the letter "X":

X- all the money

And again we read the problem. Already knowing that Petrovich X money. This is where halves will work! We write down:

0.5 x- half of all money.

The remainder will also be half, i.e. 0.5 x. And half of half can be written like this:

0.5 0.5 x = 0.25x- half of the remainder.

Now all hidden information has been revealed and recorded.

3. We are looking for a connection between the recorded data. Here you can simply read Petrovich’s suffering and write it down mathematically):

If I spend half of all the money...

Let's record this process. All the money - X. Half - 0.5 x. To spend is to take away. The phrase turns into a recording:

x - 0.5 x

yes half the rest...

Let's subtract another half of the remainder:

x - 0.5 x - 0.25x

then I'll only have one bag of money left...

And here we have found equality! After all the subtractions, one bag of money remains:

x - 0.5 x - 0.25x = 1

Here it is, a mathematical model! This is again a linear equation, we solve it, we get:

Question for consideration. What is four? Ruble, dollar, yuan? And in what units is money written in our mathematical model? In bags! That means four bag money from Petrovich. Good too.)

The tasks are, of course, elementary. This is specifically to capture the essence of drawing up a mathematical model. Some tasks may contain much more data, which can be easy to get lost in. This often happens in the so-called. competency tasks. How to extract mathematical content from a pile of words and numbers is shown with examples

One more note. In classic school problems (pipes filling a pool, boats floating somewhere, etc.), all data, as a rule, is selected very carefully. There are two rules:
- there is enough information in the problem to solve it,
- There is no unnecessary information in a problem.

This is a hint. If there is some value left unused in the mathematical model, think about whether there is an error. If there is not enough data, most likely, not all hidden information has been identified and recorded.

In competence-related and other life tasks, these rules are not strictly observed. No clue. But such problems can also be solved. If, of course, you practice on the classic ones.)

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)

You can get acquainted with functions and derivatives.

Mathematical models

Mathematical model - approximate opithe meaning of the modeling object, expressed usingof mathematical symbolism.

Mathematical models appeared along with mathematics many centuries ago. The advent of computers gave a huge impetus to the development of mathematical modeling. The use of computers has made it possible to analyze and apply in practice many mathematical models that were previously not amenable to analytical research. Implemented on a computer mathematicallysky model called computer mathematical model, A carrying out targeted calculations using a computer model called computational experiment.

Stages of computer mathematical sciencedivision are shown in the figure. Firststage - defining modeling goals. These goals can be different:

a model is needed in order to understand how a specific object works, what its structure is, its basic properties, laws of development and interaction
with the outside world (understanding);
a model is needed in order to learn how to manage an object (or process) and determine the best methods of management for given goals and criteria (management);
the model is needed in order to predict the direct and indirect consequences of the implementation of given methods and forms of influence on the object (forecasting).

Let's explain with examples. Let the object of study be the interaction of a flow of liquid or gas with a body that is an obstacle to this flow. Experience shows that the force of resistance to flow on the part of the body increases with increasing flow speed, but at some sufficiently high speed this force decreases abruptly in order to increase again with a further increase in speed. What caused the decrease in resistance force? Mathematical modeling allows us to obtain a clear answer: at the moment of an abrupt decrease in resistance, the vortices formed in the flow of liquid or gas behind the streamlined body begin to break away from it and are carried away by the flow.

An example from a completely different area: populations of two species of individuals that had peacefully coexisted with stable numbers and had a common food supply, “suddenly” begin to change their numbers sharply. And here mathematical modeling allows (with a certain degree of reliability) to establish the cause (or at least refute a certain hypothesis).

Developing a concept for managing an object is another possible goal of modeling. Which aircraft flight mode should I choose to ensure that the flight is safe and most economically profitable? How to schedule hundreds of types of work on the construction of a large facility so that it is completed in the shortest possible time? Many such problems systematically arise before economists, designers, and scientists.

