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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 largely occurs due to the division of mathematics into a number of independent areas. Mathematics has become for many branches of knowledge not only a tool of 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 areas of human activity would not be possible. 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 incorporated a large number of mathematical methods.

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

Studying the economic applications of mathematical disciplines that form the basis of modern economic mathematics allows you to acquire some solution skills economic tasks 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 important direction improving economic analysis. Specifying data or presenting it in the form of a mathematical model helps to choose the least labor-intensive solution path and increases the efficiency of analysis.

All economic problems solved using linear programming are distinguished by alternative solutions and certain limiting conditions. Solving such a problem means choosing the best, optimal one from all the permissibly possible (alternative) options. The importance and value of using the linear programming method in economics is that the optimal option is selected from a fairly significant number alternative options.

The most significant points when formulating and solving economic problems in the form of a mathematical model are:

· adequacy of the economic and mathematical model of reality;

· analysis of patterns corresponding to this process;

· identification of methods by which the problem can be solved;

· analysis of the results obtained or summing up.

Economic analysis means, first of all, factor analysis.

Let y=f(x i) be some function characterizing 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 between the indicator y and a set of factors is specified. 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 influence of labor productivity and the number of workers on the volume of products produced; analysis of the impact of the profit margin of fixed production assets and standardized working capital on the level of profitability; analysis of the impact of borrowed funds on the agility 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 necessary to functionally link a number of economic characteristics, i.e. construct a function that contains the main quality of all economic indicators under consideration.

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

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 changes in the process F, containing the main characteristics of all indicators x 1,x 2,…,x n

The main point in economic analysis is the determination of the criterion by which various solution options will be compared.

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

· availability of choice;

· choosing an option according to a certain principle.

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

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

Criteria-based choice consists of accepting a certain criterion and comparing possible options according to this criterion. The option for which the adopted 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 whose solution boils down to finding the maximum or minimum objective function, is called an extremal problem.

Management tasks are associated with 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, amount of profit, etc. are taken as the objective function. Limitations usually relate to human material and financial resources.

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

Let's consider the classification of some basic optimization problems implemented by management in production.

Classification of optimization problems by control function:

Control function	Optimization problems	Class of economic and mathematical models
Technical and organizational preparation of production	Product composition modeling; Optimization of the composition of grades, batches, mixtures; Optimization of cutting of sheet material, rolled products; Optimization of resource allocation in network models of work packages; Optimization of plant layouts, production facilities 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 modeling Dynamic programming Nonlinear programming
Technical and economic planning	Construction of a master plan and forecasting of enterprise development indicators; Optimization of the order portfolio and production program; Optimization of distribution of the production program across planning periods.	Matrix balance models “Input-output” Correlation- regression analysis Extrapolation of trends Linear programming
Operational management of main production	Optimization of calendar and planning standards; Calendar tasks; Optimization of standard plans; Optimization of short-term production plans.	Nonlinear programming Simulation modeling Linear programming Integer programming

Table 1.

Combination various elements 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 and types of economic and mathematical models necessary 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 relationships between different parameters, the indicators of the object at that time. Discrete models reflect the state of the control object at separate, fixed points in time. Simulation models are economic and mathematical models used to simulate controlled economic objects and processes using information and computer technology. Based on the type of mathematical apparatus used in the models, there are economic-statistical models, linear and nonlinear programming models, matrix models, and 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 a managed economic object depends, and on the other, parameters of the object’s state that depend on these factors. If the factors are known, then the model allows us to determine the required 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 sheet models. Balance sheet 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 quantity of products manufactured by individual economic objects and the total demand for these products. 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 sheet models:

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

· Interindustry balances;

· Matrix balance sheets of enterprises and firms.

Optimization models. A large class of economic and mathematical models form optimization models that allow you to select the best optimal option from all solutions. IN mathematical content optimality is understood as achieving the 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 achieving the maximum target effect. Mathematical programming was developed based on solving the problem of optimal cutting of plywood sheets, which ensures the most complete use of the material. Having posed such a problem, the famous Russian mathematician and economist Academician L.V. Kantorovich was considered 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 carried out 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 to lay the foundations of linear programming is John von Neumann, widely famous mathematician and the physicist who proved the fundamental theorem about matrix games. Among domestic scientists, a great contribution to the theory of linear optimization was made by Nobel Prize laureate 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 that 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, in connection with the development of industrial production, transport, the agro-industrial complex, and the banking sector, the traditional results of mathematical analysis turned out to be insufficient. The needs of practice and the development of computer technology have led to the need to determine optimal solutions when analyzing 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 using mathematical tools.

The art of mathematical modeling is to take into account the widest possible range of factors influencing the behavior of an object, using as simple relationships as possible. It is for this reason that the modeling process is often multi-stage in nature. First, a relatively simple model is built, then its research is carried out, making it possible to understand which of the integrating properties of the object are not captured by a given formal scheme, after which, by complicating the model, its greater adequacy to reality is ensured. Moreover, 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 amount economic processes is described quite completely by linear models, and therefore, linear programming as an apparatus that allows you to find conditional extremum on a 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 and mathematical.

The problem of cutting materials. Material of one sample in the amount of d units is received for processing. It is required to make k different components from it 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 j-th product (j = 1,...,k).

