Methodological development on the topic: mathematical research in mathematics lessons.

In 1774 L. Euler showed that if the function y(x) gives a minimum to the line integral j = j[y(x), then it must satisfy certain differential equations, subsequently named Euler's equations. The discovery of this fact was an important achievement of mathematical modeling (construction of mathematical models). It became clear that the same mathematical model can be represented in two equivalent forms: in the form of a functional or in the form of the Euler differential equation (a system of differential equations). In this regard, the replacement of a differential equation by a functional is called inverse problem of the calculus of variations. Thus, the solution of the problem on the extremum of a functional can be considered in the same way as the solution of the Euler differential equation corresponding to this functional. Consequently, the mathematical formulation of the same physical problem can be represented either as a functional with the corresponding boundary conditions (the extremum of this functional provides a solution to the physical problem), or as the Euler differential equation corresponding to this functional with the same boundary conditions (integration of this equation provides problem solving).

The appearance in 1908 of the publication of W. Ritz, associated with the method of minimizing functionals, which was later called the Ritz method. This method is considered the classical variational method. Its main idea is that the desired function y = y(x) y delivering the functional (A ) With boundary conditions y(a) = A, y(b) = AT minimum value, sought as a series

where cj (i = 0, y) - unknown coefficients; (p/(d) (r = 0, P) - coordinate functions (algebraic or trigonometric polypom).

The coordinate functions are in such a form that they exactly satisfy the boundary conditions of the problem.

Substituting (c) into (A), after determining the derivatives of the functional J from the unknowns C, (r = 0, r) with respect to the latter, a system of algebraic linear equations is obtained. After determining the coefficients C, the solution of the problem in closed form is found from (c).

When using a large number of members of the series (c) (P- 5 ? °o) in principle it is possible to obtain a solution of the required accuracy. However, as show the calculations of specific tasks, the matrix of coefficients C, (r = 0, P) is a filled square matrix with a large spread of coefficients in absolute value. Such matrices are close to degenerate and, as a rule, are ill-conditioned. This is because they do not satisfy any of the conditions under which matrices can be well-conditioned. Let's look at some of these conditions.

• 1. Positive definiteness of the matrix (the terms on the main diagonal must be positive and maximal).
• 2. Band view of the matrix with respect to the main diagonal with the minimum width of the tape (the coefficients of the matrix outside the tape are equal to zero).
• 3. Symmetry of the matrix with respect to the main diagonal.

In this regard, with increasing approximations in the Ritz method, the condition number of the matrix, determined by the ratio of its maximum eigenvalue to the minimum, tends to an infinitely large value. And the accuracy of the solution obtained in this case, due to the rapid accumulation of rounding errors when solving large systems of algebraic linear equations, may not improve, but worsen.

Along with Ritz's method, the related method of Galerkin was developed. In 1913, I. G. Bubnov established that algebraic linear equations with respect to the unknowns C, (/ = 0, P) from (c) can be obtained without using a functional of the form (A). The mathematical formulation of the problem in this case includes a differential equation with appropriate boundary conditions. The decision, as in the Ritz method, is taken in the form (c). Owing to the special construction of the coordinate functions φ,(x), the solution (c) exactly satisfies the boundary conditions of the problem. To determine the unknown coefficients C, (r = 0, P) the discrepancy of the differential equation is compiled and the orthogonality of the discrepancy to all coordinate functions φ 7 Cr) (/ = i = 0, P). Determining the recipient while giving the integrals with respect to the unknown coefficients C, (G= 0, r) we obtain a system of algebraic linear equations, which completely coincides with the system of similar equations of the Ritz method. Thus, when solving the same problems using the same systems of coordinate functions, the Ritz and Bubnov-Galerkin methods lead to the same results.

Despite the identity of the results obtained, an important advantage of the Bubnov-Galerkin method compared to the Ritz method is that it does not require the construction of a variational analogue (functional) of a differential equation. Note that such an analogue cannot always be constructed. In connection with this, the Bubnov-Galerkin method can be used to solve problems for which classical variational methods are inapplicable.

Another method belonging to the variational group is the Kantorovich method. His hallmark is that as unknown coefficients in linear combinations of the form (c) are taken not constants, but functions depending on one of the independent variables of the problem (for example, time). Here, as in the Bubnov-Galerkin method, the discrepancy of the differential equation is compiled and the orthogonality of the discrepancy to all coordinate functions (py(dz) (j=i= 0, P). After defining integrals with respect to unknown functions fj(x) we will have a system of ordinary differential equations of the first order. Methods for solving such systems are well developed (there are standard computer programs).

