How to make a mathematical model of the situation. Mathematical modeling

Basis for decision economic tasks are mathematical models.

mathematical model problem is a set of mathematical relationships that describe the essence of the problem.

Drawing up a mathematical model includes:

• 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:

Let's take linear inequality a 1 x 1 +a 2 x 2 +...+anxn ≤b and add some value x n+1 to its left side, such that the inequality becomes the equality a 1 x 1 +a 2 x 2 +... +anxn +x n+1 =b. Moreover, this value x n+1 is non-negative.

Let's consider everything with an example.

Example 26.1

Reduce 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.

To build a mathematical model, you need:

1. carefully analyze the real object or process;
2. highlight its most significant features and properties;
3. define variables, i.e. parameters whose values ​​affect the main features and properties of the object;
4. describe the dependence of the basic properties of an object, process or system on the value of variables using logical and mathematical relationships (equations, equalities, inequalities, logical and mathematical constructions);
5. highlight internal communications object, process or system with the help of restrictions, equations, equalities, inequalities, logical and mathematical constructions;
6. determine external relations and describe them using constraints, equations, equalities, inequalities, logical and mathematical constructions.

Mathematical modeling, in addition to studying an object, process or system and compiling their mathematical description, also includes:

1. construction of an algorithm that models the behavior of an object, process or system;
2. verification of the adequacy of the model and object, process or system based on computational and natural experiment;
3. model adjustment;
4. using the model.

The mathematical description of the processes and systems under study depends on:

1. nature real process or systems and is compiled on the basis of the laws of physics, chemistry, mechanics, thermodynamics, hydrodynamics, electrical engineering, the theory of plasticity, the theory of elasticity, etc.
2. the required reliability and accuracy of the study and study of real processes and systems.

The construction of a mathematical model usually begins with the construction and analysis of the simplest, most rough mathematical model of the object, process or system under consideration. In the future, if necessary, the model is refined, its correspondence to the object is made more complete.

Let's take a simple example. You need to determine the surface area of ​​the desk. Usually, for this, its length and width are measured, and then the resulting numbers are multiplied. Such an elementary procedure actually means the following: the real object (table surface) is replaced by an abstract mathematical model - a rectangle. The dimensions obtained as a result of measuring the length and width of the table surface are attributed to the rectangle, and the area of ​​​​such a rectangle is approximately taken as the desired area of ​​\u200b\u200bthe table. However, the desk rectangle model is the simplest, most crude model. With a more serious approach to the problem, before using the rectangle model to determine the table area, this model needs to be checked. Checks can be carried out as follows: measure the lengths of the opposite sides of the table, as well as the lengths of its diagonals and compare them with each other. If, with the required degree of accuracy, the lengths of the opposite sides and the lengths of the diagonals are pairwise equal, then the surface of the table can indeed be considered as a rectangle. Otherwise, the rectangle model will have to be rejected and replaced by a general quadrilateral model. With a higher requirement for accuracy, it may be necessary to refine the model even further, for example, to take into account the rounding of the corners of the table.

With the help of this a simple example it was shown that the mathematical model is not uniquely determined by the investigated object, process or system.

OR (to be confirmed tomorrow)

Ways to solve mat. Models:

1, Construction of m. on the basis of the laws of nature (analytical method)

2. Formal way with the help of statistical. Processing and measurement results (statistical approach)

3. Construction of a meter based on a model of elements (complex systems)

1, Analytical - use with sufficient study. General pattern Izv. models.

2. experiment. In the absence of information

3. Imitation m. - explores the properties of the object sst. Generally.

An example of building a mathematical model.

Mathematical model- this mathematical representation reality.

Mathematical modeling is the process of constructing and studying mathematical models.

All natural and social sciences that use the mathematical apparatus are, in fact, engaged in mathematical modeling: they replace an object with its mathematical model and then study the latter. The connection of a mathematical model with reality is carried out with the help of a chain of hypotheses, idealizations and simplifications. Via mathematical methods describes, as a rule, an ideal object built at the stage of meaningful modeling.

Why are models needed?

Very often, when studying an object, difficulties arise. The original itself is sometimes unavailable, or its use is not advisable, or the involvement of the original is costly. All these problems can be solved with the help of simulation. The model in a certain sense can replace the object under study.

The simplest examples of models

§ A photograph can be called a model of a person. In order to recognize a person, it is enough to see his photograph.

