Example 1.5.1.
Let some economic region produce several (n) types of products exclusively on its own and only for the population of this region. It is assumed that the technological process has been worked out, and the demand of the population for these goods has been studied. It is necessary to determine the annual volume of output of products, taking into account the fact that this volume must provide both final and industrial consumption.
Let's make a mathematical model of this problem. According to its condition, the following are given: types of products, demand for them and the technological process; find the volume of output for each type of product.
Let us denote the known quantities:
c i- public demand for i-th product ( i=1,...,n); a ij- number i-th product required to produce a unit of the j -th product using this technology ( i=1,...,n ; j=1,...,n);
X i - volume of output i-th product ( i=1,...,n); totality from =(c 1 ,..., c n ) is called the demand vector, the numbers a ij– technological coefficients, and the set X =(X 1 ,..., X n ) is the release vector.
By the condition of the problem, the vector X is divided into two parts: for final consumption (vector from ) and reproduction (vector x-s ). Calculate that part of the vector X which goes to reproduction. According to our designations for production X j quantity of the j-th product goes a ij · X j quantities i-th product.
Then the sum a i1 · X 1 +...+ a in · X n shows the value i-th product, which is needed for the entire output X =(X 1 ,..., X n ).
Therefore, the equality must hold:
Extending this reasoning to all types of products, we arrive at the desired model:
Solving this system of n linear equations with respect to X 1 ,...,X n and find the required output vector.
In order to write this model in a more compact (vector) form, we introduce the notation:
Square ( 
) -the matrix BUT called the technology matrix. It is easy to check that our model will now be written like this: x-s=Ah or
(1.6)
We got the classic model " Input - Output ”, the author of which is the famous American economist V. Leontiev.
Example 1.5.2.
An oil refinery has two grades of oil: grade BUT in the amount of 10 units, grade IN- 15 units. When processing oil, two materials are obtained: gasoline (we denote B) and fuel oil ( M). There are three options for the processing technology:
I: 1 unit BUT+ 2 units IN gives 3 units. B+ 2 units M
II: 2 units BUT+ 1 unit IN gives 1 unit. B+ 5 units M
III: 2 units BUT+ 2 units IN gives 1 unit. B+ 2 units M
The price of gasoline is $10 per unit, fuel oil is $1 per unit.
It is required to determine the most advantageous combination of technological processes for processing the available amount of oil.
Before modeling, we clarify the following points. It follows from the conditions of the problem that the “profitability” of the technological process for the plant should be understood in the sense of obtaining the maximum income from the sale of its finished products (gasoline and fuel oil). In this regard, it is clear that the "choice (making) decision" of the plant is to determine which technology and how many times to apply. Obviously, there are many such possibilities.
Let us denote the unknown quantities:
X i- amount of use i-th technological process (i=1,2,3). Other parameters of the model (reserves of oil grades, prices of gasoline and fuel oil) known.
Now one specific decision of the plant is reduced to the choice of one vector X =(x 1 ,X 2 ,X 3 ) , for which the plant's revenue is equal to (32x 1 +15x 2 +12x 3 ) dollars. Here, 32 dollars is the income received from one application of the first technological process (10 dollars 3 units. B+ $1 2 units M= $32). Coefficients 15 and 12 have a similar meaning for the second and third technological processes, respectively. Accounting for the oil reserve leads to the following conditions:
for variety BUT:
for variety IN:,
where in the first inequality the coefficients 1, 2, 2 are the consumption rates of grade A oil for a one-time application of technological processes I,II,III respectively. The coefficients of the second inequality have a similar meaning for grade B oil.
The mathematical model as a whole has the form:
Find such a vector x = (x 1 ,X 2 ,X 3 ) to maximize
f(x) = 32x 1 +15x 2 +12x 3
when the conditions are met:
The abbreviated form of this entry is as follows:
under restrictions
(1.7)
We got the so-called linear programming problem.
