Mathematical models constitute an abstract part of the spectrum (Fig. 7.2), for the convenience of their use in various industries, including logistics, they are classified according to the six most representative features:

The method of obtaining the model;

The way of describing or representing an object or its properties;

The method of formalizing an object or its properties;

Membership at the hierarchical level;

The scale of the description of an object or its properties;

The degree of complexity of describing an object or its properties.

Bymethod of obtaining models are divided into theoretical , neural (perceptrons) and empirical .

Theoretical models are derived mathematically based on knowledge of the primary laws of classical mechanics, electrodynamics, chemistry, etc. Real life models based on statistical processing results of observations, form a group of empirical. The problem of constructing an empirical model includes the choice of the form of this model, suitable, as well as a reasonable degree of its complexity, compatible with the available experimental data.

In recent years, neural models (perceptrons) have become increasingly important in the field of modeling economic processes. The neural model (perceptron) consists of binary neural-like elements and has a simple topology.

The perceptron itself includes a matrix of binary inputs (sensory neurons or retina, where input images are fed), a set of binary neural-like elements with fixed connections to subsets of the retina, a binary Neuro-like element with modified connections in these predicates (elements, decide).

Previously, the perceptron was used to solve the problem of automatic classification, in general, it consists in dividing the feature space between a given number of classes. In today's conditions, at the level of neural networks, it is possible to solve the problem of logistic forecasting, which is formalized through the problem of pattern recognition.

Consider the following example. There is data on the current demand for the company's products for six years (Ac = 6): 71, 80, 101, 84, 60, 73.

To formalize the task, we use the window method. Set the size of the windows η = 3, T= 1 and the level of excitation of the Neuro-like element s = 1. Next, using the method of windows with already fixed parameters n, t, s the following training sample is generated for the neural network:

As you can see, each subsequent vector is formed as a result of shifting the windows W and and W 0 to the right one element (s= 1). In this case, it is assumed that there are hidden dependencies in the time sequence as a set of observations.

The neural network, learning on these observations and adjusting its coefficients accordingly, tries to extract these patterns and form, as a result, the expected forecast function, that is, "build" model . Forecasting is carried out according to the same principle as the formation of a training sample.

By the way the object is described models are divided as follows:

1) algebraic;

2) regression-correlation;

3) probabilistic-statistical, combining models of the theory of queues, models of stocks and statistical models;

4) mathematical programming - linear programming, network (stream).

Regarding the first group of models - algebraic , it is necessary to immediately make a reservation that they, in essence, for the logistician are auxiliary in order to make the right decision. Algebraic models are commonly used in tasks such as tipping point analysis and cost-benefit analysis.

Regression-correlation models , representing the second group, is a generalization of extrapolation and statistical models and are used to describe the specifics of an object or its properties.

The third group consists of probabilistic statistical models , based on phenological phenomena and hypotheses. These models can be deterministic or stochastic. So, for example, the dependence V = φ (Χ), which is established according to the results of observations of random variables X and V the least squares method is a deterministic model. If we take into account the random deviations of the experimental points from the curve observed as a result of the experiments Y = φ (X) and write the dependence of B on X in the form B = φ (Χ)+ Ζ (where Ζ - some random variable), then we get a stochastic model in its ideal expression.

In this case, the quantities X and V can be both scalar and vector. Function φ (Χ) can be either a linear combination of these functions or a given nonlinear function, the parameters of which are determined by the least squares method.

Models linear programming are increasingly used to solve logistics problems.

Anyone who is familiar with mathematical programming knows that it is practically impossible to solve it in general form. However, the most developed in mathematical programming are linear programming problems.

In linear programming problems, the objective function is linear, and the constraint conditions include linear equalities and linear inequalities; variables may or may not be subject to the immutability requirement.

To demonstrate the simplicity of solving logistic problems using linear programming, we turn to two well-known problems:

The first is about the grandmother who is going to the market to sell the animals that have grown in her yard during the year;

The second is about nutrition.

The first task (about the grandmother)

The essence of this problem comes down to obtaining an answer to a simple question: "How much grandmother should be taken to sell live geese, ducks and chickens on the market so that it receives the greatest revenue, provided that it can deliver livestock weighing no more than R kg? ". In this case, the following are known:

Mass of chicken (t,), duck ( T 2 ) and goose (t3)

Cost of chicken (c7), duck (c2) and goose (c3).

Consider an algorithm for solving the problem.

1. To solve the problem, we denote the number, respectively, of chickens - X 1 ducks - X 2, geese - X 3 taken by the grandmother to sell to the market.

2. Let's compose the objective function of this task:

3. Let us describe the constraints on the solution of the problem.

The mass of goods, the grandmother can simultaneously deliver to the market, should not exceed R kilogram:

The value, and must be positive integers (), that is:

After completing the three described steps, we get a linear programming problem. Substituting the original values x, t, s and R, we find the answer to the question posed.

Second task (about nutrition)

Cafe "Bistro" buys food products from the store every day for preparing certain dishes for its visitors. The diet contains three different nutrients ( b) and need them, respectively, at least b 1, b 2, b 3 units. The store sells five types of different products X 1 - X 5 for the price, respectively S-I - s 5.

Each unit of the product i-th of the form ( X i) contains a and j units j-th nutrient, that is, for example, a 2 With shows that in units of the second product of the third nutrient there will be a 23 units.

Since the cafe operates surrounded by competitors, it is necessary to correctly determine the number of products of each type X 1 - x 5 worth buying. In this case, the following conditions must be met:

1) that the cost of products is minimal;

2) so that the diet contains all the necessary nutrients in the right amount.

The mathematical formulation of the solution to the problem will be as follows:

1. The objective function of this task is to minimize the cost of products X 1 - X 5. Mathematically, it will look like this:

2. Conditions for limiting the solution of the problem:

a) the amount of the first nutrient must be at least b 1 ,:

b) the amount of the second nutrient must be at least b 2 :

c) the amount of the third nutrient must be at least b 3:

It should be borne in mind that the number of products cannot have a negative number, that is:

For a correct understanding of the solution to the given problem, consider the following example.

Let in this problem we have the following initial data:

The objective function will look like this:

The minimum value of the function must be determined subject to the following restrictions:

Bearing in mind that the number of products cannot be negative, we assume that

As a result of solving the problem according to the presented initial data, we have the following answer: and. With these values, the objective function will have the following meaning:

Network (stream) models.

An important class of mathematical programming problems are the so-called network (flow) problems, in terms of which linear programming problems can be formulated.

Let us consider as an example the so-called transport problem (Fig. 7.3), which is one of the first flow problems, which was solved in 1941 by F.L. Hitchcock.

Suppose there are two factories (1 and 2) and three trains (A, B, C). The factories produce s1 and s2 units, respectively. Warehouses have the ability to store d1, d2 and d3 units of products, that is:

The challenge is to minimize the cost of transporting products from manufacturing plants to warehouses. Let's set the following initial conditions. Let's pretend that X ij - volume of products to be transported from i-th plant on j-th compound; с - - the cost of transportation of a unit of production with i-th plant on j-th compound. Then the objective function of the problem, the cost of transportation, will have the following form:

Rice. 7.3.

The condition that all products will be transported from each plant:

Equality data can be written in short form, namely:

The condition for filling the warehouses is as follows: moreover

This model can be described using a network, assuming that the nodes of the network are factories and warehouses, and the arcs are roads for the transportation of goods (Fig. 7.3). The formulated transport problem is a special case of the problem of finding the minimum cost flow within the network.

Network tasks are used in the design and improved large and complex systems, as well as in the search for ways of their most rational use. First of all, this is due to the fact that with the help of networks it is quite simple to build a model of the system. The latter is based on the idea of ​​a critical path (CPM method) and assessment and observation tools (for example, the PERT-Program Evalution Research Task system).

In addition, networks allow you to:

Formalization of a model of a complex system as a set of simple systems (in this case, a logistics system as a set of its subsystems and links - procurement, warehouses, transportation, stocks, production, distribution and sales);

Drawing up formal procedures to determine the quality characteristics of the system;

Determination of the mechanism of interaction between the components of the control system in order to describe the latter in terms of its main characteristics;

Determination of the data required for the study of the logistics system and its main subsystems;

Initial investigation of the control system, drawing up a preliminary schedule for the operation of its components.

