Mathematical models in economics and programming. The goals of the organization are realized in the external environment

Any real process characteristic random fluctuations caused by the physical variability of any factors over time. In addition, there may be random external influences on the system. Therefore, with an equal average value of the input parameters at different times the output parameters will be different. Therefore, if random impacts on the system under study are significant, it is necessary to develop probabilistic (stochastic) model of the object, taking into account the statistical laws of distribution of system parameters and choosing the appropriate mathematical apparatus.

When building deterministic models random factors are neglected, taking into account only the specific conditions of the problem being solved, the properties and internal connections of the object (almost all branches of classical physics are built on this principle)

The idea of ​​deterministic methods- in the use of the model’s own dynamics during the evolution of the system.

In our course these methods are presented: molecular dynamics method, the advantages of which are: accuracy and certainty of the numerical algorithm; The disadvantage is that it is labor intensive due to the calculation of the interaction forces between particles (for a system of N particles, at each step you need to perform
operations of counting these forces).

At deterministic approach equations of motion are specified and integrated over time. We will consider systems of many particles. The positions of the particles contribute potential energy to the total energy of the system, and their velocities determine the contribution of kinetic energy. The system moves along a trajectory with constant energy in phase space (further explanations will follow). For deterministic methods, a microcanonical ensemble is natural, the energy of which is the integral of motion. In addition, it is possible to study systems for which the integral of motion is temperature and (or) pressure. In this case, the system is not closed, and it can be represented in contact with a thermal reservoir (canonical ensemble). To model it, we can use an approach in which we limit a number of degrees of freedom of the system (for example, we set the condition
).

As we have already noted, in the case when processes in a system occur unpredictably, such events and quantities associated with them are called random, and algorithms for modeling processes in the system - probabilistic (stochastic). Greek stoohastikos- literally means “one who can guess.”

Stochastic methods use a slightly different approach than deterministic ones: they only need to calculate the configuration part of the problem. The equations for the momentum of a system can always be integrated. The problem that then arises is how to conduct transitions from one configuration to another, which in the deterministic approach are determined by momentum. Such transitions in stochastic methods are carried out with probabilistic evolution in Markov process. The Markov process is a probabilistic analogue of the model’s own dynamics.

This approach has the advantage that it allows one to model systems that do not have any inherent dynamics.

Unlike deterministic methods, stochastic methods on a PC are simpler and faster to implement, but to obtain values ​​close to the true ones, good statistics are required, which requires modeling a large ensemble of particles.

An example of a completely stochastic method is Monte Carlo method. Stochastic methods use the important concept of a Markov process (Markov chain). The Markov process is a probabilistic analogue of the process in classical mechanics. The Markov chain is characterized by the absence of memory, i.e. the statistical characteristics of the near future are determined only by the present, without taking into account the past.

More practical than busy 2.

Random walk model

Example(formal)

Let us assume that particles are placed in arbitrary positions at the nodes of a two-dimensional lattice. At each time step, the particle “jumps” to one of the idle positions. This means that the particle has the ability to choose the direction of its jump to any of the four nearest places. After a jump, the particle “does not remember” where it jumped from. This case corresponds to a random walk and is a Markov chain. The result at each step is a new state of the particle system. The transition from one state to another depends only on the previous state, i.e., the probability of the system being in state i depends only on state i-1.

What physical processes in a solid body remind us (similar to) the described formal model of a random walk?

Of course, diffusion, that is, the very processes, the mechanisms of which we considered in the course of heat and mass transfer (3rd course). As an example, let us recall the usual classical self-diffusion in a crystal, when, without changing their visible properties, atoms periodically change places of temporary residence and wander around the lattice, using the so-called “vacancy” mechanism. It is also one of the most important mechanisms of diffusion in alloys. The phenomenon of migration of atoms in solids plays a decisive role in many traditional and non-traditional technologies - metallurgy, metalworking, the creation of semiconductors and superconductors, protective coatings and thin films.

It was discovered by Robert Austen in 1896 by observing the diffusion of gold and lead. Diffusion- the process of redistribution of atomic concentrations in space through chaotic (thermal) migration. Causes, from the point of view of thermodynamics, there can be two: entropy (always) and energy (sometimes). The entropic reason is the increase in chaos when mixing atoms of the carved variety. Energy - promotes the formation of an alloy, when it is more advantageous to have atoms of different types nearby, and promotes diffusion decomposition, when the energy gain is ensured by placing atoms of the same type together.