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

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

Considering that total quantity material is equal to the sum of its units, cut in different ways, we get:

The completeness condition will be expressed by the equations:

It's obvious that

x i 0 (i=1,…,n)(3)

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

Table 1

Let us denote by x i the number of logs sawn using the i-th method (i = 1.2, 3, 4); x is the number of sets of beams.

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 the following form x > max with restrictions:

x 1 +x 2 +x 3 +x 4 =200

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

The problem of choosing the optimal production program for an enterprise. Let an enterprise 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) at 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 production of one unit of product of type i;

Zp are conditionally constant costs, which we will assume to be 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 use of equipment over the interval must be met.

Let us denote by Lj(j = l,...,M) the volume of reserves of material and raw materials of type j, and by φ k (k = 1,..., N) the time during which equipment of the type can be used k. We know the consumption of material and raw material resources 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 production of one unit of product of type i (i = 1,..., n; k = 1,..., N). Let m k denote the number of units of equipment of type k (k=l,...,N).

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

Constraints on production capacity are given by the following inequalities

In addition, the variables

x i ?0 i=1,…,n (7)

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

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

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

Diet problem. Let's consider the problem of compiling a per capita food ration of the minimum cost, which would contain certain nutrients in the required volumes. We will assume that there is a known list of products of n items (bread, sugar, butter, milk, meat, etc.), which we will denote by the letters F 1,...,F n. In addition, food characteristics (nutrients) such as proteins, fats, vitamins, minerals and others. Let us denote these components by the letters N 1,...,N m. Let us assume that for each product F i the quantitative content of the above components in one unit of product is known (i = 1,...,n). In this case, you can create 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 us 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 a month x, units (kilograms) of product F 1, x 2 units of product F 2, etc. It is not difficult to calculate the amount of 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 product F 1 according to the nutritional matrix contain a 11 x 1 units of component N 1; to this quantity is added a portion of a 12 x 2 substance N 1 of x 2 units of product F 2, etc. Similarly, you can determine the amount of all other substances N i in the prepared diet (x 1,..., x n).

Let us assume that there are certain physiological requirements regarding required quantity nutrients in N i (i/ = 1,..., N) within the planned period. Let these requirements be specified 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 restrictions:

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

x 1 ?0; x 2 ?0;… x n ?0; (9)

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

Let prices for products F 1,...,F n be equal to 1,...,c n, respectively

Therefore, the cost of the entire ration 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 constraints (8) and (9) the one for which the objective function (10) takes the minimum value.

Transport problem. There are m points S 1 ,..., S m for the production of a homogeneous product (coal, cement, oil, etc.), and the volume of production at point 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 draw up a transportation plan from points S i (i = 1,...,m) to points Q j (j = 1,..., n) in order to satisfy the needs for product b j, minimizing transport 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, transport costs are equal to c ij x ij.

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

x ij ?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 (x ij) that satisfy constraints (11), find a set that minimizes (12).

2.1.3 Optimal resource allocation

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

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

Generally speaking, the overwhelming 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 using 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 a DP model are met in the problem. Let's describe typical task distribution of resources in general.

Problem 1. There is an initial amount of funds that must be distributed over n years between s enterprises. Funds (k=1, 2,…,n; i=1,…, s) allocated in kth year to the i-th enterprise, generate income in the amount and return in amount 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 method of resource distribution (the amount of funds allocated to each enterprise in each plan year) so that the total income from s enterprises for n years is maximum.

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

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

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

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

It is required to determine ns non-negative variables that satisfy conditions (4.2) and maximize function (4.1).

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

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 represents an optimization problem for a function depending on s variables. Thus, a problem with ns variables is reduced to a sequence of n problems, each of which contains s variables. In this general setting the problem is still complex (due to its multidimensionality) and in this case it is impossible to simplify it by considering it as an ns-step problem. In fact, let's try to do this. Let's number the steps by enterprise numbers, first in the 1st year, then in the 2nd, etc.:

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

During the kth year, the state "at the beginning of any step s(k-1)_+i (i=1,2,…,s) will be determined from the previous state using simple equation. However, after a year, i.e. by the beginning of the next year, funds will need to be added to the available funds 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 for the previous year. As a result, we get a process with an aftereffect. To eliminate aftereffects, several state parameters have to be introduced; the task at each step remains complex due to its multidimensionality.

Problem 2. The activities of two enterprises (s=2) are planned for n years. The initial funds are: Funds x invested in enterprise I generate income f 1 (x) by the end of the year and are returned in the same amount; funds x invested in enterprise II generate income f 2 (x) and are returned in the same amount. At the end of the year, all remaining funds are redistributed between enterprises I and II, no new funds are received and income is not invested in production.

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

We will consider the process of distributing funds as an n-step one, in which the step number corresponds to the year number. The 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 enterprises I and II, respectively. Since the funds are redistributed in full every year, then). For each step the problem becomes one-dimensional. Let us denote by, then

The efficiency indicator of the kth 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 over 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 us write down 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 choosing the step Dx in changing the control variables. This step can be specified or determined based on the required accuracy of calculations and the accuracy of the source data. IN general case This task is complex and requires interpolation from tables in previous calculation steps. Sometimes a preliminary analysis of the equation of state allows one to select the appropriate step Dx, as well as set the limiting values for which tabulation must be performed at each step.

