System modeling. System, Model, Simulation

Modeling is based on the theory of similarity, which states that absolute similarity can only take place when one object is replaced by another exactly the same. When modeling, absolute similarity does not take place and they strive to ensure that the model reflects the studied side of the object's functioning well enough.

Classification signs. As one of the first features of the classification of types of modeling, you can choose the degree of completeness of the model and divide the models according to this feature into complete, incomplete and approximate. Full simulation is based on complete similarity, which manifests itself both in time and in space. Incomplete modeling is characterized by incomplete similarity of the model to the object under study. Approximate modeling is based on approximate similarity, in which some aspects of the functioning of a real object are not modeled at all.

Depending on the nature of the processes under study in the system S, all types of modeling can be divided into deterministic and stochastic, static and dynamic, discrete, continuous and discrete-continuous. Deterministic modeling displays deterministic processes, i.e. processes in which the absence of any random influences is assumed; stochastic modeling displays probabilistic processes and events. In this case, a number of implementations of a random process are analyzed and the average characteristics are evaluated, i.e., a set of homogeneous implementations. Static modeling is used to describe the behavior of an object at any point in time, while dynamic modeling reflects the behavior of an object over time. Discrete modeling serves to describe processes that are assumed to be discrete, respectively, continuous modeling allows you to reflect continuous processes in systems, and discrete-continuous modeling is used for cases where you want to highlight the presence of both discrete and continuous processes.

Depending on the form of representation of the object (system J, mental and real modeling can be distinguished.

Mental modeling is often the only way to model objects that are either practically unrealizable in a given time interval or exist outside the conditions possible for their physical creation. For example, on the basis of mental modeling, many situations of the microworld that are not amenable to physical experiment can be analyzed. Mental modeling can be implemented in the form of visual, symbolic and mathematical.

Analog modeling is based on the use of analogies at various levels. The highest level is complete analogy, which takes place only for fairly simple objects. With the complication of the object, analogies of subsequent levels are used, when the analog model displays several or only one side of the object's functioning.

Prototyping occupies an essential place in mental visual modeling. A mental layout can be used in cases where the processes occurring in a real object are not amenable to physical modeling, or it can precede other types of modeling. The construction of mental layouts is also based on analogies, but usually based on cause-and-effect relationships between phenomena and processes in an object. If you introduce a symbol of individual concepts, i.e. signs, as well as certain operations between these signs, then you can implement sign modeling and use signs to display a set of concepts - to make separate chains of words and sentences. Using the operations of union, intersection and addition of set theory, it is possible to give a description of some real object in separate symbols.

At the heart of language modeling is a certain thesaurus. The latter is formed from a set of incoming concepts, and this set must be fixed. It should be noted that there are fundamental differences between a thesaurus and a regular dictionary. Thesaurus is a dictionary that is cleared of ambiguity, that is, in it only a single concept can correspond to each word, although in an ordinary dictionary several concepts can correspond to one word.

Symbolic modeling is an artificial process of creating a logical object that replaces the real one and expresses the main properties of its relations with the help of a certain system of signs or symbols.

Math modeling. To study the characteristics of the process of functioning of any system S by mathematical methods, including machine methods, this process must be formalized, i.e., a mathematical model has been built.

By mathematical modeling we will understand the process of establishing correspondence to a given real object of some mathematical object, called a mathematical model, and the study of this model, which allows obtaining the characteristics of the real object under consideration. The type of mathematical model depends on both the nature of the real object and the tasks of researching the object and the required reliability and accuracy of solving this problem. Any mathematical model, like any other,

Fig 1.

describes a real object only with some degree of approximation to reality. Mathematical modeling for studying the characteristics of the process of functioning of systems can be divided into analytical, simulation and combined.

For analytical modeling, it is characteristic that the processes of functioning of the elements of the system are written in the form of some functional relations (algebraic, integro-differential, finite-difference, etc.) or logical conditions. An analytical model can be studied by the following methods: a) analytical, when one strives to obtain in general terms explicit dependencies for the desired characteristics; b) numerical, when, not being able to solve equations in a general form, they strive to obtain numerical results with specific initial data; c) qualitative, when, without having an explicit solution, you can find some properties of the solution (for example, evaluate the stability of the solution).

In some cases, studies of the system can also satisfy the conclusions that can be drawn when using the qualitative method of analyzing a mathematical model. Such qualitative methods are widely used, for example, in the theory of automatic control to evaluate the effectiveness of various options for control systems.

Modeling is based on similarity theory, which states that absolute similarity can take place only when one object is replaced by another exactly the same. When modeling, absolute similarity does not take place and they strive to ensure that the model reflects the studied side of the object's functioning well enough.

Classification signs. As one of the first features of the classification of types of modeling, one can choose the degree of completeness of the model and divide the models according to this feature into complete, incomplete and approximate. Full simulation is based on complete similarity, which manifests itself both in time and in space. Incomplete modeling is characterized by incomplete similarity of the model to the object under study. Approximate modeling is based on approximate similarity, in which some aspects of the functioning of a real object are not modeled at all. Classification of types of system modeling S shown in fig. 1.2.

Rice. 1.2 - Classification of types of system modeling

Depending on the nature of the studied processes in the system S all types of modeling can be divided into deterministic and stochastic, static and dynamic, discrete, continuous and discrete-continuous. Deterministic Simulation displays deterministic processes, i.e. processes in which the absence of any random influences is assumed; stochastic modeling displays probabilistic processes and events. In this case, a number of implementations of a random process are analyzed and the average characteristics are estimated, i.e., a set of homogeneous implementations. Static Simulation is used to describe the behavior of an object at some point in time, and dynamic simulation reflects the behavior of an object over time.

Discrete Simulation serves to describe processes that are assumed to be discrete, respectively, continuous modeling allows you to reflect continuous processes in systems, and discretely - continuous simulation is used for cases when they want to highlight the presence of both discrete and continuous processes.

Depending on the form of representation of the object (system S) it is possible to distinguish mental and real modeling.

mental modeling often is the only way to model objects that are either practically unrealizable in a given time interval, or exist outside the conditions possible for their physical creation. For example, on the basis of mental modeling, many situations of the microworld that are not amenable to physical experiment can be analyzed. Mental modeling can be implemented in the form of visual, symbolic and mathematical.