The main advantage of the network approach is that it can be successfully applied to virtually any problem where a network model can be accurately constructed.

A generalized characteristic of mathematical models classified according to the method of object description is given in table. 7.3. The table shows the most suitable areas of application of these models with a preliminary indicated accuracy of the estimates obtained. This information useful for logisticians at the stage of building models or choosing the latter to solve a problem.

By the nature of the displayed properties of the object models are classified into structural and functional, which together reflect the relationship and mutual influence of individual elements on the processes occurring in the object during its operation or manufacture.

Structural models are intended to display the structural properties of the composition object, the relationship and relative position, as well as the shape of the components.

Functional models are intended to a greater extent for displaying the processes occurring in an object during its operation or manufacture, and, as a rule, contain algorithms that link phase variables, internal, external or output parameters.

Table 7.3

Characteristic features of mathematical models

By the way the object is formalized with the complexity of existing situations, it becomes necessary to simplify their description using analytical and algorithmic models, properly

"Abstracts" selected "essential" properties of objects and situations. Computer simulation of real objects is a valuable tool for analyzing complex service systems, service policies and investment choices.

The distribution of objects into hierarchical levels leads to certain levels of modeling, the hierarchy of which is determined by both the complexity of the objects and the ability of the controls. Therefore, according to belonging to the hierarchical level, mathematical models are divided into micro-, macro- and metamodels. The difference between these models lies in the fact that at a higher level of the hierarchy, the components of the model take the form of rather complex collections of elements of the previous level. The same qualities determine the division of models by the degree of scale and complexity of the description of the object.

The above classification of models is designed to help logisticians in more efficient and correct decision-making in order to implement the mission of the organization.

Imagine an airplane: wings, fuselage, tail unit, all this together - a real huge, immense, whole airplane. Or you can make a model of an airplane, small, but everything is in fact, the same wings, etc., but compact. So is the mathematical model. There is a word problem, cumbersome, you can look at it, read it, but not quite understand, and even more so it is not clear how to solve it. But what if we make a small model of a large verbal problem, a mathematical model? What does math mean? This means, using the rules and laws of mathematical notation, to remake the text into a logically correct representation using numbers and arithmetic signs. So, a mathematical model is a representation of a real situation using a mathematical language.

Let's start with a simple one: The number is greater than the number by. We need to write this down, not using words, but only the language of mathematics. If more by, then it turns out that if we subtract from, then the same difference of these numbers remains equal. Those. or. Understood the essence?

Now it's more complicated, now there will be a text that you should try to represent in the form of a mathematical model, until you read how I will do it, try it yourself! There are four numbers:, and. The piece is larger than the piece and doubled.

What happened?

In the form of a mathematical model, it will look like this:

Those. the product is related as two to one, but this can still be simplified:

Well, okay, with simple examples you get the point, I suppose. Let's move on to full-fledged problems in which these mathematical models still need to be solved! Here's the challenge.

Mathematical model in practice

Problem 1

After rain, the water level in the well may rise. The boy measures the time of falling of small stones into the well and calculates the distance to the water using the formula, where is the distance in meters and is the time of falling in seconds. Before the rain, the time for the stones to fall was s. How much should the water level rise after rain for the measured time to change by s? Express your answer in meters.

Oh God! What formulas, what kind of well, what is happening, what to do? Did I read your mind? Relax, in problems of this type conditions are even worse, the main thing is to remember that in this problem you are interested in formulas and relations between variables, and what all this means in most cases is not very important. What do you see useful here? I personally see. The principle for solving these problems is as follows: take all known quantities and substitute them.BUT, sometimes you need to think!

Following my first advice, and substituting all known into the equation, we get:

It was I who substituted the time of a second, and found the height that the stone flew before the rain. And now we need to count after the rain and find the difference!

Now listen to the second advice and think about it, the question specifies "how much the water level should rise after the rain so that the measured time changes by s." Immediately it is necessary to estimate, soooo, after the rain the water level rises, which means that the time of the stone falling to the water level is shorter and here the ornate phrase “so that the measured time changes” takes on a specific meaning: the fall time does not increase, but decreases by the specified seconds. This means that in the case of a throw after rain, we just need to subtract c from the initial time c, and we get the equation for the height that the stone will fly after the rain:

And finally, to find how much the water level should rise after the rain, so that the measured time changes by s., You just need to subtract the second from the first fall height!

We get the answer: by the meter.

As you can see, there is nothing complicated, the main thing is, do not bother too much where such an incomprehensible and sometimes complex equation came from in the conditions and what everything in it means, take my word for it, most of these equations are taken from physics, and there is a jungle worse than in algebra. Sometimes it seems to me that these problems were invented in order to intimidate the student on the exam with an abundance of complex formulas and terms, and in most cases they do not require almost any knowledge. Just read the condition carefully and plug the known values ​​into the formula!

Here's another problem, no longer in physics, but from the world of economic theory, although knowledge of sciences other than mathematics is not required here again.

Task 2

The dependence of the volume of demand (units per month) for the products of the monopolist enterprise on the price (thousand rubles) is given by the formula

The company's revenue per month (in thousand rubles) is calculated using the formula. Determine the highest price at which the monthly revenue will be at least thousand rubles. Give your answer in thousand rubles.

Guess what I'm going to do now? Yeah, I'll start substituting what we know, but, again, I'll have to think a little. Let's go from the end, we need to find at which. So, there is, equal to someone, we find what else is equal to, and equally it is, and we will write it down. As you can see, I don't bother too much about the meaning of all these values, I just look from the conditions that what is equal, so you need to do it. Let's return to the problem, you already have it, but as you remember from one equation with two variables, none of them can be found, what to do? Yeah, we still have an unused piece in the condition. Now, there are already two equations and two variables, which means that now both variables can be found - great!

Can you solve such a system?

We solve by substitution, we have already expressed it, which means we substitute it in the first equation and simplify.

It turns out here is such a quadratic equation:, we solve, the roots are like this,. In the task, it is required to find the highest price at which all the conditions that we took into account when the system was compiled will be met. Oh, it turns out that was the price. Cool, so we found prices: and. The highest price, you say? Okay, the largest of them, obviously, is the answer and we write. Well, is it difficult? I think not, and there is no need to delve into it too much!

And here's the frightening physics, or rather, another challenge:

Problem 3

To determine the effective temperature of stars, the Stefan-Boltzmann law is used, according to which, where is the radiation power of the star, is constant, is the surface area of ​​the star, and is the temperature. It is known that the surface area of ​​some star is equal, and the power of its radiation is equal to W. Find the temperature of this star in degrees Kelvin.

Where did it come from? Yes, the condition says what is equal. Previously, I recommended substituting all unknowns at once, but here it is better to first express the unknown sought. Look how simple everything is: there is a formula and it is known in it, and (this is the Greek letter "sigma". In general, physicists love Greek letters, get used to it). And the temperature is unknown. Let's express it as a formula. I hope you know how to do this? Such tasks for the GIA in grade 9 usually give:

Now it remains to substitute numbers instead of letters on the right side and simplify:

Here's the answer: degrees Kelvin! And what a terrible task it was, eh!

We continue to torment the problems in physics.

Problem 4

The height above the ground of a ball thrown upward changes according to the law, where is the height in meters, is the time in seconds elapsed since the throw. How many seconds will the ball stay at least three meters high?

Those were all the equations, but here it is necessary to determine how much the ball was at a height of at least three meters, that means at a height. What are we going to compose? Inequality, exactly! We have a function that describes how the ball flies, where is the very same height in meters, we need the height. Means

And now you just solve the inequality, the main thing is, do not forget to change the sign of inequality from greater than or equal to less or equal, when you multiply by both sides of the inequality in order to get rid of the minus beforehand.