The most common diffusion mechanisms are:

vacancy

internodal

displacement mechanism

To implement the vacancy mechanism, at least one vacancy is required. Migration of vacancies is carried out by moving to an unoccupied site of one of the neighboring atoms. An atom can make a diffusion jump if there is a vacancy next to it. Vacancy cm, with a period of thermal vibrations of an atom in a lattice site, at a temperature T = 1330 K (by 6 K< точки плавления), число скачков, которое совершает вакансия в 1с, путь за одну секунду-см=3 м (=10 км/ч). По прямой же путь, проходимый вакансиейсм, т. е. в 300 раз короче пути по ломаной.

Nature needed it. so that the vacancy changes its place of residence within 1 s, passes along a broken line 3 m, and moves along a straight line by only 10 microns. Atoms behave calmer than vacancies. But they also change their place of residence a million times per second and move at a speed of approximately 1 m/hour.

So. that one vacancy per several thousand atoms is enough to move atoms at a micro level at a temperature close to melting.

Let us now form a random walk model for the phenomenon of diffusion in a crystal. The process of wandering of an atom is chaotic and unpredictable. However, for an ensemble of wandering atoms, statistical regularities should appear. We will consider uncorrelated jumps.

This means that if
And
is the movement of atoms during the i and j jumps, then after averaging over the ensemble of wandering atoms:

(average product = product of averages. If the walk is completely random, all directions are equal and
=0.)

Let each particle of the ensemble make N elementary jumps. Then its total displacement is:

;

and the average square of displacement

Since there is no correlation, the second term =0.

Let each jump have the same length h and random direction, and the average number of jumps per unit time is v. Then

It's obvious that

Let's call the quantity
- diffusion coefficient of wandering atoms. Then
;

For the three-dimensional case -
.

We got parabolic diffusion law- the mean square of the displacement is proportional to the wandering time.

This is exactly the problem we have to solve in the next laboratory work - modeling one-dimensional random walks.

Numerical model.

We define an ensemble of M particles, each of which takes N steps, independently of each other, to the right or to the left with the same probability. Step length = h.

For each particle we calculate the square of the displacement
in N steps. Then we perform averaging over the ensemble -
. Magnitude
, If
, i.e. The mean square of the displacement is proportional to the random walk time
- average time of one step) - parabolic law of diffusion.

Modeling is one of the most important tools in modern life when one wants to anticipate the future. And this is not surprising, because the accuracy of this method is very high. Let's take a look at what a deterministic model is in this article.

general information

Deterministic models of systems have the peculiarity that they can be studied analytically if they are simple enough. In the opposite case, when using a significant number of equations and variables, electronic computers can be used for this purpose. Moreover, computer assistance, as a rule, comes down solely to solving them and finding answers. Because of this, it is necessary to change the systems of equations and use a different discretization. And this entails an increased risk of error in calculations. All types of deterministic models are characterized by the fact that knowledge of the parameters on a certain studied interval allows us to fully determine the dynamics of development of known indicators beyond the border.

Peculiarities

Factor modeling

References to this could be seen throughout the article, but we have not yet discussed what it is. Factor modeling implies that the main provisions for which quantitative comparison is necessary are identified. To achieve the stated goals, the research transforms the form.

If a strictly deterministic model has more than two factors, then it is called multifactorial. Its analysis can be carried out through various techniques. As an example, let us give In this case, she considers the assigned tasks from the point of view of pre-established and worked out a priori models. The choice among them is carried out according to their content.

To construct a qualitative model, it is necessary to use theoretical and experimental studies of the essence of the technological process and its cause-and-effect relationships. This is precisely the main advantage of the subjects we are considering. Deterministic models allow for accurate forecasting in many areas of our lives. Thanks to their quality parameters and versatility, they have become so widespread.

Cybernetic deterministic models

They are of interest to us due to the analysis-based transient processes that arise with any, even the most insignificant, changes in the aggressive properties of the external environment. For simplicity and speed of calculations, the existing state of affairs is replaced by a simplified model. The important thing is that it satisfies all basic needs.

The performance of the automatic control system and the effectiveness of the decisions it makes depend on the unity of all necessary parameters. In this case, it is necessary to solve the following problem: the more information is collected, the higher the probability of error and the longer the processing time. But if you limit your data collection, you can expect a less reliable result. Therefore, it is necessary to find a middle ground that will allow obtaining information of sufficient accuracy, and at the same time it will not be unnecessarily complicated by unnecessary elements.