One of the directions in solving boundary value problems is the joint use of exact (Fourier, integral transformations, etc.) and approximate (variational, weighted residuals, collocations, etc.) analytical methods. This integrated approach allows the best way to use the positive aspects of these two most important apparatuses of applied mathematics, since it becomes possible, without carrying out subtle and cumbersome mathematical calculations, in a simple form to obtain expressions equivalent to the main part of the exact solution, consisting of an infinite functional series. For practical calculations, as a rule, it is this partial sum of several terms that is used. When using such methods, in order to obtain more accurate results on the initial section of the parabolic coordinate, it is necessary to perform a large number of approximations. However, with a large P coordinate functions with neighboring indices lead to algebraic equations connected by an almost linear relationship. The matrix of coefficients in this case, being a filled square matrix, is close to degenerate and, as a rule, turns out to be ill-conditioned. And at P- 3? °° an approximate solution may not converge even to a weakly exact solution. Solving systems of algebraic linear equations with ill-conditioned matrices presents significant technical difficulties due to the rapid accumulation of rounding errors. Therefore, such systems of equations must be solved with high accuracy of intermediate calculations.

A special place among the approximate analytical methods that make it possible to obtain analytical solutions on the initial segment of the time (parabolic) coordinate is occupied by methods that use the concept temperature perturbation front. According to these methods, the entire process of heating or cooling bodies is formally divided into two stages. The first of them is characterized by a gradual spread of the temperature perturbation front from the surface of the body to its center, and the second is characterized by a change in temperature over the entire volume of the body up to the onset of steady state. Such a division of the thermal process in time into two stages makes it possible to carry out a step-by-step solution of the problems of non-stationary heat conduction and, as a rule, for each of the stages separately, as a rule, already in the first approximation, to find satisfactory in accuracy, fairly simple and convenient in engineering applications, calculation formulas. These methods also have a significant drawback, which consists in the need to a priori choose the coordinate dependence of the desired temperature function. Usually quadratic or cubic parabolas are accepted. This ambiguity of the solution gives rise to the problem of accuracy, since, assuming in advance one or another profile of the temperature field, each time we will obtain different final results.

Among the methods that use the idea of ​​a finite velocity of the temperature perturbation front, the most widely used is the integral method heat balance. With its help, the partial differential equation can be reduced to an ordinary one. differential equation with given initial conditions, the solution of which can quite often be obtained in a closed analytical form. The integral method, for example, can be used for the approximate solution of problems when the thermophysical properties are not constant, but are determined by a complex functional dependence, and problems in which, together with thermal conductivity, convection must also be taken into account. The integral method also has the disadvantage noted above - the a priori choice of the temperature profile, which gives rise to the problem of the uniqueness of the solution and leads to its low accuracy.

Numerous examples of the application of the integral method to solving problems of heat conduction are given in the work of T. Goodman. In this work, along with illustrations great opportunities its limitations are also shown. So, despite the fact that many problems are successfully solved by the integral method, there is a whole class of problems for which this method is practically not applicable. These are, for example, tasks with an impulsive change in input functions. The reason is that the temperature profile in the form of a quadratic or cubic parabola does not correspond to the true temperature profile for such applications. Therefore, if the true temperature distribution in the body under study takes the form of a nonmonotonic function, then under no circumstances can a satisfactory solution be obtained that is consistent with the physical meaning of the problem.

An obvious way to increase the accuracy of the integral method is to use polynomial temperature functions of a higher order. In this case, the main boundary conditions and smoothness conditions at the temperature perturbation front are not sufficient to determine the coefficients of such polynomials. In this regard, it becomes necessary to search for the missing boundary conditions, which, together with the given ones, would make it possible to determine the coefficients of the optimal temperature profile of a higher order, taking into account all the physical features of the problem under study. Such additional boundary conditions can be obtained from the main boundary conditions and the original differential equation by differentiating them at the boundary points in terms of the spatial coordinate and in terms of time.

When researching various tasks of heat transfer, it is assumed that the thermophysical properties do not depend on temperature, and linear conditions are taken as boundary conditions. However, if the temperature of the body varies over a wide range, then due to the dependence of thermophysical properties on temperature, the heat conduction equation becomes nonlinear. Its solution becomes much more complicated, and the known exact analytical methods turn out to be inefficient. The integral heat balance method makes it possible to overcome some of the difficulties associated with the nonlinearity of the problem. For example, with its help, a partial differential equation with nonlinear boundary conditions is reduced to an ordinary differential equation with given initial conditions, the solution of which can often be obtained in a closed analytical form.

