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 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 level surface map groundwater 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. Main features open systems- 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.

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 a basis for adoption. management decision when searching optimal option. How justified are these decisions, are they the best possible, are all the factors that determine them taken into account and weighed? optimal solution, what is the criterion to determine that this decision really the best - this is the range of questions that have 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. Yes, on manufacturing process influence a large number of factors that 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, often when talking about mathematical modeling economic processes, they 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 certain factors with an effective indicator as functional dependence, i.e. in deterministic models, the effective indicator of the model is presented in the form of a product, quotient, algebraic sum of factors, or in the form of any other function. This type mathematical models are 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 proportions are 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 mathematical analysis 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.

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

Systematic research of linear programming problems and development common methods their solutions 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.

, Where

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 satisfying this inequality must either lie on the 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’s easier to take the point (0;0).

Conditions for non-negativity

also define half-planes corresponding to the 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). At 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.

at every point perpendicular to the line

, therefore the value of f will increase as the line moves in the direction of the gradient (decreases in the direction of the antigradient). To do this, parallel to the straight line

draw straight lines, 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 vector

Build a straight line

Construct parallel 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 unboundedness from above (from below) of the function 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 certain numbers; the first index indicates the product number, the second - the element number (proteins, carbohydrates, fats).

Stochastic models

As mentioned above, stochastic models are probabilistic models. Moreover, as a result of calculations, it is possible to say with a sufficient degree of probability what the value of the analyzed indicator will be if the factor changes. The most common application of stochastic models is forecasting.

Stochastic modeling is, to a certain extent, a complement and deepening of deterministic factor analysis. IN factor analysis these models are used for three main reasons:

it is necessary to study the influence of factors by which it is impossible to construct a strictly determined factor model(for example, the level of financial leverage);

it is necessary to study the influence of complex factors that cannot be combined in the same strictly determined model;
it is necessary to study the influence of complex factors that cannot be expressed by one quantitative indicator (for example, the level of scientific and technological progress).

In contrast to the strictly deterministic approach, the stochastic approach requires a number of prerequisites for implementation:

the presence of a population;
sufficient volume of observations;
randomness and independence of observations;
uniformity;
the presence of a distribution of characteristics close to normal;
the presence of a special mathematical apparatus.

The construction of a stochastic model is carried out in several stages:

qualitative analysis (setting the purpose of the analysis, defining the population, determining the effective and factor characteristics, choosing the period for which the analysis is carried out, choosing the analysis method);
preliminary analysis of the simulated population (checking the homogeneity of the population, excluding anomalous observations, clarifying the required sample size, establishing distribution laws for the indicators being studied);
construction of a stochastic (regression) model (clarification of the list of factors, calculation of estimates of the parameters of the regression equation, enumeration of competing model options);
assessing the adequacy of the model (checking the statistical significance of the equation as a whole and its individual parameters, checking the compliance of the formal properties of the estimates with the objectives of the study);
economic interpretation and practical use models (determining the spatio-temporal stability of the constructed relationship, assessing the practical properties of the model).

Basic concepts of correlation and regression analysis

Correlation analysis - a set of methods of mathematical statistics that make it possible to estimate coefficients characterizing the correlation between random variables and test hypotheses about their values based on the calculation of their sample analogues.

Correlation analysis is a method of processing statistical data that involves studying coefficients (correlation) between variables.

Correlation(which is also called incomplete, or statistical) manifests itself on average, for mass observations, when the given values of the dependent variable correspond to a certain number of probable values of the independent variable. The explanation for this is the complexity of the relationships between the analyzed factors, the interaction of which is influenced by unaccounted random variables. Therefore, the connection between the signs appears only on average, in the mass of cases. In a correlation connection, each argument value corresponds to function values randomly distributed in a certain interval.

In the most general view the task of statistics (and, accordingly, economic analysis) in the field of studying relationships consists of quantitatively assessing their presence and direction, as well as characterizing the strength and form of influence of some factors on others. To solve it, two groups of methods are used, one of which includes methods of correlation analysis, and the other - regression analysis. At the same time, a number of researchers combine these methods into correlation-regression analysis, which has some basis: the presence of a number of general computational procedures, complementarity in the interpretation of results, etc.