These are the roots, we build intervals for inequality:

We are interested in the interval where the minus sign is, since inequality takes negative values ​​there, this is from to both inclusive. And now we turn on the brain and think carefully: for inequality we used the equation describing the flight of the ball, it somehow flies in a parabola, i.e. it takes off, reaches a peak and falls, how to understand how long it will be at an altitude of at least meters? We found 2 tipping points, i.e. the moment when he soars above meters and the moment when he, falling, reaches the same mark, these two points are expressed by us in the form of time, i.e. we know at what second of the flight he entered the zone of interest to us (above meters) and into which one he left it (fell below the mark of meters). How many seconds was he in this zone? It is logical that we take the time of leaving the zone and subtract the time of entering this zone from it. Accordingly: - so much he was in the zone above meters, this is the answer.

You are so lucky that most of the examples on this topic can be taken from the category of problems in physics, so catch one more, it is the final one, so push yourself, there are very few left!

Problem 5

For a heating element of a certain device, the temperature dependence on the operating time was experimentally obtained:

Where is the time in minutes,. It is known that at a temperature of the heating element above the device may deteriorate, therefore it must be turned off. Find the longest time after starting work you need to turn off the device. Express your answer in minutes.

We act according to a debugged scheme, everything that is given, first we write out:

Now we take the formula and equate it to the temperature value to which the device can be heated as much as possible until it burns out, that is:

Now we substitute numbers instead of letters where they are known:

As you can see, the temperature during the operation of the device is described by a quadratic equation, which means that it is distributed along a parabola, i.e. the device heats up to a certain temperature, and then cools down. We received answers and, therefore, with and with minutes of heating, the temperature is equal to the critical one, but between and minutes - it is even higher than the limiting one!

This means that you need to turn off the device in minutes.

MATHEMATICAL MODELS. BRIEFLY ABOUT THE MAIN

Most often, mathematical models are used in physics: after all, you probably had to memorize dozens of physical formulas. And the formula is the mathematical representation of the situation.

In the OGE and the Unified State Exam there are tasks just on this topic. In the exam (profile), this is problem number 11 (formerly B12). In the OGE - task number 20.

The solution scheme is obvious:

1) It is necessary to "isolate" useful information from the text of the condition - what we write under the word "Given" in physics problems. This useful information is:

• Formula
• Known physical quantities.

That is, each letter from the formula must be associated with a certain number.

2) You take all known quantities and substitute them into the formula. The unknown value remains in the form of a letter. Now all you have to do is solve the equation (usually a fairly simple one), and the answer is ready.

Model and modeling concept.

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

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

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

During research, a complex real phenomenon is replaced by some simplified copy or diagram, sometimes such a copy serves only to remember and at the next meeting to recognize the necessary phenomenon. Sometimes the constructed scheme reflects some essential features, makes it possible to understand the mechanism of the phenomenon, makes it possible to predict its change. Different models can correspond to the same phenomenon.

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

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

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

Mathematical models play a special role in science, the building material and tools of these models - mathematical concepts. They have been accumulating and improving over the course of millennia. Modern mathematics provides extremely powerful and versatile research tools. Almost every concept in mathematics, every mathematical object, starting from the concept of a number, is a mathematical model. When constructing a mathematical model of the object or phenomenon under study, those features, features and details are distinguished that, on the one hand, contain more or less complete information about the object, and on the other hand, allow mathematical formalization. Mathematical formalization means that the features and details of the object can be associated with appropriate adequate mathematical concepts: numbers, functions, matrices, and so on. Then the connections and relationships found and assumed in the object under study between its individual parts and components can be written using mathematical relations: equalities, inequalities, equations. The result is a mathematical description of the studied process or phenomenon, that is, its mathematical model.

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

Building a mathematical model is a central stage in the research or design of any system. All subsequent analysis of the object depends on the quality of the model. Model building is not a formal procedure. It strongly depends on the researcher, his experience and taste, always relies on certain experimental material. The model must be reasonably accurate, adequate and comfortable to use.

Mathematical modeling.

Classification of mathematical models.

Mathematical models can bedeterministic and stochastic .

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

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

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

V dynamicmodelthe relationships between the main variables of the modeled object during the transition from one mode to another are described.

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

Linear models- all functions and relations describing the model linearly depend on the variables andnot linearotherwise.

Mathematical modeling.

Requirements , n announced to the models.

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

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

Mathematical modeling.

The main stages of modeling.

1. Statement of the problem.

Determination of the goal of the analysis and ways to achieve it and develop a general approach to the problem under study. This stage requires a deep understanding of the essence of the task at hand. Sometimes, setting a task correctly is no less difficult than solving it. Setting is not a formal process, there are no general rules.

2. Studying the theoretical foundations and collecting information about the original object.

At this stage, a suitable theory is selected or developed. If it does not exist, cause-and-effect relationships are established between the variables describing the object. The inputs and outputs are defined, and simplifying assumptions are made.

3. Formalization.

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

4. Choice of a solution method.

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

5. Implementation of the model.

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

6. Analysis of the information received.

The obtained and expected solutions are compared, and the simulation error is monitored.

7. Checking the adequacy of the real object.

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

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

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

Mathematical modeling and the associated computer experiment are indispensable in cases where a natural experiment is impossible or difficult for one reason or another. For example, it is impossible to set up a natural experiment in history to check "what would have happened if ..." It is impossible to verify the correctness of one or another cosmological theory. In principle, it is possible, but hardly reasonable, to experiment with the spread of a disease, such as plague, or to carry out a nuclear explosion to study its consequences. However, all this can be done on a computer, having previously built mathematical models of the studied phenomena.

1.1.2 2. The main stages of mathematical modeling

1) Building the model. At this stage, a certain "non-mathematical" object is set - a natural phenomenon, design, 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 connections 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 stage of modeling.

2) Solution of the mathematical problem to which the model leads... 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 a reasonable 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 the given field.

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

5) Modification of the model.At this stage, there is either a complication of the model so that it is more adequate to reality, or its simplification in order to achieve a practically acceptable solution.

1.1.3 3. Model classification

Models can be classified according to various criteria. For example, according to the nature of the problems being solved, the models can be divided into functional and structural. In the first case, all quantities that characterize a phenomenon or object are expressed quantitatively. In this case, some of them are considered as independent variables, while others - 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. It is convenient to use graph theory to build such models. A graph is a mathematical object that is a set of points (vertices) on a plane or in space, some of which are connected by lines (edges).

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

MATHEMATICAL SIMULATION AND UNIVERSAL COMPUTERIZATION OR SIMULATION MODELS

Now, when almost universal computerization is taking place in the country, we have to hear statements from specialists of various professions: "If we introduce a computer, then all tasks will be solved immediately." This point of view is completely wrong, computers by themselves without mathematical models of certain processes will not be able to do anything, and one can only dream of general computerization.

In support of the above, we will try to substantiate the need for modeling, including mathematical modeling, we will reveal its advantages in human cognition and transformation of the external world, identify the existing shortcomings and go ... to simulation, i.e. computer simulation. But everything is in order.

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

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

A well-constructed model is more accessible for research than a real object. For example, it is unacceptable to experiment with the country's economy for educational purposes; here you cannot do without a model.

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

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

What is positive about any model? It allows you to gain new knowledge about the object, but, unfortunately, to one degree or another, it is incomplete.

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

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

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

How is a mathematical model built?

• First, the goal and subject of the research are formulated.
• Secondly, the most important characteristics corresponding to this goal are highlighted.
• Thirdly, the relationship between the elements of the model is verbally described.
• Further, the relationship is formalized.
• And the calculation is made according to the mathematical model and the analysis of the obtained solution.

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

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

Optimization models, including multicriteria ones, have a common property - there is a known goal (or several goals) for the achievement of which it is often necessary to deal with complex systems, where it is not so much about solving optimization problems as about studying and predicting states depending on selectable management strategies. And here we are faced with the difficulties of implementing the previous plan. They are as follows:

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

In connection with the listed difficulties arising in the study of complex systems, practice demanded a more flexible method, and it appeared - simulation modeling "Simujation modeling".