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

Task 3. Draw up an optimal plan for the annual distribution of funds between two enterprises over 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 are returned 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. 1:

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

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

A functional equations in the form

Let's try to determine the maximum possible values for which it is necessary to tabulate at the kth 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 change interval coincide with the table one, i.e. Dx = 50. Let's create a table of the total profit at this step:

This will make further calculations easier. Since the cells located diagonally in the table correspond to the same value indicated in the 1st row (1st column) of the table. 2. The 2nd line of the table contains the values of f 1 (x), and the 2nd column contains the values of 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), located 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 carry out conditional optimization according to the usual scheme. 3rd step. Basic equation (4.9)

As stated above, . Let's look at the numbers on the diagonals corresponding to =0; 50; 100; 150 and choose the largest on each diagonal. This is In the 1st line we find the corresponding conditional optimal control. At the 3rd step, we will place the optimization data in the main table (Table 4). It introduces the column Dx, which is subsequently used for 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 can be obtained equal to Zmax=99,l. 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 explained by a linear interpolation error.

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

Conclusion

In this course work The types of mathematical models used in economics and management, as well as their classification, are considered.

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

The principle of constructing linear programming models is studied, and models of the following problems are also given:

· Problem of cutting materials;

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

· Diet problem;

· Transport task.

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

Three tasks were selected to build models:

· The problem of optimal resource allocation;

· The problem of optimal inventory management;

· Replacement problem.

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

References:

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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Introduction

Modeling in scientific research began to be used in ancient times and gradually captured new areas of scientific knowledge: technical design, construction and architecture, astronomy, physics, chemistry, biology and, finally, social science. Great success and recognition in almost all branches of modern science brought the modeling method of the twentieth century. However, modeling methodology has been developed independently by individual sciences for a long time. There was no unified system of concepts, no unified terminology. Only gradually the role of modeling as a universal method of scientific knowledge began to be realized.

The term "model" is widely used in various fields of human activity and has many semantic meanings. Let us consider only such “models” that are tools for obtaining knowledge.

A model is a material or mentally imagined object that, in the process of research, replaces the original object so that its direct study provides new knowledge about the original object.

Modeling refers to the process of constructing, studying and applying models. It is closely related to such categories as abstraction, analogy, hypothesis, etc. The modeling process necessarily includes the construction of abstractions, inferences by analogy, and the construction of scientific hypotheses.

The main feature of modeling is that it is a method of indirect cognition using proxy objects. The model acts as a kind of cognition tool that the researcher places between himself and the object and with the help of which he studies the object of interest to him. It is this feature of the modeling method that determines the specific forms of using abstractions, analogies, hypotheses, and other categories and methods of cognition.

The need to use the modeling method is determined by the fact that many objects (or problems related to these objects) are either impossible to directly study, or this research requires a lot of time and money.

Modeling is a cyclical process. This means that the first four-step cycle may be followed by a second, third, etc. At the same time, knowledge about the object under study is expanded and refined, and the initial model is gradually improved. Deficiencies discovered after the first modeling cycle, due to poor knowledge of the object and errors in model construction, can be corrected in subsequent cycles. Thus, the modeling methodology contains great opportunities for self-development.

The goal of mathematical modeling of economic systems is to use mathematical methods to most effectively solve problems arising in the field of economics, using, as a rule, modern computer technology.

The process of solving economic problems is carried out in several stages:

Substantial (economic) formulation of the problem. First you need to understand the task and clearly formulate it. At the same time, objects that relate to the problem being solved are also determined, as well as the situation that needs to be realized as a result of its solution. This is the stage of meaningful formulation of the problem. In order for a problem to be described quantitatively and to use computer technology in solving it, it is necessary to carry out a qualitative and quantitative analysis of objects and situations related to it. In this case, complex objects are divided into parts (elements), the connections of these elements, their properties, quantitative and qualitative values of properties, quantitative and logical relationships between them, expressed in the form of equations, inequalities, etc. are determined. This is the stage of system analysis of the problem, as a result of which the object is presented in the form of a system.

The next stage is the mathematical formulation of the problem, during which a mathematical model of the object is constructed and methods (algorithms) are determined for obtaining a solution to the problem. This is the stage of system synthesis (mathematical formulation) of the problem. It should be noted that at this stage it may turn out that the previously conducted system analysis has led to a set of elements, properties and relationships for which there is no acceptable method for solving the problem, as a result we have to return to the system analysis stage. As a rule, problems solved in economic practice are standardized, system analysis is carried out based on a well-known mathematical model and an algorithm for solving it, the problem is only in choosing an appropriate method.

The next step is to develop a program for solving the problem on a computer. For complex objects consisting of a large number of elements with a large number of properties, it may be necessary to compile a database and tools for working with it, methods for retrieving data necessary for calculations. For standard tasks it is not development that is carried out, but the selection of a suitable package application programs and database management systems.

At the final stage, the model is operated and results are obtained.

Thus, solving the problem includes the following steps:

2. System analysis.

3. System synthesis (mathematical formulation of the problem)

4. Development or selection of software.

5. Solving the problem.

Consistent use of operations research methods and their implementation on modern information and computing technology makes it possible to overcome subjectivity and eliminate so-called volitional decisions based not on a strict and accurate account of objective circumstances, but on random emotions and personal interest of managers at various levels, who, moreover, do not can coordinate these volitional decisions.