Model (1.7.) is an example of an optimization model of a deterministic type (with well-defined elements).
Example 1.5.3.
The investor needs to determine the best set of stocks, bonds and other securities to purchase them for a certain amount in order to obtain a certain profit with minimal risk to himself. Return on every dollar invested in a security j-th type, characterized by two indicators: expected profit and actual profit. For an investor, it is desirable that the expected profit per dollar of investments for the entire set of securities is not lower than a given value b.
Note that for the correct modeling of this problem, a mathematician requires certain basic knowledge in the field of portfolio theory of securities.
Let us denote the known parameters of the problem:
n- the number of types of securities; but j– actual profit (random number) from the j-th type of security; 
is the expected profit from j th type of security.
Denote the unknown quantities :
y j - funds allocated for the purchase of securities of the type j.
In our notation, the total amount invested is expressed as 
. To simplify the model, we introduce new quantities
.
In this way, X i- this is the share of all funds allocated for the purchase of securities of the type j.
It's clear that 
It can be seen from the condition of the problem that the goal of the investor is to achieve a certain level of profit with minimal risk. Essentially, risk is a measure of deviation of actual profit from expected one. Therefore, it can be identified with the profit covariance for securities of type i and type j. Here M is the designation of the mathematical expectation.
The mathematical model of the original problem has the form:
under restrictions
,
,
,
. (1.8)
We have obtained the well-known Markowitz model for optimizing the structure of a securities portfolio.
Model (1.8.) is an example of an optimization model of a stochastic type (with elements of randomness).
Example 1.5.4.
On the basis of a trade organization, there are n types of one of the products of the assortment minimum. Only one of the types of this product must be delivered to the store. It is required to choose the type of goods that it is advisable to bring to the store. If the product type j will be in demand, then the store will profit from its sale R j, if it is not in demand - a loss q j .
Before modeling, we will discuss some fundamental points. In this problem, the decision maker (DM) is the store. However, the outcome (getting the maximum profit) depends not only on his decision, but also on whether the imported goods will be in demand, i.e. whether they will be bought out by the population (it is assumed that for some reason the store does not have the opportunity to study the demand of the population ). Therefore, the population can be considered as the second decision maker, choosing the type of goods according to their preferences. The worst "decision" of the population for the store is: "the imported goods are not in demand." So, in order to take into account all kinds of situations, the store needs to consider the population as its “opponent” (conditionally), pursuing the opposite goal - to minimize the store’s profit.
So, we have a decision problem with two participants pursuing opposite goals. Let us clarify that the store chooses one of the types of goods for sale (there are n solutions), and the population chooses one of the types of goods that is in the greatest demand ( n solution options).
To compile a mathematical model, we draw a table with n lines and n columns (total n 2 cells) and agree that the rows correspond to the choice of the store, and the columns correspond to the choice of the population. Then the cage (i, j) corresponds to the situation when the store chooses i-th type of goods ( i-th line), and the population chooses j-th type of goods ( j- th column). In each cell, we write a numerical assessment (profit or loss) of the corresponding situation from the point of view of the store:
Numbers q i written with a minus to reflect the loss of the store; in each situation, the “payoff” of the population is (conditionally) equal to the “payoff” of the store, taken with the opposite sign.
An abbreviated view of this model is as follows:
(1.9)
We got the so-called matrix game. Model (1.9.) is an example of game decision making models.
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 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 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.
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. 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.
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.
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, 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) 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.
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 external 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? In order to
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:
How is a mathematical model built?
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:
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.
1.2.1
|
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 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... |
Lecture 1
METHODOLOGICAL BASES OF MODELING
The current state of the problem of system modeling
Concepts of Modeling and Simulation
Modeling can be considered as a replacement of the investigated object (original) by its conditional image, description or another object, called model and providing behavior close to the original within certain assumptions and acceptable errors. Modeling is usually performed with the aim of knowing the properties of the original by examining its model, and not the object itself. Of course, modeling is justified in the case when it is simpler than creating the original itself, or when the latter, for some reason, is better not to create at all.