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

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

–Great closeness to the real system than mathematical models;

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

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

The listed advantages determine the disadvantages

–Build a simulation model longer, more difficult and more expensive;

- to work with the simulation system, it is necessary to have a computer suitable for the class;

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

–Building a simulation model requires a deeper study of the real process than mathematical modeling.

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

So, neither a computer, nor a mathematical model, nor an algorithm for its study, separately, can solve a sufficiently complex problem. But together they represent the force that allows you to know the world around you, to manage it in the interests of man.

1.2 Model classification

Let's consider the concept: “Models. Classification of models "from a scientific point of view.

Classification

Currently, there is a division of them into separate groups. Depending on the intended purpose, the following classification of economic and mathematical models is implied:

• theoretical and analytical types associated with the study of general characteristics and patterns;
• applied models aimed at solving certain economic problems. These include models of forecasting, economic analysis, management.

The classification of economic and mathematical models is associated with the scope of their practical application.

Depending on the content of the problem, such models are subdivided into groups:

• production models in general;
• separate options for regions, subsystems, industries;
• complexes of models of consumption, production, distribution and formation of labor resources, income, financial ties.

The classification of models of these groups implies the allocation of structural, subsystems.

When conducting research at the economic level, structural models are explained by the interconnection of individual subsystems. Models of intersectoral systems can be singled out as common options.

Functional options are used for the economic regulation of commodity-money relations. One and the same object can be presented in the form of functional, structural forms at the same time.

The use of structural models in research at the economic level is substantiated by the interconnection of subsystems. Models of intersectoral relations are typical in this case.

Functional models are widely used in the field of economic regulation. Typical in this case are the models of consumer behavior in the conditions of commodity-money relations.

Differences between models

Let's analyze different models. The classification of models currently used in economics involves the selection of normative and descriptive options. Using descriptive models, one can explain the analyzed facts, predict the possibility of the existence of certain facts.

The purpose of the descriptive hike

It involves the empirical identification of various dependencies in a modern economy. For example, the statistical patterns of various social groups are established, the probable ways of development of certain processes under constant conditions or without external influences are studied. Based on the results obtained in the course of a sociological survey, it is possible to build a model of consumer demand.

Regulatory models

With their help, one can assume purposeful activity. An example is the optimal planning model.

It can be both normative and descriptive. If the model is used in the analysis of the proportions of the past period, it is descriptive. When calculating with its help the optimal ways of economic development, it is normative.

Model features

The classification of models involves taking into account individual functions that help clarify controversial points. The descriptive approach is most widely used in simulation.

Depending on the nature of the detection of cause-and-effect relationships, there is a classification of models into variants, including individual elements of uncertainty and randomness, as well as rigidly deterministic models. It is important to distinguish between uncertainty, which is based on the theory of probability, and uncertainty that goes beyond the boundaries of the law.

Division of models according to the ways of reflecting the time factor

It is assumed that the models are classified according to this factor into dynamic and static types. Static models imply the consideration of all patterns in a certain period of time. Dynamic options are characterized by changes over time. Depending on the duration of use, it is allowed to classify models into the following options:

• short-term, the duration of which does not exceed a year;
• medium-term, calculated for a period of one to five years;
• long-term, calculated for a period of more than five years.

Depending on the specifics of the project, it is allowed to make changes in the process of using the model.

By the form of mathematical dependencies

The basis for the classification of models is the form of mathematical dependencies chosen for work. They mainly use the class of linear models for calculations and analysis. Consider the economic types of models. Classification of models of this type helps to study changes in consumption and demand of the population in the event of an increase in their material incomes. In addition, with the help of the analysis of changes in the needs of the population in the event of an increase in production, the efficiency of the use of resources in a specific situation is assessed.

Depending on the ratio of endogenous and exogenous variables that are included in the model, the models of these species are classified into closed and open systems.

Any model should include at least one endogenous variable, and therefore it is very problematic to find completely open systems. Models that do not include exogenous variables (closed variants) are also practically uncommon. In order to create such a variant, it will be necessary to completely abstract from the environment, to allow serious coarsening of the real economic system, which has external connections.

As the achievements of mathematical and economic research increase, the classification of models, systems, becomes significantly more complicated. Mixed types as well as complex model designs are currently in use. Unified classification information models currently not installed. At the same time, about ten parameters can be noted, according to which the types of models are built.

Model types

A monographic or verbal model assumes a description of a process or phenomenon. Often we are talking about rules, a law, a theorem, or a combination of several parameters.

The graphic model is drawn up in the form of a drawing, a geographic map, or a picture. For example, the relationship between consumer demand and product value can be represented using coordinate axes... The graph clearly demonstrates the relationship between the two values.

Real or physical models are created for objects that do not yet exist in reality.

The degree of aggregation of objects

There is a classification of information models on this basis for:

• local, with the help of which the analysis and forecast of certain indicators of the development of the industry is carried out;
• on microeconomic, intended for a serious analysis of the structure of production;
• macroeconomic, based on the study of the economy.

There is also a separate classification of management models for macroeconomic types. They are subdivided into one-, two-, multi-sector options.

Depending on the purpose of creation and use, the following options are distinguished:

• deterministic, having unambiguously understandable results;
• stochastic, which assume probabilistic outcomes.

In the modern economy, balance models are distinguished, which reflect the requirement for the conformity of the resource base and their use. To write them, they use the form of square chess matrices.

There are also econometric types, for which the methods of mathematical statistics are used. On such models, the development of the main indicators of the created economic system is expressed through a long-term trend (trend). They are in demand in the analysis and forecasting of certain economic situations associated with real statistical information.

Optimization models make it possible to choose the optimal variant of production, consumption or distribution of resources from a variety of alternative (possible) options. The use of limited resources in such a situation will be the most effective means of achieving the goal.

Participation in the project is assumed not only for an expert, but also for specialized software and computers. The resulting expert database is designed to solve one or more tasks by simulating human activity.

Network models are a set of operations and events interconnected over time. Most often, such a model is intended to carry out work in such a sequence as to achieve the minimum project completion time.

Depending on the selected type of mathematical apparatus, models are distinguished:

• matrix;
• correlation-regressive;
• network;
• inventory management;
• queuing service.

Stages of economic and mathematical modeling

This process is purposeful, it obeys a certain logical program of action. Among the main stages of creating such a model are:

• formulation of an economic problem and its qualitative analysis;
• development of a mathematical model;
• preparation of initial information;
• numerical solution;
• analysis of the results obtained, their use.

When posing an economic problem, it is necessary to clearly formulate the essence of the problem, note the important features and parameters of the modeled object, analyze the relationship of individual elements in order to explain the development and behavior of the object under consideration.

When developing a mathematical model, the relationship between equations, inequalities, functions is revealed. First of all, the type of model is determined, the possibility of using it in a specific task is analyzed, and a specific list of parameters and variables is formed. When considering complex objects, multi-aspect models are built, so that each one characterizes separate aspects of the object.

Conclusion

Currently, there is no separate concept of the model. The classification of the models is conditional, but this does not diminish their relevance.

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ANNOTATION

This course work will consider the types of mathematical models, their classification, the main types of mathematical models, their schemes. Examples of constructing mathematical models using several examples will be given. This work will help students to understand all the variety of types and types of mathematical models, to understand by what principle it is possible to classify mathematical models, which determines the choice of a particular mathematical model. Here we will find out what are the schemes of mathematical models and what are their features.

ABSTRACT

In given term paper will are considered types of the mathematical models, their categorization. The Main types of the mathematical models, their schemes. Will cite an instance buildings of the mathematical models on several examples.

