Mathematical modeling in economics. Chelyabinsk Institute of Communications

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Content
Introduction
1. Mathematical models

1.1 Classification of economic and mathematical models

2. Optimization modeling

2.1 Linear programming

2.1.1 Linear programming as a tool for mathematical modeling of the economy
2.1.2 Examples of linear programming models
2.2.3 Optimal resource allocation

Conclusion

Introduction

Modern mathematics is characterized by intensive penetration into other sciences, this process is largely due to the division of mathematics into a number of independent areas. Mathematics has become for many branches of knowledge not only a tool for quantitative calculation, but also a method of precise research and a means of extremely clear formulation of concepts and problems. Without modern mathematics, with its developed logical and computing apparatus, progress in various fields would not be possible. human activity. economic mathematical linear modeling

Economics as a science about the objective reasons for the functioning and development of society uses a variety of quantitative characteristics, and therefore has absorbed big number mathematical methods.

The relevance of this topic lies in the fact that in the modern economy optimization methods are used, which form the basis of mathematical programming, game theory, network planning, queuing theory and other applied sciences.

The study of economic applications of mathematical disciplines, which form the basis of current economic mathematics, allows you to acquire some skills in solving economic problems and expand knowledge in this area.

The purpose of this work is to study some optimization methods used in solving economic problems.

1. Mathematical models

Mathematical models in economics. The widespread use of mathematical models is an important direction in improving economic analysis. Concretization of data or their presentation in the form of a mathematical model helps to choose the least labor-intensive solution path, increases the efficiency of analysis.

All economic problems solved using linear programming are distinguished by alternative solutions and certain limiting conditions. To solve such a problem means to choose the best, optimal one from all feasible (alternative) options. The importance and value of using the linear programming method in economics lies in the fact that the optimal option is selected from a sufficiently significant number of alternative options.

The most significant points in the formulation and solution of economic problems in the form of a mathematical model are:

· the adequacy of the economic and mathematical model of reality;

analysis of regularities corresponding to this process;

Determination of methods by which it is possible to solve the problem;

Analysis of the results obtained or summing up.

Under the economic analysis is understood, first of all, factor analysis.

Let y=f(x i) be some function that characterizes the change in an indicator or process; x 1 ,x 2 ,…,x n - factors on which the function y=f(x i) depends. A functional deterministic relationship of the indicator y with a set of factors is given. Let the indicator y change over the analyzed period. It is required to determine what part of the numerical increment of the function y=f(x 1 ,x 2 ,…,x n) is due to the increment of each factor.

It can be distinguished in economic analysis - analysis of the impact of labor productivity and the number of employees on the volume of output; analysis of the impact of the value of profit of fixed production assets and normalized working capital on the level of profitability; analysis of the impact of borrowed funds on the flexibility and independence of the enterprise, etc.

In economic analysis, in addition to tasks that boil down to breaking it down into its component parts, there is a group of tasks where it is required to functionally link a number of economic characteristics, i.e. build a function that contains the main quality of all the considered economic indicators.

In this case, an inverse problem is posed - the so-called inverse problem. factor analysis.

Let there be a set of indicators x 1 ,x 2 ,…,x n characterizing some economic process F. Each of the indicators characterizes this process. It is required to construct a function f(x i) of the process F change, containing the main characteristics of all indicators x 1 ,x 2 ,…,x n

The main point in economic analysis is the definition of a criterion by which different solutions will be compared.

Mathematical models in management. Decision making plays an important role in all spheres of human activity. To set up a decision-making problem, two conditions must be met:

the presence of a choice;

selection of an option according to a certain principle.

There are two principles for choosing a solution: volitional and criterial.

Volitional choice, the most commonly used, is used in the absence of formalized models as the only possible one.

The criterion choice consists in accepting a certain criterion and comparing possible options according to this criterion. The variant for which the accepted criterion makes the best decision is called optimal, and the problem of making the best decision is called an optimization problem.

The optimization criterion is called the objective function.

Any problem, the solution of which is reduced to finding the maximum or minimum of the objective function, is called an extremal problem.

Management tasks are related to finding the conditional extremum of the objective function under known restrictions imposed on its variables.

When solving various optimization problems, the quantity or cost of manufactured products, production costs, the amount of profit, etc. are taken as the objective function. Restrictions usually concern human material, financial resources.

Management optimization tasks, different in their content and implemented using standard software products, correspond to one or another class of economic and mathematical models.

Consider the classification of some of the main optimization tasks implemented by management in production.

Classification of optimization problems by control function:

Control function	Optimization problems	Class of economic-mathematical models
Technical and organizational preparation of production	Modeling the composition of products; Optimization of the composition of grades, charge, mixtures; Optimization of cutting sheet material, rolled products; Optimization of resource allocation in network models of work packages; Optimization of layouts of enterprises, industries and equipment; Optimization of the product manufacturing route; Optimization of technologies and technological regimes.	graph theory Discrete Programming Linear programming Network planning and management Simulation Dynamic programming Nonlinear programming
Technical and economic planning	Construction of a master plan and forecasting of enterprise development indicators; Optimization of the portfolio of orders and production program; Optimization of the distribution of the production program for planning periods.	Matrix balance models “Input-output” Correlation- regression analysis Extrapolation of trends Linear programming
Operational management of the main production	Optimization of calendar and planning standards; Calendar tasks; Optimization of standard plans; Optimization of short-term production plans.	Nonlinear programming Simulation Linear programming Integer programming

Table 1.

The combination of different elements of the model leads to different classes of optimization problems:

Table 2.

1.1 Classification of economic and mathematical models

There is a significant variety of types, types of economic and mathematical models required for use in the management of economic objects and processes. Economic and mathematical models are divided into: macroeconomic and microeconomic, depending on the level of the modeled control object, dynamic, which characterize changes in the control object over time, and static, which describe the relationship between different parameters, indicators of the object at that time. Discrete models display the state of the control object at separate, fixed points in time. Imitation is called economic and mathematical models used to simulate controlled economic objects and processes using information and computer technology. Type mathematical apparatus used in the models, economic-statistical, linear and non-linear programming, matrix models, network models.

factor models. The group of economic-mathematical factor models includes models that, on the one hand, include economic factors on which the state of the managed economic object depends, and on the other hand, parameters of the state of the object that depend on these factors. If the factors are known, then the model allows you to determine the desired parameters. Factor models are most often provided by mathematically simple linear or static functions that characterize the relationship between factors and the parameters of an economic object that depend on them.

balance models. Balance models, both statistical and dynamic, are widely used in economic and mathematical modeling. The creation of these models is based on the balance method - a method of mutual comparison of material, labor and financial resources and the needs for them. Describing the economic system as a whole, its balance model is understood as a system of equations, each of which expresses the need for a balance between the amount of production produced by individual economic objects and the total need for this product. With this approach, the economic system consists of economic objects, each of which produces a certain product. If instead of the concept of "product" we introduce the concept of "resource", then the balance model must be understood as a system of equations that satisfy the requirements between a certain resource and its use.

The most important types of balance models:

· Material, labor and financial balances for the economy as a whole and its individual sectors;

· Intersectoral balances;

· Matrix balance sheets of enterprises and firms.

optimization models. A large class of economic and mathematical models is formed by optimization models that allow you to choose the best optimal option from all solutions. In the mathematical content, optimality is understood as the achievement of an extremum of the optimality criterion, also called the objective function. Optimization models are most often used in problems of finding the best way to use economic resources, which allows you to achieve the maximum target effect. Mathematical programming was formed on the basis of solving the problem of optimal cutting of plywood sheets, which provides the most full use material. Having posed such a problem, the famous Russian mathematician and economist academician L.V. Kantorovich was recognized as worthy of the Nobel Prize in Economics.

2. Optimization modeling

2.1 Linear programming

2.1.1 Linear programming as a tool for mathematical modeling of the economy

Properties research common system linear inequalities has been conducted since the 19th century, and the first optimization problem with a linear objective function and linear constraints was formulated in the 30s of the 20th century. One of the first foreign scientists who laid the foundations of linear programming is John von Neumann, a well-known mathematician and physicist who proved the main theorem about matrix games. Among domestic scientists, a great contribution to the theory of linear optimization was made by Nobel Prize winner L.V. Kantorovich, N.N. Moiseev, E.G. Holstein, D.B. Yudin and many others.

Linear programming is traditionally considered one of the branches of operations research, which studies methods for finding the conditional extremum of functions of many variables.

In classical mathematical analysis, the general formulation of the problem of determining a conditional extremum is studied, however, due to the development industrial production, transport, agro-industrial complex, banking sector traditional results of mathematical analysis was not enough. The needs of practice and the development of computer technology have led to the need to determine the optimal solutions in the analysis of complex economic systems. The main tool for solving such problems is mathematical modeling, i.e. a formalized description of the process under study and its study with the help of a mathematical apparatus.

The art of mathematical modeling is to take into account the widest possible range of factors influencing the behavior of an object, while using as simple as possible relationships. It is in connection with this that the modeling process often has a multi-stage character. First, a relatively simple model is built, then its study is carried out, which makes it possible to understand which of the integrating properties of the object are not captured by this formal scheme, after which, due to the complication of the model, its greater adequacy to reality is ensured. At the same time, in many cases, the first approximation to reality is a model in which all dependencies between the variables characterizing the state of the object are linear. Practice shows that a significant number economic processes described quite well linear models, and consequently, linear programming as an apparatus that allows you to find conditional extremum on the set defined by linear equations and inequalities plays an important role in the analysis of these processes.