Under model a physical or abstract object is understood, the properties of which are in a certain sense similar to the properties of the object under study. In this case, the requirements for the model are determined by the problem being solved and the available means. There are a number of general requirements for models:
2) completeness - providing the recipient with all the necessary information
about the object;
3) flexibility - the ability to reproduce different situations in everything
range of changing conditions and parameters;
4) the complexity of development should be acceptable for the existing
time and software.
Modeling is the process of building a model of an object and studying its properties by examining the model.
Thus, modeling involves 2 main stages:
1) model development;
2) study of the model and drawing conclusions.
At the same time, at each stage, different tasks are solved and
essentially different methods and means.
In practice, various modeling methods are used. Depending on the method of implementation, all models can be divided into two large classes: physical and mathematical.
Mathematical modeling It is customary to consider it as a means of studying processes or phenomena with the help of their mathematical models.
Under physical modeling is understood as the study of objects and phenomena on physical models, when the process under study is reproduced with the preservation of its physical nature or another physical phenomenon similar to the one under study is used. Wherein physical models As a rule, they assume the real embodiment of those physical properties of the original that are essential in a particular situation. For example, when designing a new aircraft, its model is created that has the same aerodynamic properties; when planning a building, architects make a layout that reflects the spatial arrangement of its elements. In this regard, physical modeling is also called prototyping.
HIL Modeling is a study of controlled systems on simulation complexes with the inclusion of real equipment in the model. Along with real equipment, the closed model includes impact and interference simulators, mathematical models of the external environment and processes for which a sufficiently accurate mathematical description is not known. The inclusion of real equipment or real systems in the circuit for modeling complex processes makes it possible to reduce a priori uncertainty and investigate processes for which there is no exact mathematical description. With the help of semi-natural simulation, studies are performed taking into account small time constants and non-linearities inherent in real equipment. In the study of models with the inclusion of real equipment, the concept is used dynamic simulation, in the study of complex systems and phenomena - evolutionary, imitation And cybernetic simulation.
Obviously, the real benefit of modeling can only be obtained if two conditions are met:
1) the model provides a correct (adequate) display of properties
the original, significant from the point of view of the operation under study;
2) the model makes it possible to eliminate the problems listed above, which are inherent
conducting research on real objects.
The solution of practical problems by mathematical methods is consistently carried out by formulating the problem (development of a mathematical model), choosing a method for studying the obtained mathematical model, and analyzing the obtained mathematical result. The mathematical formulation of the problem is usually presented in the form of geometric images, functions, systems of equations, etc. The description of an object (phenomenon) can be represented using continuous or discrete, deterministic or stochastic and other mathematical forms.
Theory of mathematical modeling ensures the identification of regularities in the course of various phenomena of the surrounding world or the operation of systems and devices by their mathematical description and modeling without field tests. In this case, the provisions and laws of mathematics are used that describe the simulated phenomena, systems or devices at a certain level of their idealization.
Mathematical Model (MM) is a formalized description of a system (or operation) in some abstract language, for example, in the form of a set of mathematical relations or an algorithm scheme, i.e. e. such a mathematical description that provides an imitation of the operation of systems or devices at a level sufficiently close to their real behavior obtained during full-scale testing of systems or devices.
Any MM describes a real object, phenomenon or process with some degree of approximation to reality. The type of MM depends both on the nature of the real object and on the objectives of the study.
Mathematical modeling social, economic, biological and physical phenomena, objects, systems and various devices is one of the most important means of understanding nature and designing a wide variety of systems and devices. There are known examples of the effective use of modeling in the creation of nuclear technologies, aviation and aerospace systems, in the forecast of atmospheric and oceanic phenomena, weather, etc.