System analysis allows you to take into account and use in management all available information about the managed object, to coordinate decisions made from the point of view of an objective, rather than subjective, efficiency criterion. Saving on calculations when controlling is the same as saving on aiming when firing. However, a computer not only allows you to take into account all the information, but also relieves the manager of unnecessary information, and bypasses all the necessary information, presenting him only with the most generalized information, the quintessence. Systematic approach in economics it is effective on its own, without the use of a computer, as a research method, while it does not change previously discovered economic laws, but only teaches how to best use them.

The complexity of processes in the economy requires the decision maker to be highly qualified and have extensive experience. This, however, does not guarantee errors; mathematical modeling allows you to give a quick answer to the question posed, or carry out experimental studies that are impossible or require large costs and time on a real object.

Mathematical modeling allows you to make an optimal, that is, the best decision. It may differ slightly from correctly decision taken without the use of mathematical modeling (about 3%). However, with large production volumes, such a “minor” error can lead to huge losses.

The mathematical methods used to analyze the mathematical model and make the optimal decision are very complex and their implementation without the use of a computer is difficult. As part of the programs Excel And Mathcad There are tools that allow you to carry out mathematical analysis and find the optimal solution.

Part No. 1 "Study of the mathematical model"

Statement of the problem.

The company has the ability to produce 4 types of products. To produce a unit of each type of product, it is necessary to spend a certain amount of labor, financial, and raw material resources. In stock limited quantity each resource. Sales of a unit of production bring profit. The parameter values are given in Table 1. Additional condition: financial costs for the production of products No. 2 and No. 4 should not exceed 50 rubles. (each type).

Based on mathematical modeling by means Excel determine what products and in what quantities it is advisable to produce from the point of view of obtaining the greatest profit, analyze the results, answer questions, and draw conclusions.

To study various economic phenomena economists use simplified formal descriptions of them, called economic models . When constructing economic models, essential factors are eliminated and details that are not essential for solving the problem are discarded.

Economic models may include the following models:

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

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

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

The nature of the models may vary. Models are divided into: real, symbolic, verbal and tabular description, etc.

Economic and mathematical model

In managing business processes highest value have first of all economic and mathematical models, often combined into model systems.

Economic and mathematical model(EMM) - a mathematical description of an economic object or process for the purpose of studying and managing them. This mathematical notation economic problem being solved.

Main types of models

Extrapolation models
Factor econometric models
Optimization models
Balance models, Inter-Industry Balance (IOB) model
Expert assessments
Note that game theory
Network models
Models of queuing systems

Economic and mathematical models and methods used in economic analysis

Currently, in the analysis of the economic activities of organizations, they are increasingly used. mathematical methods research. This helps to improve economic analysis, deepen it and increase its effectiveness.

As a result of the use of mathematical methods, a more complete study of the influence of individual factors on the general economic indicators of organizations' activities is achieved, the time required for analysis is reduced, the accuracy of economic calculations is increased, and multidimensional analytical problems are solved that cannot be performed by traditional methods. In the process of using economic and mathematical methods in economic analysis, the construction and study of economic and mathematical models are carried out, describing the influence of individual factors on the general economic indicators of organizations.

There are four main types of economic and mathematical models used in analyzing the influence of individual factors:

additive models;
multiplicative models;
multiple models;
mixed models.

Additive Models can be defined as algebraic sum individual indicators. It must be remembered that such models can be characterized using the following formula:

Example additive model there will be a balance of marketable products.

Multiplicative models can be defined as the product of individual factors.

It is important to note that one example of such a model could be a two-factor model, expressing the relationship between the volume of output, the number of units of equipment used and output per unit of equipment:

P = K V,

P— volume of production;
TO— number of equipment units;
IN— production output per unit of equipment.

Multiple models— ϶ᴛᴏ correlation of individual factors. It is worth noting that they are characterized by the following formula:

OP = x/y

Here OP is a general economic indicator that is influenced by individual factors x And y. An example of a multiple model is a formula expressing the relationship between the duration of turnover of current assets in days, the average value of these assets for a given period and one-day sales volume:

P = OA/OP,

P- duration of turnover;
OA— average value of current assets;
OP— one-day sales volume.

Finally, mixed models— ϶ᴛᴏ a combination of the types of models we have already considered. For example, such a model can describe the return on assets indicator, the level of which is influenced by three factors: net profit (NP), the value of non-current assets (VA), the value of current assets (CA):

R a = PE / VA + OA,

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

Thus, first you should build an economic and mathematical model that describes the influence of individual factors on the general economic indicators of the organization’s activities. It's important to know that widespread in the analysis of economic activity received multifactor multiplicative models, since they make it possible to study the influence of a significant number of factors on generalizing indicators and thereby achieve greater depth and accuracy of analysis.

After this, you need to choose a method for solving this model. Traditional methods: method of chain substitutions, methods of absolute and relative differences, balance method, index method, as well as methods of correlation-regression, cluster, analysis of variance, etc. Along with these methods and methods, specific ones can be used in economic analysis mathematical methods and methods.

Integral method of economic analysis

It is important to note that one of these methods (methods) will be integral. It is worth noting that it is used in determining the influence of individual factors using multiplicative, multiple, and mixed (multiple-additive) models.

