The basis for solving economic problems are mathematical models.
Drawing up a mathematical model includes:mathematical model problem is a set of mathematical relationships that describe the essence of the problem.
- task variable selection
- drawing up a system of restrictions
- choice of objective function
Task variables are called quantities X1, X2, Xn, which fully characterize the economic process. Usually they are written as a vector: X=(X 1 , X 2 ,...,X n).
The system of restrictions tasks are a set of equations and inequalities that describe the limited resources in the problem under consideration.
target function task is called a function of task variables that characterizes the quality of the task and the extremum of which is required to be found.
In general, a linear programming problem can be written as follows:
This entry means the following: find the extremum of the objective function (1) and the corresponding variables X=(X 1 , X 2 ,...,X n) provided that these variables satisfy the system of constraints (2) and non-negativity conditions (3) .
Acceptable Solution(plan) of a linear programming problem is any n-dimensional vector X=(X 1 , X 2 ,...,X n) that satisfies the system of constraints and non-negativity conditions.
The set of feasible solutions (plans) of the problem forms range of feasible solutions(ODR).
The optimal solution(plan) of a linear programming problem is such a feasible solution (plan) of the problem, in which the objective function reaches an extremum.
An example of compiling a mathematical model
The task of using resources (raw materials)
Condition: For the manufacture of n types of products, m types of resources are used. Make a mathematical model.
Known:
- b i (i = 1,2,3,...,m) are the reserves of each i-th type of resource;
- a ij (i = 1,2,3,...,m; j=1,2,3,...,n) are the costs of each i-th type of resource for the production of a unit volume of the j-th type of product;
- c j (j = 1,2,3,...,n) is the profit from the sale of a unit volume of the j-th type of product.
It is required to draw up a plan for the production of products that provides maximum profit with given restrictions on resources (raw materials).
Solution:
We introduce a vector of variables X=(X 1 , X 2 ,...,X n), where x j (j = 1,2,...,n) is the volume of production of the j-th type of product.
The costs of the i-th type of resource for the production of a given volume x j of products are equal to a ij x j , therefore, the restriction on the use of resources for the production of all types of products has the form:
The profit from the sale of the j-th type of product is equal to c j x j , so the objective function is equal to:
Answer- The mathematical model looks like:
Canonical form of a linear programming problem
In the general case, a linear programming problem is written in such a way that both equations and inequalities are constraints, and variables can be either non-negative or arbitrarily changing.
In the case when all constraints are equations and all variables satisfy the non-negativity condition, the linear programming problem is called canonical.
It can be represented in coordinate, vector and matrix notation.
The canonical linear programming problem in coordinate notation has the form:
The canonical linear programming problem in matrix notation has the form:
- A is the matrix of coefficients of the system of equations
- X is a column matrix of task variables
- Ao is the matrix-column of the right parts of the constraint system
Often, linear programming problems are used, called symmetric ones, which in matrix notation have the form:
Reduction of a general linear programming problem to canonical form
In most methods for solving linear programming problems, it is assumed that the system of constraints consists of equations and natural conditions for the non-negativity of variables. However, when compiling models of economic problems, constraints are mainly formed in the form of a system of inequalities, so it is necessary to be able to move from a system of inequalities to a system of equations.
This can be done like this:Take a linear inequality a 1 x 1 +a 2 x 2 +...+a n x n ≤b and add to its left side some value x n+1 such that the inequality becomes the equality a 1 x 1 +a 2 x 2 + ...+a n x n +x n+1 =b. Moreover, this value x n+1 is non-negative.
Let's consider everything with an example.
Example 26.1Reduce the linear programming problem to canonical form:
Solution:
Let's move on to the problem of finding the maximum of the objective function.
To do this, we change the signs of the coefficients of the objective function.
To convert the second and third inequalities of the constraint system into equations, we introduce non-negative additional variables x 4 x 5 (this operation is marked with the letter D on the mathematical model).
The variable x 4 is entered on the left side of the second inequality with a "+" sign, since the inequality has the form "≤".
The variable x 5 is entered on the left side of the third inequality with the "-" sign, since the inequality has the form "≥".
Variables x 4 x 5 are entered into the objective function with a coefficient. equal to zero.
We write the problem in canonical form.