However, such serious areas of modeling often require supercomputers and years of work by large teams of scientists to prepare data for modeling and its debugging. Nevertheless, in this case, too, mathematical modeling of complex systems and devices not only saves money on research and testing, but can also eliminate environmental disasters - for example, it makes it possible to abandon the testing of nuclear and thermonuclear weapons in favor of their mathematical modeling or testing aerospace systems before their real flights. Meanwhile, mathematical modeling at the level of solving simpler problems, for example, from the field of mechanics, electrical engineering, electronics, radio engineering and many other areas of science and technology, has now become available to perform on modern PCs. And when using generalized models, it becomes possible to model quite complex systems, for example, telecommunication systems and networks, radar or radio navigation systems.
The purpose of mathematical modeling is the analysis of real processes (in nature or technology) by mathematical methods. In turn, this requires the formalization of the MM process to be investigated. The model can be a mathematical expression containing variables whose behavior is similar to the behavior of a real system. The model can include elements of randomness that take into account the probabilities of possible actions of two or more "players", games; or it may represent the real variables of the interconnected parts of the operating system.
Mathematical modeling for studying the characteristics of systems can be divided into analytical, simulation and combined. In turn, MM are divided into simulation and analytical.
Analytical Modeling
For analytical modeling it is characteristic that the processes of functioning of the system are written in the form of some functional relations (algebraic, differential, integral equations). The analytical model can be investigated by the following methods:
1) analytical, when they strive to obtain in general terms explicit dependencies for the characteristics of systems;
2) numerical, when it is not possible to find a solution to equations in general form and they are solved for specific initial data;
3) qualitative, when, in the absence of a solution, some of its properties are found.
Analytical models can be obtained only for relatively simple systems. For complex systems, large mathematical problems often arise. To apply the analytical method, one goes to a significant simplification of the original model. However, a study on a simplified model helps to obtain only indicative results. Analytical models mathematically correctly reflect the relationship between input and output variables and parameters. But their structure does not reflect the internal structure of the object.
In analytical modeling, its results are presented in the form of analytical expressions. For example, by connecting RC- circuit to a constant voltage source E(R, C And E are the components of this model), we can make an analytical expression for the time dependence of the voltage u(t) on the capacitor C:
This is a linear differential equation (DE) and is an analytical model of this simple linear circuit. Its analytical solution, under the initial condition u(0) = 0 , meaning a discharged capacitor C at the beginning of the simulation, allows you to find the required dependence - in the form of a formula:
u(t) = E(1− exp(- t/RC)). (2)
However, even in this simplest example, certain efforts are required to solve differential equation (1) or to apply computer mathematics systems(SCM) with symbolic calculations - computer algebra systems. For this quite trivial case, the solution of the problem of modeling a linear RC-circuit gives an analytical expression (2) of a rather general form - it is suitable for describing the operation of the circuit for any component ratings R, C And E, and describes the exponential charge of the capacitor C through a resistor R from a constant voltage source E.
Undoubtedly, finding analytical solutions in analytical modeling turns out to be extremely valuable for revealing the general theoretical laws of simple linear circuits, systems and devices. However, its complexity increases sharply as the impact on the model becomes more complex and the order and number of equations of state that describe the modeled object increase. You can get more or less visible results when modeling objects of the second or third order, but even with a higher order, analytical expressions become excessively cumbersome, complex and difficult to comprehend. For example, even a simple electronic amplifier often contains dozens of components. However, many modern SCMs, such as systems of symbolic mathematics Maple, Mathematica or Wednesday MATLAB are capable of automating to a large extent the solution of complex problems of analytical modeling.
One type of modeling is numerical simulation, which consists in obtaining the necessary quantitative data about the behavior of systems or devices by any suitable numerical method, such as the Euler or Runge-Kutta methods. In practice, the modeling of nonlinear systems and devices using numerical methods is much more efficient than the analytical modeling of individual private linear circuits, systems or devices. For example, to solve DE (1) or DE systems in more complex cases, the solution in an analytical form is not obtained, but numerical simulation data can provide sufficiently complete data on the behavior of the simulated systems and devices, as well as plot graphs describing this behavior of dependencies.