Multiplicative deterministic model

It is built by dividing factors into many. As an example, we can consider the process of forming the volume of manufactured products (PP). So, for this you need to have labor (PC), materials (M) and energy (E). In this case, the PP factor can be divided into a set (RS;M;E). This option reflects the multiplicative form of the factor system and the possibility of its division. In this case, you can use the following transformation methods: expansion, formal decomposition and lengthening. The first option has found wide application in analysis. It can be used to calculate an employee's performance, and so on.

When lengthening, one value is replaced by other factors. But in the end it should be the same number. An example of elongation was discussed above. All that remains is the formal decomposition. It involves the use of lengthening the denominator of the original factor model due to the replacement of one or more parameters. Let's consider this example: we calculate the profitability of production. To do this, the amount of profit is divided by the amount of costs. When multiplying, instead of a single value, we divide by the summed up expenses for materials, personnel, taxes, and so on.

Probabilities

Oh, if only everything went exactly as planned! But this rarely happens. Therefore, in practice, deterministic and What can be said about the latter are often used together? Their peculiarity is that they also take into account various probabilities. Take the following for example. There are two states. The relationship between them is very bad. A third party decides whether to invest in businesses in one of the countries. After all, if a war breaks out, profits will suffer greatly. Or you can give the example of building a plant in an area with high seismic activity. There are natural factors at work here that cannot be accurately taken into account; this can only be done approximately.

Conclusion

We examined what deterministic analysis models are. Alas, in order to fully understand them and be able to apply them in practice, you need to study very well. The theoretical foundations are already there. Also within the framework of the article, some simple examples were presented. Next, it is better to follow the path of gradually complicating the working material. You can make your task a little easier and start learning about software that can carry out the corresponding simulation. But whatever the choice, understanding the basics and being able to answer questions about what, how and why is still necessary. You should first learn how to select the right input data and choose the right actions. Then the programs will be able to successfully complete their tasks.

Mathematical models in economics and programming

1. Deterministic and probabilistic mathematical models in economics. Advantages and disadvantages

Methods for studying economic processes are based on the use of mathematical - deterministic and probabilistic - models representing the process, system or type of activity being studied. Such models provide a quantitative description of the problem and serve as the basis for making management decisions when searching for the optimal option. How justified are these decisions, are they the best possible, are all the factors that determine the optimal solution taken into account and weighed, what is the criterion to determine that this solution is really the best - these are the range of questions that are of great importance for production managers, and the answer to which can be found using operations research methods [Chesnokov S.V. Deterministic analysis of socio-economic data. - M.: Nauka, 1982, p. 45].

One of the principles of forming a control system is the method of cybernetic (mathematical) models. Mathematical modeling occupies an intermediate position between experiment and theory: there is no need to build a real physical model of the system; it will be replaced by a mathematical model. The peculiarity of the formation of a control system lies in the probabilistic, statistical approach to control processes. In cybernetics, it is accepted that any control process is subject to random, disturbing influences. Thus, the production process is influenced by a large number of factors, which cannot be taken into account in a deterministic manner. Therefore, the production process is considered to be influenced by random signals. Because of this, enterprise planning can only be probabilistic.

For these reasons, when speaking about mathematical modeling of economic processes, they often mean probabilistic models.

Let us describe each type of mathematical model.

Deterministic mathematical models are characterized by the fact that they describe the relationship of some factors with an effective indicator as a functional dependence, i.e. in deterministic models, the effective indicator of the model is presented in the form of a product, a quotient, an algebraic sum of factors, or in the form of any other function. This type of mathematical models is the most common, since, being quite simple to use (compared to probabilistic models), it allows one to understand the logic of the action of the main factors in the development of the economic process, quantify their influence, understand which factors and in what proportion it is possible and advisable to change to increase production efficiency.

Probabilistic mathematical models are fundamentally different from deterministic ones in that in probabilistic models the relationship between factors and the resulting attribute is probabilistic (stochastic): with a functional dependence (deterministic models), the same state of factors corresponds to a single state of the resulting attribute, whereas in probabilistic models one and the same state of factors corresponds to a whole set of states of the resulting attribute [Tolstova Yu. N. Logic of mathematical analysis of economic processes. - M.: Nauka, 2001, p. 32-33].

The advantage of deterministic models is their ease of use. The main drawback is the low adequacy of reality, since, as noted above, most economic processes are probabilistic in nature.

The advantage of probabilistic models is that, as a rule, they are more consistent with reality (more adequate) than deterministic ones. However, the disadvantage of probabilistic models is the complexity and labor-intensive nature of their application, so in many situations it is sufficient to limit ourselves to deterministic models.