In the article brought to your attention, we offer examples of mathematical models. In addition, we will pay attention to the stages of creating models and analyze some of the problems associated with mathematical modeling.
Another issue of ours is mathematical models in economics, examples of which we will consider a definition a little later. We propose to start our conversation with the very concept of “model”, briefly consider their classification and move on to our main questions.
The concept of "model"
We often hear the word "model". What is it? This term has many definitions, here are just three of them:
- a specific object that is created to receive and store information, reflecting some properties or characteristics, and so on, of the original of this object (this specific object can be expressed in different forms: mental, description using signs, and so on);
- a model also means a display of any specific situation, life or management;
- a small copy of an object can serve as a model (they are created for a more detailed study and analysis, since the model reflects the structure and relationships).
Based on everything that was said earlier, we can draw a small conclusion: the model allows you to study in detail a complex system or object.
All models can be classified according to a number of criteria:
- by area of use (educational, experimental, scientific and technical, gaming, simulation);
- by dynamics (static and dynamic);
- by branch of knowledge (physical, chemical, geographical, historical, sociological, economic, mathematical);
- according to the method of presentation (material and informational).
Information models, in turn, are divided into sign and verbal. And iconic - on computer and non-computer. Now let's move on to a detailed consideration of examples of a mathematical model.
Mathematical model
As you might guess, a mathematical model reflects some features of an object or phenomenon using special mathematical symbols. Mathematics is needed in order to model the laws of the world in its own specific language.
The method of mathematical modeling originated quite a long time ago, thousands of years ago, along with the advent of this science. However, the impetus for the development of this modeling method was given by the appearance of computers (electronic computers).
Now let's move on to classification. It can also be carried out according to some signs. They are presented in the table below.
We propose to stop and take a closer look at the last classification, since it reflects the general patterns of modeling and the goals of the models being created.
Descriptive Models
In this chapter, we propose to dwell in more detail on descriptive mathematical models. In order to make everything very clear, an example will be given.
To begin with, this view can be called descriptive. This is due to the fact that we simply make calculations and forecasts, but we cannot influence the outcome of the event in any way.
A striking example of a descriptive mathematical model is the calculation of the flight path, speed, distance from the Earth of a comet that invaded the expanses of our solar system. This model is descriptive, since all the results obtained can only warn us of some kind of danger. Unfortunately, we cannot influence the outcome of the event. However, based on the calculations obtained, it is possible to take any measures to preserve life on Earth.
Optimization Models
Now we will talk a little about economic and mathematical models, examples of which can be various situations. In this case, we are talking about models that help to find the right answer in certain conditions. They must have some parameters. To make it very clear, consider an example from the agrarian part.
We have a granary, but the grain spoils very quickly. In this case, we need to choose the right temperature regime and optimize the storage process.
Thus, we can define the concept of "optimization model". In a mathematical sense, this is a system of equations (both linear and not), the solution of which helps to find the optimal solution in a particular economic situation. We have considered an example of a mathematical model (optimization), but I would like to add one more thing: this type belongs to the class of extreme problems, they help to describe the functioning of the economic system.
We note one more nuance: models can be of a different nature (see the table below).
Multicriteria models
Now we invite you to talk a little about the mathematical model of multiobjective optimization. Before that, we gave an example of a mathematical model for optimizing a process according to any one criterion, but what if there are a lot of them?
A striking example of a multicriteria task is the organization of proper, healthy and at the same time economical nutrition of large groups of people. Such tasks are often encountered in the army, school canteens, summer camps, hospitals and so on.
What criteria are given to us in this task?
- Food should be healthy.
- Food expenses should be kept to a minimum.
As you can see, these goals do not coincide at all. This means that when solving a problem, it is necessary to look for the optimal solution, a balance between the two criteria.
Game models
Speaking about game models, it is necessary to understand the concept of "game theory". Simply put, these models reflect mathematical models of real conflicts. It is only worth understanding that, unlike a real conflict, a game mathematical model has its own specific rules.
Now I will give a minimum of information from game theory, which will help you understand what a game model is. And so, in the model there are necessarily parties (two or more), which are usually called players.
All models have certain characteristics.