§ The architect created the layout of the new residential area. He can move a high-rise building from one part to another with a movement of his hand. In reality, this would not be possible.

Model types

Models can be divided into material" And ideal. the above examples are material models. Ideal models often have an iconic shape. At the same time, real concepts are replaced by some signs, which can be easily fixed on paper, in computer memory, etc.

Mathematical modeling

Mathematical modeling belongs to the class of sign modeling. At the same time, models can be created from any mathematical objects: numbers, functions, equations, etc.

Building a mathematical model

§ There are several stages of constructing a mathematical model:

1. Understanding the task, highlighting the most important qualities, properties, values ​​and parameters for us.

2. Introduction of notation.

3. Drawing up a system of restrictions that must be satisfied by the entered values.

4. Formulation and recording of the conditions that the desired optimal solution must satisfy.

The modeling process does not end with the compilation of the model, but only begins with it. Having compiled a model, they choose a method for finding the answer, solve the problem. after the answer is found, compare it with reality. And it is possible that the answer does not satisfy, in which case the model is modified or even a completely different model is chosen.

Example of a mathematical model

A task

The production association, which includes two furniture factories, needs to upgrade its machine park. Moreover, the first furniture factory needs to replace three machines, and the second seven. Orders can be placed at two machine tool factories. The first factory can produce no more than 6 machines, and the second factory will accept an order if there are at least three of them. It is required to determine how to place orders.

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;

Let us consider examples of specific economic problems of each type, and dwell in detail on building a model for each problem.

Transport task

On two trading bases BUT And IN 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 IN 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 IN to shops FROM And D respectively. Then the amount of furniture removed from the warehouse BUT, equals ( X 1 + X 2) well from stock IN - (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 IN in FROM - 2y 1 day units, from IN 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 we have considered can be represented in a more general view, 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 is based on the idea of ​​modeling scientific research, at the same time, in theoretical methods, various kinds of sign, abstract models are used, in experimental ones - subject models.

In the study of a complex real phenomenon, it is replaced by some simplified copy or scheme, sometimes such a copy serves only to remember and at the next meeting to find out the desired phenomenon. Sometimes the constructed scheme reflects some essential features, allows you to understand the mechanism of the phenomenon, makes it possible to predict its change. The same phenomenon can correspond different models.

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 deemed unsuitable. 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 singled out, 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 details and constituent parts can be written using mathematical relations: 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.

Mathematical 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 . IN stochastic model, the relationship between variables is random, sometimes it happens fundamentally. Impact huge amount 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.

IN 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. IN 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.

Mathematical 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.

1. Adequacy - the ability to reflect desired properties object with an error not higher than the specified one.
2. 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.
3. economy - is determined by the cost of computer memory resources and time for its implementation and operation.

Mathematical 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, general rules no.

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 set up an experiment on the spread of some disease, such as the plague, or to carry out nuclear explosion to study its implications. 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, a structure, economic plan, manufacturing process etc. At the same time, 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 the allowable 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 different type(differential, algebraic, etc.), establishing 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, and reveal its advantages in human cognition and transformation outside world, we will identify the 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? In order 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 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 the 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. Usage random variables makes it necessary to repeatedly conduct experiments with a simulation system (on a computer) and subsequent statistical analysis 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 enough difficult task. But together they represent the power that allows you to know the world, manage it in the interests of man.

1.2 Model classification

1.2.1 Classification taking into account the time factor and the area of ​​\u200b\u200buse (Makarova N.A.) Static model - it is like a one-time slice of information on the object (the result of one survey) Dynamic model-allows see changes in the object over time (Card in the clinic) Models can be classified according to what field of knowledge do they belong to(biological, historical, ecological, etc.) Return to start

1.2.2 Classification by area of ​​​​use (Makarova N.A.) Training- visual aids, trainers , oh thrashing programs Experienced models-reduced copies (car in a wind tunnel) Scientific and technical synchrophasotron, stand for testing electronic equipment Game- economic, sports, business games simulation- not they simply reflect reality, but imitate it (drugs are tested on mice, experiments are carried out in schools, etc.. This modeling method is called trial and error Return to start

1.2.3 Classification according to the method of presentation Makarova N.A.) material models- otherwise can be called subject. They perceive geometric and physical properties original and always have a real embodiment Informational models-not allowed touch or see. They are based on information. .Information model is a set of information that characterizes the properties and states of an object, process, phenomenon, as well as the relationship with the outside world. Verbal model - information model in a mental or conversational form. Iconic model-informational model expressed by signs , i.e.. by means of any formal language. Computer model - m A model implemented by means of a software environment.