At visual modeling on the basis of human ideas about real objects, various visual models are created that display the phenomena and processes occurring in the object. The basis hypothetical simulation the researcher lays down some hypothesis about the patterns of the process in a real object, which reflects the level of knowledge of the researcher about the object and is based on cause-and-effect relationships between the input and output of the object under study. Hypothetical modeling is used when knowledge about the object is not enough to build formal models.

Analog simulation is based on the application of analogies of various levels. The highest level is a complete analogy, which takes place only for fairly simple objects.

With the complication of the object, analogies of subsequent levels are used, when the analog model displays several or only one side of the object's functioning.

An important place in mental visual modeling is occupied by prototyping. A mental model can be used in cases where the processes occurring in a real object are not amenable to physical modeling, or it can precede other types of modeling. The construction of mental models is also based on analogies, however, they are usually based on cause-and-effect relationships between phenomena and processes in an object. If we introduce a symbol for individual concepts, i.e. signs, as well as certain operations between these signs, then we can implement iconic modeling and using signs to display a set of concepts - to make separate chains of words and sentences. Using the operations of union, intersection and addition of set theory, it is possible to give a description of some real object in separate symbols.

At the core language modeling lies some thesaurus. The latter is formed from a set of incoming concepts, and this set must be fixed. It should be noted that there are fundamental differences between a thesaurus and a regular dictionary. Thesaurus is a dictionary that is cleared of ambiguity, i.e. in it only a single concept can correspond to each word, although in a regular dictionary several concepts can correspond to one word.

Symbolic modeling is an artificial process of creating a logical object that replaces the real one and expresses the main properties of its relations using a certain system of signs or symbols.

Math modeling. To study the characteristics of the process of functioning of any system S using mathematical methods, including machine methods, this process should be formalized, i.e., a mathematical model should be built.

Under mathematical modeling we will understand the process of establishing correspondence to a given real object of some mathematical object, called a mathematical model, and the study of this model, which makes it possible to obtain the characteristics of the real object under consideration. The type of mathematical model depends on both the nature of the real object and the tasks of studying the object and the required reliability and accuracy of solving this problem. Any mathematical model, like any other, describes a real object only with a certain degree of approximation to reality. Mathematical modeling for studying the characteristics of the process of functioning of systems can be divided into analytical, simulation and combined.

For analytical modeling it is characteristic that the processes of functioning of the elements of the system are written in the form of some functional relations (algebraic, integro-differential, finite-difference, etc.) or logical conditions. An analytical model can be studied by the following methods: a) analytical, when one strives to obtain in general terms explicit dependencies for the desired characteristics; b) numerical, when, not being able to solve equations in a general form, they strive to obtain numerical results with specific initial data; c) qualitative, when, without having a solution in an explicit form, it is possible to find some properties of the solution (for example, to estimate the stability of the solution).

The most complete study of the process of system functioning can be carried out if explicit dependencies are known that connect the desired characteristics with the initial conditions, parameters and variables of the system S. However, such dependencies can be obtained only for relatively simple systems. As systems become more complex, their study by the analytical method encounters significant difficulties, which are often insurmountable. Therefore, wishing to use the analytical method, in this case they go to a significant simplification of the original model in order to be able to study at least the general properties of the system. Such a study on a simplified model by the analytical method helps to obtain indicative results for determining more accurate estimates by other methods. The numerical method allows us to study a wider class of systems compared to the analytical method, but the solutions obtained are of a particular nature. The numerical method is especially effective when using a computer.

In some cases, studies of the system can also satisfy the conclusions that can be drawn using the qualitative method of analyzing a mathematical model. Such qualitative methods are widely used, for example, in the theory of automatic control to evaluate the effectiveness of various options for control systems.

At present, methods of machine implementation of the study of the characteristics of the process of functioning of large systems are widespread. To implement a mathematical model on a computer, it is necessary to build an appropriate modeling algorithm.

At simulation modeling the algorithm that implements the model reproduces the process of the system functioning S in time, and the elementary phenomena that make up the process are simulated, with the preservation of their logical structure and the sequence of flow in time, which allows, according to the initial data, to obtain information about the states of the process at certain points in time, making it possible to evaluate the characteristics of the system S.

The main advantage of simulation modeling compared to analytical modeling is the ability to solve more complex problems. Simulation models make it possible to easily take into account such factors as the presence of discrete and continuous elements, non-linear characteristics of system elements, numerous random effects, etc., which often create difficulties in analytical studies. Currently, simulation modeling is the most effective method for studying large systems, and often the only practically accessible method for obtaining information about the behavior of a system, especially at the stage of its design.

When the results obtained by reproducing on the simulation model of the system functioning process S, are realizations of random variables and functions, then to find the characteristics of the process, it is required to reproduce it multiple times with subsequent statistical processing of information, and it is advisable to use the method of statistical modeling as a method of machine implementation of the simulation model. Initially, a statistical test method was developed, which is a numerical method that was used to simulate random variables and functions whose probabilistic characteristics coincided with the solutions of analytical problems (this procedure was called the Monte Carlo method). Then this technique began to be used for machine simulation in order to study the characteristics of the processes of functioning of systems subject to random influences, i.e., the method of statistical modeling appeared. Thus, the method of statistical modeling will henceforth be called the method of machine implementation of the simulation model, and the method of statistical tests (Monte Carlo) is the numerical method for solving an analytical problem.

The simulation method allows solving the problems of analyzing large systems S, including evaluation tasks: options for the structure of the system, the effectiveness of various algorithms for managing the system, the impact of changing various system parameters. Simulation modeling can also be used as the basis for structural, algorithmic and parametric synthesis of large systems, when it is required to create a system with specified characteristics under certain restrictions, which is optimal according to certain criteria for evaluating efficiency.

When solving problems of machine synthesis of systems based on their simulation models, in addition to developing modeling algorithms for analyzing a fixed system, it is also necessary to develop algorithms for finding the optimal system variant. Further, in the methodology of machine modeling, we will distinguish two main sections: statics and dynamics, the main content of which are, respectively, the issues of analysis and synthesis of systems specified by modeling algorithms.