The game model can be paired or multiple. If we have two subjects, then the conflict is paired, if more - multiple. An antagonistic game can also be distinguished, it is also called a zero-sum game. This is a model in which the gain of one of the participants is equal to the loss of the other.
simulation models
In this section, we will focus on simulation mathematical models. Examples of tasks are:
- model of the dynamics of the number of microorganisms;
- model of molecular motion, and so on.
In this case, we are talking about models that are as close as possible to real processes. By and large, they imitate any manifestation in nature. In the first case, for example, we can model the dynamics of the number of ants in one colony. In this case, you can observe the fate of each individual. In this case, the mathematical description is rarely used, more often there are written conditions:
- after five days, the female lays eggs;
- after twenty days the ant dies, and so on.
Thus, are used to describe a large system. Mathematical conclusion is the processing of the received statistical data.
Requirements
It is very important to know that there are some requirements for this type of model, among which are those given in the table below.
Versatility | This property allows you to use the same model when describing groups of objects of the same type. It is important to note that universal mathematical models are completely independent of the physical nature of the object under study. |
Adequacy | Here it is important to understand that this property allows the most correct reproduction of real processes. In operation problems, this property of mathematical modeling is very important. An example of a model is the process of optimizing the use of a gas system. In this case, calculated and actual indicators are compared, as a result, the correctness of the compiled model is checked. |
Accuracy | This requirement implies the coincidence of the values that we obtain when calculating the mathematical model and the input parameters of our real object |
economy | The requirement of economy for any mathematical model is characterized by implementation costs. If the work with the model is carried out manually, then it is necessary to calculate how much time it will take to solve one problem using this mathematical model. If we are talking about computer-aided design, then indicators of time and computer memory are calculated |
Modeling steps
In total, it is customary to distinguish four stages in mathematical modeling.
- Formulation of laws linking parts of the model.
- Study of mathematical problems.
- Finding out the coincidence of practical and theoretical results.
- Analysis and modernization of the model.
Economic and mathematical model
In this section, we will briefly highlight the issue. Examples of tasks can be:
- formation of a production program for the production of meat products, ensuring the maximum profit of production;
- maximizing the profit of the organization by calculating the optimal number of tables and chairs to be produced in a furniture factory, and so on.
The economic-mathematical model displays an economic abstraction, which is expressed using mathematical terms and signs.
Computer mathematical model
Examples of a computer mathematical model are:
- hydraulics tasks using flowcharts, diagrams, tables, and so on;
- problems on solid mechanics, and so on.
A computer model is an image of an object or system, presented as:
- tables;
- block diagrams;
- diagrams;
- graphics, and so on.
At the same time, this model reflects the structure and interconnections of the system.
Building an economic and mathematical model
We have already talked about what an economic-mathematical model is. An example of solving the problem will be considered right now. We need to analyze the production program to identify the reserve for increasing profits with a shift in the assortment.
We will not fully consider the problem, but only build an economic and mathematical model. The criterion of our task is profit maximization. Then the function has the form: Л=р1*х1+р2*х2… tending to the maximum. In this model, p is the profit per unit, x is the number of units produced. Further, based on the constructed model, it is necessary to make calculations and summarize.
An example of building a simple mathematical model
A task. The fisherman returned with the following catch:
- 8 fish - inhabitants of the northern seas;
- 20% of the catch - the inhabitants of the southern seas;
- not a single fish was found from the local river.
How many fish did he buy at the store?
So, an example of constructing a mathematical model of this problem is as follows. We denote the total number of fish as x. Following the condition, 0.2x is the number of fish living in southern latitudes. Now we combine all the available information and get a mathematical model of the problem: x=0.2x+8. We solve the equation and get the answer to the main question: he bought 10 fish in the store.
MATHEMATICAL MODEL - representation of a phenomenon or process studied in concrete scientific knowledge in the language of mathematical concepts. At the same time, a number of properties of the phenomenon under study are supposed to be obtained on the path of studying the actual mathematical characteristics of the model. Construction of M.m. most often dictated by the need to have a quantitative analysis of the studied phenomena and processes, without which, in turn, it is impossible to make experimentally verifiable predictions about their course.
The process of mathematical modeling, as a rule, goes through the following stages. At the first stage, the links between the main parameters of the future M.m. First of all, we are talking about a qualitative analysis of the phenomena under study and the formulation of patterns that link the main objects of research. On this basis, the identification of objects that allow a quantitative description is carried out. The stage ends with the construction of a hypothetical model, in other words, a record in the language of mathematical concepts of qualitative ideas about the relationships between the main objects of the model, which can be quantitatively characterized.