It is known that exact analytical solutions are currently obtained only for problems in a simplified mathematical formulation, when many factors are not taken into account. important characteristics processes (nonlinearity, variability of properties and boundary conditions, etc.). All this leads to a significant deviation of mathematical models from real ones. physical processes flowing in specific power plants. In addition, the exact solutions are expressed by complex infinite functional series, which converge slowly in the vicinity of the boundary points and for small values ​​of the time coordinate. Such solutions are of little use for engineering applications, and especially in cases where the solution of a temperature problem is an intermediate step in solving some other problems (problems of thermal flexibility, inverse problems, control problems, etc.). In this regard, of great interest are the methods of applied mathematics listed above, which make it possible to obtain solutions, although approximate, but in an analytical form, with an accuracy sufficient in many cases for engineering applications. These methods make it possible to significantly expand the range of problems for which analytical solutions can be obtained in comparison with classical methods.

INTRODUCTION DISCIPLINE OPERATIONS RESEARCH AND WHAT IT DOES

The formation of operations research as an independent branch of applied mathematics dates back to the period of the 40s and 50s. The next decade and a half were marked by the wide application of the obtained fundamental theoretical results to various practical problems and the rethinking of the potential possibilities of the theory associated with this. As a result, operations research has acquired the features of a classical scientific discipline, without which basic economic education is unthinkable.

Turning to the tasks and problems that are the subject of operations research, one cannot help but recall the contribution made to their solution by representatives of the national scientific school, among which L. V. Kantorovich, who became a laureate in 1975 Nobel Prize for his work on the optimal use of resources in the economy.

The beginning of the development of operations research as a science is traditionally associated with the forties of the twentieth century. Among the first studies in this direction, the work of L. V. Kantorovich "Mathematical Methods of Organization and Planning of Production", published in 1939, can be named. tasks.

It should be noted that there is no rigid, well-established and generally accepted definition of the subject of operations research. Often, when answering this question, it is said that " operations research is a set of scientific methods for solving the problems of effective management of organizational systems.

Second definition: Operations research - this is the scientific preparation of the decision being made - this is a set of methods proposed for preparing and finding the most effective or most economical solutions.

The nature of the systems appearing in the above definition under the name "organizational" can be very different, and their general mathematical models are used not only in solving industrial and economic problems, but also in biology, sociological research and other practical areas. By the way, the very name of the discipline is associated with the use of mathematical methods to manage military operations.

Despite the variety of tasks of organizational management, when solving them, one can single out a certain general sequence of stages through which any operational research passes. As a rule, this is:

1. Statement of the problem.

2. Construction of a meaningful (verbal) model of the considered object (process). At this stage, the goal of managing the object is formalized, the selection of possible control actions that affect the achievement of the formulated goal, as well as the description of the system of restrictions on control actions.

3. Construction of a mathematical model, i.e., translation of the constructed verbal model into the form in which the mathematical apparatus can be used to study it.

4. Solving problems formulated on the basis of the constructed mathematical model.

5. Verification of the results obtained for their adequacy to the nature of the system under study, including the study of the influence of the so-called out-of-model factors, and possible correction of the original model.

6. Implementation of the obtained solution in practice.

The focus of this course is on questions related to the fourth point of the above diagram. This is done not because it is the most important, complex or interesting, but because the remaining points depend significantly on the specific nature of the system under study, which is why universal and meaningful recommendations cannot be formulated for the actions that should be carried out within their framework. .

In the most diverse areas of human activity, tasks similar to each other are encountered: the organization of production, the operation of transport, military operations, the placement of personnel, telephone communications, etc. The problems arising in these areas are similar in formulation, have a number of common features and are solved by similar methods.

Example :

Some purposeful event (a system of actions) is organized, which can be organized in one way or another. It is necessary to choose a certain solution from a number of possible options. Each option has advantages and disadvantages - it is not immediately clear which one is preferable. In order to clarify the situation and compare different options for a number of features, a series of mathematical calculations is organized. The results of the calculations show which option will stop.

Math modeling in operations research is, on the one hand, a very important and complex process, and on the other hand, a process that is practically not amenable to scientific formalization. It should be noted that repeated attempts to identify the general principles for creating mathematical models have led either to the declaration of recommendations of the most general nature, which are difficult to apply to solving specific problems, or, conversely, to the emergence of recipes that are actually applicable only to a narrow range of problems. Therefore, it is more useful to get acquainted with the technique of mathematical modeling using specific examples.