Combined(analytical and simulation) modeling in the analysis and synthesis of systems allows you to combine the advantages of analytical and simulation modeling. When building combined models, a preliminary decomposition of the process of functioning of an object into constituent subprocesses is carried out, and for those of them, where possible, analytical models are used, and simulation models are built for the remaining subprocesses. Such a combined approach makes it possible to cover qualitatively new classes of systems that cannot be studied using only analytical and simulation modeling separately.

Other types of modeling. At real simulation the possibility of studying various characteristics either on a real object as a whole or on its part is used. Such studies can be carried out both on objects operating in normal modes and when organizing special modes to assess the characteristics of interest to the researcher (for other values ​​of variables and parameters, on a different time scale, etc.). Real simulation is the most adequate, but at the same time, its capabilities, taking into account the characteristics of real objects, are limited. For example, carrying out a real simulation of an automated control system by an enterprise will require, firstly, the creation of such an automated control system, and secondly, experiments with a controlled object, i.e. an enterprise, which is impossible in most cases. Consider the varieties of real simulation.

Full-scale modeling called conducting a study on a real object with subsequent processing of the results of the experiment based on the theory of similarity. When the object is functioning in accordance with the goal, it is possible to identify the patterns of the real process. It should be noted that such types of natural experiment as production experiment and complex tests, have a high degree of reliability.

With the development of technology and penetration into the depths of the processes occurring in real systems, the technical equipment of modern scientific experiment. It is characterized by the widespread use of automation tools, the use of very diverse information processing tools, the possibility of human intervention in the process of conducting an experiment, and in accordance with this, a new scientific direction has appeared - automation of scientific experiments.

The difference between the experiment and the real course of the process lies in the fact that individual critical situations may appear in it and the boundaries of the stability of the process can be determined. During the experiment, new factors and disturbing influences are introduced during the operation of the object. One of the varieties of the experiment is complex tests, which can also be attributed to full-scale modeling, when, as a result of repeated testing of products, general patterns are revealed about the reliability of these products, about quality characteristics, etc. In this case, modeling is carried out by processing and summarizing the information passing in group of similar events. Along with specially organized tests, it is possible to implement full-scale simulation by summarizing the experience gained during the production process, i.e., we can talk about a production experiment. Here, on the basis of the theory of similarity, statistical material on the production process is processed and its generalized characteristics are obtained.

Another kind of real simulation is physical, which differs from natural in that the study is carried out on installations that preserve the nature of phenomena and have a physical similarity. In the process of physical modeling, some characteristics of the external environment are set and the behavior of either a real object or its model is studied under given or artificially created environmental influences. Physical modeling can proceed in real and unreal(pseudo-real) time scales, and can also be considered without regard to time. In the latter case, the so-called "frozen" processes, which are fixed at some point in time, are subject to study. The greatest complexity and interest in terms of the fidelity of the results obtained is physical modeling in real time.

From the point of view of the mathematical description of the object and depending on its nature, the models can be divided into analog (continuous), digital (discrete) and analog-to-digital (combined) models. Under analog A model is a model that is described by equations relating continuous quantities. Under digital understand a model that is described by equations relating discrete quantities presented in digital form. Under analog-digital refers to a model that can be described by equations relating continuous and discrete quantities.

A special place in modeling is occupied by cybernetic modeling, in which there is no direct similarity of physical processes occurring in models to real processes. In this case, they tend to display only some function and consider the real object as a “black box” with a number of inputs and outputs, and model some connections between outputs and inputs. Most often, when using cybernetic models, an analysis of the behavioral side of an object is carried out under various environmental influences.

Thus, cybernetic models are based on the reflection of some information management processes, which makes it possible to evaluate the behavior of a real object. To build a simulation model in this case, it is necessary to isolate the function of the real object under study, try to formalize this function in the form of some communication operators between the input and output, and reproduce this function on the simulation model, moreover, on the basis of completely different mathematical relationships and, of course, a different physical implementation of the process .

1.2 Applied aspects of modeling 13

1.3. Basic properties of the model and simulation 18

2. Mathematical and computer modeling 22

2.1. Classification of types of modeling 22

2.2. Mathematical modeling of complex systems 24

2.3. Simulation of random variables and processes 27

2.4. Fundamentals of mathematical modeling 28

2.5.Computer simulation 34

3.Evolutionary modeling and genetic algorithms 41

3.1 Main attributes of evolutionary modeling 41

3.2.Basic research on the evolution of systems 42

3.3. Genetic Algorithms 50

4. Fundamentals of decision making and situational modeling 53

4.1. Fundamentals of Decision Making 53

4.2. Formalizable solutions 56

Literature 63

^

1. System Modeling Fundamentals

   1. Models and Simulation

Model and modeling- universal concepts, attributes of one of the most powerful methods of cognition in any professional field, cognition of a system, process, phenomenon.

View models and the methods of its research depend more on the information-logical connections of the elements and subsystems of the modeled system, resources, connections with the environment, and not on the specific content of the system.

At models, especially mathematical ones, there is a feature - the development of a model style of thinking that allows you to delve into the structure and internal logic of the system being modeled.

Building models- a system task that requires analysis and synthesis of initial data, hypotheses, theories, knowledge of specialists. A systematic approach allows not only to build model real system, but also use this model to evaluate (e.g. management effectiveness, performance) of the system.

Model - this is an object or description of an object, a system for replacing one system (original) with another system for better study of the original or reproduction of any of its properties.

For example, by mapping a physical system onto a mathematical system, we obtain the mathematical model physical system. Any model is constructed and investigated under certain assumptions, hypotheses.

Example. Consider a physical system: a body with mass m rolling down an inclined plane with acceleration a , which is affected by the force F .

Investigating such systems, Newton obtained a mathematical relation: F = m*a. This is a physical and mathematical model systems or mathematical model physical system.

When describing this system, the following hypotheses were adopted:

• 
  the surface is ideal (i.e. the coefficient of friction is zero);
• 
  the body is in a vacuum (i.e. air resistance is zero);
• 
  body weight is unchanged;
• 
  the body moves with the same constant acceleration at any point.