At the second stage, the study of the actual mathematical problems, to which the constructed hypothetical model leads, takes place. The main thing at this stage is to obtain empirically verifiable theoretical consequences (solution of the direct problem) as a result of the mathematical analysis of the model. At the same time, cases are not uncommon when, for the construction and study of M.m. in different areas of concrete scientific knowledge, the same mathematical apparatus is used (for example, differential equations) and mathematical problems of the same type, although very non-trivial in each specific case, arise. In addition, at this stage, the use of high-speed computing technology (computer) becomes of great importance, which makes it possible to obtain an approximate solution of problems, often impossible in the framework of pure mathematics, with a previously unavailable (without the use of a computer) degree of accuracy.
The third stage is characterized by activities to identify the degree of adequacy of the constructed hypothetical M.m. those phenomena and processes for the study of which it was intended. Namely, in the event that all model parameters have been specified, the researchers try to find out how, within the accuracy of observations, their results are consistent with the theoretical consequences of the model. Deviations beyond the accuracy of observations indicate the inadequacy of the model. However, there are often cases when, when building a model, a number of its parameters remain unchanged.
indefinite. Problems in which the parametric characteristics of the model are established in such a way that the theoretical consequences are comparable within the accuracy of observations with the results of empirical tests are called inverse problems.
At the fourth stage, taking into account the identification of the degree of adequacy of the constructed hypothetical model and the emergence of new experimental data on the phenomena under study, the subsequent analysis and modification of the model takes place. Here, the decision taken varies from an unconditional rejection of the applied mathematical tools to the adoption of the constructed model as a foundation for constructing a fundamentally new scientific theory.
The first M.m. appeared in ancient science. So, to model the solar system, the Greek mathematician and astronomer Eudoxus gave each planet four spheres, the combination of the movement of which created a hippopede - a mathematical curve similar to the observed movement of the planet. Since, however, this model could not explain all the observed anomalies in the motion of the planets, it was later replaced by the epicyclic model of Apollonius from Perge. Hipparchus used the latest model in his studies, and then, subjecting it to some modification, Ptolemy. This model, like its predecessors, was based on the belief that the planets make uniform circular motions, the overlap of which explained the apparent irregularities. At the same time, it should be noted that the Copernican model was fundamentally new only in a qualitative sense (but not as M.M.). And only Kepler, based on the observations of Tycho Brahe, built a new M.m. The solar system, proving that the planets move not in circular, but in elliptical orbits.
At present, the most adequate are the MMs constructed to describe mechanical and physical phenomena. On the adequacy of M.m. outside of physics one can, with a few exceptions, speak with a fair amount of caution. Nevertheless, fixing the hypotheticality, and often simply the inadequacy of M.m. in various fields of knowledge, their role in the development of science should not be underestimated. There are frequent cases when even models that are far from adequate to a large extent organized and stimulated further research, along with erroneous conclusions, contained those grains of truth that fully justified the efforts expended on developing these models.
Literature:
Math modeling. M., 1979;
Ruzavin G.I. Mathematization of scientific knowledge. M., 1984;
Tutubalin V.N., Barabasheva Yu.M., Grigoryan A.A., Devyatkova G.N., Uger E.G. Differential equations in ecology: historical and methodological reflection // Problems of the history of natural science and technology. 1997. No. 3.
Dictionary of philosophical terms. Scientific edition of Professor V.G. Kuznetsova. M., INFRA-M, 2007, p. 310-311.
The tasks solved by LP methods are very diverse in content. But their mathematical models are similar and are conditionally combined into three large groups of problems:
- transport tasks;
- planning tasks;
Transport task
On two trading bases BUT and AT There are 30 sets of furniture, 15 for each. All furniture needs to be delivered to two furniture stores, FROM and D and in FROM you need to deliver 10 headsets, and in D- 20. It is known that the delivery of one headset from the base BUT to the store FROM costs one monetary unit, to the store D- in three monetary units. According to the base AT to shops FROM and D: two and five monetary units. Make a transportation plan so that the cost of all transportation is the least.For convenience, we mark these tasks in a table. At the intersection of rows and columns are numbers characterizing the cost of the respective transportation (Table 3.1).