Therefore, in this context, we can talk about correlation analysis in a broad sense - when the relationship is comprehensively characterized. At the same time, there is a correlation analysis in the narrow sense - when the strength of the connection is examined - and regression analysis, during which its form and the impact of some factors on others are assessed.

The tasks themselves correlation analysis are reduced to measuring the closeness of the connection between varying characteristics, determining unknown causal relationships and assessing the factors that have the greatest influence on the resulting characteristic.

Tasks regression analysis lie in the area of establishing the form of the dependence, determining the regression function, and using an equation to estimate the unknown values of the dependent variable.

The solution to these problems is based on appropriate techniques, algorithms, and indicators, which gives grounds to talk about the statistical study of relationships.

It should be noted that traditional methods correlations and regressions are widely represented in various kinds statistical software packages for computers. The researcher can only prepare the information correctly, select a software package that meets the analysis requirements and be ready to interpret the results obtained. There are many algorithms for calculating communication parameters, and at present it is hardly advisable to carry out such complex look manual analysis. Computational procedures are of independent interest, but knowledge of the principles of studying relationships, capabilities and limitations of certain methods of interpreting results is prerequisite research.

Methods for assessing the strength of a connection are divided into correlation (parametric) and nonparametric. Parametric methods are based on the use, as a rule, of estimates of the normal distribution and are used in cases where the population under study consists of values that obey the law of normal distribution. In practice, this position is most often accepted a priori. Actually, these methods are parametric and are usually called correlation methods.

Nonparametric methods do not impose restrictions on the distribution law of the studied quantities. Their advantage is the simplicity of calculations.

Autocorrelation- statistical relationship between random variables from the same series, but taken with a shift, for example, for random process- with a time shift.

Pairwise correlation

The simplest technique for identifying the relationship between two characteristics is to construct correlation table:

\Y\X\	Y 1	Y2	...	Y z	Total	Y i
X 1	f 11		...	f 1z
X 1	f 21		...	f 2z
...	...	...	...	...	...	...
Xr	f k1	k2	...	f kz
Total			...		n
			...			-

The grouping is based on two characteristics studied in relationship - X and Y. Frequencies f ij show the number of corresponding combinations of X and Y.

If f ij are located randomly in the table, we can talk about the lack of connection between the variables. In the case of the formation of any characteristic combination f ij, it is permissible to assert a connection between X and Y. Moreover, if f ij is concentrated near one of the two diagonals, a direct or inverse linear connection takes place.

A visual representation of the correlation table is correlation field. It is a graph where X values are plotted on the abscissa axis, Y values are plotted on the ordinate axis, and the combination of X and Y is shown with dots. By the location of the dots and their concentrations in a certain direction, one can judge the presence of a connection.

Correlation field is called a set of points (X i, Y i) on the XY plane (Figures 6.1 - 6.2).

If the points of the correlation field form an ellipse, the main diagonal of which has a positive angle of inclination (/), then a positive correlation occurs (an example of such a situation can be seen in Figure 6.1).

If the points of the correlation field form an ellipse, the main diagonal of which has a negative angle of inclination (\), then a negative correlation occurs (an example is shown in Figure 6.2).

If there is no pattern in the location of the points, then they say that in this case there is a zero correlation.

In the results of the correlation table, two distributions are given in rows and columns - one for X, the other for Y. Let us calculate the average value of Y for each Xi, i.e. , How

The sequence of points (X i, ) gives a graph that illustrates the dependence of the average value of the effective attribute Y on the factor X, – empirical regression line, clearly showing how Y changes as X changes.

Essentially, both the correlation table, the correlation field, and the empirical regression line already preliminarily characterize the relationship when the factor and resultant characteristics are selected and it is necessary to formulate assumptions about the form and direction of the relationship. At the same time, quantitative assessment of the tightness of the connection requires additional calculations.

January 23, 2017

The stochastic model describes a situation where there is uncertainty. In other words, the process is characterized by some degree of randomness. The adjective “stochastic” itself comes from Greek word"guess". Because uncertainty is a key characteristic Everyday life, then such a model can describe anything.

However, each time we use it, we will get a different result. Therefore, deterministic models are more often used. Although they are not as close as possible to the real state of affairs, they always give the same result and make it easier to understand the situation, simplify it by introducing a set of mathematical equations.