Finally, predicting the consequences of certain impacts on an object can be both a relatively simple matter in simple physical systems, and extremely complex - on the verge of feasibility - in biological, economic, and social systems. While it is relatively easy to answer the question about changes in the mode of heat distribution in a thin rod due to changes in its constituent alloy, it is incomparably more difficult to trace (predict) the environmental and climatic consequences of the construction of a large hydroelectric power station or the social consequences of changes in tax legislation. Perhaps here, too, mathematical modeling methods will provide more significant assistance in the future.

Second phase: determination of input and output parameters of the model; division of input parameters according to the degree of importance of the influence of their changes on the output. This process is called ranking, or separation by rank (see. "Formalizationtion and modeling").

Third stage: construction of a mathematical model. At this stage, there is a transition from an abstract formulation of the model to a formulation that has a specific mathematical representation. A mathematical model is equations, systems of equations, systems of inequalities, differential equations or systems of such equations, etc.

Fourth stage: choosing a method for studying a mathematical model. Most often, numerical methods are used here, which lend themselves well to programming. As a rule, several methods are suitable for solving the same problem, differing in accuracy, stability, etc. The success of the entire modeling process often depends on the correct choice of method.

Fifth stage: developing an algorithm, compiling and debugging a computer program is a difficult process to formalize. Among the programming languages, many professionals prefer FORTRAN for mathematical modeling: both due to traditions and due to the unsurpassed efficiency of compilers (for calculation work) and the availability of huge, carefully debugged and optimized libraries of standard programs for mathematical methods written in it. Languages such as PASCAL, BASIC, C are also in use, depending on the nature of the task and the inclinations of the programmer.

Sixth stage: program testing. The operation of the program is tested on a test problem with a previously known answer. This is just the beginning of a testing procedure that is difficult to describe in a formally comprehensive manner. Typically, testing ends when the user, based on his professional characteristics, considers the program correct.

Seventh stage: the actual computational experiment, during which it is determined whether the model corresponds to a real object (process). The model is sufficiently adequate to the real process if some characteristics of the process obtained on a computer coincide with the experimentally obtained characteristics with a given degree of accuracy. If the model does not correspond to the real process, we return to one of the previous stages.

Classification of mathematical models

The classification of mathematical models can be based on various principles. You can classify models by branches of science (mathematical models in physics, biology, sociology, etc.). Can be classified according to the mathematical apparatus used (models based on the use of ordinary differential equations, partial differential equations, stochastic methods, discrete algebraic transformations, etc.). Finally, if we proceed from the general problems of modeling in different sciences, regardless of the mathematical apparatus, the following classification is most natural:

descriptive (descriptive) models;
optimization models;
multicriteria models;
game models.

Let's explain this with examples.

Descriptive (descriptive) models. For example, modeling the motion of a comet that has invaded the solar system is carried out to predict its flight path, the distance at which it will pass from the Earth, etc. In this case, the modeling goals are descriptive in nature, since there is no way to influence the movement of the comet or change anything in it.

Optimization models are used to describe processes that can be influenced in an attempt to achieve a given goal. In this case, the model includes one or more parameters that can be influenced. For example, when changing the thermal regime in a granary, you can set the goal of choosing a regime that will achieve maximum grain safety, i.e. optimize the storage process.

Multicriteria models. It is often necessary to optimize a process along several parameters simultaneously, and the goals can be quite contradictory. For example, knowing the prices of food and a person’s need for food, it is necessary to organize nutrition for large groups of people (in the army, children’s summer camp, etc.) physiologically correctly and, at the same time, as cheaply as possible. It is clear that these goals do not coincide at all, i.e. When modeling, several criteria will be used, between which a balance must be sought.

Game models may relate not only to computer games, but also to very serious things. For example, before a battle, a commander, if there is incomplete information about the opposing army, must develop a plan: in what order to introduce certain units into battle, etc., taking into account the possible reaction of the enemy. There is a special branch of modern mathematics - game theory - that studies methods of decision-making under conditions of incomplete information.

In the school computer science course, students receive an initial understanding of computer mathematical modeling as part of the basic course. In high school, mathematical modeling can be studied in depth in a general education course for physics and mathematics classes, as well as as part of a specialized elective course.