Table 3.1
Let's make a mathematical model of the problem.
Variables must be entered. The wording of the question says that it is necessary to draw up a transportation plan. Denote by X 1 , X 2 number of headsets transported from the base BUT to shops FROM and D respectively, and through at 1 , at 2 - the number of headsets transported from the base AT to shops FROM and D respectively. Then the amount of furniture removed from the warehouse BUT, equals ( X 1 + X 2) well from stock AT - (at 1 + at 2). Store need FROM is equal to 10 headsets, and they brought it ( X 1 + at 1) pieces, i.e. X 1 + at 1 = 10. Similarly, for the store D we have X 2 + at 2 = 20. Note that the needs of stores are exactly equal to the number of headsets in stock, so X 1 + at 2 = 15 and at 1 + at 2 = 15. If you took away less than 15 sets from the warehouses, then the stores would not have enough furniture to meet their needs.
So the variables X 1 , X 2 , at 1 , at 2 are non-negative in the meaning of the problem and satisfy the system of constraints:
(3.1)
Denoting through F shipping costs, let's count them. for the transportation of one set of furniture from BUT in FROM spend one day. units, for transportation x 1 sets - x 1 day units Likewise, for transportation x 2 sets of BUT in D cost 3 x 2 days units; from AT in FROM - 2y 1 day units, from AT in D - 5y 2 days units
So,
F = 1x 1 + 3x 2 + 2y 1 + 5y 2 → min (3.2)
(we want the total cost of shipping to be as low as possible).
Let's formulate the problem mathematically.
On the set of solutions of the constraint system (3.1), find a solution that minimizes the objective function F(3.2), or find the optimal plan ( x 1 , x 2, y 1 , y 2) determined by the system of constraints (3.1) and the objective function (3.2).
The problem that we have considered can be represented in a more general form, with any number of suppliers and consumers.
In the problem we have considered, the availability of cargo from suppliers (15 + 15) is equal to the total need of consumers (10 + 20). Such a model is called closed, and the corresponding task is balanced transport task.
In economic calculations, the so-called open models, in which the indicated equality is not observed, also play a significant role. Either the supply of suppliers is greater than the demand of consumers, or demand exceeds the availability of goods. note that then the system of constraints of the unbalanced transport problem, along with the equations, will also include inequalities.
Questions for self-control
1. Statement of the transport problem. describe the construction of a mathematical model.
2. What is a balanced and unbalanced transport problem?
3. What is calculated in the objective function of the transport task?
4. What does each inequality of the system of constraints of the plan problem reflect?
5. What does each inequality of the system of constraints of the mixture problem reflect?
6. What do the variables mean in the plan problem and the mixture problem?
The concept of model and simulation.
Model in a broad sense- this is any image, analogue of a mental or established image, description, diagram, drawing, map, etc. of any volume, process or phenomenon, used as its substitute or representative. The object, process or phenomenon itself is called the original of this model.
Modeling - this is the study of any object or system of objects by building and studying their models. This is the use of models to determine or refine the characteristics and rationalize the ways of constructing newly constructed objects.
Any method of scientific research is based on the idea of modeling, at the same time, various kinds of sign, abstract models are used in theoretical methods, and subject models are used in experimental ones.
In the study, a complex real phenomenon is replaced by some simplified copy or scheme, sometimes such a copy serves only to remember and to recognize the desired phenomenon at the next meeting. Sometimes the constructed scheme reflects some essential features, allows you to understand the mechanism of the phenomenon, makes it possible to predict its change. Different models can correspond to the same phenomenon.
The task of the researcher is to predict the nature of the phenomenon and the course of the process.
Sometimes, it happens that an object is available, but experiments with it are expensive or lead to serious environmental consequences. Knowledge about such processes is obtained with the help of models.
An important point is that the very nature of science involves the study of not one specific phenomenon, but a wide class of related phenomena. It implies the need to formulate some general categorical statements, which are called laws. Naturally, with such a formulation, many details are neglected. In order to more clearly identify the pattern, they deliberately go for coarsening, idealization, schematicity, that is, they study not the phenomenon itself, but a more or less exact copy or model of it. All laws are laws about models, and therefore it is not surprising that, over time, some scientific theories are found to be unusable. This does not lead to the collapse of science, since one model has been replaced by another. more modern.