Example. Physiological system (human circulatory system) - obeys some laws of thermodynamics. Describing this system in the physical (thermodynamic) language of balance laws, we obtain a physical, thermodynamic model physiological system. If we write these laws in mathematical language, i.e. the corresponding thermodynamic equations, then we already obtain the mathematical model circulatory systems.

Example . The set of enterprises operates in the market, exchanging goods, raw materials, services, information. If we describe economic laws, the rules of their interaction in the market with the help of mathematical relations, for example, a system of algebraic equations, where the unknowns will be the profits received from the interaction of enterprises, and the coefficients of the equation will be the values ​​of the intensities of such interactions, then we will get the economic and mathematical model enterprise systems in the market.

If the bank has developed a lending strategy, was able to describe it with the help of economic and mathematical models and predicts its lending tactics, then it has greater stability and viability.

Word " model"(lat. modelium) means "measure", "method", "resemblance to some thing".

Modeling is based on the mathematical theory of similarity, according to which absolute similarity can take place only when one object is replaced by another exactly the same.

At modeling most systems, absolute similarity is impossible, and the main goal modeling - model should reflect the functioning of the simulated system quite well.

By level, "depth" modeling models there are:

• 
  empirical - based on empirical facts, dependencies;
• 
  theoretical - based on mathematical descriptions;
• 
  mixed, semi-empirical - based on empirical dependencies and mathematical descriptions.

Problem modeling consists of three tasks:

• 
  construction models(this problem is less formalizable and constructive, in the sense that there is no algorithm for constructing models);
• 
  study models(this task is more formalizable, there are methods for studying various classes models);
• 
  usage models(constructive and concretized task).

Model M, describing the system S(x 1 , x 2 , ..., x n ; R), has the form: M = (z 1 , z 2 , ..., z m ; Q), where z i Z, i = 1, 2, ..., n, Q, R - sets of relations over X - a set of input, output signals and system states, Z - a set of descriptions, representations of elements and subsets of X.

Model building scheme M systems S with input signals X and output signals Y shown in fig. 1.1.

Rice. 1.1. Model building scheme

If signals from X arrive at the input M and signals Y appear at the input, then the law is given (the rule f functioning of the model) of the system.

Modeling is a universal method for obtaining a description of the functioning of an object and using knowledge about it. Modeling is used in any professional activity

classification models carried out according to different criteria.

Model called static , if there is no time parameter among the parameters participating in its description. ^ Static model at each moment of time gives only a "photo" of the system, its slice.

Example. Newton's law F=a*m is static model moving with acceleration a material point mass m. This model does not take into account the change in acceleration from one point to another.

^ Model dynamic , if among its parameters there is a time parameter, i.e. it displays the system (processes in the system) in time.

Example. Dynamic Model Newton's law will be:

F(t)=a(t)*m(t).

Model discrete , if it describes the behavior of the system only at discrete times.

Example. If we consider only t=0, 1, 2, …, 10 (sec), then model S t =gt 2 /2 or a numerical sequence S 0 =0, S 1 =g/2, S 2 =2g, S 3 =9g/2, :, S 10 =50g can serve as discrete model motion of a freely falling body.

^ Model continuous , if it describes the behavior of the system for all times of a certain time interval.

Example. Model S=gt 2 /2, 0< t < 100 непрерывна на промежутке времени (0;100).

Model imitation if it is intended to test or study the possible ways of development and behavior of an object by varying some or all of the parameters models.

Example. Let model of the economic system for the production of goods of two types 1 and 2, in the amount of x 1 and x 2 units and the cost of each unit of goods a 1 and a 2 at the enterprise is described as the ratio:

A 1 x 1 + a 2 x 2 = S,

Where S is the total cost of all products produced by the enterprise (types 1 and 2). Can be used as simulation model, by which it is possible to determine (variate) the total cost S depending on certain values ​​of the volumes of goods produced.

Model deterministic, if each input set of parameters corresponds to a well-defined and uniquely determined set of output parameters; otherwise - model non-deterministic, stochastic(probabilistic).

Example. The above physical models- deterministic. If in models S=gt2/2,0< t < 100 мы учли бы случайный параметр - порыв ветра с силой p when the body falls:

S(p) = g(p) t 2 / 2, 0< t < 100,

Then we would get stochastic model(no longer free!) fall.

Model functional , if it can be represented as a system of some functional relations.

^ Model set-theoretic , if it is representable with the help of some sets and relations of belonging to them and between them.

Example . Let the set X = (Nikolai, Peter, Nikolaev, Petrov, Elena, Ekaterina, Mikhail, Tatiana) be given and the relations: Nikolai - Elena's husband, Ekaterina - Peter's wife, Tatiana - daughter of Nikolai and Elena, Mikhail - son of Peter and Ekaterina, families Michael and Petra are friends with each other. Then the set X and the set of enumerated relations Y can serve as set-theoretic model two friendly families.

Model logical, if it is representable by predicates, logical functions.

For example, a set of two logical functions of the form:

Z = x y x y, p = x y

It can serve as a mathematical model of a single-digit adder.

Model game, if it describes, implements some game situation between game participants (persons, coalitions).

Example. Let player 1 be a conscientious tax inspector and player 2 be an unscrupulous taxpayer. There is a process (game) on tax evasion (on the one hand) and on revealing the concealment of tax payments (on the other hand). Players choose positive integers i and j (i, j n), which can be identified, respectively, with the fine of player 2 for non-payment of taxes upon discovery of the fact of non-payment by player 1 and with the temporary benefit of player 2 from tax evasion. Consider a matrix game with a payoff matrix of order n. Each element of this matrix A is determined by the rule a ij = |i - j|. Model the game is described by this matrix and the strategy of evasion and capture. This game is antagonistic.

Model algorithmic, if it is described by some algorithm or a set of algorithms that determine its functioning, development.

It must be remembered that not all models can be explored or implemented algorithmically.