2.1.2 Examples of linear programming models

Below we will consider several situations, the study of which is possible using linear programming tools. Since the main indicator in these situations is economic - cost, the corresponding models are economic-mathematical.

The problem of cutting materials. The material of one sample is supplied for processing in the amount of d units. It is required to make from it k different components in quantities proportional to the numbers a 1 ,..., a k. Each unit of material can be cut in n different ways, while using the i-th method (i=1,…,n) gives b ij , units of the jth product (j = 1,...,k).

It is required to find a cutting plan that provides the maximum number of sets.

The economic-mathematical model of this problem can be formulated as follows. Let x i be the number of units of materials cut i-th way, and x is the number of manufactured sets of products.

Given that the total amount of material is equal to the sum of its units cut in various ways, we get:

The completeness condition is expressed by the equations:

It's obvious that

x i 0 (i=1,…,n)(3)

The goal is to determine such a solution X= (x 1 ,…,x n) that satisfies the constraints (1)-(3), in which the function F = x takes the maximum value. Let's illustrate the considered problem with the following example. For the manufacture of beams with a length of 1.5 m, 3 m and 5 m in a ratio of 2:1:3, 200 logs with a length of 6 m are fed to the cut. Determine the cutting plan that provides the maximum number of sets. To formulate the corresponding linear programming optimization problem, we define all possible ways sawing logs, indicating the corresponding number of beams obtained in this case (Table 1).

Table 1

Let x i denote the number of logs sawn in the i-th way (i = 1.2, 3, 4); x - the number of sets of bars.

Taking into account the fact that all logs must be sawn, and the number of beams of each size must satisfy the condition of completeness, the optimization economic and mathematical model will take next view x > max under restrictions:

x 1 + x 2 + x 3 + x 4 \u003d 200

x i 0 (i=1,2,3,4)

The problem of choosing the optimal production program of the enterprise. Let a company produce n different types of products. To produce these types of products, the enterprise uses M types of material and raw materials and N types of equipment. It is necessary to determine the production volumes of the enterprise (i.e. its production program) for a given planning interval in order to maximize the gross profit of the enterprise.

where a i is the selling price of products of type i;

b i -- variable costs for the release of one unit of product type i;

Zp -- conditionally fixed costs, which we will assume independent of the vector x = (x 1 ,..., x n).

At the same time, restrictions on the volumes of material and raw materials used and the time of using the equipment in the interval must be met.

Let us denote by Lj(j = l,...,M) the volume of stocks of material and raw materials of the type j, and by f k (k = 1,..., N) the time during which the equipment of the type k. We know the consumption of material and raw materials of type j for the production of one unit of product of type i, which we denote by l ij (i = 1,..., n; j = 1,...,M). It is also known t ik -- the loading time of one unit of equipment of type k for the manufacture of one unit of production of type i (i = 1,..., n; k = 1,..., N). We denote by m k the number of pieces of equipment of the form k (k=l,...,N).

With the introduced notation, restrictions on the volume of consumed material and raw materials can be set as follows:

The constraints on production capacity are given by the following inequalities

In addition, the variables

x i ?0 i=1,…,n (7)

Thus, the problem of choosing a production program that maximizes profit is to choose such an output plan x = (x 1 ..., x n) that would satisfy constraints (5)-(7) and maximize function (4).

In some cases, an enterprise must supply predetermined volumes of production Vt to other economic entities, and then in the model under consideration, instead of constraint (1.7), a constraint of the form can be included:

x t > Vt i= 1,...,n.

Diet issue. Consider the problem of compiling a minimum cost per capita diet that would contain certain nutrients in required volumes. We will assume that there is a known list of products from n items (bread, sugar, butter, milk, meat, etc.), which we will denote by the letters F 1 ,...,F n . In addition, such characteristics of products (nutrients) as proteins, fats, vitamins, minerals and others are considered. Let's designate these components by letters N 1 ,...,N m . Suppose that for each product F i it is known (i = 1,...,n) the quantitative content of the above components in one unit of the product. In this case, you can make a table containing the characteristics of the products:

F 1 ,F 2 ,…F j …F n

N 1 a 11 a 12 …a 1j …a 1N

N 2 a 21 a 22 …a 2j …a 2N

N i a i1 a i2 …a ij …a iN

N m a m1 a m2 …a mj …a mN

The elements of this table form a matrix with m rows and n columns. Let's denote it by A and call it the nutritional matrix. Suppose that we have compiled a diet x = (x 1, x 2, ..., x n) for a certain period (for example, a month). In other words, we plan for each person for month x, units (kilograms) of product F 1, x 2 units of product F 2, etc. It is easy to calculate how many vitamins, fats, proteins and other nutrients a person will receive during this period. For example, component N 1 is present in this diet in an amount

a 11 x 1 + a 12 x 2+…+ a 1n x n

since, according to the condition, x 1 units of the product F 1 according to the nutritional matrix contain a 11 x 1 units of the component N 1; to this quantity is added portion a 12 x 2 of substance N 1 from x 2 units of product F 2, etc. Similarly, you can determine the amount of all other substances N i in the diet (x 1 ,..., x n).

Let's assume that there are certain physiological requirements regarding the required amount of nutrients in N i (i/ = 1,..., N) at the planned time. Let these requirements be given by the vector b = (b 1 ...,b n), the i-th component of which b i indicates the minimum required content of component N i in the diet. This means that the coefficients x i of the vector x must satisfy the following system of constraints:

a 11 x 1 + a 12 x 2+…+ a 1n x n ?b 1

a 21 x 1 + a 22 x 2+…+ a 2n x n?b 2 (8)

a m1 x 1 + a m2 x 2+…+ a mn x n ?b m

In addition, from the meaningful meaning of the problem, it is obvious that all the variables x 1 ,..., x n are non-negative and therefore the inequalities are added to the constraints (8)

x1?0; x 2 ?0;… x n ?0; (9)

Taking into account that in most cases the constraints (8) and (9) are satisfied by an infinite number of rations, we will choose one of them, the cost of which is minimal.

Let the prices of products F 1 ,...,F n be equal to 1 ,…,c n

Therefore, the cost of the entire diet x = (x 1 ..., x n) can be written as

c 1 x 1 + c 2 x 2 +…+ c n x n >min (10)

The final formulation of the diet problem is to choose among all vectors x = (x 1 ,..., x n) satisfying the constraints (8) and (9) the one for which the objective function (10) takes the minimum value.

transport task. There are m production sites S 1 ,..., S m of a homogeneous product (coal, cement, oil, etc.), while the production volume at the site S i is equal to a i units. The produced product is consumed at points Q 1 ...Q n and the need for it at point Q j is k j units (j = 1,...,n). It is required to make a transportation plan from points S i (i = 1,...,m) to points Q j (j = 1,..., n) in order to satisfy the demand for product b j , minimizing transportation costs.

Let the cost of transporting one unit of product from point S i to point Q i be equal to c ij . We will further assume that when transporting x ij units of product from S i to Q j, transportation costs are equal to c ij x ij.

Let's call a transportation plan a set of numbers х ij c i = 1,..., m; j = 1,..., n satisfying the constraints:

xij?0, i=1,2,…,m; j=1,…,n (11)

With a transportation plan (x ij), transportation costs will amount to

The final formation of the transport problem is as follows: among all sets of numbers (х ij) that satisfy the constraints (11), find a set that minimizes (12).

2.1.3 Optimal resource allocation

The class of problems considered in this chapter has numerous practical applications.

AT general view these tasks can be described as follows. There is a certain amount of resources, which can be understood as cash, material resources (for example, raw materials, semi-finished products, labor resources, different kinds equipment, etc.). These resources must be distributed between different objects of their use at separate intervals of the planning period or at different intervals for various objects so as to obtain the maximum total efficiency from the chosen method of distribution. An indicator of efficiency can be, for example, profit, marketable output, capital productivity (tasks of maximization) or total costs, cost, time to complete a given amount of work, etc. (tasks of minimization).

Generally speaking, the vast majority of mathematical programming problems fit into the general formulation of the problem of optimal resource allocation. Naturally, when considering models and computational schemes for solving such problems by the DP method, it is necessary to specify the general form of the resource allocation problem.

In what follows, we will assume that the conditions necessary for constructing the DP model are satisfied in the problem. Let us describe a typical resource allocation problem in general terms.

Task 1. Available initial amount funds to be distributed within n years among s enterprises. Funds (k=1, 2,…,n; i=1,…, s) allocated in k-th year i-th enterprise, bring income in the amount and return in quantity by the end of the year. In the subsequent distribution, income can either participate (partially or completely), or not participate.

It is required to determine such a way of distributing resources (the amount of funds allocated to each enterprise in each planning year) so that the total income from s enterprises over n years is maximum.

Therefore, as an indicator of the efficiency of the resource allocation process for n years, the total income received from s enterprises is taken:

The amount of resources at the beginning of the kth year will be characterized by the value (state parameter). Management on k-th step consists in the choice of variables denoting the resources allocated in the k-th year to the i-th enterprise.

If we assume that income does not participate in the further distribution, then the equation of the state of the process has the form

If, on the other hand, some part of the income participates in further distribution in some year, then the corresponding value is added to the right side of equality (4.2).

It is required to determine ns nonnegative variables satisfying conditions (4.2) and maximizing function (4.1).