A special role in science is played by mathematical models, the building material and tools of these models - mathematical concepts. They have accumulated and improved over thousands of years. Modern mathematics provides exceptionally powerful and universal means of research. Almost every concept in mathematics, every mathematical object, starting from the concept of a number, is a mathematical model. When constructing a mathematical model of an object or phenomenon under study, those of its features, features and details are distinguished, which, on the one hand, contain more or less complete information about the object, and, on the other hand, allow mathematical formalization. Mathematical formalization means that the features and details of an object can be associated with appropriate adequate mathematical concepts: numbers, functions, matrices, and so on. Then the connections and relationships found and assumed in the object under study between its individual parts and components can be written using mathematical relationships: equalities, inequalities, equations. The result is a mathematical description of the process or phenomenon under study, that is, its mathematical model.
The study of a mathematical model is always associated with some rules of action on the objects under study. These rules reflect the relationships between causes and effects.
Building a mathematical model is a central stage in the study or design of any system. The whole subsequent analysis of the object depends on the quality of the model. Building a model is not a formal procedure. It strongly depends on the researcher, his experience and taste, always relies on certain experimental material. The model should be accurate enough, adequate and should be convenient for use.
Math modeling.
Classification of mathematical models.
Mathematical models can bedetermined and stochastic .
Deterministic model and - these are models in which a one-to-one correspondence is established between the variables describing an object or phenomenon.
This approach is based on knowledge of the mechanism of functioning of objects. The object being modeled is often complex and deciphering its mechanism can be very laborious and time-consuming. In this case, they proceed as follows: experiments are carried out on the original, the results are processed, and, without delving into the mechanism and theory of the modeled object, using the methods of mathematical statistics and probability theory, they establish relationships between the variables describing the object. In this case, getstochastic model . AT stochastic model, the relationship between variables is random, sometimes it happens fundamentally. The impact of a huge number of factors, their combination leads to a random set of variables describing an object or phenomenon. By the nature of the modes, the model isstatistical and dynamic.
Statisticalmodelincludes a description of the relationships between the main variables of the simulated object in the steady state without taking into account the change in parameters over time.
AT dynamicmodelsdescribes the relationship between the main variables of the simulated object in the transition from one mode to another.
Models are discrete and continuous, as well as mixed type. AT continuous variables take values from a certain interval, indiscretevariables take isolated values.
Linear Models- all functions and relations that describe the model are linearly dependent on the variables andnot linearotherwise.
Math modeling.
Requirements , presented to the models.
1. Versatility- characterizes the completeness of the display by the model of the studied properties of the real object.
- Adequacy - the ability to reflect the desired properties of the object with an error not higher than the specified one.
- Accuracy - is estimated by the degree of coincidence of the values of the characteristics of a real object and the values of these characteristics obtained using models.
- economy - is determined by the cost of computer memory resources and time for its implementation and operation.
Math modeling.
The main stages of modeling.
1. Statement of the problem.
Determining the purpose of the analysis and ways to achieve it and develop a common approach to the problem under study. At this stage, a deep understanding of the essence of the task is required. Sometimes, it is not less difficult to correctly set a task than to solve it. Staging is not a formal process, there are no general rules.
2. The study of the theoretical foundations and the collection of information about the object of the original.
At this stage, a suitable theory is selected or developed. If it is not present, causal relationships are established between the variables describing the object. Input and output data are determined, simplifying assumptions are made.
3. Formalization.
It consists in choosing a system of symbols and using them to write down the relationship between the components of the object in the form of mathematical expressions. A class of tasks is established, to which the resulting mathematical model of the object can be attributed. The values of some parameters at this stage may not yet be specified.
4. Choice of solution method.
At this stage, the final parameters of the models are set, taking into account the conditions for the operation of the object. For the obtained mathematical problem, a solution method is selected or a special method is developed. When choosing a method, the knowledge of the user, his preferences, as well as the preferences of the developer are taken into account.
5. Implementation of the model.
Having developed an algorithm, a program is written that is debugged, tested, and a solution to the desired problem is obtained.
6. Analysis of the received information.
The received and expected solution is compared, the modeling error is controlled.