Example. The model for calculating the sum of an infinite decreasing series of numbers can be an algorithm for calculating the finite sum of a series up to a certain specified degree of accuracy. algorithmic model the square root of the number x can serve as an algorithm for calculating its approximate arbitrarily exact value using a well-known recursive formula.

^ Model structural, if it can be represented by a data structure or data structures and relationships between them.

For example, with structural model may serve as a description (tabular, graphical, functional or other) of the structure of the ecosystem.

^ Model graph, if it is representable by a graph or graphs and relations between them.

Model hierarchical(tree-like) if it is represented by some hierarchical structure (tree).

Example. To solve the problem of finding a route in a search tree, you can build, for example, a tree model (rice. 1.2):

Rice. 1.2. Hierarchical structure model

Model network, if it is represented by some network structure.

Example. The construction of a new house includes the operations shown in the following table.

^ Table of works during the construction of a house
№	Operation	Lead time (days)	^ Previous Operations	Count Arcs
1	Site clearing	1	No	-
2	Foundation laying	4	Site clearing (1)	1-2
3	Walling	4	Foundation laying (2)	2-3
4	Installation of electrical wiring	3	Building walls (3)	3-4
5	Plaster work	4	Electrical wiring (4)	4-5
6	Landscaping	6	Building walls (3)	3-6
7	Finishing work	4	Plastering (5)	5-7
8	Roof decking	5	Building walls (3)	3-8

network model(network diagram) of building a house is given in fig. 1.3.

Rice. 1.3. Network schedule of construction works

Two jobs corresponding to arc 4-5 are parallel, they can either be replaced by one representing a joint operation (wiring and roofing) with a new operation of duration 3+5=8, or a dummy event can be introduced on one arc.

^ Model language, linguistic, if it is represented by some linguistic object, formalized language system or structure.

Sometimes such models called verbal, syntactic.

For example, the rules of the road - linguistic, structural model traffic and pedestrians on the roads.

Let B be the set of generating noun stems, C the set of suffixes, P the adjectives, "+" the word concatenation operation, ":=" the assignment operation, "=>" the output operation (the derivation of new words), Z the set of values (semantic) adjectives. Language model M word formation:

= + <с i >. With b i - "fish (a)", with i - "n (th)", we get from this models p i - "fish", z i - "made from fish".

^ Model visual, if it allows you to visualize the relationships and connections of the simulated system, especially in dynamics.

For example, on a computer screen, visual model of this or that object, for example, the keyboard in the program - a simulator for learning to work on the keyboard.

^ Model natural, if it is a material copy of the object modeling.

For example, a globe is a natural geographical model the globe.

^ Model geometric, graphic, if it can be represented by geometric images and objects.

For example, the layout of the house is full-scale geometric model house under construction. A polygon inscribed in a circle gives model circles. It is she who is used when depicting a circle on a computer screen. The straight line is model the numerical axis, and the plane is often depicted as a parallelogram.

^ Model cellular automaton if it represents the system using a cellular automaton or a system of cellular automata.

A cellular automaton is a discrete dynamic system, an analogue of a physical (continuous) field. Cellular automata geometry is an analogue of Euclidean geometry. An indivisible element of Euclidean geometry is a point; segments, straight lines, planes, etc. are built on its basis.

An indivisible element of the cellular-automaton field is a cell, on the basis of which clusters of cells and various configurations of cellular structures are built. The cellular automaton is represented by a uniform network of cells ("cells") of this field. The evolution of a cellular automaton unfolds in a discrete space - a cellular field.

The change of states in the cellular automaton field occurs simultaneously and in parallel, and time passes discretely. Despite the apparent simplicity of their construction, cellular automata can exhibit a variety and complex behavior.

Recently, they have been widely used in modeling not only physical, but also socio-economic processes.

The classification of types of modeling can be carried out for various reasons. One of the classification options is shown in the figure.

Rice. - An example of the classification of types of modeling

In accordance with the classification sign of completeness, modeling is divided into: complete, incomplete, approximate.

At complete modeling models are identical to the object in time and space.

For incomplete modeling this identity is not preserved.

At the core approximate Simulation lies in the similarity, in which some aspects of the real object are not modeled at all. The theory of similarity states that absolute similarity is possible only when one object is replaced by another exactly the same. Therefore, when modeling, absolute similarity does not take place. Researchers strive to ensure that the model well reflects only the studied aspect of the system. For example, to assess the noise immunity of discrete information transmission channels, the functional and information models of the system may not be developed. To achieve the goal of modeling, the event model described by the matrix of conditional probabilities of transitions of the i-th character of the alphabet to the j-th one is quite sufficient.

Depending on the type of media and the signature of the model, the following types of modeling are distinguished: deterministic and stochastic, static and dynamic, discrete, continuous and discrete-continuous.

deterministic modeling displays processes in which the absence of random influences is assumed.

Stochastic modeling takes into account probabilistic processes and events.

Static Simulation serves to describe the state of an object at a fixed point in time, and dynamic - to study the object in time. At the same time, they operate with analog (continuous), discrete and mixed models.

Depending on the form of implementation of the carrier and signature, modeling is classified into mental and real.

mental modeling is used when models are not realizable in a given time interval or there are no conditions for their physical creation (for example, the situation of the microworld). Mental modeling of real systems is realized in the form of visual, symbolic and mathematical. A significant number of tools and methods have been developed to represent functional, informational and event models of this type of modeling.

At visual modeling on the basis of human ideas about real objects, visual models are created that display the phenomena and processes occurring in the object. An example of such models are educational posters, drawings, charts, diagrams.

The basis hypothetical modeling, a hypothesis is laid about the patterns of the process in a real object, which reflects the level of knowledge of the researcher about the object and is based on cause-and-effect relationships between the input and output of the object under study. This type of modeling is used when knowledge about the object is not enough to build formal models. Analog modeling is based on the application of analogies of various levels. For sufficiently simple objects, the highest level is complete analogy. With the complication of the system, analogies of subsequent levels are used, when the analog model displays several (or only one) aspects of the object's functioning.

Prototyping is used when the processes occurring in a real object are not amenable to physical modeling or may precede other types of modeling. The construction of mental layouts is also based on analogies, usually based on causal relationships between phenomena and processes in an object.