The computational procedure of the DP begins with the introduction of a function denoting the income received for n - k + 1 years, starting from the kth year until the end of the period under consideration, with the optimal distribution of funds among s enterprises, if funds were distributed in the kth year. The functions for k=1, 2, ...n-1 satisfy the functional equations (2.2), which will be written as:

For k=n, according to (2.2), we obtain

Next, it is necessary to sequentially solve equations (4.4) and (4.3) for all possible (k = n--1, n--2, 1). Each of these equations is an optimization problem for a function that depends on s variables. Thus, a problem with ns variables is reduced to a sequence of n problems, each containing s variables. In this general setting, the problem is still difficult (because of the multidimensionality) and to simplify it, considering it as an ns-step problem, in this case it is forbidden. In fact, let's try to do it. We number the steps according to the numbers of enterprises, first in the 1st year, then in the 2nd, etc.:

and we will use one parameter to characterize the balance of funds.

During the k-th year, the state "by the beginning of any step s(k-1)_+i (i=1,2,…,s) will be determined from the previous state using a simple equation. However, after a year, i.e. by the beginning of the next year, it will be necessary to add funds to the cash and, therefore, the state at the beginning of the (ks+1)-th step will depend not only on the previous ks-th state, but also on all s states and controls over the past year. As a result, we get a process with aftereffect.To eliminate the aftereffect, we have to introduce several state parameters, the task at each step remains still difficult due to multidimensionality.

Task 2. The activity of two enterprises (s=2) is planned for n years. Initial funds are. Funds x invested in enterprise I bring income f 1 (x) by the end of the year and return in the same amount, funds x invested in enterprise II give income f 2 (x) and return in the amount. At the end of the year, all the remaining funds are redistributed anew between enterprises I and II, no new funds are received and no income is invested in production.

It is required to find the optimal way to distribute the available funds.

We will consider the process of distributing funds as an n-step process, in which the step number corresponds to the year number. A managed system is two enterprises with funds invested in them. The system is characterized by one state parameter - the amount of funds that should be redistributed at the beginning of the k-th year. There are two control variables at each step: - the amount of funds allocated to enterprise I and II, respectively. Since the funds are redistributed annually in full, then). For each step, the problem becomes one-dimensional. Denote by, then

The efficiency indicator of the k-th step is equal to. This is the income received from two enterprises during the k-th year.

The performance indicator of the task - the income received from two enterprises for n years - is

The equation of state expresses the balance of funds after the kth step and has the form

Let be the conditional optimal income received from the distribution of funds between two enterprises for n--k+1 years, starting from the k-th year until the end of the period under consideration. Let's write the recurrence relations for these functions:

where - is determined from the equation of state (4.6).

With a discrete investment of resources, the question may arise about the choice of the step Dx in changing the control variables. This step can be set or determined based on the required accuracy of calculations and the accuracy of the initial data. In the general case, this task is difficult and requires interpolation from tables at previous calculation steps. Sometimes a preliminary analysis of the equation of state allows one to choose an appropriate step Dx, as well as to set the limit values for which tabulation must be performed at each step.

Let us consider a two-dimensional problem similar to the previous one, in which a discrete model of the DP of the resource allocation process is constructed.

Task 3. Draw up an optimal plan for the annual distribution of funds between two enterprises during a three-year planning period under the following conditions:

1) the initial amount is 400;

2) invested funds in the amount of x bring income f 1 (x) at enterprise I and return in the amount of 60% of x, and at enterprise II - f2 (x) and 20%, respectively;

3) all cash received from the returned funds is distributed annually:

4) functions f 1 (x) and f2 (x) are given in Table. one:

The dynamic programming model of this problem is similar to the model compiled in problem 1.

The management process is three-step. The parameter is the funds to be distributed in the k-th year (k=l, 2, 3). The control variable is the funds invested in enterprise I in the kth year. The funds invested in enterprise II in the kth year are Therefore, the control process at the kth step depends on one parameter (one-dimensional model). The equation of state will be written in the form

And functional equations in the form

Let's try to determine the maximum possible values for which it is necessary to tabulate at the k-th step (k=l, 2, 3). At =400 from equation (4.8) we determine the maximum possible value we have = 0.6 * 400 = 2400 (all funds are invested in enterprise I). Similarly, for we obtain the limit value 0.6 * 240 = 144. Let the interval of change coincide with the table, i.e. Dx \u003d 50. Let's make a table of total profit at this step:

This will make further calculations easier. Since the cells located along the diagonal of the table correspond to the same value indicated in the 1st row (in the 1st column) of Table. 2. The 2nd row of the table contains the values f 1 (x), and the 2nd column contains the values f 2 (y) taken from the table. 1. The values in the remaining cells of the table are obtained by adding the numbers f 1 (x) and f 2 (y) in the 2nd row and in the 2nd column and corresponding to the column and row at the intersection of which this cell is located. For example, for =150 we get a series of numbers: 20 - for x = 0, y=150; 18 --for x=50, y=100; 18-- for x--100, y=50; 15 -- for x=150, y=0.

Let's spend conditional optimization in the usual way. 3rd step. Basic equation (4.9)

As stated above, . Let's look at the numbers on the diagonals corresponding to =0; fifty; 100; 150 and choose the largest on each diagonal. This is what we find in the 1st line of the corresponding conditional optimal control. The optimization data at the 3rd step will be placed in the main table (Table 4). It introduces the column Dx, which is further used in interpolation.

Optimization of the 2nd step is carried out in Table. 5 according to an equation of the form (4.10):

In this case, the maximum income equal to Zmax=99,l can be obtained. Direct calculation of income according to the table. 2 for the found optimal control gives 97.2. The discrepancy in the results by 1.9 (about 2%) is due to a linear interpolation error.

We have considered several variants of the problem of optimal allocation of resources. There are other versions of this problem, the features of which are taken into account by the corresponding dynamic model.

Conclusion

In this term paper the types of mathematical models used in economics and management, as well as their classification, are considered.

Particular attention in the course work is given to optimization modeling.

The principle of constructing linear programming models has been studied, models of the following tasks are also given:

· The task of cutting materials;

· The task of choosing the optimal production program of the enterprise;

· Diet task;

transport task.

The paper presents the general characteristics of discrete programming problems, describes the principle of optimality and the Bellman equation, gives general description modeling process.

Three tasks were chosen for building models:

· The problem of optimal allocation of resources;

· The task of optimal control reserves;

The problem of replacement.

In turn, for each of the tasks, various dynamic programming models are built. For individual tasks, numerical calculations are given, in accordance with the constructed models.

Bibliography:

1. Vavilov V.A., Zmeev O.A., Zmeeva E.E. Electronic manual"Operations research"

2. Kalikhman I.L., Voitenko M.A. “Dynamic Programming in Examples and Problems”, 1979

3. Kosorukov O.A., Mishchenko A.V. Operations Research, 2003

4. Materials from the Internet.

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Faculty of Economics and Law

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Timonina T.S.

Math modeling

One of the types of formalized sign modeling is mathematical modeling, carried out by means of the language of mathematics and logic. To study any class of phenomena of the external world, its mathematical model is built, i.e. an approximate description of this class of phenomena, expressed with the help of mathematical symbols.

The process of mathematical modeling can be divided into four main stages:

Istage: Formulation of laws linking the main objects of the model, i.e. a record in the form of mathematical terms of the formulated qualitative ideas about the relationships between the objects of the model.

IIstage: The study of mathematical problems to which mathematical models lead. The main issue is the solution of the direct problem, i.e. obtaining output data (theoretical consequences) as a result of the analysis of the model for their further comparison with the results of observations of the studied phenomena.

IIIstage: Correction of the accepted hypothetical model according to the criterion of practice, i.e. clarification of the question of whether the results of observations are consistent with the theoretical consequences of the model within the accuracy of observations. If the model was completely defined - all its parameters were given - then the determination of the deviations of the theoretical consequences from observations gives solutions to the direct problem, followed by an estimate of the deviations. If the deviations are outside the accuracy of the observations, then the model cannot be accepted. Often, when building a model, some of its characteristics remain undefined. The application of the criterion of practice to the evaluation of a mathematical model makes it possible to conclude that the assumptions underlying the (hypothetical) model to be studied are correct.

IVstage: Subsequent analysis of the model in connection with the accumulation of data on the studied phenomena and modernization of the model. With the advent of computers, the method of mathematical modeling has taken a leading place among other research methods. This method plays a particularly important role in modern economic science. The study and forecasting of any economic phenomenon by mathematical modeling allows you to design new technical means predict the impact of certain factors on a given phenomenon, plan these phenomena even in the presence of an unstable economic situation.

Essence of economic analysis

Analysis (decomposition, dismemberment, parsing) is a logical technique, a research method, the essence of which is that the subject being studied is mentally divided into constituent elements, each of which is then examined separately as part of a dismembered whole, in order to identify the elements identified during the analysis. combine with the help of another logical technique - synthesis - into a whole, enriched with new knowledge.

Under economic analysis understand applied scientific discipline, which is a system special knowledge, allowing to evaluate the effectiveness of the activities of a particular subject of a market economy.

Theory of economic analysis allows you to rationally justify, predict for the near future the development of the control object and evaluate the feasibility of making a management decision.

Main directions of economic analysis:

Formulation of a system of indicators characterizing the work of the analyzed object;

Qualitative analysis of the studied phenomenon (result);

Quantitative analysis of this phenomenon (result):

For the development and adoption of a management decision, it is important that it is a means of solving the main task of identifying reserves for increasing the efficiency of economic activity in improving the use of production resources, reducing costs, increasing profitability and increasing profits, i.e. is aimed at the ultimate goal of implementing a management decision.