Simulation
At imitation In modeling, the algorithm that implements the model reproduces the process of the system functioning in time. The elementary phenomena that make up the process are imitated, with the preservation of their logical structure and the sequence of flow in time.
The main advantage of simulation models compared to analytical ones is the ability to solve more complex problems.
Simulation models make it easy to take into account the presence of discrete or continuous elements, non-linear characteristics, random effects, etc. Therefore, this method is widely used at the design stage of complex systems. The main tool for the implementation of simulation modeling is a computer that allows digital modeling of systems and signals.
In this regard, we define the phrase " computer modelling”, which is increasingly used in the literature. We will assume that computer modelling- this is mathematical modeling using computer technology. Accordingly, computer simulation technology involves the following actions:
1) definition of the purpose of modeling;
2) development of a conceptual model;
3) formalization of the model;
4) software implementation of the model;
5) planning of model experiments;
6) implementation of the experiment plan;
7) analysis and interpretation of simulation results.
At simulation modeling the used MM reproduces the algorithm (“logic”) of the functioning of the system under study in time for various combinations of values of the parameters of the system and the environment.
An example of the simplest analytical model is the equation of uniform rectilinear motion. When studying such a process using a simulation model, observation of the change in the path traveled over time should be implemented. Obviously, in some cases, analytical modeling is more preferable, in others - simulation (or a combination of both). To make a good choice, two questions must be answered.
What is the purpose of modeling?
To what class can the simulated phenomenon be assigned?
Answers to both of these questions can be obtained during the execution of the first two stages of modeling.
Simulation models not only in properties, but also in structure correspond to the object being modeled. In this case, there is an unambiguous and explicit correspondence between the processes obtained on the model and the processes occurring on the object. The disadvantage of simulation modeling is that it takes a long time to solve the problem in order to obtain good accuracy.
The results of simulation modeling of the work of a stochastic system are realizations of random variables or processes. Therefore, to find the characteristics of the system, multiple repetition and subsequent data processing are required. Most often, in this case, a type of simulation is used - statistical
modeling(or the Monte Carlo method), i.e. reproduction in models of random factors, events, quantities, processes, fields.
According to the results of statistical modeling, estimates of probabilistic quality criteria, general and particular, characterizing the functioning and efficiency of the controlled system are determined. Statistical modeling is widely used to solve scientific and applied problems in various fields of science and technology. Methods of statistical modeling are widely used in the study of complex dynamic systems, evaluation of their functioning and efficiency.
The final stage of statistical modeling is based on the mathematical processing of the obtained results. Here, methods of mathematical statistics are used (parametric and non-parametric estimation, hypothesis testing). An example of a parametric assessment is the sample mean of a performance measure. Among the nonparametric methods, the most widely used histogram method.
The considered scheme is based on multiple statistical tests of the system and methods of statistics of independent random variables. This scheme is far from always natural in practice and optimal in terms of costs. Reduction of system testing time can be achieved by using more accurate estimation methods. As is known from mathematical statistics, effective estimates have the highest accuracy for a given sample size. Optimal filtering and the maximum likelihood method provide a general method for obtaining such estimates. In statistical modeling problems, the processing of realizations of random processes is necessary not only for the analysis of output processes.
It is also very important to control the characteristics of input random effects. The control consists in checking whether the distributions of the generated processes correspond to the given distributions. This task is often formulated as hypothesis testing task.
The general trend in computer-assisted simulation of complex controlled systems is the desire to reduce the simulation time, as well as to conduct research in real time. Computational algorithms are conveniently represented in a recurrent form that allows their implementation at the pace of current information.