The main forms of teaching computer mathematical modeling in high school are lectures, laboratory and test classes. Typically, the work of creating and preparing to study each new model takes 3-4 lessons. During the presentation of the material, problems are set that must be solved by students independently in the future, and ways to solve them are outlined in general terms. Questions are formulated, the answers to which must be obtained when completing tasks. Additional literature is indicated that allows you to obtain auxiliary information for more successful completion of tasks.

The form of organization of classes when studying new material is usually a lecture. After completing the discussion of the next model students have at their disposal the necessary theoretical information and a set of tasks for further work. In preparation for completing a task, students choose an appropriate solution method and test the developed program using some well-known private solution. In case of quite possible difficulties when completing tasks, consultation is given, and a proposal is made to study these sections in more detail in literary sources.

The most appropriate for the practical part of teaching computer modeling is the project method. The task is formulated for the student in the form of an educational project and is carried out over several lessons, with the main organizational form being computer laboratory work. Teaching modeling using the educational project method can be implemented at different levels. The first is a problematic presentation of the process of completing the project, which is led by the teacher. The second is the implementation of the project by students under the guidance of a teacher. The third is for students to independently complete an educational research project.

The results of the work must be presented in numerical form, in the form of graphs and diagrams. If possible, the process is presented on the computer screen in dynamics. Upon completion of the calculations and receipt of the results, they are analyzed, compared with known facts from the theory, reliability is confirmed and a meaningful interpretation is carried out, which is subsequently reflected in a written report.

If the results satisfy the student and teacher, then the work counts completed, and its final stage is the preparation of a report. The report includes brief theoretical information on the topic under study, a mathematical formulation of the problem, a solution algorithm and its justification, a computer program, the results of the program, analysis of the results and conclusions, and a list of references.

When all the reports have been compiled, during the test lesson, students give brief reports on the work done and defend their project. This is an effective form of report from the group carrying out the project to the class, including setting the problem, building a formal model, choosing methods for working with the model, implementing the model on a computer, working with the finished model, interpreting the results, and making predictions. As a result, students can receive two grades: the first - for the elaboration of the project and the success of its defense, the second - for the program, the optimality of its algorithm, interface, etc. Students also receive grades during theory quizzes.

An essential question is what tools to use in a school computer science course for mathematical modeling? Computer implementation of models can be carried out:

using a spreadsheet processor (usually MS Excel);
by creating programs in traditional programming languages (Pascal, BASIC, etc.), as well as in their modern versions (Delphi, Visual
Basic for Application, etc.);
using special application packages for solving mathematical problems (MathCAD, etc.).

At the basic school level, the first method seems to be more preferable. However, in high school, when programming is, along with modeling, a key topic in computer science, it is advisable to use it as a modeling tool. During the programming process, details of mathematical procedures become available to students; Moreover, they are simply forced to master them, and this also contributes to mathematical education. As for the use of special software packages, this is appropriate in a specialized computer science course as a supplement to other tools.

Exercise :

Make a diagram of key concepts.

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

Model classification

Formal classification of models

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

Classification according to the way the object is represented

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

Structural or functional models

Content and formal models

Content classification of models

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

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

Type 2: Phenomenological model (we behave as if…)

The idea of simplification is very popular when building models. But simplification comes in different forms. Peierls identifies three types of simplifications in modeling.

Type 3: Approximation (we consider something very big or very small)

Here comes Type 8, which is widespread in mathematical models of biological systems.

Type 8: Feature Demonstration (the main thing is to show the internal consistency of the possibility)