Introduction

1. Simulation

1.1 Goals and objectives of modeling

1.2 Model requirements

2. Classification of models

3. Mathematical modeling

3.1 Continuously deterministic models (D - schemes)

3.2 Discrete-deterministic models (F-circuits)

3.3 Methods of queuing theory

4. Choosing a mathematical model

4.1 Comparison of methods for constructing mathematical models

4.2 Validity and simplicity of the model

4.3 Validation and model identification

4.4 Choosing a mathematical model

5. Examples of compiling mathematical models

Conclusion

List of sources of information

INTRODUCTION

At the present stage of the economic and social development of the republic, high requirements are imposed on the level of economic work at all levels. Today, qualitative changes in the economy are especially needed, a significant increase in the efficiency of all links of the economic system: enterprises, associations, industries. Of particular importance, in the context of the expanding rights of enterprises in the field of production and economic activities, their independence in making managerial decisions, is acquiring a deep knowledge of the latest achievements of economic science, methods of mathematical modeling and forecasting of economic processes based on information technologies of optimal decisions. These circumstances put forward increased requirements for the quality of training of specialists who must own the latest achievements of science and be able, using their rich arsenal of methods, to find the most effective management solutions, and this, in turn, determines the role and place of mathematical optimization methods in the educational process. simulation service deterministic

Methods of mathematical modeling, being a powerful tool for researching economic processes, plays a very important role in the analysis and synthesis of economic development, the definition provides multi-level optimization that captures the relationship of industries, regions and enterprises.

In science, technology and economics, models are used that, in a generally accepted, formal way, describe the characteristic features of systems and allow for a fairly reliable prediction of their behavior. The simplest models can be tables or graphs connecting the magnitude of the impact on the system with the values ​​that reflect its response to these actions. A higher level of models is equations reflecting a similar relationship (algebraic, differential, integral, etc.). the properties of a complex system are reflected by a set of different equations. Such models are called mathematical models and describe classes of systems. Regardless of the method of creating a mathematical model, it always approximately reflects the system under study. This is due to the incompleteness of our knowledge about the nature of the processes occurring in the system, with the impossibility of taking into account all the processes and their features (an excessively cumbersome mathematical model), with an inaccurate representation of data about the system and its elements. Having a mathematical model of the system, it is possible to predict its behavior in various situations (to carry out mathematical modeling of the system).

1. MODELING

Modeling - it is the study of an object by building and researching its model, carried out with a specific purpose and consists in replacing the experiment with the original experiment on the model. The model should be built in such a way that it most fully reproduces those qualities of the object that need to be studied in accordance with the set goal. In all respects, the model should be simpler than the object and more convenient to study. Thus, for the same object, there may be different models, classes of models corresponding to different purposes of its study. A necessary condition for modeling is the similarity of the object and its model. Those. modeling is the replacement of one object (original) with another (model) and the fixation and study of the properties of the model. The substitution is made with the aim simplifications, cheapening, accelerating the study of the properties of the original.

In the general case, the original object can be a natural or artificial, real or imaginary system. It has many parameters and is characterized by certain properties. A quantitative measure of the properties of a system is a set of characteristics; the system manifests its properties under the influence of external influences. From a specialist engaged in building models, the following basic qualities are required:

o a clear understanding of the essence of physical and chemical phenomena occurring in the object;

o the ability to mathematically describe ongoing processes and apply modeling methods;

o be able to provide meaningful results on the model.

1.1 Goals and objectives of modeling

The main goals and objectives of modeling are as follows:

1. Optimal design of new and intensification of existing technological processes.

2. Control over the course of the process, obtaining the necessary information about it and processing the information received in order to control the course of the technological process.

3. Solving the problems of studying objects where it is impossible to conduct active experiments - operating modes of reactors, trajectories of space objects, etc.

4. Maximum acceleration of the transfer of laboratory research results to industrial scale.

1.2 Model Requirements

1. The cost of creating a model should be significantly less than the cost of creating an original.

2. The rules for interpreting the results of a computational experiment must be clearly defined.

3. The main requirement - the model must be substantial. This requirement is that the model must reflect the properties of the object that are essential for solving a specific problem. For one and the same object, it is difficult to create a generalized model that reflects all of its properties. Therefore, it is important to ensure the materiality of the model.

Modeling is advisable when the model lacks those features of the original that impede its study.

Modeling theory is an interconnected set of provisions, definitions, methods and tools for creating models. The models themselves are the subject of modeling theory.

Modeling theory is the main component of the general theory of systems - systemology, where feasible models are postulated as the main principle: the system is represented by a finite set of models, each of which reflects a certain facet of its essence.

2 . MODEL CLASSIFICATION

Models can be classified according to different types of attributes:

- by the method of cognition: scientific and technical, artistic, everyday;

- by the nature of the models: objective (physical / material), sign (mental).

Fig. 1 Classification of models by nature

- in relation to time, static and dynamic models are distinguished;

- by the nature of the dependence of the output parameters on the input, the models are divided into deterministic and stochastic.

Material models - reduced (enlarged) reflection of the original while preserving the physical essence (reactor - test tube). The mental model is a reflection of the original, reflecting the essential features and arising in the consciousness of a person in the process of cognition. Image models are descriptive. Sign models are mathematic descriptions of processes, phenomena, objects and are usually called mathematic models. Signature models can also include diagrams and drawings.

Vmodel ides in relation to timeand by the nature of the output parameters

Fig. 2.

Physical models. The classification is based on the degree of abstraction of the model from the original. All models can be preliminarily divided into 2 groups - physical and abstract (mathematical).

A physical model is usually called a system equivalent or similar to the original, but possibly having a different physical nature. Types of physical models:

natural;

quasi-natural;

large-scale;

analog.

Natural models are real investigated systems (mock-ups, prototypes). They have full adequacy (correspondence) with the original system, but they are expensive.

Quasi-natural models are a collection of natural and mathematical models. This type is used when the model of a part of the system cannot be mathematical due to the complexity of its description (model of a human operator) or when a part of the system must be investigated in interaction with other parts, but they do not yet exist or their inclusion is very expensive (computational polygons , automated control systems).

A scale model is a system of the same physical nature as the original, but differs from it in scale. The methodological basis for large-scale modeling is the theory of similarity. In the design of computing systems, scale models can be used to analyze options for layout solutions.

Analog models are systems that have a physical nature that differs from the original, but processes of functioning similar to the original. To create an analog model, a mathematical description of the system under study is required. Mechanical, hydraulic, pneumatic and electrical systems are used as analog models. Analog modeling is used in the study of computer technology at the level of logical elements and electrical circuits, as well as at the system level, when the functioning of the system is described, for example, by differential or algebraic equations.

Mathematical models represent a formalized representation of a system using an abstract language, using mathematical relationships that reflect the process of the system's functioning. To compile mathematical models, you can use any mathematical means - algebraic, differential, integral calculus, set theory, theory of algorithms, etc. In essence, all mathematics is created for the compilation and study of models of objects and processes.

The means of abstract description of systems also include the languages ​​of chemical formulas, diagrams, drawings, maps, diagrams, etc. The choice of the type of model is determined by the characteristics of the system under study and the goals of modeling, since the study of the model allows you to get answers to a specific group of questions. Other information may require a different type of model. Mathematical models can be classified into deterministic and probabilistic, analytical, numerical and simulation.

An analytical model is a formalized description of a system that allows one to obtain an explicit solution to an equation using a well-known mathematical apparatus.

The numerical model is characterized by dependence (1.2) of a form that allows only particular solutions for specific initial conditions and quantitative parameters of the models.

A simulation model is a set of descriptions of the system and external influences, algorithms for the functioning of the system or the rules for changing the state of the system under the influence of external and internal disturbances. These algorithms and rules do not make it possible to use the available mathematical methods of analytical and numerical solution, but they allow simulating the process of the system's functioning and making calculations of the characteristics of interest. Simulation models can be created for a much wider class of objects and processes than analytical and numerical ones. Since VS are used to implement simulation models, universal and special algorithmic languages ​​serve as the means of formalized description of IM. MI are most suitable for the study of VS at the systemic level.

3 . MATHEMATICAL MODELING

This is the most important method of modern scientific research, the main apparatus of system analysis. Mathematical modeling is the study of the behavior of an object in certain conditions by solving the equations of its mathematic model. In chemical technology, mathematical modeling is used practically at all levels of research, development and implementation. This method is based on mathematic similarity. In mathematically similar objects, the processes have a different physical nature, but are described by identical equations.