When using the integral method, it is possible to obtain more substantiated results for calculating the influence of individual factors than when using the method of chain substitutions and its variants. The method of chain substitutions and its variants, as well as the index method, have significant disadvantages: 1) the results of calculations of the influence of factors depend on the accepted sequence of replacing the basic values of individual factors with actual ones; 2) the additional increase in the general 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, the increase is divided equally between all factors.

The integral method establishes a general approach to solving models of various types, regardless of the number of elements that are included in a given model, as well as regardless of the form of connection between these elements.

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

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

d y / d x = const

When using the integral method, calculus definite integral for a given integrand and a given integration interval is carried out using an existing standard program using modern computer technology.

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

ΔZ(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 a given period. Examples of such tasks include analyzing the implementation of business plans or analyzing changes in economic indicators compared to the previous period. The dynamic type of tasks occurs in the presence of information about changes in the analyzed factors during a given period. This type of problem includes calculations related to the study of time series of economic indicators.

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

Logarithm method

In addition to this method, the logarithm method (method) is also used in analysis. It is worth noting that it is used when conducting factor analysis when multiplicative models are solved. The essence of the method under consideration is essentially that when it is used, there is a logarithmically proportional distribution of the quantity joint action factors between the latter, that is, this value is distributed among the factors in proportion to the share of influence of each individual factor on the sum of the general indicator. With the integral method, the mentioned value is distributed equally among the factors. Therefore, the logarithm method makes calculations of the influence of factors more reasonable compared to the integral method.

In the process of logarithms, they are not used absolute values growth of economic indicators, as occurs with the integral method, but relative ones, that is, indices of changes in these indicators. For example, a general 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 general economic indicator. Thus, the influence of the first factor can be determined by the following formula:

Δf x = Δf log(x 1 / x 0) / log(f 1 / f 0)

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

Δf y = Δf log(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 = Δf log(z 1 / z 0)/ log(f 1 / f 0)

Based on all of the above, we come to the conclusion that 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.

Differential calculus method

When carrying out factor analysis, the method of differential calculus is also used. The latter assumes that the overall change in the function, that is, the generalizing indicator, is divided into individual 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. It is appropriate to note that we will determine the influence of individual factors on the general indicator, using as an example a function of two variables.

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

Let's explain individual elements϶ᴛᴏth formula:

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

Δx = (x 1 - x 0)— the magnitude of change in one factor;

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

- an infinitesimal quantity of a higher order than

IN in this example influence of individual factors x And y to change function Z(general indicator) is calculated as follows:

ΔZ x = δZ / δx Δx; ΔZ y = δZ / δy · Δy.

The sum of the influence of both these factors is the main one, linear with respect to the increment this factor part of the increment of the differentiable function, that is, the general indicator.

Participation method

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

We will proceed from the assumption that we determine the influence of three factors - A,b And With to a general indicator y. Then for the factor, and determining its share and multiplying it by the total amount of change in the generalizing indicator can be done using the following formula:

Δy a = Δa/Δa + Δb + Δc*Δy

For factor b, the formula under consideration will have the following form:

Δy b =Δb/Δa + Δb +Δc*Δy

Finally, for factor c we have:

Δy c =Δc/Δa +Δb +Δc*Δy

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

Linear programming method

See further: Linear programming method

Note that queuing theory

See further: Note that queuing theory

Note that game theory

Game theory is also used. Just like queuing theory, game theory is one of the branches of applied mathematics. Note that game theory studies the optimal solutions possible in gaming situations. This includes 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 can be used, which are based on a system of linear equations and inequalities, iterative methods, as well as methods for reducing this problem to a specific system of differential equations.

It is important to note that one of the economic and mathematical methods used in the analysis of the economic activities of organizations is the so-called sensitivity analysis. The material was published on http://site
This method often used in the process of analyzing investment projects, as well as for the purpose of predicting the amount of profit remaining at the disposal of a given organization.

For optimal planning and forecasting of an organization’s activities, it is extremely important to foresee in advance those changes that may occur in the future with the analyzed economic indicators.

For example, you should predict in advance changes in the values of those factors that affect the profit margin: the level of purchase prices for purchased material resources, the level of sales prices for the products of a given organization, changes in customer demand for these products.

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

For example, they establish by what amount profit will change in the future, subject to a change in the number of products sold per unit. By doing this, we analyze the sensitivity of net profit to changes in one of the factors influencing it, that is, in this case, the sales volume factor.
It is worth noting that the remaining factors influencing the amount of profit will remain unchanged. It is also possible to determine the amount of profit if the influence of several factors changes simultaneously in the future. Thus, sensitivity analysis makes it possible to establish the strength of the response of a general economic indicator to changes in individual factors that influence this indicator.

Matrix method

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

Network planning method

See further: Network planning method

Extrapolation Analysis

In addition to the methods discussed, extrapolation analysis is also used. It is worth noting that it contains a consideration of changes in the state of the analyzed system and extrapolation, that is, the extension of the existing characteristics of the system for future periods. In the process of carrying out this type of analysis, the following main stages can be distinguished: primary processing and transformation of the initial series of available data; choosing the type of empirical functions; determination of the main parameters of these functions; extrapolation; establishing the degree of reliability of the analysis performed.