2. Statement of the linear programming problem using the example of the food ration problem

For the first time, the formulation of a linear programming problem in the form of a proposal for drawing up an optimal transportation plan; allowing to minimize the total mileage was given in the work of the Soviet economist A. N. Tolstoy in 1930.

Systematic studies of linear programming problems and the development of general methods for solving them were further developed in the works of Russian mathematicians L. V. Kantorovich, V. S. Nemchinov and other mathematicians and economists. Also, many works by foreign and, above all, American scientists are devoted to linear programming methods.

The linear programming problem is to maximize (minimize) a linear function.

under restrictions

and all

Comment. Inequalities can also have opposite meanings. By multiplying the corresponding inequalities by (-1) one can always obtain a system of the form (*).

If the number of variables in the system of constraints and the objective function in the mathematical model of the problem is 2, then it can be solved graphically.

So, we need to maximize the function to a satisfying system of constraints.

Let us turn to one of the inequalities of the system of restrictions.

From a geometric point of view, all points that satisfy this inequality must either lie on a line or belong to one of the half-planes into which the plane of this line is divided. In order to find out, you need to check which of them contains a dot ().

Remark 2. If , then it is easier to take the point (0;0).

Non-negativity conditions also define half-planes, respectively, with boundary lines. We will assume that the system of inequalities is consistent, then the half-planes, intersecting, form a common part, which is a convex set and represents a set of points whose coordinates are a solution to this system - this is the set of admissible solutions. The set of these points (solutions) is called a solution polygon. It can be a point, a ray, a polygon, or an unbounded polygonal area. Thus, the task of linear programming is to find a point in the decision polygon at which the objective function takes on the maximum (minimum) value. This point exists when the solution polygon is not empty and the objective function on it is bounded from above (from below). Under the specified conditions, at one of the vertices of the solution polygon, the objective function takes on the maximum value. To determine this vertex, we construct a straight line (where h is some constant). Most often a straight line is taken. It remains to find out the direction of movement of this line. This direction is determined by the gradient (antigradient) of the objective function.

The vector at each point is perpendicular to the line, so the value of f will increase as the line moves in the direction of the gradient (decrease in the direction of the antigradient). To do this, draw straight lines parallel to the straight line, shifting in the direction of the gradient (anti-gradient).

We will continue these constructions until the line passes through the last vertex of the solution polygon. This point determines the optimal value.

So, finding a solution to a linear programming problem using the geometric method includes the following steps:

Lines are constructed, the equations of which are obtained by replacing the inequality signs in the restrictions with exact equality signs.

Find the half-planes defined by each of the constraints of the problem.

Find a solution polygon.

Build a vector.

They are building a straight line.

They construct parallel straight lines in the direction of the gradient or antigradient, as a result of which they find the point at which the function takes on the maximum or minimum value, or establish that the function is unbounded from above (from below) on the admissible set.

The coordinates of the maximum (minimum) point of the function are determined and the value of the objective function at this point is calculated.

Problem about rational nutrition (problem about food ration)

Formulation of the problem

The farm fattens livestock for commercial purposes. For simplicity, let’s assume that there are only four types of products: P1, P2, P3, P4; The unit cost of each product is equal to C1, C2, C3, C4, respectively. From these products you need to create a diet that should contain: proteins - at least b1 units; carbohydrates - at least b2 units; fat - at least b3 units. For products P1, P2, P3, P4, the content of proteins, carbohydrates and fats (in units per unit of product) is known and specified in the table, where aij (i=1,2,3,4; j=1,2,3) - some specific numbers; the first index indicates the product number, the second - the element number (proteins, carbohydrates, fats).

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Technical systems. The parameters of technical objects are moving objects, energy objects, chemical industry objects, mechanical engineering objects, household appliances and many others. Objects of technical systems are well studied in control theory.

Economic objects. Economic objects are: workshop, plant, enterprises of various industries. One of the variables in them are economic indicators - for example, profit.

Biological systems. Living systems maintain their vital functions thanks to the control mechanisms embedded in them.

Deterministic and stochastic systems

If the external influences applied to the system (controlling and disturbing) are certain known functions of time u=f(t). In this case, the state of the system described by ordinary differential equations at any time t can be unambiguously described by the state of the system at the previous point in time. Systems for which the state of the system is uniquely determined by the initial values ​​and can be predicted for any moment in time are called deterministic.

Stochastic systems are systems in which changes are random in nature. For example, the impact on the power system of various users. With random influences, data on the state of the system is not enough to make a prediction at a subsequent point in time.