Symbolic modeling is an artificial process of creating a logical object that replaces the real one and expresses its main properties using a certain system of signs and symbols.

At the core linguistic modeling lies some thesaurus, which is formed from a set of concepts of the studied subject area, and this set must be fixed. A thesaurus is a dictionary that reflects the relationships between words or other elements of a given language, designed to search for words by their meaning.

A traditional thesaurus consists of two parts: a list of words and set phrases grouped according to semantic (thematic) headings; an alphabetical dictionary of keywords that define classes of conditional equivalence, an index of relationships between keywords, where for each word the corresponding headings are indicated. Such construction allows defining semantic (semantic) relations of hierarchical (genus/species) and non-hierarchical (synonymy, antonymy, associations) type.

There are fundamental differences between a thesaurus and a regular dictionary. Thesaurus is a dictionary that has been cleared of ambiguity, i.e. in it, only a single concept can correspond to each word, although in an ordinary dictionary, several concepts can correspond to one word.

If we introduce a symbol for individual concepts, i.e. signs, as well as certain operations between these signs, then you can implement iconic modeling and using signs to display a set of concepts - to make separate chains of words and sentences. Using the operations of union, intersection and addition of set theory, it is possible to give a description of some real object in separate symbols.

Mathematical modeling is the process of establishing correspondence to a given real object of some mathematical object, called a mathematical model. In principle, to study the characteristics of any system by mathematical methods, including machine methods, this process must be formalized, i.e. a mathematical model is built. The type of mathematical model depends both on the nature of the real object and on the tasks of studying the object, on the required reliability and accuracy of solving the problem. Any mathematical model, like any other, describes a real object with a certain degree of approximation.

Various notation forms can be used to represent mathematical models. The main ones are invariant, analytical, algorithmic and circuit (graphic).

An invariant form is a record of model relations using a traditional mathematical language, regardless of the method for solving model equations. In this case, the model can be represented as a set of inputs, outputs, state variables and global equations of the system. Analytical form - recording the model as a result of solving the initial equations of the model. Typically, models in analytical form are explicit expressions of output parameters as functions of inputs and state variables.

For analytical modeling is characterized by the fact that basically only the functional aspect of the system is modeled. In this case, the global equations of the system that describe the law (algorithm) of its functioning are written in the form of some analytical relations (algebraic, integro-differential, finite-difference, etc.) or logical conditions. The analytical model is studied by several methods:

• analytical, when they strive to obtain explicit dependencies in a general form, connecting the desired characteristics with the initial conditions, parameters and state variables of the system;
• numerical, when, not being able to solve equations in a general form, they strive to obtain numerical results with specific initial data (recall that such models are called digital);
• qualitative, when, without having a solution in an explicit form, you can find some properties of the solution (for example, evaluate the stability of the solution).

At present, computer methods for studying the characteristics of the process of functioning of complex systems are widespread. To implement a mathematical model on a computer, it is necessary to build an appropriate modeling algorithm.

Algorithmic form - a record of the relationship between the model and the selected numerical solution method in the form of an algorithm. Among the algorithmic models, an important class is made up of simulation models designed to simulate physical or informational processes under various external influences. Actually, the imitation of these processes is called simulation modeling.

At imitation Simulation reproduces the algorithm of the system functioning in time - the behavior of the system, and the elementary phenomena that make up the process are simulated, with the preservation of their logical structure and sequence of flow, which allows, according to the initial data, to obtain information about the states of the process at certain points in time, making it possible to evaluate the characteristics of the system. The main advantage of simulation modeling compared to analytical modeling is the ability to solve more complex problems. Simulation models make it possible to easily take into account such factors as the presence of discrete and continuous elements, non-linear characteristics of system elements, numerous random effects, and others that often create difficulties in analytical studies. Currently, simulation modeling is the most effective method for studying systems, and often the only practically accessible method for obtaining information about the behavior of a system, especially at the stage of its design.

In simulation, a distinction is made between the method of statistical tests (Monte Carlo) and the method of statistical modeling.

The Monte Carlo method is a numerical method that is used to simulate random variables and functions whose probabilistic characteristics coincide with the solutions of analytical problems. It consists in multiple reproduction of processes that are realizations of random variables and functions, with subsequent processing of information by methods of mathematical statistics.

If this technique is used for machine simulation in order to study the characteristics of the processes of functioning of systems subject to random influences, then this method is called the method of statistical modeling.

The simulation method is used to evaluate options for the system structure, the effectiveness of various system control algorithms, and the impact of changing various system parameters. Simulation modeling can be used as the basis for the structural, algorithmic and parametric synthesis of systems, when it is required to create a system with specified characteristics under certain restrictions.

Combined (analytical and simulation) modeling allows you to combine the advantages of analytical and simulation modeling. When building combined models, a preliminary decomposition of the Object Functioning process into constituent subprocesses is carried out, and for those of them, where possible, analytical models are used, and simulation models are built for the remaining subprocesses. This approach makes it possible to cover qualitatively new classes of systems that cannot be studied using analytical or simulation modeling separately.

informational (cybernetic) modeling is associated with the study of models in which there is no direct similarity of the physical processes occurring in the models to real processes. In this case, they seek to display only some function, consider the real object as a “black box” with a number of inputs and outputs, and model some connections between outputs and inputs. Thus, information (cybernetic) models are based on the reflection of some information management processes, which makes it possible to evaluate the behavior of a real object. To build a model in this case, it is necessary to isolate the investigated function of a real object, try to formalize this function in the form of some communication operators between the input and output, and reproduce this function on a simulation model, moreover, in a completely different mathematical language and, of course, a different physical implementation of the process. So, for example, expert systems are models of decision makers.

Structural modeling of system analysis is based on some specific features of structures of a certain type, which are used as a means of studying systems or serve to develop specific approaches to modeling based on them using other methods of formalized representation of systems (set-theoretic, linguistic, cybernetic, etc.) . The development of structural modeling is object-oriented modeling.

Structural modeling of system analysis includes:

• network modeling methods;
• combination of structuring methods with linguistic ones;
• a structural approach in the direction of formalizing the construction and study of structures of various types (hierarchical, matrix, arbitrary graphs) based on set-theoretic representations and the concept of a nominal scale of measurement theory.