PRINCIPLES OF A SYSTEM APPROACH IN MODELING
Fundamentals of systems theory
The main provisions of the theory of systems arose in the course of the study of dynamic systems and their functional elements. A system is understood as a group of interrelated elements acting together to perform a predetermined task. Systems analysis allows you to determine the most realistic ways to complete the task, ensuring maximum satisfaction of the requirements.
The elements that form the basis of systems theory are not created with the help of hypotheses, but are discovered experimentally. In order to start building a system, it is necessary to have general characteristics of technological processes. The same is true for the principles of creating mathematically formulated criteria that a process or its theoretical description must satisfy. Modeling is one of the most important methods of scientific research and experimentation.
When building models of objects, a systematic approach is used, which is a methodology for solving complex problems, which is based on the consideration of an object as a system operating in a certain environment. The system approach involves the disclosure of the integrity of the object, the identification and study of its internal structure, as well as connections with the external environment. In this case, the object is presented as a part of the real world, which is identified and studied in connection with the problem of building a model being solved. In addition, the systematic approach involves a consistent transition from the general to the particular, when the consideration is based on the design goal, and the object is considered in relation to the environment.
A complex object can be divided into subsystems, which are parts of the object that meet the following requirements:
1) the subsystem is a functionally independent part of the object. It is connected with other subsystems, exchanges information and energy with them;
2) for each subsystem, functions or properties that do not coincide with the properties of the entire system can be defined;
3) each of the subsystems can be further subdivided to the level of elements.
In this case, an element is understood as a subsystem of the lower level, the further division of which is inexpedient from the standpoint of the problem being solved.
Thus, a system can be defined as a representation of an object in the form of a set of subsystems, elements, and relationships for the purpose of its creation, research, or improvement. At the same time, an enlarged representation of the system, which includes the main subsystems and connections between them, is called a macrostructure, and a detailed disclosure of the internal structure of the system to the level of elements is called a microstructure.
Along with the system, there is usually a supersystem - a system of a higher level, which includes the object under consideration, and the function of any system can be determined only through the supersystem.
It is necessary to single out the concept of the environment as a set of objects of the external world that significantly affect the efficiency of the system, but are not part of the system and its supersystem.
In connection with the systematic approach to building models, the concept of infrastructure is used, which describes the relationship of the system with its environment (environment). In this case, the selection, description and study of the properties of an object that are essential within a specific task is called the stratification of an object, and any model of an object is its stratified description.
For a systematic approach, it is important to determine the structure of the system, i.e. set of links between the elements of the system, reflecting their interaction. To do this, we first consider the structural and functional approaches to modeling.
With a structural approach, the composition of the selected elements of the system and the links between them are revealed. The totality of elements and relationships makes it possible to judge the structure of the system. The most general description of a structure is a topological description. It allows you to define the components of the system and their relationships using graphs. Less general is the functional description when individual functions are considered, i.e., algorithms for the behavior of the system. At the same time, a functional approach is implemented that determines the functions that the system performs.
On the basis of a systematic approach, a sequence of model development can be proposed, when two main stages of design are distinguished: macro-design and micro-design.
At the macro-design stage, a model of the external environment is built, resources and constraints are identified, a system model and criteria for assessing adequacy are selected.
The stage of microdesign largely depends on the specific type of model chosen. In the general case, it involves the creation of information, mathematical, technical and software support for the modeling system. At this stage, the main technical characteristics of the created model are established, the time of working with it and the cost of resources to obtain the specified quality of the model are estimated.
Regardless of the type of model, when building it, it is necessary to be guided by a number of principles of a systematic approach:
1) consistent progress through the stages of creating a model;
2) coordination of information, resource, reliability and other characteristics;
3) the correct ratio of different levels of model building;
4) the integrity of the individual stages of model design.
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:
Table 3.1
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?
Mathematical models
Mathematical model - approximate opidescription of the object of modeling, expressed usingschyu mathematical symbolism.