Developers of the theory of economic analysis emphasize it characteristic peculiarities:

1. The dialectical approach to the study of economic processes, which are characterized by: the transition of quantity into quality, the emergence of a new quality, the negation of negation, the struggle of opposites, the withering away of the old and the emergence of the new.

2. Conditionality of economic phenomena by causal relationships and interdependence.

3. Identification and measurement of interrelations and interdependencies of indicators are based on knowledge of objective patterns of development of production and circulation of goods.

Economic analysis, first of all, is factorial, i.e., determining the influence of a complex of economic factors on the performance indicator of an enterprise.

The influence of various factors on the economic indicator of the functioning of an enterprise, firm is carried out using stochastic analysis.

In turn, deterministic and stochastic analyzes provide:

Establishment of causal or probabilistic relationships of factors and performance indicators;

Identification of economic patterns of the influence of factors on the functioning of the enterprise and their expression with the help of mathematical dependencies;

The ability to build models (primarily mathematical) of the impact of factor systems on performance indicators and study, with their help, the impact on the final result of a managerial decision .

In practice, various types of economic analysis are used. For management decisions made, analyzes are especially important: operational, current, prospective (by time intervals); partial and complex (by volume); to identify reserves, improve quality, etc. (by appointment); predictive analysis. Forecasts allow you to economically justify strategic, operational (functional) or tactical management decisions .

Historically, two groups of methods and techniques have developed: traditional and mathematical. Let us consider in more detail the application of mathematical methods in economic analysis.

Mathematical methods in economic analysis

The use of mathematical methods in the field of management is the most important direction in improving management systems. Mathematical methods speed up economic analysis, contribute to a more complete account of the influence of factors on performance, improve the accuracy of calculations. The application of mathematical methods requires:

* a systematic approach to the study of a given object, taking into account the relationships and relationships with other objects (enterprises, firms);

* development of mathematical models that reflect the quantitative indicators of the systemic activity of the employees of the organization, the processes taking place in complex systems ah, what are the enterprises;

* system improvements information support enterprise management with the use of electronic computers.

Solving problems of economic analysis by mathematical methods is possible if they are formulated mathematically, i.e. real economic relationships and dependencies are expressed using mathematical analysis. This necessitates the development of mathematical models.

In management practice, various methods are used to solve economic problems. Figure 1 shows the main mathematical methods used in economic analysis.

The selected features of the classification are rather conditional. For example, in network planning and management, various mathematical methods are used, and many authors put different content into the meaning of the term "operations research".

Methods of elementary mathematics are used in traditional economic calculations when substantiating resource needs, developing a plan, projects, etc.

Classical methods of mathematical analysis are used independently (differentiation and integration) and within the framework of other methods (mathematical statistics, mathematical programming).

Statistical Methods - the main means of investigating mass recurring phenomena. They are used when it is possible to represent changes in the analyzed indicators as a random process. If the relationship between the analyzed characteristics is not deterministic, but stochastic, then statistical and probabilistic methods become practically the only research tool. In economic analysis, the methods of multiple and paired correlation analysis are best known.

To study simultaneous statistical aggregates, the distribution law, the variation series, and the sampling method are used. For multidimensional statistical aggregates, correlations, regressions, dispersion, covariance, spectral, component, factorial types of analysis are used.

Economic Methods are based on the synthesis of three areas of knowledge: economics, mathematics and statistics. The basis of econometrics is an economic model, i.e. a schematic representation of an economic phenomenon or processes, a reflection of their characteristic features using scientific abstraction. The most common method of economic analysis is "costs - output". The method represents matrix (balance) models built according to a chess scheme and clearly illustrating the relationship between costs and production results.

Mathematical programming methods - the main means of solving problems of optimization of production and economic activities. In fact, the methods are means of planned calculations, and they make it possible to assess the intensity of planned targets, the scarcity of results, to determine the limiting types of raw materials, groups of equipment.

Under Operations Research refers to the development of methods of purposeful actions (operations), the quantitative evaluation of solutions and the choice of the best of them. The goal of operations research is a combination of structural interrelated elements of the system, to the greatest extent providing the best economic indicator.

Game theory as a section of operations research, it is a theory of mathematical models for making optimal decisions under conditions of uncertainty or conflict of several parties with different interests.

Methods of mathematical statistics

Rice. 1. Classification of the main mathematical methods used in economic analysis.

Queuing theory based on probability theory explores mathematical methods for quantifying queuing processes. A feature of all tasks related to queuing is the random nature of the phenomena under study. The number of requests for service and the time intervals between their receipts are random, but in the aggregate they obey statistical patterns, the quantitative study of which is the subject of queuing theory.

Economic cybernetics analyzes economic phenomena and processes as complex systems from the point of view of the laws of control and the movement of information in them. Methods of modeling and system analysis are most developed in this area.

The application of mathematical methods in economic analysis is based on the methodology of economic and mathematical modeling of economic processes and scientifically substantiated classification of methods and tasks of analysis. All economic and mathematical methods (tasks) are divided into two groups: optimization solutions according to a given criterion and non-optimization(solutions without optimality criterion).

On the basis of obtaining an exact solution, all mathematical methods are divided into precise(with or without a criterion, a unique solution is obtained) and approximate(based on stochastic information).

Optimal exact methods include methods of the theory of optimal processes, some methods of mathematical programming and methods of operations research, optimization approximations - part of the methods of mathematical programming, operations research, economic cybernetics, heuristic.

Methods of elementary mathematics and classical methods of mathematical analysis, economic methods belong to non-optimization exact methods, and the method of statistical tests and other methods of mathematical statistics belong to non-optimization approximate ones.

Particularly often used are mathematical models of queues and inventory management. For example, the theory of queues is based on the one developed by scientists A.N. Kolmogorov and A.L. Khanchin queuing theory.

Queuing Theory

This theory allows you to study systems designed to serve the mass flow of requirements of a random nature. Random can be both the moments of the appearance of requirements and the time spent on their maintenance. The purpose of the methods of the theory is to find a reasonable organization of service that ensures its given quality, to determine the optimal (from the point of view of the accepted criterion) standards of on-duty service, the need for which arises unplanned, irregularly.

Using the method of mathematical modeling, it is possible to determine, for example, the optimal number of automatically operating machines that can be serviced by one worker or a team of workers, etc.

A typical example of objects of the theory of queuing can serve as automatic telephone exchanges - automatic telephone exchanges. The PBX randomly receives “requests” - calls from subscribers, and “service” consists in connecting subscribers to other subscribers, maintaining communication during a conversation, etc. The problems of the theory, formulated mathematically, are usually reduced to the study of a special type of random processes.

Based on the data on the probabilistic characteristics of the incoming call flow and service duration, and taking into account the scheme of the service system, the theory determines the corresponding characteristics of the quality of service (failure probability, average waiting time for the start of service, etc.).

Mathematical models of numerous problems of technical and economic content are also problems of linear programming. Linear programming is a discipline dedicated to the theory and methods for solving problems about extrema linear functions on sets defined by systems of linear equalities and inequalities.

The task of planning the work of the enterprise

For the production of homogeneous products, it is necessary to spend various production factors - raw materials, labor, machine park, fuel, transport, etc. Usually there are several proven technological methods of production, and in these methods the costs of production factors per unit of time for the release of products are different.

The number of consumed production factors and the number of manufactured products depends on how long the enterprise will work according to one or another technological method.

The task is to rationally distribute the time of the enterprise's work according to various technological methods, i.e. the one at which the maximum number of products will be produced for a given limited cost of each production factor.

Based on the method of mathematical modeling in operational research, many important tasks are also solved that require specific methods solutions. These include:

The task of product reliability.

· Equipment replacement task.

scheduling theory (the so-called scheduling theory scheduling).

· Resource allocation problem.

The problem of pricing.

· The theory of network planning.

The task of product reliability

The reliability of products is determined by a set of indicators. For each type of product, there are recommendations for choosing reliability indicators.

To evaluate products that can be in two possible states - operable and failure, the following indicators are used: average time to failure (time to first failure), time to failure, failure rate, failure rate parameter, average recovery time of a working state, probability of non-failure operation during time t, availability factor.

Resource Allocation Problem

The issue of resource allocation is one of the main ones in the process of production management. To address this issue, operational research uses the construction of a linear statistical model.

Pricing Challenge

For the enterprise, the issue of pricing for products plays an important role. How the pricing is carried out at the enterprise depends on its profit. In addition, in the current conditions of a market economy, price has become an essential factor in the competitive struggle.

Network planning theory

Network planning and management is a management planning system for the development of large economic complexes, design and technological preparation for the production of new types of goods, construction and reconstruction, major repairs of fixed assets through the use of network schedules.

The essence of network planning and management is the compilation of a mathematical model of a managed object in the form of a network diagram or a model located in the computer's memory, which reflects the relationship and duration of a certain set of works. The network diagram after its optimization by means of applied mathematics and computer technology is used for operational management of work.

The solution of economic problems using the method of mathematical modeling makes it possible to effectively manage both individual production processes at the level of forecasting and planning economic situations and making management decisions based on this, and the entire economy as a whole. Consequently, mathematical modeling as a method is closely related to the theory of decision making in management.

Stages of economic and mathematical modeling

The main stages of the modeling process have already been discussed above. In various branches of knowledge, including in the economy, they acquire their own specific features. Let us analyze the sequence and content of the stages of one cycle of economic and mathematical modeling.