In the early stages of its development, mathematical modeling was called analog. Moreover, the use of the method of analogy has led to the emergence of analog computing machines - ACM. These are electronic devices consisting of integrators, differentiators, summers and amplifiers. The ABM simulates physical phenomena that are analogous to the effects of an electrical nature. Compared to physical modeling, mathematical modeling is a more universal method.

Mathematical modeling:

- allows to carry out with the help of one device (computer) the solution of a whole class of problems having the same mathematical description;

- provides ease of transition from one task to another, allows you to enter variable parameters, disturbances and different initial conditions;

- makes it possible to carry out modeling in parts ("elementary processes"), which is especially important in the study of complex objects of chemical technology;

- more economical than the method of physical modeling, both in terms of costs, as well as in terms of cost.

The initial information in the construction of a mathematical model of the systems functioning processes is the data on the purpose and operating conditions of the investigated (projected) system S. This information determines the main purpose of modeling, the requirements for the mathematical model, the level of abstraction, and the choice of the mathematical modeling scheme.

Concept mathematical scheme allows us to consider mathematics not as a method of calculation, but as a method of thinking, a means of formulating concepts, which is most important in the transition from a verbal description to a formalized representation of the process of its functioning in the form of some mathematical model.

When using a mathematical scheme, first of all, the system researcher should be interested in the question of the adequacy of the mapping in the form of specific schemes of real processes in the system under study, and not the possibility of obtaining an answer (solution result) to a specific research question.

A mathematical scheme can be defined as a link in the transition from a meaningful to a formalized description of the process of the system's functioning, taking into account the impact of the external environment. Those. there is a chain: a descriptive model - a mathematical scheme - a simulation model.

As deterministic models, when a random fact is not taken into account in the study, differential, integral and other equations are used to represent systems operating in continuous time, and finite automata and finite difference schemes are used to represent systems operating in discrete time.

At the beginning of stochastic models (taking into account a random factor), probabilistic automata are used to represent systems with discrete time, and queuing systems (QS) are used to represent systems with continuous time. The so-called aggregate models are of great practical importance in the study of complex individual management systems, which include automated control systems.

Aggregate models (systems) make it possible to describe a wide range of research objects with a reflection of the systemic nature of these objects. It is with an aggregate description that a complex object is divided into a finite number of parts (subsystems), while maintaining connections, ensuring the interaction of parts.

3 .1 Continuously deterministic m O delhi (D - schemes)

Let us consider the features of the continuously deterministic approach using an example using differential equations as a mathematical model.

Differential equations are those equations in which functions of one variable or several variables are unknown, and the equation includes not only their functions, but their derivatives of various orders.

If the unknowns are functions of several variables, then the equations are called partial differential equations. If the unknown functions are of one independent variable, then ordinary differential equations take place.

General mathematical relationship for deterministic systems:

For example, the process of small oscillations of a pendulum is described by an ordinary differential equation where m 1, l 1 is the mass, the length of the suspension of the pendulum, is the angle of deviation of the pendulum from the equilibrium position. From this equation, you can find estimates of the characteristics of interest, for example, the oscillation period

Differential equations, D - circuits are the mathematical apparatus of the theory of automatic regulation and control systems.

When designing and operating automatic control systems (ACS), it is necessary to select such system parameters that would provide the required control accuracy.

It should be noted that systems of differential equations often used in ACS are determined by linearizing the control of an object (system) of a more complex form with nonlinearities:

3 .2 Discrete - deterministic models ( F -scheme)

Discrete - deterministic models (DDM) are the subject of consideration of the theory of automata (TA). TA is a section of theoretical cybernetics that studies devices that process discrete information and change their internal states only at acceptable times.

A state machine has many internal states and input signals that are finite sets. The automaton is set by the F-scheme:

F = ,

where z, x, y are, respectively, finite sets of input and output signals (alphabets) and a finite set of internal states (alphabet). z 0 Z - initial state; (z, x) - transition function; (z, x) - exit function. The automaton operates in discrete automaton time, the moments of which are ticks, i.e. adjacent to each other equal time intervals, each of which corresponds to constant values ​​of the input, output signal and internal state. An abstract automaton has one input and one output channel.

At the moment t, being in the state z (t), the automaton is able to perceive the signal x (t) and issue the signal y (t) =, passing into the state z (t + 1) =, z (t) Z; y (t) Y; x (t) X. An abstract spacecraft in the initial state z 0 accepting signals x (0), x (1), x (2)… gives out signals y (0), y (1), y (2)… (output word).

There is an F-automaton of the 1st kind (Mile), operating according to the scheme:

z (t + 1) =, t = 0,1,2 ... (1)

y (t) =, t = 0,1,2 ... (2)

automatic machine of the 2nd kind:

z (t + 1) =, t = 0,1,2 ... (3)

y (t) =, t = 1,2,3 ... (4)

An automaton of the 2nd kind, for which y (t) =, t = 0,1,2, ... (5)

those. the function of outputs does not depend on the input variable x (t), it is called a Moore automaton.

That. equations 1-5 completely setting the F-automaton are a special case of the equation

(6)

where is the state vector, is the vector of independent input variables, is the vector of external environment influences, is the vector of the intrinsic internal parameters of the system, is the vector of the initial state, t is the time; and the equation, (7)

when the system S is denominated and a discrete signal x arrives at its input.

According to the number of states, finite state machines can be with memory and without memory. Automata with memory have more than one state, and automata without memory (combinational or logic circuits) have only one state. In this case, according to (2), the operation of the combinational circuit is that it associates each input signal x (t) with a certain output signal y (t), i.e. implements a logical function of the form:

y (t) =, t = 0,1,2, ...

This function is called boolean if the alphabets X and Y, to which the values ​​of signals x and y belong, consist of 2 letters.

By the nature of the timing (discrete), F-machines are divided into synchronous and asynchronous. In synchronous automata, the times at which the automaton "reads" the input signals are determined forcibly by synchronizing signals. The response of the machine to each value of the input signal ends in one clock cycle. An asynchronous F-automaton reads the input signal continuously and therefore, responding to a sufficiently long water signal of constant value x, it can, as follows from 1-5, change its state several times, issuing a corresponding number of output signals until it becomes stable.

To define an F-automaton, it is necessary to describe all elements of the set F = , i.e. input, internal and output alphabets, as well as transition and output functions. To set the work of F-automata, the tabular, graphical and matrix methods are most often used.

In the tabular way of setting, transition and output tables are used, the rows of which correspond to the input signals of the automaton, and the columns - to its states. In this case, usually the 1st column on the left corresponds to the initial state z 0. At the intersection of the i-th row and j-th column of the transition table, the corresponding value (z k, x i) of the transition function is placed, and in the output table - (z k, x i) of the output function. For the F-Moore automaton, both tables can be combined, having received the so-called. the marked transition table, in which above each state z k of the automaton, denoting a column of the table, there is an output signal corresponding to this state, according to (5), (z i).

The description of the operation of the F-Mealy automaton by the transition and exit tables is illustrated in Table 3.1., And the description of the F-Moore automaton - by the transition table 3.2 ..

Table 3.1.Description of the work of the Miles machine

Table 3.2.Description of the operation of the Moore machine

Examples of the tabular way of specifying the F-Mealy machine F1 with three states, two input and two output signals are given in table 3.3, and for the F-Moore machine F2 - in table 3.4.

Table 3.3.Method of setting a Miles machine with three states

Table 3.4.A way to define a Moore automaton with three states

Another way of defining a finite state machine uses the concept of a directed graph. The automaton graph is a set of vertices corresponding to different states of the automaton and connecting the vertices of the graph arcs corresponding to certain transitions of the automaton. If the input signal x k causes a transition from the state z i to the state z j, then on the automaton graph the arc connecting the vertex z i with the vertex z j is denoted by x k. In order to set the transition function, the graph arcs must be marked with the corresponding output signals. For Mealy automata, this marking is done as follows: if the input signal x k acts on the state z i, then according to what has been said, an arc is obtained, outgoing from z i and marked with x k; this arc is additionally marked with the output signal y = (z i, x k). For a Moore automaton, a similar labeling of the graph is as follows: if the input signal x k, acting on a certain state of the automaton, causes a transition to the state z j, then the arc directed to z j and labeled x k is additionally marked with the output signal y = (z j, x k). In fig. 3 shows the F-automata of Mealy F1 and Moore F2, given earlier by the tables, respectively.