7. Checking the adequacy of a real object.
The results obtained by the model are comparedeither with the information available about the object, or an experiment is carried out and its results are compared with the calculated ones.
The modeling process is iterative. In case of unsatisfactory results of the stages 6. or 7. a return to one of the early stages, which could lead to the development of an unsuccessful model, is carried out. This stage and all subsequent stages are refined, and such refinement of the model occurs until acceptable results are obtained.
A mathematical model is an approximate description of any class of phenomena or objects of the real world in the language of mathematics. The main purpose of modeling is to explore these objects and predict the results of future observations. However, modeling is also a method of cognition of the surrounding world, which makes it possible to control it.
Mathematical modeling and the associated computer experiment are indispensable in cases where a full-scale experiment is impossible or difficult for one reason or another. For example, it is impossible to set up a full-scale experiment in history to check “what would happen if...” It is impossible to check the correctness of this or that cosmological theory. In principle, it is possible, but hardly reasonable, to experiment with the spread of some kind of disease, such as the plague, or to carry out a nuclear explosion in order to study its consequences. However, all this can be done on a computer, having previously built mathematical models of the phenomena under study.
1.1.2 2. Main stages of mathematical modeling
1) Model building. At this stage, some "non-mathematical" object is specified - a natural phenomenon, construction, economic plan, production process, etc. In this case, as a rule, a clear description of the situation is difficult. First, the main features of the phenomenon and the relationship between them at a qualitative level are identified. Then the found qualitative dependencies are formulated in the language of mathematics, that is, a mathematical model is built. This is the most difficult part of the modeling.
2) Solving the mathematical problem that the model leads to. At this stage, much attention is paid to the development of algorithms and numerical methods for solving the problem on a computer, with the help of which the result can be found with the required accuracy and within an acceptable time.
3) Interpretation of the obtained consequences from the mathematical model.The consequences derived from the model in the language of mathematics are interpreted in the language accepted in this field.
4) Checking the adequacy of the model.At this stage, it is found out whether the results of the experiment agree with the theoretical consequences from the model within a certain accuracy.
5) Model modification.At this stage, either the model becomes more complex so that it is more adequate to reality, or it is simplified in order to achieve a practically acceptable solution.
1.1.3 3. Model classification
Models can be classified according to different criteria. For example, according to the nature of the problems being solved, models can be divided into functional and structural ones. In the first case, all quantities characterizing a phenomenon or object are expressed quantitatively. At the same time, some of them are considered as independent variables, while others are considered as functions of these quantities. A mathematical model is usually a system of equations of various types (differential, algebraic, etc.) that establish quantitative relationships between the quantities under consideration. In the second case, the model characterizes the structure of a complex object, consisting of separate parts, between which there are certain connections. Typically, these relationships are not quantifiable. To build such models, it is convenient to use graph theory. A graph is a mathematical object, which is a set of points (vertices) on a plane or in space, some of which are connected by lines (edges).
According to the nature of the initial data and prediction results, the models can be divided into deterministic and probabilistic-statistical. Models of the first type give definite, unambiguous predictions. Models of the second type are based on statistical information, and the predictions obtained with their help are of a probabilistic nature.
MATHEMATICAL MODELING AND GENERAL COMPUTERIZATION OR SIMULATION MODELS
Now, when almost universal computerization is taking place in the country, one can hear statements from specialists of various professions: "Let's introduce a computer in our country, then all tasks will be solved immediately." This point of view is completely wrong, computers themselves cannot do anything without mathematical models of certain processes, and one can only dream of universal computerization.
In support of the foregoing, we will try to justify the need for modeling, including mathematical modeling, reveal its advantages in the knowledge and transformation of the outside world by a person, identify existing shortcomings and go ... to simulation modeling, i.e. modeling using computers. But everything is in order.
First of all, let's answer the question: what is a model?
A model is a material or mentally represented object that, in the process of cognition (study), replaces the original, retaining some typical properties that are important for this study.
A well-built model is more accessible for research than a real object. For example, experiments with the country's economy for educational purposes are unacceptable; here one cannot do without a model.
Summarizing what has been said, we can answer the question: what are models for? To
- understand how an object works (its structure, properties, laws of development, interaction with the outside world).
- learn to manage an object (process) and determine the best strategies
- predict the consequences of the impact on the object.