Economic analysis also uses the principal component method. It is worth noting that they are used for comparative analysis individual components, that is, the parameters of the analysis of the organization’s activities. The main components are the most important characteristics linear combinations of components, that is, parameters of the analysis that have the most significant dispersion values, namely, the largest absolute deviations from the average values.

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A model is, first of all, a simplified representation of a real object or phenomenon that preserves its basic, essential features. The process of developing the model itself, i.e. modeling can be carried out in various ways, of which the most common are physical and mathematical modeling. However, using each of these methods, different models can be obtained, since their specific implementation depends on which features of the real object the creator of the model considers to be the main ones. Therefore, in engineering practice and in scientific research, various models of the same object can be used, since their diversity allows a more thorough study of the most diverse aspects of a real object or phenomenon.

Widespread in engineering practice and natural sciences physical models, which differ from the object being studied, as a rule, in smaller sizes, and serve to conduct experiments, the results of which are used to study the original object and to draw conclusions about the choice of one or another option for its development or design, if we're talking about about the engineering construction project. The path of physical modeling turns out to be unproductive for the analysis of economic objects and phenomena. Due to this The main method of modeling in economics is the method of mathematical modeling , i.e. description of the main features of a real process using a system of mathematical formulas.

How do we proceed when creating a mathematical model? What are the types of mathematical models? What features arise when modeling economic phenomena? Let's try to clarify these issues.

When creating a mathematical model, we start from a real problem. First, the situation is clarified, important and minor characteristics, parameters, properties, qualities, connections, etc. are identified. Then one of the existing mathematical models is selected or a new mathematical model is created to describe the object being studied.

Notations are introduced. The constraints that the variables must satisfy are written down. The goal is determined - the target function is selected (if possible). The choice of objective function is not always straightforward. There may be situations when you want this, that, and much more... But different goals lead to various solutions. In this case, the problem belongs to the class of multicriteria problems.

Economics is one of the most complex fields of activity. Economic objects can be described by hundreds and thousands of parameters, many of which are random. In addition, the human factor operates in the economy.

Predicting human behavior can be difficult, sometimes impossible.

The complexity of a system of any nature (technical, biological, social, economic) is determined by the number of elements included in it, the connections between

these elements, as well as the relationships between the system and the environment. The economy has all the hallmarks of a very complex system. It unites a huge number of elements, is distinguished by a variety of internal connections and connections with other systems (natural environment, economic activities of other entities, social relations etc.). IN national economy natural, technological, social processes, objective and subjective factors. The economy depends on the social structure of society, on politics and on many, many other factors.

The complexity of economic relations was often used to justify the impossibility of modeling the economy and studying it using mathematics. And yet, modeling economic phenomena, objects, and processes is possible. You can model an object of any nature and any complexity. To model the economy, not one model is used, but a system of models. This system contains models that describe different aspects of the economy. There are models of a country's economy (they are called macroeconomic), there are models of economic models for a separate enterprise, or even a model of one economic event (they are called microeconomic). When compiling a model of the economy of a complex object, so-called aggregation is performed. In this case, a number of related parameters are combined into one parameter, thereby reducing the total number of parameters. At this stage, experience and intuition play an important role. You can select not all characteristics as parameters, but the most important ones.

After a mathematical problem has been compiled, a method for solving it is chosen. At this stage, as a rule, a computer is used. After receiving a solution, it is compared with reality. If the results obtained are confirmed by practice, then the model can be applied and forecasts can be made with its help. If the answers obtained based on the model do not correspond to reality, then the model is no good. It is necessary to create a more complex model that better matches the object being studied.

Which model is better: simple or complex? The answer to this question cannot be unambiguous.

If the model is too simple, then it does not fit well real object. If the model is too complex, then it may turn out that even though a good model exists, we are not able to obtain an answer based on it. There may be a good model and an algorithm for solving the corresponding problem. But the solution time will be so long that all other advantages of the model will be crossed out. Therefore, when choosing a model, you need a “golden mean”.

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"Economic and mathematical methods and modeling"

Introduction

1. Mathematical modeling in economics

1.1 Development of modeling methods

1.2 Modeling as a method of 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 15th century. Leonardo da Vinci worked on the justification of modeling methods: he made an attempt to derive general patterns of similarity, used mechanical and geometric similarity in analyzing 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.

Leonardo da Vinci's ideas about mechanical similarity were developed by Galileo in the 17th century and were used in the construction of galleys in Venice.

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

The first strict scientific formulations of the conditions of similarity and clarification 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 checked and approved by L. Euler.

1. Mathematical modeling in economics

1.1 Development of modeling methods

The successes of mathematics stimulated the use of formalized methods in non-traditional areas of science and practice. Thus, O. Cournot (1801–1877) introduced the concept of supply and demand functions, and even earlier the German economist I.G. Thunen (1783–1850) began to apply mathematical methods in economics and proposed the theory of production location, anticipating the theory of marginal productivity of labor. The pioneers of using the modeling method include F. Quesnay (1694–1774), the author of the “Economic Table” (Quesnay’s 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 refers to the ability to satisfy human needs, which is the basis of goods and prices. Jevons distinguished:

– abstract utility, which is devoid of concrete form;

– utility in general as the pleasure a person receives from consuming goods;

– marginal utility – the least utility among the entire 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 a price system in which the total demand for all goods and markets would be equal to the total supply. According For Walras, the 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 price system
other goods.