1. Statement of the economic problem and its qualitative analysis. The main thing here is to clearly articulate the essence of the problem, the assumptions made and the questions that need to be answered. This stage includes highlighting the most important features and properties of the object being modeled and abstracting from minor ones; studying the structure of the object and the main dependencies connecting its elements; formulation of hypotheses explaining the behavior and development of the object.

2. Building a mathematical model. This is the stage of formalizing the economic problem, expressing it in the form of specific mathematical dependencies and relationships (functions, equations, inequalities, etc.). Usually, the main construction (type) of the mathematical model is first determined, and then the details of this construction are specified (a specific list of variables and parameters, the form of relationships). Thus, the construction of the model is subdivided in turn into several stages.

It is wrong to assume that the more facts the model takes into account, the better it "works" and gives better results. The same can be said about such characteristics of the complexity of the model as the forms of mathematical dependencies used (linear and non-linear), taking into account the factors of randomness and uncertainty, etc. The excessive complexity and cumbersomeness of the model complicate the research process. It is necessary to take into account not only real opportunities information and mathematical support, but also to compare the costs of modeling with the effect obtained (as the complexity of the model increases, the increase in costs may exceed the increase in effect).

One of the important features of mathematical models is the potential possibility of their use for solving problems of different quality. Therefore, even when faced with a new economic challenge, one should not strive to "invent" a model; First, it is necessary to try to apply already known models to solve this problem.

In the process of building a model, the two systems are compared scientific knowledge- economic and mathematical. It is natural to strive to obtain a model that belongs to a well-studied class of mathematical problems. Often this can be done by some simplification of the initial assumptions of the model that do not distort the essential features of the modeled object. However, it is also possible that the formalization of an economic problem leads to a previously unknown mathematical structure. The needs of economic science and practice in the middle of the twentieth century. contributed to the development of mathematical programming, game theory, functional analysis, and computational mathematics. It is likely that in the future the development of economic science will become an important stimulus for the creation of new branches of mathematics.

3. Mathematical analysis of the model. The purpose of this step is to clarify the general properties of the model. Here purely mathematical methods of research are used. The most important point is the proof of the existence of solutions in the formulated model (existence theorem). If it is possible to prove that a mathematical problem has no solution, then the need for further work on original version the model disappears; either the formulation of the economic problem or the methods of its mathematical formalization should be corrected. During the analytical study of the model, such questions are clarified as, for example, is the solution unique, what variables (unknowns) can be included in the solution, what will be the relationships between them, within what limits and depending on what initial conditions they change, what are the trends of their change and etc. The analytical study of the model compared to the empirical (numerical) one has the advantage that the conclusions obtained remain valid for various specific values of the external and internal parameters of the model.

Knowledge of the general properties of the model is so important that often, in order to prove such properties, researchers deliberately go for the idealization of the original model. And yet, models of complex economic objects lend themselves to analytical research with great difficulty. In those cases when analytical methods fail to determine the general properties of the model, and simplifications of the model lead to unacceptable results, they switch to numerical methods of investigation.

4. Preparation of initial information. Modeling imposes strict requirements on the information system. At the same time, the real possibilities of obtaining information limit the choice of models intended for practical use. This takes into account not only the fundamental possibility of preparing information (for a certain period of time), but also the costs of preparing the relevant information arrays. These costs should not exceed the effect of using additional information.

In the process of preparing information, methods of probability theory, theoretical and mathematical statistics are widely used. With systemic economic and mathematical modeling background information, used in some models, is the result of the operation of other models.

5. Numerical solution. This stage includes the development of algorithms for the numerical solution of the problem, the compilation of computer programs and direct calculations. The difficulties of this stage are due, first of all, to the large dimension of economic problems, the need to process significant amounts of information.

Usually, calculations based on the economic-mathematical model are of a multivariate nature. Due to the high speed of modern computers, it is possible to conduct numerous "model" experiments, studying the "behavior" of the model under various changes in certain conditions. A study carried out by numerical methods can significantly supplement the results of an analytical study, and for many models it is the only feasible one. The class of economic problems that can be solved by numerical methods is much wider than the class of problems accessible to analytical research.

6. Analysis of numerical results and their application. At this final stage of the cycle, the question arises about the correctness and completeness of the simulation results, about the degree of practical applicability of the latter.

Mathematical verification methods can detect incorrect model constructions and thereby narrow the class of potentially correct models. An informal analysis of the theoretical conclusions and numerical results obtained by means of the model, their comparison with the available knowledge and facts of reality also make it possible to detect the shortcomings of the formulation of the economic problem, the constructed mathematical model, its information and mathematical support.

References

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When constructing economic models, significant factors are identified and details that are not essential for solving the problem are discarded.

Economic models may include models:

economic growth
consumer choice
equilibrium in the financial and commodity markets and many others.

Model is a logical or mathematical description of the components and functions that reflect the essential properties of the modeled object or process.

The model is used as a conditional image designed to simplify the study of an object or process.

The nature of the models may be different. Models are divided into: real, sign, verbal and tabular description, etc.

Economic and mathematical model

In the management of business processes, the most important are, first of all, economic and mathematical models, often combined into model systems.

Economic and mathematical model(EMM) is a mathematical description of an economic object or process for the purpose of their study and management. This is a mathematical record of the economic problem being solved.

Main types of models

Extrapolation Models
Factorial econometric models
Optimization Models
Balance models, Inter-Industry Balance Model (ISB)
Expert assessments
Game theory
network models
Models of queuing systems

Economic and mathematical models and methods used in economic analysis

R a \u003d PE / VA + OA,

In a generalized form, the mixed model can be represented by the following formula:

So, first you need to build an economic-mathematical model that describes the influence of individual factors on the general economic indicators of the organization. Widespread in the analysis of economic activity received multifactorial multiplicative models, since they allow us to study the influence of a significant number of factors on generalizing indicators and thereby achieve greater depth and accuracy of analysis.

After that, you need to choose a way to solve this model. Traditional ways : the method of chain substitutions, the methods of absolute and relative differences, the balance method, the index method, as well as the methods of correlation-regression, cluster, dispersion analysis, etc. Along with these methods and methods, economic analysis also uses specifically mathematical ways and methods.

Integral method of economic analysis

One of these methods (methods) is integral. It finds application in determining the influence of individual factors using multiplicative, multiple, and mixed (multiple additive) models.

Under the conditions of applying the integral method, it is possible to obtain more reasonable results for calculating the influence of individual factors than when using the chain substitution method and its variants. The chain substitution method and its variants, as well as the index method, have significant drawbacks: 1) the results of calculating the influence of factors depend on the accepted sequence of replacing the basic values of individual factors with actual ones; 2) an additional increase in the generalizing indicator, caused by the interaction of factors, in the form of an indecomposable remainder, is added to the sum of the influence of the last factor. When using the integral method, this increase is divided equally between all factors.

The integral method sets general approach to solving models of various types, regardless of the number of elements that are included in this model, and also regardless of the form of connection between these elements.

The integral method of factor economic analysis is based on the summation of the increments of a function defined as a partial derivative, multiplied by the increment of the argument over infinitely small intervals.

In the process of applying the integral method, several conditions must be met. First, the condition of continuous differentiability of the function must be observed, where some economic indicator is taken as an argument. Secondly, the function between the start and end points of the elementary period must change in a straight line G e. Finally, thirdly, there must be a constancy of the ratio of the rates of change in the values of the factors

dy / dx = const

When using the integral method, the calculation of a definite integral over a given integrand and given interval integration is carried out according to the existing standard program using modern computer technology.

If we are solving a multiplicative model, then the following formulas can be used to calculate the influence of individual factors on a general economic indicator:

∆Z(x) = y 0 * Δ x + 1/2Δ x *Δ y

Z(y)=x 0 * Δ y +1/2 Δ x* Δ y

When solving a multiple model to calculate the influence of factors, we use the following formulas:

Z=x/y;

Δ Z(x)= Δ x/Δ y Lny1/y0

Δ Z(y)=Δ Z- Δ Z(x)

There are two main types of problems solved using the integral method: static and dynamic. In the first type, there is no information about changes in the analyzed factors during this period. Examples of such tasks are the analysis of the implementation of business plans or the analysis of changes in economic indicators compared to the previous period. The dynamic type of tasks takes place in the presence of information about the change in the analyzed factors during a given period. This type of tasks includes calculations related to the study of time series of economic indicators.

These are the most important features of the integral method of factorial economic analysis.

Log method

In addition to this method, the method (method) of logarithm is also used in the analysis. It is used in factor analysis when solving multiplicative models. The essence of the method under consideration lies in the fact that when using it, there is a logarithmically proportional distribution of the value of the joint action of factors between the latter, that is, this value is distributed between the factors in proportion to the share of influence of each individual factor on the sum of the generalizing indicator. With the integral method, the mentioned value is distributed among the factors equally. Therefore, the logarithm method makes the calculation of the influence of factors more reasonable than the integral method.

In the process of taking logarithms, not absolute values of the growth of economic indicators are used, as is the case with the integral method, but relative ones, that is, indices of changes in these indicators. For example, a generalizing economic indicator is defined as the product of three factors - factors f = x y z.

Let us find the influence of each of these factors on the generalizing economic indicator. So, the influence of the first factor can be determined by the following formula:

Δf x \u003d Δf lg (x 1 / x 0) / log (f 1 / f 0)

What was the impact next factor? To find its influence, we use the following formula:

Δf y \u003d Δf lg (y 1 / y 0) / log (f 1 / f 0)

Finally, in order to calculate the influence of the third factor, we apply the formula:

Δf z \u003d Δf lg (z 1 / z 0) / log (f 1 / f 0)

Thus, the total amount of change in the generalizing indicator is divided between individual factors in accordance with the proportions of the ratios of the logarithms of individual factor indices to the logarithm of the generalizing indicator.