Main features

A stochastic model always includes one or more random variables. She strives to reflect real life in all its manifestations. Unlike a deterministic model, a stochastic one does not have the goal of simplifying everything and reducing it to known values. Therefore, uncertainty is its key characteristic. Stochastic models are suitable for describing anything, but they all have the following common features:

Any stochastic model reflects all aspects of the problem it was created to study.
The outcome of each event is uncertain. Therefore, the model includes probabilities. The correctness of the overall results depends on the accuracy of their calculation.
These probabilities can be used to predict or describe the processes themselves.

Deterministic and stochastic models

For some, life seems like a series of random events, for others - processes in which the cause determines the effect. In fact, it is characterized by uncertainty, but not always and not in everything. Therefore, it is sometimes difficult to find clear differences between stochastic and deterministic models. Probabilities are a fairly subjective indicator.

For example, consider a coin toss situation. At first glance, it seems that the probability of landing “tails” is 50%. Therefore, a deterministic model must be used. However, in reality it turns out that a lot depends on the sleight of hand of the players and the perfection of balancing the coin. This means that you need to use a stochastic model. There are always parameters that we don't know. IN real life The cause always determines the effect, but there is also a certain degree of uncertainty. The choice between using deterministic and stochastic models depends on what we are willing to sacrifice - ease of analysis or realism.

Video on the topic

In chaos theory

IN Lately the concept of which model is called stochastic has become even more blurred. This is due to the development of the so-called chaos theory. It describes deterministic models that can produce different results with slight changes in the initial parameters. This is like an introduction to uncertainty calculation. Many scientists even admitted that this is already a stochastic model.

Lothar Breuer explained everything gracefully with poetic imagery. He wrote: “A mountain stream, a beating heart, an epidemic of smallpox, a column of rising smoke - all this is an example of a dynamic phenomenon that sometimes seems to be characterized by chance. In reality, such processes are always subordinated a certain order, which scientists and engineers are just beginning to understand. This is the so-called deterministic chaos." The new theory sounds very plausible, which is why many modern scientists are its supporters. However, it still remains poorly developed and is quite difficult to apply in statistical calculations. Therefore, stochastic or deterministic models are often used.

Construction

The stochastic mathematical model begins with the choice of a space of elementary outcomes. This is what statistics call a list of possible results of the process or event being studied. The researcher then determines the probability of each of the elementary outcomes. This is usually done based on a specific methodology.

However, probabilities are still a rather subjective parameter. The researcher then determines which events seem most interesting to solve the problem. After that, he simply determines their probability.

Example

Let's consider the process of constructing the simplest stochastic model. Let's say we're rolling a dice. If “six” or “one” comes up, our winnings will be ten dollars. The process of building a stochastic model in this case will look like this:

Let us define the space of elementary outcomes. The die has six sides, so the rolls can be “one”, “two”, “three”, “four”, “five” and “six”.
The probability of each outcome will be 1/6, no matter how many times we roll the dice.
Now we need to determine the outcomes we are interested in. This is the fall of the edge with the number “six” or “one”.
Finally, we can determine the probability of the event we are interested in. It is 1/3. We sum up the probabilities of both elementary events of interest to us: 1/6 + 1/6 = 2/6 = 1/3.

Concept and result

Stochastic modeling is often used in gambling. But it is also indispensable in economic forecasting, as they allow us to understand the situation more deeply than deterministic ones. Stochastic models in economics are often used when making investment decisions. They allow you to make assumptions about the profitability of investments in certain assets or groups of assets.

Modeling makes financial planning more effective. With its help, investors and traders optimize the allocation of their assets. Using stochastic modeling always has benefits in the long run. In some industries, refusal or inability to apply it can even lead to bankruptcy of the enterprise. This is due to the fact that in real life new important parameters appear daily and if left unchecked can have catastrophic consequences.

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.

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).

Stochastic model in economics. Deterministic and stochastic models

Deterministic and stochastic systems

Open and closed systems

Good and bad systems

Stochastic models

Main features

Deterministic and stochastic models

Video on the topic

In chaos theory

Construction

Example

Concept and result