The end of the 19th and beginning of the 20th centuries were marked by the widespread use of mathematics in economics. In the 20th 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 an increasingly high level of formalization, intellectualization and the 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 (statistical test method), methods of reliability theory, random processes, Markov chains,modeling and similarity theory.

Formalized, simplified descriptions of economic phenomena are called economic models. Models are used to detect the most significant factors in the phenomena and processes of the functioning of economic objects, to make a forecast of 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 connections;

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

e) assessing the relative weights of objects and connections using the expert method;

e) building the system in the most important elements in verbal, graphic or symbolic form;

g) collecting the necessary data and checking the accuracy of the modeling 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 of either additional research for the possible replacement of some data with others, or special experiments to obtain missing data.

Mathematical models used in economics 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 GDP, consumption, investment, employment, 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 among the others. The main objects of application of modeling in microeconomics are supply, demand, elasticity, costs, production, competition, consumer choice, pricing, monopoly theory, theory of the firm, etc.

By nature, models can be theoretical (abstract), applied, static, dynamic, deterministic, stochastic, equilibrium, optimization, full-scale, physical.

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

Application models allow you to evaluate the operating parameters of an economic entity. They operate with numerical knowledge of economic variables. Most often, these models use statistical or actual observed data.

Equilibrium models describe 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 state of equilibrium at the micro level is maintained.

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 presence of a functional connection between the studied characteristics and, as a rule, use the apparatus of differential equations.

Full-scale modeling carried out realistically existing facilities under specially selected conditions, for example, an experiment conducted during the production process at an existing enterprise, while meeting the objectives of the production itself. The natural research method arose out of needs material production then, when science did not yet exist. It coexists on a par with natural science experiment at the present time, demonstrating the unity of theory and practice. A type of full-scale modeling is modeling by generalizing production experience. The difference is that instead of a specially formed experiment in production conditions, they use the available material, processing it in the appropriate criterion relationships, 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 function of transition from parameters characterizing one of the objects to parameters characterizing another object is known.

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

Similarity theory is used when:

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

b) processing the results of experimental studies in 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, research is carried out on installations that have physical similarity, i.e. when the nature of the phenomenon is essentially preserved. For example, connections in economic systems are modeled by an electrical circuit/network. Physical modeling can be temporary, in which phenomena occurring only in time are studied; spatiotemporal - when non-stationary phenomena distributed in time and space are studied; spatial, or object - when studied equilibrium states, independent of other objects or time.

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

Similarity laws are formulated in the form of two theorems that establish relationships between the parameters of similar phenomena, without indicating ways to implement similarity when constructing models. The third or converse theorem determines the necessary and sufficient conditions similarity of phenomena, requiring similarity of uniqueness conditions (selecting a given process from the variety of processes) and such a selection of parameters under which the similarity criteria containing initial and boundary conditions become identical.

First theorem

Phenomena that are similar in one sense or another have the same combinations of parameters.

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

Second theorem

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

The dependence is complete if we take into account all the connections between the quantities included in it. This dependence cannot change when changing the units of measurement of physical quantities.

Third theorem

For the similarity of phenomena, the defining criteria of similarity and the conditions of unambiguity must be similar.

Defining parameters are understood as criteria containing those parameters of processes and systems that can be considered independent in a given task (time, capital, resources, etc.); Under the conditions of unambiguity we mean a group of parameters whose values, specified in the form of functional dependencies or numbers, distinguish a specific phenomenon from a possible variety of phenomena.

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

Similarity nonlinear systems is preserved if the conditions for the coincidence of the relative characteristics of similar parameters that are nonlinear or variable are met.

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

Similarity with probabilistic nature the phenomena being studied. All similarity conditions theorems related to deterministic systems, turn out to be valid provided that the probability densities of similar parameters, presented in the form of 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 condition for similarity is the fulfillment of the requirement of physical realizability of a similar correlation between stochastically given parameters included in the unambiguity condition.

There are two ways to determine similarity criteria:

a) bringing the process equations to dimensionless form;

b) the use of parameters that describe the process, despite the fact that the process equation 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 selected basic values. The most significant parameters, expressed in shares of the basic ones, can be considered as similarity criteria operating in specific conditions.

Thus, economic-mathematical models and methods are not only an apparatus for obtaining economic patterns, but also a widely used toolkit practical solution problems in management, forecasting, business, banking and other areas 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 they are used various methods, 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 the use of models to determine or clarify the characteristics and rationalize the methods of constructing newly constructed objects.

“Modeling is one of the main categories of the theory of knowledge; Any method of scientific knowledge, both theoretical and experimental, is essentially based on the idea of modeling.” Modeling began to be used in scientific research back in ancient times and gradually covered all 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 developing for a long time in relation to specific sciences, independently of one another. Under these conditions, there was no unified system of knowledge or terminology. Then the role of modeling began to emerge 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 a certain tool - a model, which is placed between the researcher and the object of study. Modeling is used either when the object cannot be studied directly (the Earth’s core, solar system etc.), or when the object does not yet exist (future state of the economy, future demand, expected supply, etc.), or when research requires a lot of time and money, or, finally, to test various kinds of hypotheses. Modeling is most often part of general process knowledge. Currently there are many different definitions and classifications of models in relation to problems of various sciences. Let's accept the definition given by the economist V.S. Nemchinov, known, in particular, for his work on the development of models of planned economy: “A model is a means of identifying any objectively operating system of regular connections and relationships that take place in the reality being studied.”