Rice.3 . Automata graphs of Mealy (a) and Moore (b)

When solving modeling problems, a matrix definition of a finite state machine is often a more convenient form. In this case, the matrix of connections of the automaton is a square matrix C = || c ij ||, the rows of which correspond to the initial states, and the columns - to the transition states. The element c ij = x k / y S in the case of the Mealy automaton corresponds to the input signal x k, causing the transition from the state z i to the state z j and the output signal y S issued during this transition. For the Mealy automaton F1, considered above, the matrix of connections has the form:

If the transition from state z i to state z j occurs under the action of several signals, the element of the matrix c ij is a set of "input / output" pairs for this transition, connected by a disjunction sign.

For an F-Moore automaton, the element c ij is equal to the set of input signals at the transition (z i z j), and the output is described by a vector of outputs:

the i-th component of which the output signal indicating the state z i

Example. For the previously considered Moore automaton F2, we write the state matrix and the output vector:

;

For deterministic automata, the transitions are unambiguous. As applied to the graphical method of defining an F-automaton, this means that in the graph of an F-automaton, 2 or more edges marked with the same input signal cannot leave any vertex. Similarly, in the matrix of connections of the automaton C in each row, any input signal should not occur more than once.

Consider the view of the transition table and the graph of an asynchronous state machine. For an F-automaton, the state z k is called stable , if for any input x i X for which (z k, x i) = z k (z k x i) = y k. That. An F-automaton is called asynchronous if each of its states z k Z is stable.

In practice, automata are always asynchronous, and the stability of their states is ensured in one way or another, for example, by introducing synchronization signals. At the level of abstract theory, it is often convenient to operate with synchronous finite state machines.

Example. Consider an asynchronous Moore F-automaton, which is described in table. 3.5 and shown in Fig. 4.

Table 3.5.Moore's asynchronous automaton

Rice.4 . Moore's asynchronous automaton graph

If in the transition table of an asynchronous automaton some state z k stands at the intersection of row x S and column z S (Sk), then this state z k must necessarily occur in the same row in column z k.

With the help of F-diagrams, nodes and elements of electronic computing systems, monitoring, regulation and control devices, systems of time and space switching in information exchange technology are described. The breadth of application of F-circuits does not mean their universality. This approach is unsuitable for describing decision-making processes, processes in dynamic systems with the presence of transient processes and stochastic elements.

3.3 Continuous stochastic models (Q - schemes)

These include queuing systems, which are called Q-schemes.

The subject of queuing theory is queuing systems (QS) and queuing networks. QS is understood as a dynamic system designed to efficiently service a random flow of applications with limited system resources. The generalized structure of the QS is shown in Figure 5.

Rice.5 . CMO scheme

Homogeneous claims arriving at the input of the QS are divided into types depending on the generating cause, the flow rate of claims of type i (i = 1 ... M) is denoted by i. The totality of applications of all types is the incoming flow of the QS.

Service of applications is carried out m channels. Distinguish between universal and specialized service channels. For a universal channel of type j, the distribution functions F ji () of the duration of servicing of claims of an arbitrary type are considered known. For specialized channels, the distribution functions for the service duration of channels of certain types of claims are undefined, the assignment of these claims to this channel.

As a service process, various in their physical nature processes of functioning of economic, production, technical and other systems can be represented, for example, product supply flows to a certain enterprise, flows of parts and components on the assembly line of a shop, requests for information processing of electronic computing systems from remote terminals, etc. In this case, a characteristic feature of the operation of such objects is the random behavior of claims (claims) for servicing and completion of servicing at random times.

Q - circuits can be investigated analytically and by simulation models. The latter provides great versatility.

Let's consider the concept of queuing.

In any elementary act of servicing, two main components can be distinguished: the expectation of service by the claim and the actual servicing of the claim. This can be displayed in the form of some i-th service device P i, consisting of a claim accumulator, in which there can be simultaneously li = 0 ... L i H claims, where L i H is the capacity of the i-th accumulator, and a claim service channel, ki ...

Rice.6 . Schematic diagram of the CMO device

Each element of the servicing device P i receives streams of events: the stream of claims w i to the storage device H i, the stream of servicing u i to the channel k i.

A stream of events (FL) is a sequence of events that occur one after another at some random moments in time. Distinguish between streams of homogeneous and heterogeneous events. A homogeneous PS (OPS) is characterized only by the moments of arrival of these events (causing moments) and is given by the sequence (t n) = (0t 1 t 2… t n…), where t n is the moment of arrival of the nth event - a non-negative real number. The MPS can also be specified as a sequence of time intervals between the n-th and n-1-th events (n).

Inhomogeneous PS is a sequence (t n, f n), where t n - causing moments; f n - a set of event attributes. For example, the affiliation to one or another source of claims, the presence of a priority, the ability to serve one or another type of channel, etc. can be specified.

Consider an OPS for which i (n) - random variables, independent from each other. Then the PS is called a flow with limited aftereffect.

An SS is called ordinary if the probability that more than one event P 1 (t, t) falls on a small time interval t adjacent to the time t is negligible.

If for any interval t event P 0 (t, t) + P 1 (t, t) + P 1 (t, t) = 1, P 1 (t, t) is the probability of hitting exactly one event on the interval t. As the sum of the probabilities of events that form a complete group and are inconsistent, then for an ordinary stream of events P 0 (t, t) + P 1 (t, t) 1, P 1 (t, t) = (t), where (t) - a quantity of order of smallness which is higher than t, i.e. lim ((t)) = 0 for t0.

A stationary PS is a flow for which the probability of occurrence of a particular number of events in a time interval depends on the length of this segment and does not depend on where this segment is taken on the time axis 0 - t. For the OPS, 0 * P 0 (t, t) + 1 * P 1 (t, t) = P 1 (t, t) is the average number of events in the interval t. The average number of events occurring in the segment t per unit time is P 1 (t, t) / t. Consider the limit of this expression at t0

lim P 1 (t, t) / t = (t) * (1 / single time).

If this limit exists, then it is called the intensity (density) of the OPS. For standard PS (t) == const.

With regard to the elementary service channel k i, we can assume that the flow of claims w i W, i.e. the time intervals between the moments of appearance of claims at the input k i form a subset of uncontrollable variables, and the service flow u i U, i.e. the time intervals between the beginning and the end of servicing a claim form a subset of controlled variables.

The claims served by the channel k i and the claims that left the server П i for various reasons not served form the output stream y i Y.

The process of functioning of the service device P i can be represented as a process of changing the states of its elements in time Z i (t). The transition to a new state for P i means a change in the number of requests that are in it (in the channel k i and the accumulator H i). That. the vector of states for П i has the form:, where are the states of the storage, (= 0 - the storage is empty, = 1 - there is one customer in the storage ..., = - the storage is fully occupied; - the state of the channel ki (= 0 - the channel is free, = 1 channel busy).

Q-diagrams of real objects are formed by the composition of many elementary service devices P i. If k i different service devices are connected in parallel, then multichannel service takes place (multichannel Q-circuit), and if devices P i and their parallel compositions are connected in series, then multiphase service takes place (multiphase Q-circuit).

That. to define a Q-scheme, a conjugation operator R is required, which reflects the interconnection of structure elements.

Links in the Q-diagram are depicted as arrows (flow lines reflecting the direction of movement of applications). A distinction is made between open and closed Q-circuits. In open loop, the output stream cannot enter any element again, i.e. Feedback missing.

The intrinsic (internal) parameters of the Q-circuit will be the number of phases L Ф, the number of channels in each phase, L kj, j = 1 ... L Ф, the number of accumulators of each phase L kj, k = 1 ... L Ф, capacity i- th drive L i H. It should be noted that in the queuing theory, depending on the storage capacity, the following terminology is used:

lossy systems (L i H = 0, no storage device);

waiting systems (L i H);

systems with limited storage capacity H i (mixed).