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 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. These three models differ - memoirs, atlas and timetable and the nature of the presentation of information. In the first case, the model is presented verbal description information (descriptive model), in the second - like a photograph from nature (natural model), in the third - a table containing symbols: time of departure and arrival, day of the week, ticket price (the so-called sign model) However, this division is very conditional - maps and diagrams (elements of a full-scale model) can be found in memoirs, there are symbols on the maps (elements of a symbolic model), a decoding of symbols (elements of a descriptive model) is given in the schedule. So this classification of models ... in our opinion is unproductive" In my opinion, this fragment demonstrates the descriptive (wonderful language and style of presentation) common to all Gein's books and, as it were, the Socratic style of teaching (Everyone thinks that this is so. I completely agree with you, but if you look closely, then ...). In such books it is quite difficult to find a clear system of definitions (it is not intended by the author). In the textbook edited by N.A. Makarova demonstrates a different approach - the definitions of concepts are clearly distinguished and somewhat static.

1.2.5 Classification of models given in the manual of A.I. Bochkin There 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... Every sign gives a certain knowledge about the properties of both the model and the modeled reality. The sign can serve as a hint about the way the simulation has been performed or is to be done. Discreteness and continuity discreteness - a characteristic feature of computer models .After all the computer may be in the final, albeit very in large numbers states. Therefore, even if the object is continuous (time), in the model it will change in jumps. It could be considered continuity a sign of non-computer type models. Randomness and determinism . Uncertainty, accident initially opposed to the computer world: The algorithm launched again must repeat itself and give the same results. But to simulate random processes, pseudo-random number sensors are used. Introducing randomness into deterministic problems leads to powerful and interesting models (Random Tossing Area Calculation). Matrix - scalar. Availability of parameters matrix model indicates its greater complexity and, possibly, accuracy compared to scalar. For example, if we do not single out all age groups in the country's population, considering its change as a whole, we get a scalar model (for example, the Malthus model), if we single out, a matrix (gender and age) model. It was the matrix model that made it possible to explain the fluctuations in the birth rate after the war. static dynamism. These properties of the model are usually predetermined by the properties of the real object. There is no freedom of choice here. Just static model can be a step towards dynamic, or some of the model variables can be considered unchanged for the time being. For example, a satellite moves around the Earth, its movement is influenced by the Moon. If we consider the Moon to be stationary during the satellite's revolution, we obtain a simpler model. Analytical Models. Description of processes analytically, formulas and equations. But when trying to build a graph, it is more convenient to have tables of function values ​​​​and arguments. simulation models. simulation models appeared a long time ago in the form of large-scale copies of ships, bridges, etc. appeared a long time ago, but in connection with computers they are considered recently. Knowing how connected model elements analytically and logically, it is easier not to solve a system of certain relationships and equations, but to map the real system into computer memory, taking into account the links between memory elements. Information Models. Informational It is customary to oppose models to mathematical ones, more precisely algorithmic ones. The data/algorithm ratio is important here. If there is more data or they are more important, we have an information model, otherwise - mathematical. Subject Models. This is primarily a children's model - a toy. Figurative-sign models. It is primarily a model in the human mind: figurative, if graphic images predominate, and iconic, if there are more than words and/or numbers. Figurative-sign models are built on a computer. scale models. TO large-scale models are those of the subject or figurative models that repeat the shape of the object (map).

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 phenomena and processes being studied, 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. It's about first of all about qualitative analysis 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.

The second stage is the study of math problems, to which the constructed hypothetical model leads. The main thing at this stage is to obtain, as a result of mathematical analysis of the model, empirically verifiable theoretical consequences (solution of the direct problem). At the same time, cases are not uncommon when, for the construction and study of M.m. in various fields specifically- scientific knowledge the same mathematical apparatus is used (for example, differential equations) and the same type arises, although very non-trivial in each specific case, math problems. 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 the parameters of the model have been specified, the researchers try to find out how, within the limits of observational accuracy, 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 building a fundamentally new scientific theory.

The first M.m. appeared in ancient science. So, for modeling solar system the Greek mathematician and astronomer Eudoxus gave each planet four spheres, the combination of the movement of which created the 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, M.m. 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 the development of these models.

Literature:

Mathematical 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 // Issues 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.