At the same time, the term "model structure" can be applied both to functions and to system elements. The corresponding structures are called functional and morphological. Object-oriented modeling combines structures of both types into a class hierarchy that includes both elements and functions.

In structural modeling, a new CASE technology has emerged over the past decade. The abbreviation CASE has a double interpretation, corresponding to two areas of use of CASE systems. The first of them - Computer-Aided Software Engineering - translates as computer-aided software design. The corresponding CASE systems are often referred to as Rapid Application Development (RAD) tooling environments. The second - Computer-Aided System Engineering - emphasizes the focus on supporting the conceptual modeling of complex systems, mostly semi-structured. Such CASE systems are often referred to as BPR (Business Process Reengineering) systems. In general, CASE technology is a set of methodologies for analyzing, designing, developing and maintaining complex automated systems, supported by a set of interconnected automation tools. CASE is a toolkit for system analysts, developers and programmers that allows you to automate the process of designing and developing complex systems, including software.

situational modeling is based on the model theory of thinking, within which it is possible to describe the main mechanisms for regulating decision-making processes. At the center of the model theory of thinking lies the idea of ​​the formation of an information model of an object and the external world in the structures of the brain. This information is perceived by a person on the basis of the knowledge and experience he already has. Expedient human behavior is built by forming the target situation and mentally transforming the initial situation into the target one. The basis for constructing the model is the description of the object in the form of a set of elements interconnected by certain relationships that reflect the semantics of the subject area. The object model has a multi-level structure and represents the information context against which management processes proceed. The richer the information model of the object and the higher the possibility of manipulating it, the better and more diverse the quality of decisions made in management.

At real modeling uses the possibility of studying the characteristics either on a real object as a whole or on its part. Such studies are carried out both on objects operating in normal modes and when organizing special modes to assess the characteristics of interest to the researcher (for other values ​​of variables and parameters, on a different time scale, etc.). Real simulation is the most adequate, but its possibilities are limited.

Natural modeling is called conducting a study on a real object with subsequent processing of the results of the experiment based on the theory of similarity. Full-scale simulation is divided into a scientific experiment, complex tests and a production experiment. scientific experiment characterized by the widespread use of automation tools, the use of very diverse means of information processing, the possibility of human intervention in the process of conducting an experiment. One type of experiment complex tests, during which, as a result of repeated testing of objects as a whole (or large parts of the system), general patterns are revealed about the quality characteristics and reliability of these objects. In this case, modeling is carried out by processing and generalizing information about a group of homogeneous phenomena. Along with specially organized tests, it is possible to implement full-scale simulation by summarizing the experience gained during the production process, i.e. can talk about production experiment. Here, on the basis of the theory of similarity, statistical material on the production process is processed and its generalized characteristics are obtained. It is necessary to remember about the difference between the experiment and the real course of the process. It lies in the fact that individual critical situations may appear in the experiment and the boundaries of the stability of the process can be determined. In the course of the experiment, new factors of perturbing influences are introduced into the process of the object's functioning.

Another kind of real simulation is physical, which differs from natural in that the study is carried out in installations that preserve the nature of phenomena and have a physical similarity. In the process of physical modeling, some characteristics of the external environment are set and the behavior of either a real object or its model is studied under given or artificially created environmental influences. Physical modeling can proceed in real and model (pseudo-real) time scales or be considered without taking time into account. In the latter case, the so-called "frozen" processes, fixed at some point in time, are subject to study.

INTRODUCTION

MODELING AS A METHOD OF SCIENTIFIC KNOWLEDGE

Methodological basis of modeling. Everything that human activity is aimed at is called object(lat. objection - subject). The development of a methodology is aimed at streamlining the receipt and processing of information about objects that exist outside of our consciousness and interact with each other and the external environment.

play an important role in scientific research hypotheses that is, certain predictions based on a small amount of experimental data, observations, conjectures. A quick and complete test of the put forward hypotheses can be carried out in the course of a specially designed experiment. When formulating and testing the correctness of hypotheses, analogy is of great importance as a method of judgment.

by analogy called a judgment about some particular similarity of two objects, and such similarity can be significant and insignificant. It should be noted that the concepts of materiality and insignificance of the similarity or difference of objects are conditional and relative. The significance of similarity (difference) depends on the level of abstraction and is generally determined by the ultimate goal of the study. A modern scientific hypothesis is created, as a rule, by analogy with scientific provisions tested in practice. Thus, the analogy connects the hypothesis with the experiment.

Hypotheses and analogies reflecting the real, objectively existing world should be visual or reduced to logical schemes convenient for research; such logical schemes that simplify reasoning and logical constructions or allow experiments to be carried out that clarify the nature of phenomena are called models. In other words, model (lat. modulus - measure) is an object-substitute of the original object, providing the study of some properties of the original.

Computer model - this is a software implementation of a mathematical model, supplemented by various utility programs (for example, those that draw and change graphic images in time). The computer model has two components - software and hardware. The software component is also an abstract sign model. This is just another form of an abstract model, which, however, can be interpreted not only by mathematicians and programmers, but also by a technical device - a computer processor.

Modeling called the replacement of one object by another in order to obtain information about the properties of the original object by studying the model object.

Thus, modeling can be defined as the representation of an object by a model in order to obtain information about this object by conducting experiments with its model. The theory of replacing some objects (originals) with other objects (models) and studying the properties of objects on their models is called modeling theory.

BASIC CONCEPTS OF SYSTEM MODELING THEORY

At present, in the analysis and synthesis of complex (large) systems, the systems approach, which differs from the classical (or inductive - by passing from private to general and synthesizes (constructs) the system by merging its components, developed separately) approach. In contrast to this systems approach assumes a sequential transition From general to specific when the consideration is based on the goal, and the object under study stands out from the environment.

The concept of a system and an element of a system. Specialists in the design and operation of complex systems deal with control systems of various levels that have a common property - the desire to achieve some goal. This feature will be taken into account in the following definitions of the system.

SystemS - purposeful set of interconnected elements of any nature.