What is positive in any model? It allows you to get new knowledge about the object, but, unfortunately, it is not complete to one degree or another.
Modelformulated in the language of mathematics using mathematical methods is called a mathematical model.
The starting point for its construction is usually some task, for example, an economic one. Widespread, both descriptive and optimization mathematical, characterizing various economic processes and events such as:
- resource allocation
- rational cutting
- transportation
- consolidation of enterprises
- network planning.
How is a mathematical model built?
- First, the purpose and subject of the study are formulated.
- Secondly, the most important characteristics corresponding to this goal are highlighted.
- Thirdly, the relationships between the elements of the model are verbally described.
- Further, the relationship is formalized.
- And the calculation is carried out according to the mathematical model and the analysis of the obtained solution.
Using this algorithm, you can solve any optimization problem, including a multicriteria one, i.e. one in which not one, but several goals, including contradictory ones, are pursued.
Let's take an example. Queuing theory - the problem of queuing. You need to balance two factors - the cost of maintaining service devices and the cost of staying in line. Having built a formal description of the model, calculations are made using analytical and computational methods. If the model is good, then the answers found with its help are adequate to the modeling system; if it is bad, then it must be improved and replaced. The criterion of adequacy is practice.
Optimization models, including multicriteria ones, have a common property - a goal (or several goals) is known to achieve which one often has to deal with complex systems, where it is not so much about solving optimization problems, but about researching and predicting states depending on chosen control strategies. And here we are faced with difficulties in implementing the previous plan. They are as follows:
- a complex system contains many connections between elements
- the real system is influenced by random factors, it is impossible to take them into account analytically
- the possibility of comparing the original with the model exists only at the beginning and after the application of the mathematical apparatus, because intermediate results may not have analogues in a real system.
In connection with the listed difficulties that arise when studying complex systems, the practice required a more flexible method, and it appeared - simulation modeling " Simujation modeling".
Usually, a simulation model is understood as a set of computer programs that describes the functioning of individual blocks of systems and the rules of interaction between them. The use of random variables makes it necessary to repeatedly conduct experiments with a simulation system (on a computer) and subsequent statistical analysis of the results obtained. A very common example of the use of simulation models is the solution of a queuing problem by the MONTE CARLO method.
Thus, work with the simulation system is an experiment carried out on a computer. What are the benefits?
– Greater proximity to the real system than mathematical models;
– The block principle makes it possible to verify each block before it is included in the overall system;
– The use of dependencies of a more complex nature, not described by simple mathematical relationships.
The listed advantages determine the disadvantages
– to build a simulation model is longer, more difficult and more expensive;
– to work with the simulation system, you must have a computer that is suitable for the class;
– interaction between the user and the simulation model (interface) should not be too complicated, convenient and well known;
- the construction of a simulation model requires a deeper study of the real process than mathematical modeling.
The question arises: can simulation modeling replace optimization methods? No, but conveniently complements them. A simulation model is a program that implements some algorithm, to optimize the control of which an optimization problem is first solved.
So, neither a computer, nor a mathematical model, nor an algorithm for studying it separately can solve a rather complicated problem. But together they represent the force that allows you to know the world around you, manage it in the interests of man.
1.2 Model classification
1.2.1
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1.2.2 Classification by area of use (Makarova N.A.)
Training- visual aids, trainers , oh thrashing programs |
1.2.3 Classification according to the method of presentation Makarova N.A.)
Material
models- otherwise
can be called subject. They perceive the geometric and physical properties of the original and always have a real embodiment. |
1.2.4 Classification of models given in the book "Land of Informatics" (Gein A.G.))"...here is a seemingly simple task: how long will it take to cross the Karakum desert? Answer, of course depends on the mode of travel. If a travel on camels, then one term will be required, another if you go by car, a third if you fly by plane. And most importantly, different models are required to plan a trip. For the first case, the required model can be found in the memoirs of famous desert explorers: after all, one cannot do without information about oases and camel trails. In the second case, irreplaceable information contained in the atlas of roads. In the third - you can use the flight schedule. |
1.2.5 Classification of models given in the manual of A.I. BochkinThere are many ways to classify .We present just a few of the more well-known foundations and signs: discreteness and continuity, matrix and scalar models, static and dynamic models, analytical and information models, subject and figurative-sign models, large-scale and non-scale... |