1) Enterprise supply plan.

There are a number of enterprises using various types of raw materials; there are a number of raw material bases. Bases are connected with enterprises by various means of communication ( railways, motor transport, water, air transport). Each transport has its own tariffs. It is required to develop such a plan for supplying enterprises with raw materials so that the needs for raw materials are met with minimal transportation costs.

2) Construction of a section of the highway.

A section of the railway line is under construction. We have a certain amount of resources at our disposal: people, equipment, etc. It is required to prioritize the work, distribute people and equipment along the sections of the track in such a way as to complete the construction in the shortest possible time.

A certain type of product is produced. To ensure high quality of products, it is required to organize a sampling control system: determine the size of the control lot, a set of tests, rejection rules, etc. It is required to provide a given level of product quality with minimal control costs.

4) Military operations.

The goal in this case is to destroy the enemy object.

Such problems are often encountered in practice. They have common features. Each task has a goal - these goals are similar; certain conditions are set - within the framework of these conditions, a decision must be made so that this event is the most profitable. In accordance with these common features common methods apply.

1. GENERAL CONCEPTS

1.1. Purpose and basic concepts in operations research

Operation - this is any system of actions (event), united by a single plan and directed towards the achievement of some goal. This is a managed event, that is, it depends on us how to choose some of the parameters that characterize its organization.

Each specific choice of parameters depending on us is called decision.

The purpose of operations research is a preliminary quantitative justification of optimal solutions.

Those parameters, the totality of which forms a solution, are called solution elements. Various numbers, vectors, functions, physical attributes, etc. can be used as elements of the solution.

Example : transportation of homogeneous cargo.

There are departure points: BUT 1 , BUT 2 , BUT 3 ,…, BUT m .

There are destinations: AT 1 , AT 2 , AT 3 ,…, AT n .

The elements of the solution here will be numbers x ij , showing how many goods will be sent from the i-th point of departure to j-th destination.

The totality of these numbers: x 11 , x 12 , x 13 ,…, x 1 m ,…, x n 1 , x n 2 ,…, x nm forms a solution.

In order to compare different options with each other, it is necessary to have some kind of quantitative criterion - an indicator of efficiency ( W). This indicator is called target function.

This indicator is chosen so that it reflects the target orientation of the operation. When choosing a solution, we strive for this indicator to tend to a maximum or a minimum. If W is income, then W max; and if W is the flow rate, then W min.

If the choice depends on random factors (weather, equipment failure, supply and demand fluctuations), then the average value is chosen as the performance indicator - the mathematical expectation - .

As an indicator of efficiency, the probability of achieving the goal is sometimes chosen. Here, the purpose of the operation is accompanied by random factors and works according to the YES-NO scheme.

To illustrate the principles of choosing a performance indicator, let's return to the examples considered earlier:

1) Enterprise supply plan.

The performance indicator is visible in the target. R– number – transportation cost, . In this case, all restrictions must be met.

2) Construction of a section of the highway.

In the task big role random factors play. The average expected construction completion time is chosen as a performance indicator.

3) Selective control of production.

The natural performance indicator suggested by the problem statement is the average expected control cost per unit of time, assuming that the system controls the achievement of a given level of quality.

Accompanied by physical or mathematical modeling. Physical modeling ... layouts and their laborious study. Mathematical simulation is carried out using ... for simulation it is necessary to do the following operations: 1. enter the menu...

Let's compare the methodology of applying mathematics in practical research with the methodology of other natural sciences. Such sciences as physics, chemistry, biology directly study the real object itself (perhaps on a reduced scale and in laboratory conditions). Scientific results, after the necessary verification, can also be directly applied in practice. Mathematics does not study the objects themselves, but their models. The description of the object and the formulation of the problem are translated from ordinary language into the "language of mathematics" (formalized), resulting in a mathematical model. Further, this model is studied as a mathematical problem. The obtained scientific results are not immediately applied in practice, as they are formulated in mathematical language. Therefore, the reverse process is carried out - a meaningful interpretation (in the language of the original problem) of the obtained mathematical results. Only after that the question of their application in practice is decided.

An integral part of the methodology of applied mathematics is a comprehensive analysis of a real problem that precedes it. mathematical modeling. In general, a system analysis of the problem involves the following steps:

Humanitarian (pre-mathematical) analysis of the problem;

· mathematical study of the problem;

application of the obtained results in practice.