When applying the method under consideration, any types of logarithms can be used - both natural and decimal.

Method of differential calculus

When conducting factor analysis, the method is also used differential calculus. The latter assumes that general change function, that is, a generalizing indicator, is divided into separate terms, the value of each of which is calculated as the product of a certain partial derivative and the increment of the variable by which this derivative is determined. Let's determine the influence of individual factors on the generalizing indicator, using as an example a function of two variables.

Function is set Z = f(x,y). If this function is differentiable, then its change can be expressed by the following formula:

Let us explain the individual elements of this formula:

ΔZ = (Z 1 - Z 0)- the magnitude of the function change;

Δx \u003d (x 1 - x 0)- the magnitude of the change in one factor;

Δ y = (y 1 - y 0)- the amount of change of another factor;

is an infinitesimal value of a higher order than

In this example, the influence of individual factors x and y to change the function Z(generalizing indicator) is calculated as follows:

ΔZx = δZ / δx Δx; ΔZy = δZ / δy Δy.

The sum of the influence of both of these factors is the main, linear part of the increment of the differentiable function, that is, the generalizing indicator, relative to the increment of this factor.

Equity method

In the conditions of solving additive, as well as multiple-additive models, the method is also used to calculate the influence of individual factors on the change in the generalizing indicator equity participation. Its essence lies in the fact that the share of each factor in the total amount of their changes is first determined. This fraction is then multiplied by overall value changes in the summary indicator.

Suppose we are determining the influence of three factors − a,b and With for a summary y. Then, for the factor a, determining its share and multiplying it by the total value of the change in the generalizing indicator can be carried out according to the following formula:

Δy a = Δa/Δa + Δb + Δc*Δy

For the factor in the considered formula will have the following form:

Δyb =Δb/Δa + Δb +Δc*Δy

Finally, for the factor c we have:

∆y c =∆c/∆a +∆b +∆c*∆y

This is the essence of the equity method used for the purposes of factor analysis.

Linear programming method

See further:

Queuing Theory

See further:

Game theory

Game theory also finds application. Just like queuing theory, game theory is one of the branches of applied mathematics. Game theory studies the optimal solutions that are possible in situations of a game nature. This includes such situations that are associated with the choice of optimal management decisions, with the choice of the most appropriate options for relationships with other organizations, etc.

To solve such problems in game theory, algebraic methods, which are based on the system linear equations and inequalities iterative methods, as well as methods for reducing this problem to certain system differential equations.

One of the economic and mathematical methods used in the analysis of the economic activity of organizations is the so-called sensitivity analysis. This method is often used in the process of analyzing investment projects, as well as in order to predict the amount of profit remaining at the disposal of this organization.

In order to optimally plan and forecast the activities of the organization, it is necessary to foresee those changes that may occur in the future with the analyzed economic indicators.

For example, it is necessary to predict in advance the change in the values of those factors that affect the amount of profit: the level of purchase prices for acquired material resources, the level of selling prices for the products of a given organization, changes in customer demand for these products.

Sensitivity analysis consists in determining the future value of the generalizing economic indicator provided that the value of one or more factors influencing this indicator changes.

So, for example, they establish by what amount the profit will change in the future, subject to a change in the quantity of products sold per unit. Thus, we analyze the sensitivity of net profit to a change in one of the factors affecting it, that is, in this case, the sales volume factor. The rest of the factors affecting the profit margin remain unchanged. It is possible to determine the amount of profit also with a simultaneous change in the future of the influence of several factors. Thus, sensitivity analysis makes it possible to establish the strength of the response of a generalizing economic indicator to changes in individual factors that affect this indicator.

Matrix method

Along with the above economic and mathematical methods, they are also used in the analysis of economic activity. These methods are based on linear and vector-matrix algebra.

Network planning method

See further:

Extrapolation Analysis

In addition to the considered methods, extrapolation analysis is also used. It includes consideration of changes in the state of the analyzed system and extrapolation, that is, the extension of the existing characteristics of this system for future periods. In the process of implementing this type of analysis, the following main stages can be distinguished: primary processing and transformation of the initial series of available data; choice of the type of empirical functions; determination of the main parameters of these functions; extrapolation; establishing the degree of reliability of the analysis.

In economic analysis, the method of principal components is also used. They are used for the purpose of a comparative analysis of individual components, that is, the parameters of the analysis of the organization's activities. Principal components are the most important characteristics of linear combinations of constituent parts, that is, the parameters of the analysis performed, which have the most significant values of dispersion, namely, the largest absolute deviations from the average values.

Methods of economic theory

The study of human economic life has been in the sphere of interests of scientists since ancient times. The gradual complication of economic relations required the development of economic thought. Leaps in science have always been accompanied by tasks facing humanity at various stages of evolution. Initially, people got food, then they began to exchange it. Over time, agriculture arose, which contributed to the division of labor and the emergence of the first craft professions. An important stage in the economic life of mankind was the industrial revolution, which gave impetus to the rapid growth of production, and also influenced social changes in society.

Modern economic science was formed relatively recently, when scientists moved from solving problems facing the dominant class to studying the processes occurring in systems regardless of the interests of society.

The subject of economic theory is the optimization of the ratio of increasing demand in conditions when the volume of supply is limited due to limited resources.

It is worth noting that for a long time economic systems were considered in short-term periods, that is, in statics. Although the new trends of the twentieth century demanded from economists a new approach, focused on the dynamic development of economic structures.

Economic systems are enough complex formations, in which each subject simultaneously enters into many relationships. They can be considered in terms of macroeconomic aggregates, as well as the result of the work of an individual economic agent. The science of economics uses various methods contributing to the facilitation of the processes of research and analysis of economic phenomena. The most commonly used in practice are:

abstraction method (singling out an object from its connections and acting factors);
synthesis method (combining elements into a common one);
method of analysis (splitting the overall system into components);
deduction (study from the particular to the general) and induction (study of the subject from the general to the particular);
systematic approach (allows you to consider the object under study as a structure);
mathematical modeling (building models of processes and phenomena in mathematical language).

Modeling in economics

The essence of modeling is to replace the real model of a process, phenomenon or system with another model that can simplify its study and analysis. It is important to observe the proximity of the original model to its scientific counterpart. Modeling is used for the purpose of simplification. Often in practice there are such phenomena that cannot be studied without the use of demonstrative scientific generalizations.

The following modeling goals can be distinguished:

Search and description of the reasons for the behavior of the original model.
Predicting the future behavior of the model.
Drawing up projects, plans for systems.
Process automation.
Finding ways to optimize the original model.
For training professionals, students and others.

At its core, models can also be of various types. A verbal model is based on a verbal description of a system or process. The graphical model is a visual representation of various dependencies from each other. It can also describe the behavior of the original model in dynamics. Modeling natural is to create a layout that can partially or completely reflect the behavior of the original. The most widely used mathematical modeling. It makes it possible to use the entirety of mathematical tools and language. In mathematics, statistical models, dynamic and information models are used. Each of their types is used to achieve specific goals facing specialists.

Remark 1

The division of the economy into macro and micro levels has led to the fact that modeling also simulates systems at various levels of organization. To study economic structures, econometrics is most often used, which uses statistics and probability theory. It should be noted that it is mathematical modeling that makes it possible to take into account the time factor, which is important in the dynamic development of systems.

Mathematical models in economics

Before the start of economic and mathematical modeling, preparatory work which may include the following steps:

Setting goals and objectives.
Carrying out formalization of the studied process or phenomenon.
Finding the right solution.
Checking the obtained solution and model for adequacy.
If the test results are satisfactory, these models can be applied in practice.

Mathematical models are distinguished by the use of the language of mathematics at the stage of their construction, as well as in further calculations. This language allows you to most accurately describe relationships, dependencies and patterns. When the transition to solving models is made, different types of solutions can be used here. For example, exact or analytical gives the final indicator of the calculation. An approximate value has a certain calculation error, and is often used to build graphical models. Solution, expressed as a number, gives the final result, which is often derived using computer calculations. At the same time, it should be remembered that the accuracy of the solutions does not mean the accuracy of the calculated model.

An important step in mathematical modeling is the verification of the obtained results and simulation model for adequacy. Usually, verification work is based on data comparison real model with data built. However, in mathematical and economic modeling it is quite difficult to perform this action. Usually the adequacy of the calculations is determined later in practice.

Remark 2

Mathematical modeling in the economy allows you to simplify the phenomena and processes in economic systems, make calculations and obtain relatively correct calculation results. It is important to remember that this approach is also not universal, as it has a number of disadvantages listed above. The adequacy of modeling is often achieved through time-tested hypotheses and calculation formulas.

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TEST

by subject:

"Economic and mathematical methods and modeling"

Introduction

1. Mathematical modeling in economics

1.1 Development of modeling methods

1.2 Modeling as a method scientific knowledge

1.3 Economic and mathematical methods and models

Conclusion

Literature

Introduction

The doctrine of similarity and modeling began to be created more than 400 years ago. In the middle of the XV century. Leonardo da Vinci was engaged in the justification of modeling methods: he made an attempt to derive general patterns of similarity, used mechanical and geometric similarity in the analysis of situations in the examples he considered. He used the concept of analogy and drew attention to the need for experimental verification of the results of similar reasoning, the importance of experience, the relationship between experience and theory, and their role in cognition.