Example

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

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

Hard and soft models

Versatility of models

Direct and inverse problems of mathematical modeling

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

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

Computer simulation systems

Additional examples

Malthus' model

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

Predator-prey system

Notes

“A mathematical representation of reality” (Encyclopaedia Britanica)
Novik I. B., On philosophical issues of cybernetic modeling. M., Knowledge, 1964.
Sovetov B. Ya., Yakovlev S. A., Modeling of systems: Proc. for universities - 3rd ed., revised. and additional - M.: Higher. school, 2001. - 343 p. ISBN 5-06-003860-2
Samarsky A. A., Mikhailov A. P. Math modeling. Ideas. Methods. Examples. - 2nd ed., rev. - M.: Fizmatlit, 2001. - ISBN 5-9221-0120-X
Myshkis A. D., Elements of the theory of mathematical models. - 3rd ed., rev. - M.: KomKniga, 2007. - 192 with ISBN 978-5-484-00953-4
Sevostyanov, A.G. Modeling of technological processes: textbook / A.G. Sevostyanov, P.A. Sevostyanov. – M.: Light and food industry, 1984. - 344 p.
Wiktionary: mathematical model
CliffsNotes.com. Earth Science Glossary. 20 Sep 2010
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
“A theory is considered linear or nonlinear depending on what kind of mathematical apparatus - linear or nonlinear - and what kind of linear or nonlinear mathematical models it uses. ...without denying the latter. A modern physicist, if he had to re-create the definition of such an important entity as nonlinearity, would most likely act differently, and, giving preference to nonlinearity as the more important and widespread of the two opposites, would define linearity as “not nonlinearity.” Danilov Yu. A., Lectures on nonlinear dynamics. Elementary introduction. Series “Synergetics: from past to future.” Edition 2. - M.: URSS, 2006. - 208 p. ISBN 5-484-00183-8
“Dynamical systems modeled by a finite number of ordinary differential equations are called concentrated 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 under different 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.
“Depending on the nature of the processes being studied 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 reflects deterministic processes, that is, processes in which the absence of any random influences is assumed; stochastic modeling depicts probabilistic processes and events. ... Static modeling serves to describe the behavior of an object at any point in time, and dynamic modeling reflects the behavior of an object over time. Discrete modeling is used to describe processes that are assumed to be discrete, respectively, continuous modeling allows us to reflect continuous processes in systems, and discrete-continuous modeling is used for cases when they want to highlight the presence of both discrete and continuous processes.” Sovetov B. Ya., Yakovlev S. A. ISBN 5-06-003860-2
Typically, a mathematical model reflects the structure (device) of the modeled object, the properties and relationships of the components of this object that are essential for the purposes of research; 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 functional or, figuratively, a black box. Combined models are also possible. Myshkis A. D. ISBN 978-5-484-00953-4
“The obvious, but most important initial stage of constructing or selecting a mathematical model is obtaining as clear a picture as possible about the object being modeled and refining its meaningful model, based on informal discussions. You should not spare time and effort at this stage; the success of the entire study largely depends on it. It has happened more than once that significant work spent on solving a 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 the theory of mathematical models. - 3rd ed., rev. - M.: KomKniga, 2007. - 192 with ISBN 978-5-484-00953-4, p. 35.
« Description of the conceptual model of the system. At this substage 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 standard mathematical schemes; c) hypotheses and assumptions are finally accepted; d) the choice of procedure for approximating real processes when constructing a model is justified.” Sovetov B. Ya., Yakovlev S. A., Modeling of systems: Proc. for universities - 3rd ed., revised. and additional - M.: Higher. school, 2001. - 343 p. ISBN 5-06-003860-2, p. 93.
Blekhman I. I., Myshkis A. D.,

Mathematical models of linear programming problems. Formulation of the main types of LP problems, construction of their mathematical models Construction of a mathematical model example

Model classification

Formal classification of models

Classification according to the way the object is represented

Content and formal models

Content classification of models

Type 2: Phenomenological model (we behave as if…)

Type 3: Approximation (we consider something very big or very small)

Type 8: Feature Demonstration (the main thing is to show the internal consistency of the possibility)

Example

Hard and soft models

Versatility of models

Direct and inverse problems of mathematical modeling

Computer simulation systems

Additional examples

Malthus' model

Predator-prey system

Notes

What is a mathematical model?

The concept of a mathematical model.

Drawing up (construction) of a mathematical model of the problem.

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Model classification

Formal classification of models

Classification according to the way the object is represented

Content and formal models

Content classification of models

Type 2: Phenomenological model (we behave as if…)

Type 3: Approximation (we consider something very big or very small)

Type 8: Feature Demonstration (the main thing is to show the internal consistency of the possibility)

Example

Hard and soft models

Versatility of models

Direct and inverse problems of mathematical modeling

Computer simulation systems

Additional examples

Malthus' model

Predator-prey system

Notes

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