Conducting such a system analysis of each specific problem should be carried out research group, which includes economists (as problem makers or customers), mathematicians, lawyers, sociologists, psychologists, ecologists, etc. Moreover, mathematicians, as the main researchers, should participate not only in the “solution” of the problem, but also in its formulation, as well as in putting the results into practice.

To carry out mathematical studies of an economic problem, the following main steps are required:

1. study of the subject area and determination of the purpose of the study;

2. formulation of the problem;

3. data collection (statistical, expert and others);

4. building a mathematical model;

5. choice (or development) of a computational method and construction of an algorithm for solving the problem;

6. algorithm programming and program debugging;

7. checking the quality of the model on a control example;

8. implementation of the results in practice.

Stages 1 -3 belong to the pre-mathematical part of the study. Subject area should be thoroughly studied by economists themselves so that they, as customers, can clearly formulate the problem and set goals for researchers. Researchers should be provided with all the necessary documentary and statistical data in an exhaustive volume. Mathematicians organize, store, analyze and process data provided to them in a convenient (electronic) form by customers.

Stages 4 -7 belong to the mathematical part of the research. The result of this stage is the formulation of the original problem in the form of a strict mathematical problem. A mathematical model can rarely be "picked up" from among the available, well-known models (Fig. 1.1). The process of selecting model parameters in such a way that it corresponds to the object under study is called model identification. Based on the nature of the obtained model (task) and the purpose of the study, either a known method is chosen, or a known method is adapted (modified), or a new one is developed. After that, an algorithm (the procedure for solving the problem) and a computer program are compiled. The results obtained using this program are analyzed: they solve test problems, introduce the necessary changes and corrections to the algorithm and program.

If for "pure" mathematics it is traditional to select a mathematical model once and formulate assumptions at the very beginning of the study, then in applied work it is often useful to return to the model and make corrections to it after the first round of trial calculations has already been made. Moreover, it often turns out to be fruitful to compare models when the same phenomenon is described not by one, but by several models. If the conclusions turn out to be (approximately) the same for different models, different research methods, this is evidence of the correctness of the calculations, the adequacy of the model to the object itself, and the objectivity of the recommendations issued.

The final stage 8 carried out jointly by customers and model developers.

The results of mathematical (as well as any scientific) research are only a recommendation for use in practice. The final decision on this issue - whether to apply the model or not - depends on the customer, i.e., on the person responsible for the outcome and for the consequences that the application of the recommended results will lead to.

To build a mathematical model of a specific economic task (problem), it is recommended to perform the following sequence of work:

1. definition of known and unknown quantities, as well as existing conditions and prerequisites (what is given and what needs to be found?);

2. detection critical factors Problems;

3. identification of controlled and unmanaged parameters;

4. mathematical description by means of equations, inequalities, functions and other relationships between the elements of the model (parameters, variables), based on the content of the problem under consideration.

The known parameters of the problem relative to its mathematical model are considered external(given a priori, i.e. before building the model). In the economic literature they are called exogenous variables. The value of initially unknown variables is calculated as a result of studying the model, therefore, in relation to the model, they are considered internal. In the economic literature they are called endogenous variables.

AT § 2 The most important factors are those that play a significant role in the task itself and which, one way or another, affect the final result. AT § 3 controllable are those task parameters that can be given arbitrary numerical values based on the conditions of the task; Unmanaged parameters are those whose value is fixed and cannot be changed.

From the point of view of purpose, one can distinguish descriptive models and decision making models. Descriptive models reflect the content and main properties economic objects as such. With their help, the numerical values ​​of economic factors and indicators are calculated.

Decision models help find the best options targets or management decisions. Among them, the least complex are optimization models, which describe (simulate) tasks of the planning type, and the most complex are game models that describe problems of a conflicting nature, taking into account the intersection of various interests. These models differ from descriptive ones in that they have the ability to choose the values ​​of control parameters (which is not the case in descriptive models).

Examples of compiling mathematical models

Example 1.1. Let some economic region produce several types of products exclusively on its own and only for the population of this region. It is assumed that the technological process has been worked out, and the demand of the population for these goods has been studied. It is necessary to determine the annual volume of output of products, taking into account the fact that this volume must provide both final and industrial consumption.