The ideas of Leonardo da Vinci about mechanical similarity were developed by Galileo in the 17th century, they were used in the construction of galleys in Venice.

In 1679, Mariotte used the theory of mechanical similarity in a treatise on colliding bodies.

The first rigorous scientific formulations of the similarity conditions and clarifications of the very concept of similarity were given in late XVII century by I. Newton in the "Mathematical Principles of Natural Philosophy".

In 1775–76 I.P. Kulibin used static similarity in experiments with models of a bridge across the Neva with a span of 300 m. The models were wooden, 1/10 of their natural size and weighing over 5 tons. Kulibin's calculations were verified and approved by L. Euler.

1. Mathematical modeling in economics

1.1 Development of modeling methods

Advances in mathematics stimulated the use of formalized methods in non-traditional areas of science and practice. So, O. Cournot (1801–1877) introduced the concept of supply and demand functions, and even earlier, the German economist I.G. Thünen (1783–1850) began to apply mathematical methods in economics and proposed the theory of production location, anticipating the theory of marginal labor productivity. The pioneers of using the modeling method include F. Quesnay (1694–1774), the author of the “Economic Table” (Quesnay zigzags) - one from the first models of social reproduction, a three-sector macroeconomic model of simple reproduction.

In 1871, Williams Stanley Jevons (1835–1882) published The Theory of Political Economy, where he outlined the theory of marginal utility. Utility is understood as the ability to satisfy human needs, underlying goods and prices. Jevons distinguished:

- abstract utility, which is devoid of concrete form;

- utility in general as the pleasure received by a person from the consumption of goods;

- marginal utility - the smallest utility among the whole set of goods.

Almost simultaneously (1874) with the work of Jevons, the work “Elements of Pure Political Economy” by Leon Walras (1834–1910) appeared, in which he set the task of finding such a price system in which the aggregate demand for all goods and markets would be equal to the aggregate supply. Walrasian pricing factors are:

production costs;

Marginal utility of a good;

Ask for a product offer;

The impact on the price of a given product of the entire system of prices according to
the rest of the goods.

The end of the 19th - beginning of the 20th century was marked by the widespread use of mathematics in economics. In the XX century. mathematical modeling methods are used so widely that almost all the works awarded the Nobel Prize in Economics are related to their application (D. Hicks, R. Solow, V. Leontiev, P. Samuelson, L. Kantorovich, etc.). The development of subject disciplines in most areas of science and practice is due to the ever higher level of formalization, intellectualization and use of computers. A far from complete list of scientific disciplines and their sections includes: functions and graphs of functions, differential and integral calculus, functions of many variables, analytical geometry, linear spaces, multidimensional spaces, linear algebra, statistical methods, matrix calculus, logic, graph theory, game theory, theory utility, optimization methods, scheduling theory, operations research, queuing theory, mathematical programming, dynamic, nonlinear, integer and stochastic programming, network methods, Monte Carlo method (method of statistical testing), reliability theory methods, random processes, Markov chains, the theory of modeling and similarity.

Formalized simplified descriptions of economic phenomena are called economic models. Models are used to detect the most significant factors of phenomena and processes of functioning of economic objects, to predict the possible consequences of the impact on economic objects and systems, for various assessments and the use of these assessments in management.

The construction of the model is carried out as the implementation of the following stages:

a) formulation of the purpose of the study;

b) description of the subject of research in generally accepted terms;

c) analysis of the structure of known objects and relationships;

d) description of the properties of objects and the nature and quality of links;

e) estimation of the relative weights of objects and connections by an expert method;

f) building a system of the most important elements in verbal, graphic or symbolic form;

g) collecting the necessary data and checking the accuracy of the simulation results;

i) analysis of the structure of the model for the adequacy of the representation of the described phenomenon and making adjustments; analysis of the availability of initial information and planning either additional studies for the possible replacement of some data with others, or special experiments to obtain missing data.

Mathematical models used in the economy can be divided into classes depending on the characteristics of the objects being modeled, the purpose and methods of modeling.

Macroeconomic models are designed to describe the economy as a whole. The main characteristics used in the analysis are GNP, consumption, investment, employment, the amount of money, etc.

Microeconomic models describe the interaction of structural and functional components of the economy or the behavior of one of the components in the environment of the rest. The main objects of modeling in microeconomics are supply, demand, elasticity, costs, production, competition, consumer choice, pricing, monopoly theory, theory of the firm, etc.

By the nature of the model can be theoretical (abstract), applied, static, dynamic, deterministic, stochastic, equilibrium, optimization, natural, physical.

Theoretical models allow studying the general properties of the economy based on formal premises using the deduction method.

Applied Models allow to evaluate the parameters of the functioning of an economic object. They operate with numerical knowledge of economic variables. Most often, these models use statistical or actually observed data.

Equilibrium Models describe such a state of the economy as a system in which the sum of all forces acting on it is equal to zero.

Optimization Models operate with the concept of utility maximization, the result of which is the choice of behavior in which the equilibrium state is maintained at the micro level.

Static Models describe the instantaneous state of an economic object or phenomenon.

Dynamic Model describes the state of an object as a function of time.

Stochastic Models take into account random effects on economic characteristics and use the apparatus of probability theory.

Deterministic Models assume the existence of a functional relationship between the characteristics under study and, as a rule, use the apparatus of differential equations.

Full-scale modeling is carried out on real-life objects under specially selected conditions, for example, an experiment conducted during the production process at an existing enterprise, while meeting the tasks of the production itself. The method of natural research arose from the needs of material production at a time when science did not yet exist. It coexists on a par with the natural science experiment at the present time, demonstrating the unity of theory and practice. A kind of full-scale modeling is modeling by generalizing production experience. The difference is that instead of a specially formed experiment under production conditions, the available material is used, processing it in the appropriate criteria ratios using the theory of similarity.

The concept of a model always requires the introduction of the concept of similarity, which is defined as a one-to-one correspondence between objects. The transition function from the parameters characterizing one of the objects to the parameters characterizing the other object is known.

The model provides similarity only for those processes that satisfy the similarity criteria.

Similarity theory is applied when:

a) finding analytical dependencies, relationships and solutions to specific problems;

b) processing the results of experimental studies in those cases where the results are presented in the form of generalized criterion dependencies;

c) creating models that reproduce objects or phenomena on a smaller scale, or differ in complexity from the original ones.

In physical modeling, the study is carried out on facilities that have a physical similarity, i.e. when the nature of the phenomenon is basically preserved. For example, links in economic systems are modeled by an electrical circuit/network. Physical modeling can be temporal, when phenomena that occur only in time are studied; spatio-temporal - when non-stationary phenomena distributed in time and space are studied; spatial, or object - when equilibrium states are studied that do not depend on other objects or time.

Processes are considered similar if there is a correspondence of similar values of the systems under consideration: sizes, parameters, position, etc.

The patterns of similarity are formulated as two theorems that establish relationships between the parameters of similar phenomena, without specifying ways to implement similarity when building models. The third or inverse theorem defines the necessary and sufficient conditions for the similarity of phenomena, requiring the similarity of the uniqueness conditions (separation of a given process from the variety of processes) and such a selection of parameters under which the similarity criteria containing the initial and boundary conditions become the same.

First theorem

Similar phenomena in one sense or another have the same combinations of parameters.

Dimensionless combinations of parameters that are numerically the same for all similar processes are called similarity criteria.

Second theorem

Anything complete equation process, written in a certain system of units, can be represented by a relationship between similarity criteria, i.e. an equation relating dimensionless quantities obtained from the parameters involved in the process.

The dependence is complete if all the relationships between the quantities included in it are taken into account. Such a dependence cannot change when the units of measurement of physical quantities are changed.

Third theorem

For the similarity of phenomena, the defining criteria of similarity must be correspondingly the same and the conditions for uniqueness must be similar.

The determining parameters are understood as criteria containing the parameters of processes and systems that can be considered independent in this task (time, capital, resources, etc.); unambiguity conditions are understood as a group of parameters, the values of which, given in the form of functional dependencies or numbers, single out a specific phenomenon from a possible variety of phenomena.

The similarity of complex systems consisting of several subsystems, similar in isolation, is provided by the similarity of all similar elements that are common to subsystems.

The similarity of nonlinear systems is preserved if the conditions for the coincidence of the relative characteristics of similar parameters that are nonlinear or variable are satisfied.

Similarity heterogeneous systems. The approach to establishing similarity conditions for inhomogeneous systems is the same as the approach to nonlinear systems.

Similarity with the probabilistic nature of the studied phenomena. All similarity condition theorems related to deterministic systems turn out to be valid under the condition that the probability densities of similar parameters, represented as relative characteristics, coincide. In this case, the dispersions and mathematical expectations of all parameters, taking into account the scales, should be the same for similar systems. An additional similarity condition is the fulfillment of the requirement of physical realizability of similar correlation between stochastically given parameters included in the uniqueness condition.

There are two ways to define similarity criteria:

a) reduction of process equations to a dimensionless form;

b) the use of parameters describing the process, while the equation of the process is unknown.

In practice, they also use another method of relative units, which is a modification of the first two. In this case, all parameters are expressed as fractions of certain basic values chosen in a certain way. The most significant parameters, expressed in fractions of the base ones, can be considered as similarity criteria that operate under specific conditions.