Mathematical models appeared along with mathematics many centuries ago. A huge impetus to the development of mathematical modeling was given by the appearance of computers. The use of computers made it possible to analyze and put into practice many mathematical models that had not previously been amenable to analytical research. Computer-implemented mathematicalsky model called computer mathematical model, but carrying out targeted calculations using a computer model called computational experiment.
Stages of computer mathematical modeletion shown in the figure. Firststage - definition of modeling goals. These goals can be different:
An example from a completely different area: peacefully coexisting with stable populations of two species of individuals with a common food base, “suddenly” begin to dramatically change their numbers. And here mathematical modeling allows (with a certain degree of certainty) to establish the cause (or at least to refute a certain hypothesis).
Development of the concept of object management is another possible goal of modeling. Which aircraft flight mode should be chosen in order for the flight to be safe and most economically advantageous? How to schedule hundreds of types of work on the construction of a large facility so that it ends as soon as possible? Many such problems systematically arise before economists, designers, and scientists.
Finally, predicting the consequences of certain impacts on an object can be both a relatively simple matter in simple physical systems, and extremely complex - on the verge of feasibility - in biological, economic, social systems. If it is relatively easy to answer the question about the change in the mode of heat propagation in a thin rod with changes in its constituent alloy, then it is incomparably more difficult to trace (predict) the environmental and climatic consequences of the construction of a large hydroelectric power station or the social consequences of changes in tax legislation. Perhaps, here, too, mathematical modeling methods will provide more significant assistance in the future.
Second phase: definition of input and output parameters of the model; division of input parameters according to the degree of importance of the impact of their changes on the output. This process is called ranking, or division by rank (see below). "Formalisation and modeling").
Third stage: construction of a mathematical model. At this stage, there is a transition from the abstract formulation of the model to a formulation that has a specific mathematical representation. A mathematical model is equations, systems of equations, systems of inequalities, differential equations or systems of such equations, etc.
Fourth stage: choice of method for studying the mathematical model. Most often, numerical methods are used here, which lend themselves well to programming. As a rule, several methods are suitable for solving the same problem, differing in accuracy, stability, etc. The success of the entire modeling process often depends on the correct choice of method.
Fifth stage: the development of an algorithm, the compilation and debugging of a computer program is a process that is difficult to formalize. Of the programming languages, many professionals for mathematical modeling prefer FORTRAN: both due to tradition, and due to the unsurpassed efficiency of compilers (for computational work) and the presence of huge, carefully debugged and optimized libraries of standard programs of mathematical methods written in it. Languages such as PASCAL, BASIC, C are also in use, depending on the nature of the task and the inclinations of the programmer.
Sixth stage: program testing. The operation of the program is tested on a test problem with a known answer. This is just the beginning of a testing procedure that is difficult to describe in a formally exhaustive way. Usually, testing ends when the user, according to his professional characteristics, considers the program correct.
Seventh stage: actual computational experiment, during which it becomes clear whether the model corresponds to a real object (process). The model is sufficiently adequate to the real process if some characteristics of the process obtained on a computer coincide with the experimentally obtained characteristics with a given degree of accuracy. If the model does not correspond to the real process, we return to one of the previous stages.
Classification of mathematical models
The classification of mathematical models can be based on various principles. It is possible to classify models by branches of science (mathematical models in physics, biology, sociology, etc.). It can be classified according to the applied mathematical apparatus (models based on the use of ordinary differential equations, partial differential equations, stochastic methods, discrete algebraic transformations, etc.). Finally, if we proceed from the general tasks of modeling in different sciences, regardless of the mathematical apparatus, the following classification is most natural:
Let's explain this with examples.
Descriptive (descriptive) models. For example, simulations of the motion of a comet that invades the solar system are made to predict its flight path, the distance it will pass from the Earth, and so on. In this case, the goals of modeling are descriptive, since there is no way to influence the motion of the comet, to change something in it.