Let us denote the entire set of eigen parameters of the Q-circuit as a subset of H.

To define a Q-scheme, it is also necessary to describe the algorithms for its functioning, which determine the rules for the behavior of claims in various ambiguous situations.

Depending on the place of occurrence of such situations, there are algorithms (disciplines) for waiting for claims in the accumulator H i and for servicing claims by channel k i. The inhomogeneity of the flow of applications is taken into account by introducing a priority class.

Depending on the dynamics of priorities, Q-schemes are distinguished between static and dynamic. Static priorities are assigned in advance and do not depend on the states of the Q-circuit, i.e. they are fixed within the solution specific task modeling. Dynamic priorities arise in the simulation. Based on the rules for selecting requests from the storage drive Н i for servicing by the channel k i, it is possible to select relative and absolute priorities. Relative priority means that a claim with a higher priority arriving at storage H waits for the channel k i to service the claiming claims and only after that occupies the channel. An absolute priority means that a customer with a higher priority arriving at the storage interrupts the servicing of customers with a lower priority by the channel ki and occupies the channel themselves (in this case, a customer pushed out of ki can either leave the system or can be written again to some place in H i).

It is also necessary to know the set of rules by which applications leave Н i and k i: for Н i - either overflow rules or exit rules associated with the expiration of the waiting time for applications in Н i; for k i - the rules for choosing routes or directions of departure. In addition, for claims, it is necessary to set the rules according to which they remain in the channel k i, i.e. channel blocking rules. In this case, a distinction is made between blocking k i at the output and at the input. Such locks reflect the presence of control links in the Q_scheme, which regulate the flow of requests depending on the states of the Q_scheme. The set of possible algorithms for the behavior of claims in the Q_scheme can be represented in the form of some operator of algorithms for the behavior of claims A .

That. Q_scheme describing the process of functioning of a QS of any complexity is uniquely defined as a set of sets: Q = .

4 . SELECTING A MATHEMATICAL MODEL

4 .1 Comparisonmethods forstructures of mathematical models

The choice of method depends on the importance and degree of complexity of the process. For large multi-tonnage industries, it is necessary good models, here we use theoretical method... The same method is used when creating fundamentally new technological processes.

For small industries with a complex nature of the process, an experimental method is used. In practice, as a rule, a reasonable combination of all methods is used.

4 .2 Credibility andsimplicitymodel

The mathematical model constructed by one of the methods considered above must simultaneously satisfy the requirements of reliability and simplicity.

A reliable model that correctly describes the behavior of an object can be quite complex. The complexity of the model is determined, as a rule, by the complexity of the object under study and the degree of accuracy presented by practice to the calculation results. It is necessary that this complexity does not exceed a certain limit determined by the capabilities of the existing mathematical apparatus. Therefore, the model must be simple enough in mathematical terms so that it can be solved by the available methods and means.

4 . 3 ExaminationaDecvatity andidentificationmodel

Adequacy check is an assessment of the reliability of the constructed mathematical model, the study of its compliance with the object under study.

The verification of the adequacy is carried out on test experiments by comparing the results of the calculation according to the model with the results of the experiment on the object under study under the same conditions. This allows us to establish the limits of applicability of the constructed model.

The main stage in the construction of an adequate model is the identification of the mathematic description of the mathematic description of the object. The identification task is to determine the type of the model and find its unknown parameters - individual constants or their complexes that characterize the properties of the object. Identification is possible if the necessary experimental information about the object under study is available.

4.4 Choosing a mathematical model

The problem of choosing a model arises when there is a class of models for the same object. Model selection is one of critical milestones modeling. Ultimately, the advantage of a particular model determines the criterion of practice, understood in a broad sense. When choosing a model, one should proceed from a reasonable compromise between the complexity of the model, the completeness of the object characteristics obtained with its help, and the accuracy of these characteristics. So, if the model is not accurate enough, then it needs to be supplemented, clarified by the introduction of new factors, it may also turn out that the proposed model is too complex and the same results can be obtained using a simpler model.

Sometimes, due to the limited means available, it is necessary to simplify the mathematical description. In this case, an estimate of the error introduced in this case is necessary.

When solving the equations of mathematical description using electronic computing systems, it is necessary to create a modeling algorithm ("machine" model). The modeling algorithm is a transformed mathematical description and is a sequence of arithmetic and logical decision operations written in the form of a program.

When developing such an algorithm, first of all, it is necessary to choose a method for solving the equations of the mathematical description - analytical or numerical. Remember to check the accuracy of the selected calculation method.

5. EXAMPLES OF COMPOSITION OF MATHEMATICAL MODELS

In this section, we will consider typical examples of compiling mathematical models for solving a variety of problems, both in the national economy and in school problems in mathematics.

EXAMPLE 1

Build a mathematical model for the formation of a production plan.

Table 5.1.Initial data

Determine the volume of production, ensuring maximum profit.

Building a mathematical model

Let X 1 - the number of products of type A, and X 2 - the number of products B. Then X 1 + 4x 2 - the amount of grade 1 material required for the manufacture of products, and according to the condition of the problem, this number does not exceed 320

X 1 + 4x 2 <=320 (1)

3x 1 + 4x 2 - the amount of grade 2 material required for the manufacture of products, and according to the condition of the problem, this number does not exceed 360

3x 1 + 4x 2 <=360 (2)

X 1 + 2x 2 - the amount of grade 2 material required for the manufacture of products, and according to the condition of the problem, this number does not exceed 180

X 1 + 2x 2 <=180 (3)

moreover, since X 1 and X 2 express the volume of output, then they cannot be negative, that is

X 1 > 0, x 2 > 0 (4)

F = X 1 + 2x 2 - profit, which should be maximum. Thus, we have the following mathematical model for this problem

F = x 1 + 2x 2> max

EXAMPLE 2

Transport problem. There are n cities. Leaving one of them, the traveling salesman must go around everything and return to the starting city. Each city can be entered once, and, therefore, the traveling salesman's route must form a closed loop without loops. It is required to find the shortest closed traveling salesman route if the matrix of distances between cities is known.

The mathematical model of the problem under consideration has the form:

Here the variable x ij takes on the value 1 if the traveling salesman moves from city i to city j (i, j = 1,2,…, n, i? J) and 0 otherwise. Condition (1) is an optimized function, where c ij are the distances between cities (i, j = 1,2,…, n, i? J), and in the general case with ij? with ij; condition (2) means that the traveling salesman leaves each city only once; (3) - that he enters each city only once; (4) ensures the closedness of the route and the absence of loops, where u i and u j are some real values ​​(i, j = 1,2,…, n, i? J) (5).

EXAMPLE 3

Some enterprise manufactures products of 5 types, using component parts of 7 names A, B, C, D, E, F, G. The stock of the enterprise is limited by a certain number of component parts. It is known how many components are required for the production of a unit of each type of product and the profit from the production of a unit of product of each type. Determine how many products of each type are required in order to provide the enterprise with the greatest profit.

Table 5.2.Production data

F= 2x 1 + 3x 2 + X 3 + 5x 4 + 4x 5 -

profit that should be maximized. Thus, we have the number of components for the production of the optimal amount of products:

2x 1 + 2x 2 + x 5 ? 10

the number of components A for the production of products;

X 1 + 2x 2 + X 4 ? 7

the number of components B for the production of products;

4x 1 + X 4 ? 12

the number of components C for the production of products;

4x 4 ? 12

the number of components D for the production of products;

X 3 + 2x 4 + x 5 ? 15

the number of components E for the production of products;

X 4 + 3x 5 ? 12

the number of components F for the production of products;

2x 1 + X 4 ? 8

the number of components G for the production of products;

and all the variables X 1, X 2, X 3, X 4, X 5 - must be non-negative and integer.

Thus, we have the following mathematical model of product output to maximize profit:

2x 1 + 3x 2 + X 3 + 5x 4 + 4x 5 > max

EXAMPLE 4

There is a production for the manufacture of two types of products A and B with a limited volume of materials of three grades from which the products are made. The initial data are shown in the table.

Table 5.3.Initial data

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