Thus, economic and mathematical models and methods are not only an apparatus for obtaining economic patterns, but also a widely used toolkit for practical problem solving in management, forecasting, business, banking and other sectors of the economy.

1.2 Modeling as a method of scientific knowledge

Scientific research is a process of developing new knowledge, one of the types of cognitive activity. For scientific research, various methods are used, one of which is modeling, i.e. study of any phenomenon, process or system of objects by constructing and studying its models. Modeling also means using models to define or refine characteristics and rationalize how newly constructed objects are constructed.

“Modeling is one of the main categories of the theory of knowledge; The best idea of modeling, in essence, is based on any method of scientific knowledge, both theoretical and experimental. Modeling began to be used in scientific research in ancient times and gradually covered all new and new areas of scientific knowledge: technical design, construction, architecture, astronomy, physics, chemistry, biology and, finally, social sciences. It should be noted that modeling methodologies have been developed for a long time in relation to specific sciences, independently of one another. Under these conditions, there was no unified system knowledge, terminology. Then the role of modeling began to be revealed as a universal method of scientific knowledge, as an important epistemological category. However, it is necessary to clearly understand that modeling is a method of indirect cognition with the help of some tool - a model that is placed between the researcher and the object of study. Modeling is used either when the object cannot be studied directly (the core of the Earth, the Solar system, etc.), or when the object does not yet exist (the future state of the economy, future demand, expected supply, etc.), or when the study requires a lot of time and means, or, finally, to test various kinds of hypotheses. Modeling is often part of overall process knowledge. Currently, there are many different definitions and classifications of models in relation to the problems of different sciences. Let us accept the definition given by the economist V.S. Nemchinov, known, in particular, for his works on the development of planned economy models: “A model is a means of highlighting any objectively operating system of regular connections and relationships that take place in the studied reality.”

The main requirement for models is the adequacy of reality, although the model reproduces the object or process under study in a simplified form. When building any model, the re-investigator should difficult task: on the one hand, to simplify reality, discarding everything secondary in order to focus on the essential features of the object, on the other hand, not to simplify to such a level as to weaken the connection of the model with reality. The American mathematician R. Bellman figuratively characterized such a problem as "the trap of oversimplification and the swamp of overcomplication."

In the process of scientific research, the model can work in two directions: from observations of the real world to theory and vice versa; i.e., on the one hand, the construction of a model is an important step towards the creation of a theory, on the other hand, it is one of the means of experimental research. Depending on the choice of modeling tools, material and abstract (sign) models are distinguished. Material (physical) models are widely used in engineering, architecture and other fields. They are based on obtaining a physical image of the object or process under study. Abstract models are not related to the construction of physical images. They are some intermediate link between abstract theoretical thinking and reality. Abstract models (they are called sign models) include numerical (mathematical expressions with specific numerical characteristics), logical (block diagrams of algorithms for calculations on a computer, graphs, diagrams, drawings). Models, in the construction of which the goal is to determine the following: the state of the object, which is the best from the point of view of a certain criterion, are called normative. Models designed to explain the observed facts or predict the behavior of the object are called descriptive.

The effectiveness of the application of models is determined by the scientific validity of their prerequisites, the ability of the researcher to highlight the essential characteristics of the modeling object, select the initial information, and interpret the results of numerical calculations in relation to the system.

1.3 Economic and mathematical methods and models

Like any modeling, economic and mathematical modeling is based on the principle of analogy, i.e. the possibility of studying an object through the construction and consideration of another, similar to it, but simpler and more accessible object, its model.

The practical tasks of economic and mathematical modeling are, firstly, the analysis of economic objects; secondly, economic forecasting, foreseeing the development of economic processes and the behavior of individual indicators; thirdly, the development of managerial decisions at all levels of management.

The description of economic processes and phenomena in the form of economic and mathematical models is based on the use of one of the economic and mathematical methods. The general name of the complex of economic and mathematical disciplines - economic and mathematical methods - was introduced in the early 60s by Academician V.S. Nemchinov. With a certain degree of conventionality, the classification of these methods can be represented as follows.

1. Economic and statistical methods:

economic statistics;

· math statistics;

multivariate analysis.

2. Econometrics:

· macroeconomic models;

theory of production functions

intersectoral balances;

national accounts;

· analysis of demand and consumption;

global modeling.

3. Operations research (methods for making optimal decisions):

Mathematical programming

· network management planning;

The theory of mass service;

· game theory;

the theory of decisions;

· Methods of modeling economic processes in industries and enterprises.

4. Economic cybernetics:

· system analysis of the economy;

The theory of economic information.

5. Methods of experimental study of economic phenomena:

methods of machine simulation;

· business games;

· methods of real economic experiment.

In economic and mathematical methods, various sections of mathematics, mathematical statistics, and mathematical logic are used. Computational mathematics, theory of algorithms and other disciplines play an important role in solving economic and mathematical problems. The use of the mathematical apparatus brought tangible results in solving the problems of analyzing the processes of expanded production, matrix modeling, determining the optimal growth rates of capital investments, optimal placement, specialization and concentration of production, selection problems best ways production, determining the optimal sequence of launching into production, optimal options for cutting industrial materials and compiling mixtures, the tasks of preparing production using network planning methods, and many others.

To solve standard problems, a clear goal is characteristic, the ability to develop procedures and rules for conducting calculations in advance.

There are the following prerequisites for using the methods of economic and mathematical modeling.

The most important of these are, firstly, high level knowledge of economic theory, economic processes and phenomena, methodology of their qualitative analysis; secondly, a high level of mathematical training, mastery of economic and mathematical methods.

Before starting to develop models, it is necessary to carefully analyze the situation, identify goals and relationships, problems that need to be solved, and the initial data for their solution, introduce a notation system, and only then describe the situation in the form of mathematical relationships.

Conclusion

A characteristic feature of scientific and technological progress in developed countries is the growing role of economic science. The economy comes to the fore precisely because it determines to a decisive extent the effectiveness and priority of scientific and technological progress directions and opens wide ways to realize economically profitable achievements.

The use of mathematics in economics gave impetus to the development of both economics itself and applied mathematics, in terms of methods of economic and mathematical models. The proverb says: "Measure seven times, cut once." The use of models is time, effort, material resources. In addition, calculations based on models are opposed to volitional decisions, since they allow you to evaluate in advance the consequences of each decision, discard unacceptable options and recommend the most successful ones.

At all levels of management, in all industries, methods of economic and mathematical modeling are used. Let us conditionally single out the following areas of their practical application, in which a large economic effect has already been obtained.

The first direction is forecasting and long-term planning. The rates and proportions of economic development are predicted, on their basis the rates and factors of growth of national income, its distribution for consumption and accumulation, etc. are determined. An important point is the use of economic and mathematical methods not only in the preparation of plans, but also in the operational management of their implementation.

The second direction is the development of models that are used as a tool for coordinating and optimizing planned decisions, in particular, these are inter-industry and inter-regional balances of production and distribution of products. According to the economic content and nature of information, cost and natural product balances are distinguished, each of which can be reporting and planned.

The third direction is the use of economic and mathematical models at the industry level (calculation of the industry's optimal plans, analysis using production functions, forecasting the main production proportions of the industry development). To solve the problem of location and specialization of an enterprise, optimal attachment to suppliers or consumers, etc., two types of optimization models are used: in some, for a given volume of production, it is required to find an option for implementing the plan at the lowest cost, in others, it is required to determine the scale of production and the structure of products in order to obtain the maximum effect. In continuation of the calculations, a transition is made from statistical models to dynamic ones and from statistical models to dynamic ones, and from modeling individual industries to optimizing multi-industry complexes. If earlier there were attempts to create a single model of the industry, now the most promising is the use of complexes of models that are interconnected both vertically and horizontally.

The fourth direction is economic and mathematical modeling of the current and operational planning of industrial, construction, transport and other associations, enterprises and firms. The area of practical application of the models also includes subdivisions Agriculture, trade, communications, health care, nature protection, etc. In mechanical engineering, a large number of various models are used, the most “adjusted” of which are optimization ones, which make it possible to determine production programs and the most rational options for using resources, distribute the production program in time and effectively organize the work of intra-plant transport, significantly improve the loading of equipment and reasonably organize product control, etc.

The fifth direction is territorial modeling, which was initiated by the development of intersectoral balance sheets for some regions in the late 1950s.

As the sixth direction, one can single out the economic and mathematical modeling of logistics, including the optimization of transport and economic relations and the level of reserves.

The seventh direction includes models of functional blocks of the economic system: population movement, personnel training, the formation of cash income and demand for consumer goods, etc.

Economic and mathematical methods acquire a particularly large role as information technologies are introduced in all areas of practice.

Literature

1. Wentzel E.S. Operations research. - M: Soviet radio, 1972.

2. Greshilov A.A. How to make the best decision in the real world. - M.: Radio and communication, 1991.

3. Kantorovich L.V. Economic calculation of the best use of resources. - M.: Nauka, USSR Academy of Sciences, 1960.

4. Kofman A., Debazey G. Network planning methods and their application. – M.: Progress, 1968.

5. Kofman A., Fore R. Let's take up the study of operations. – M.: Mir, 1966.

Mathematical modeling in economics. Chelyabinsk Institute of Communications

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Introduction

1. Mathematical models

1.1 Classification of economic and mathematical models

2. Optimization modeling

2.1 Linear programming

2.1.1 Linear programming as a tool for mathematical modeling of the economy

2.1.2 Examples of linear programming models

2.1.3 Optimal resource allocation

Conclusion

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