Optimization Models are used to describe the processes that can be influenced in an attempt to achieve a given goal. In this case, the model includes one or more parameters that can be influenced. For example, by changing the thermal regime in a granary, one can set a goal to choose such a regime in order to achieve maximum grain preservation, i.e. optimize the storage process.
Multicriteria models. Often it is necessary to optimize the process in several parameters at the same time, and the goals can be very contradictory. For example, knowing food prices and a person's need for food, it is necessary to organize meals for large groups of people (in the army, children's summer camp, etc.) physiologically correctly and, at the same time, as cheaply as possible. It is clear that these goals do not coincide at all; when modeling, several criteria will be used, between which a balance must be sought.
Game models can be related not only to computer games, but also to very serious things. For example, before a battle, in the presence of incomplete information about the opposing army, a commander must develop a plan: in what order to bring certain units into battle, etc., taking into account the possible reaction of the enemy. There is a special section of modern mathematics - game theory - that studies the methods of decision making under conditions of incomplete information.
In the school course of computer science, students receive an initial idea of computer mathematical modeling as part of the basic course. In high school, mathematical modeling can be deeply studied in a general education course for classes in physics and mathematics, as well as within a specialized elective course.
The main forms of teaching computer mathematical modeling in high school are lectures, laboratory and credit classes. Usually, the work on creating and preparing for the study of each new model takes 3-4 lessons. In the course of the presentation of the material, tasks are set, which in the future must be solved by students on their own, in general terms, ways to solve them are outlined. Questions are formulated, the answers to which should be obtained when performing tasks. Additional literature is indicated, which allows obtaining auxiliary information for more successful completion of tasks.
The form of organizing classes in the study of new material is usually a lecture. After the completion of the discussion of the next model students have at their disposal the necessary theoretical information and a set of tasks for further work. In preparation for the assignment, students choose the appropriate solution method, using some known private solution, they test the developed program. In case of quite possible difficulties in completing the tasks, consultation is given, a proposal is made to work out these sections in more detail in the literature.
The most relevant to the practical part of teaching computer modeling is the method of projects. The task is formulated for the student in the form of an educational project and is carried out over several lessons, and the main organizational form in this case is computer laboratory work. Learning to model using the learning project method can be implemented at different levels. The first is a problem statement of the project implementation process, which is led by the teacher. The second is the implementation of the project by students under the guidance of a teacher. The third is the independent implementation by students of an educational research project.
The results of the work should be presented in numerical form, in the form of graphs, diagrams. If possible, the process is presented on the computer screen in dynamics. Upon completion of the calculations and the receipt of the results, they are analyzed, compared with known facts from the theory, the reliability is confirmed and a meaningful interpretation is carried out, which is subsequently reflected in a written report.
If the results satisfy the student and the teacher, then the work counts completed, and its final stage is the preparation of a report. The report includes brief theoretical information on the topic under study, a mathematical formulation of the problem, a solution algorithm and its justification, a computer program, the results of the program, analysis of the results and conclusions, a list of references.
When all the reports have been drawn up, at the test session, students make brief reports on the work done, defend their project. This is an effective form of reporting by the project team to the class, including setting the problem, building a formal model, choosing methods for working with the model, implementing the model on a computer, working with the finished model, interpreting the results, forecasting. As a result, students can receive two grades: the first - for the elaboration of the project and the success of its defense, the second - for the program, the optimality of its algorithm, interface, etc. Students also receive marks in the course of surveys on theory.
An essential question is what kind of tools to use in the school informatics course for mathematical modeling? Computer implementation of models can be carried out:
At the elementary school level, the first remedy appears to be the preferred one. However, in high school, when programming is, along with modeling, a key topic of computer science, it is desirable to involve it as a modeling tool. In the process of programming, the details of mathematical procedures become available to students; moreover, they are simply forced to master them, and this also contributes to mathematical education. As for the use of special software packages, this is appropriate in a profile computer science course as a supplement to other tools.
The task :