The main requirement for models is adequacy to reality, although the model reproduces the object or process being studied in a simplified form. When constructing any model, the researcher faces a 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 described such a problem as “the trap of oversimplification and the swamp of overcomplication.”

In progress scientific research the model can work in two directions: from observations real world to theory and back; i.e., on the one hand, building a model is an important step towards creating a theory, on the other hand, it is one of the means experimental research. Depending on the choice of modeling tools, material and abstract (symbolic) models are distinguished. Material (physical) models are widely used in technology, architecture and other fields. They are based on obtaining a physical image of the object or process under study. Abstract models are not associated with the construction of physical images. They are some kind of intermediate link between abstract theoretical thinking and actual reality. Abstract models (they are called iconic) include numerical ( mathematical expressions with specific numerical characteristics), logical (block diagrams of computer calculation algorithms, graphs, diagrams, drawings). Models, the construction of which is aimed at determining the state of an object, which is the best from the point of view of a certain criterion, are called normative. Models intended to explain observed facts or predict the behavior of an object are called descriptive.

The effectiveness of using models is determined by the scientific validity of their premises, the researcher’s ability to identify the essential characteristics of the modeling object, select 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-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 management 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 for 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 convention, the classification of these methods can be presented as follows.

1. Economic and statistical methods:

· economic statistics;

· mathematical statistics;

· multivariate analysis.

2. Econometrics:

· macroeconomic models;

production function theory

· intersectoral balances;

· national accounts;

· analysis of demand and consumption;

· global modeling.

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

· mathematical programming;

· network and management planning;

· queuing theory;

· game theory;

· decision theory;

· methods for modeling economic processes in industries and enterprises.

4. Economic cybernetics:

· system analysis of the economy;

· theory of economic information.

5. Methods for experimental study of economic phenomena:

· methods of machine imitation;

· business games;

· methods of real economic experiment.

Economic-mathematical methods use various branches of mathematics, mathematical statistics, and mathematical logic. Computational mathematics, algorithm theory and other disciplines play a major role in solving economic and mathematical problems. The use of mathematical apparatus has brought tangible results in solving problems of analyzing processes of expanded production, matrix modeling, determining the optimal growth rate of capital investments, optimal placement, specialization and concentration of production, selection problems optimal ways production, determining the optimal sequence for launching production, optimal options for cutting industrial materials and composing mixtures, tasks of preparing production using network planning methods and many others.

Solving standard problems is characterized by clarity of purpose, the ability to develop procedures and rules for conducting calculations in advance.

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

The most important of them are, firstly, a high level of knowledge of economic theory, economic processes and phenomena, and the 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 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

Characteristic feature scientific and technological progress in developed countries the role of economic science is increasing. The economy comes to the fore precisely because it is decisive degree determines the effectiveness and priority of areas of scientific and technological progress, reveals broad ways to implement economically beneficial achievements.

The use of mathematics in economic science gave impetus to the development of both economic science itself and applied mathematics, in part of the methods of economic-mathematical models. The proverb says: “Measure twice, cut once.” The use of models requires time, effort, and material resources. In addition, calculations using models are opposed strong-willed 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 tentatively highlight the following directions: practical application, for which a large economic effect has already been achieved.

The first direction is forecasting and long-term planning. The rate and proportions of economic development are predicted, and on their basis the rates and growth factors are determined national income, its distribution for consumption and accumulation, etc. An important point is the use of economic and mathematical methods not only in drawing up 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 planning decisions, in particular these inter-industry and inter-regional balances of production and distribution of products. Based on the economic content and nature of the information, they distinguish between cost and natural product balances, each of which can be reporting and planning.

The third direction is the use of economic and mathematical models at the industry level (calculation of optimal industry plans, analysis using production functions, forecasting the main production proportions of industry development). To solve the problem of location and specialization of an enterprise, optimal attachment to suppliers or consumers, etc., optimization models of two types are used: in some, for a given volume of production, it is necessary to find an option for implementing the plan at the lowest cost; in others, it is necessary to determine the scale of production and the structure of products in order to obtain the maximum effect. As calculations continue, 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 unified model of the industry, now the most promising is the use of complexes of models interconnected both vertically and horizontally.

The fourth direction is economic and mathematical modeling of the current and operational planning industrial, construction, transport and other associations, enterprises and firms. The scope of practical application of models also includes departments of agriculture, trade, communications, healthcare, nature conservation, etc. In mechanical engineering, a large number of different models are used, the most “debugged” 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 over time and effectively organize the work of intra-factory transport, significantly improve the loading of equipment and intelligently organize product control, etc.

The fifth direction is territorial modeling, which began with the development of reporting inter-industry balances of some regions in the late 50s.

As a sixth area, we can highlight economic and mathematical modeling of logistics, including optimization of transport and economic connections and inventory levels.

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

Economic and mathematical methods are becoming especially important as information technologies are introduced in all areas of practice.

Literature

1. Ventzel E.S. Operations Research. – M: Soviet radio, 1972.

2. Greshilov A.A. How to make the best decision in real-life conditions. - 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., Faure R. Let's study operations. – M.: Mir, 1966.