Random influences can be applied to the system from outside, or arise inside some elements (internal noise). The study of systems in the presence of random influences can be carried out using conventional methods, minimizing the modeling step so as not to miss the influence of random parameters. Moreover, since the maximum value of a random variable is rare (normal distribution predominates in technology), the choice of the minimum step at most points in time will not be justified.

In the overwhelming majority of cases, when designing systems, it is not the maximum, but the most probable value of a random parameter. In this case, a more rational system is learned, anticipating in advance the deterioration of the system’s performance at certain periods of time. For example, installing cathodic protection.

Calculation of systems under random influences is carried out using special statistical methods. Estimates of random parameters based on many tests are introduced. For example, a surface map of the groundwater level in St. Petersburg.

The statistical properties of a random variable are determined by its distribution function or probability density.

Open and closed systems

The concept of an open system was introduced by L. von Bertalanffy. The main distinguishing features of open systems are the ability to exchange energy and information with the external environment. Closed (closed) systems are isolated from the external environment (with the accuracy accepted in the model).

Good and bad systems

Well organized systems. To present the analyzed object or process in the form of a “well-organized system” means to determine the elements of the system, their interrelationships, the rules for combining into larger components, i.e., to determine the connections between all components and goals of the system from the point of view of which the object is considered or for the sake of which a system is created. A problem situation can be described in the form of a mathematical expression connecting the goal with the means, i.e., in the form of an efficiency criterion, a criterion for the functioning of the system, which can be represented by a complex equation or system of equations. The solution of a problem, when presented in the form of a well-organized system, is carried out by analytical methods of a formalized representation of the system.

Examples of well-organized systems: the solar system, which describes the most significant patterns of planetary motion around the Sun; display of the atom as a planetary system consisting of a nucleus and electrons; description of the operation of a complex electronic device using a system of equations that takes into account the peculiarities of its operating conditions (presence of noise, instability of power supplies, etc.).

To display an object in the form of a well-organized system, it is necessary to highlight the components that are essential and not to take into account those that are relatively unimportant for this purpose of consideration: for example, when considering the solar system, do not take into account meteorites, asteroids and other elements of interplanetary space that are small compared to planets.

The description of an object in the form of a well-organized system is used in cases where it is possible to offer a deterministic description and experimentally prove the legitimacy of its application and the adequacy of the model to the real process. Attempts to apply the class of well-organized systems to represent complex multi-component objects or multi-criteria problems are not successful: they require an unacceptably large amount of time, are practically impossible to implement and are inadequate to the models used.

Poorly organized systems. When presenting an object as a “poorly organized or diffuse system,” the task is not to determine all the components taken into account, their properties and the connections between them and the goals of the system. The system is characterized by a certain set of macro-parameters and patterns, which are found on the basis of the study not of the entire object or class of phenomena, but on the basis of certain rules for selecting components that characterize the object or process under study. Based on such a sample study, characteristics or patterns (statistical, economic) are obtained and distributed to the entire system as a whole. In this case, appropriate reservations are made. For example, when statistical regularities are obtained, they are extended to the behavior of the entire system with a certain confidence probability.

The approach to displaying objects in the form of diffuse systems is widely used in: describing queuing systems, determining the number of staff in enterprises and institutions, studying documentary information flows in management systems, etc.

Self-organizing systems. Displaying an object as a self-organizing system is an approach that allows you to explore the least studied objects and processes. Self-organizing systems have the characteristics of diffuse systems: stochastic behavior, nonstationarity of individual parameters and processes. Added to this are signs such as unpredictability of behavior; the ability to adapt to changing environmental conditions, change the structure when the system interacts with the environment, while maintaining the properties of integrity; the ability to form possible behavior options and choose the best one from them, etc. Sometimes this class is divided into subclasses, highlighting adaptive or self-adjusting systems, self-healing, self-reproducing and other subclasses corresponding to various properties of developing systems.

Examples: biological organizations, collective behavior of people, organization of management at the level of an enterprise, industry, state as a whole, i.e. in those systems where there is necessarily a human factor.

When using the mapping of an object in the form of a self-organizing system, the tasks of determining goals and choosing means are usually separated. In this case, the task of choosing goals can, in turn, be described in the form of a self-organizing system, i.e. the structure of the functional part of the automated control system, the structure of the goals, the plan can be broken down in the same way as the structure of the supporting part of the automated control system (a complex of technical means of the automated control system) or organizational management system structure.

Most examples of the application of system analysis are based on the representation of objects in the form of self-organizing systems.