Let's make a mathematical model of this problem. By condition given: types of products, demand for them and the technological process; it is required to find the volume of output of each type of product Let us designate the known values: - demand of the population for the -th product; - the amount of the i-th product required to produce a unit of the -th product according to this technology . Let's designate unknown quantities: - volume of output of the -th product . Aggregate is called the demand vector, the numbers are called technological coefficients, and the set - release vector. According to the condition of the problem, the vector is divided into two parts: for final consumption (vector ) and for reproduction (vector ). Let's calculate that part of the vector that goes to reproduction. By virtue of the notation for the production of the quantity of the -th good, the quantity of the -th good goes. Then the sum shows the value of the -th product, which is needed for the entire output . Therefore, the equality must hold:

Generalizing this reasoning to all types of products, we arrive at the desired model:

Solving the resulting system of linear equations with respect to find the required output vector.

In order to write this model in a more compact (vector) form, we introduce the notation:

square matrix A (size ) is called a technological matrix. Obviously, the model can be written as: or

We have obtained the classical “Cost-output” model, the author of which is the famous American economist V. Leontiev.

Example 1.2. The refinery has two grades of oil: a grade in the amount of 10 units, a grade - 15 units. When processing oil, two materials are obtained: gasoline () and fuel oil (). There are three options for the processing technology:

I: 1 unit BUT+ 2 units AT gives 3 units. B+ 2 units M;

II:2 units BUT+ 1 unit AT gives 1 unit. B+ 5 units M;

III:2 units BUT+ 2 units AT gives 1 unit. B+ 2 units M.

The price of gasoline is $10 per unit, fuel oil is $1 per unit. It is required to determine the most advantageous combination technological processes processing of the available amount of oil.

Before modeling, we clarify the following points. It follows from the conditions of the problem that the “profitability” of the technological process for the plant should be understood in the sense of obtaining the maximum income from the sale of its finished products (gasoline and fuel oil). In this regard, it is clear that the "choice (making) decision" of the plant is to determine which technology and how many times to apply. It is obvious that such options enough.

Let's denote the unknown values: - the amount of use of the -th technological process . Other parameters of the model (reserves of oil grades, prices of gasoline and fuel oil) known.

Then one specific decision of the plant is reduced to the choice of one vector , for which the plant's revenue is equal to dollars. Here, 32 dollars is the income received from one application of the first technological process (10 dollars 3 units. B+ $1 2 units M= $32). Coefficients 15 and 12 have a similar meaning for the second and third technological processes, respectively. Accounting for the oil reserve leads to the following conditions:

for variety BUT: ,

for variety AT: ,

where in the first inequality the coefficients 1, 2, 2 are the consumption rates of oil grade BUT for single use process technology I, II, III respectively. The coefficients of the second inequality have a similar meaning for the oil grade AT.

Mathematical model in general looks like:

Find a vector such that

maximize

when the conditions are met:

,

,

.

The abbreviated form of this entry is:

under restrictions

, (1.4.2)

,

We got the so-called linear programming problem. Model (1.4.2.) is an example of an optimization model of a deterministic type (with well-defined elements).

Example 1.3. The investor needs to determine the best set of stocks, bonds and other valuable papers to acquire them for a certain amount in order to obtain a certain profit with minimal risk to themselves. The return on every dollar invested in a security of the -th type is characterized by two indicators: the expected return and the actual return. It is desirable for an investor that the expected profit per one dollar of investment should not be lower than a given value for the entire set of securities. Note that for the correct modeling of this problem, a mathematician requires certain basic knowledge in the field of portfolio theory of securities. Let us designate the known parameters of the problem: - the number of varieties of securities; - actual profit (random number) from the -th type of security - expected profit from the -th type of security. Let's designate unknown sizes: - the means allocated for acquisition of securities of a type . By virtue of the notation, the entire invested amount is defined as . To simplify the model, we introduce new quantities

Thus, is the share of all funds allocated for the purchase of securities of the type. It's obvious that . It can be seen from the condition of the problem that the goal of the investor is to achieve a certain level of profit with minimal risk. Essentially, risk is a measure of deviation of actual profit from expected one. Therefore, it can be identified with the covariance

profits for securities of type and type . Here M- designation mathematical expectation. The mathematical model of the original problem has the form:

(1.4.3)

We have obtained the well-known Markowitz model for optimizing the structure of a securities portfolio. Model (1.4.3.) is an example of an optimization model of a stochastic type (with elements of randomness).

Example 1.4. On the base trade organization there are types of one of the products of the assortment minimum. Only one of the types of this product must be delivered to the store. It is required to choose the type of goods that it is advisable to bring to the store. If a product of the type is in demand, then the store will make a profit from its sale, if it is not in demand, a loss.