External environment E- a set of elements of any nature existing outside the system that influence the system or are under its influence.

The concept of a model.Model- representation of an object, system or concept, in some form, different from their real existence.

Modeling- firstly, building a model, secondly, studying the model, and thirdly, analyzing the system based on this model.

With a systematic approach to modeling systems, it is necessary first of all to clearly define modeling goal. With regard to modeling issues, the goal arises from the required modeling tasks, which allows you to approach the choice of criterion and evaluate which elements will be included in the model being created. M. Therefore, it is necessary to have a criterion for selecting individual elements in the model being created.

Simulation Goals:

1) grade- evaluate the actual characteristics of the designed or existing system, determine how the system of the proposed structure will meet the requirements.

2) comparison- compare competing systems of the same functional purpose or compare several options for building the same system.

3) forecast – evaluate the behavior of the system under some expected combination of operating conditions.

4) sensitivity analysis- to identify from a large number of factors acting on the system the one that to a greater extent influence its behavior and determine its performance indicators.

5) optimization- to find or establish such a combination of acting factors and their values, which provides the best performance indicators of the system as a whole.

1-4 analysis tasks, 5 - synthesis task.

Systems Research Approaches. Important for a systematic approach is the definition system structure- a set of links between the elements of the system, reflecting their interaction.

At structural approach the composition of the selected elements of the system is revealed S and connections between them. The totality of elements and links between them makes it possible to judge the structure of the system. The latter, depending on the purpose of the study, can be described at different levels of consideration. The most general description of the structure is a topological description, which makes it possible to define the constituent parts of the system in the most general terms and is well formalized on the basis of graph theory.

Less general is the functional description, when individual functions are considered, i.e., system behavior algorithms, and functional approach evaluating the functions that the system performs, and the function is understood as a property that leads to the achievement of the goal.

A simple approach to studying the relationships between the individual parts of the model involves considering them as a reflection of the relationships between the individual subsystems of the object. This classical approach can be used to create fairly simple models. Model synthesis process M based on the classical (inductive) approach is shown in fig. 1.1 a. The real object to be modeled is divided into separate subsystems, i.e., the initial data are selected D for modeling and setting goals C, displaying individual aspects of the modeling process. For a separate set of initial data D the goal is to model a separate aspect of the functioning of the system, on the basis of this goal a certain component is formed To future model. The set of components is combined into a model M.

Rice. 1.1. The process of synthesizing a model based on the classical (a) and systemic (b) approaches

Thus, the development of the model M on the basis of the classical approach means the summation of individual components into a single model, with each component solving its own tasks and isolated from other parts of the model. Therefore, the classical approach can be used to implement relatively simple models in which separation and mutually independent consideration of individual aspects of the functioning of a real object are possible. For a model of a complex object, such a disunity of the tasks to be solved is unacceptable, since it leads to significant resource costs when implementing the model on the basis of specific software and hardware. Two distinctive aspects of the classical approach can be noted: there is a movement from the particular to the general, the created model (system) is formed by summing up its individual components and the emergence of a new systemic effect is not taken into account.

Model synthesis process M on the basis of a systematic approach is conditionally presented in fig. 1.1 b. Based on initial data D, which are known from the analysis of the external system, those restrictions that are imposed on the system from above or based on the possibilities of its implementation, and based on the purpose of functioning, the initial requirements are formulated T to the system model S. On the basis of these requirements some subsystems are tentatively formed. P, elements E and the most difficult stage of synthesis is carried out - the choice AT components of the system, for which special selection criteria are used KV.

Stages of model development. On the basis of a systematic approach, a certain sequence of model development can be proposed, when two main design stages are distinguished: macro design and microdesign.

On the stage macro design based on real system data S and environment E a model of the external environment is built, resources and constraints for building a system model are identified, a system model and criteria are selected to assess the adequacy of the model M real system S.

Stage microdesign largely depends on the particular type of model chosen. In the case of a simulation model, it is necessary to ensure the creation of information, mathematical, technical and software systems for modeling.

Regardless of the type of model used M in its construction, it is necessary to be guided by a number of principles of system move :

1) proportionally sequential progress through the stages and directions of creating a model;

2) coordination of information, resource, reliability and other characteristics;

3) the correct ratio of individual levels of the hierarchy in the modeling system;

4) the integrity of individual isolated stages of model building.

CLASSIFICATION OF MODELING TYPESSYSTEMS

Classification of types of system modeling S shown in fig. 1.2.

Rice. 1.2. Classification of types of system modeling

Deterministic Simulation displays processes in which the absence of any random influences is assumed; stochastic modeling displays probabilistic processes and events. In this case, a number of implementations of a random process are analyzed and the average characteristics are estimated, i.e., a set of homogeneous implementations. Static modelsing is used to describe the behavior of an object at some point in time, and dynamic simulation reflects the behavior of an object over time. Discrete Simulation serves to describe processes that are assumed to be discrete, respectively, continuous modeling allows you to reflect continuous processes in systems, and discrete-continuous simulation is used for cases when one wants to highlight the presence of both discrete and continuous processes.

Depending on the form of representation of the object (system S) mental and real modeling can be distinguished.

mental modeling often is the only way to model objects that are either practically unrealizable in a given time interval, or exist outside the conditions possible for their physical creation. For example, on the basis of mental modeling, many situations of the microworld that are not amenable to physical experiment can be analyzed. Mental modeling can be implemented in the form visual, symbolic and mathematical.

At visual modeling on the basis of human ideas about real objects, various visual models are created that display the phenomena and processes occurring in the object. The basis hypothetical simulation the researcher lays down some hypothesis about the patterns of the process in a real object, which reflects the level of knowledge of the researcher about the object and is based on cause-and-effect relationships between the input and output of the object under study. Hypothetical modeling is used when knowledge about the object is not enough to build formal models.

Analog simulation is based on the application of analogies of various levels. The highest level is a complete analogy, which takes place only for fairly simple objects. With the complication of the object, analogies of subsequent levels are used, when the analog model displays several or only one side of the object's functioning.

An important place in mental visual modeling is occupied by layout. A mental model can be used in cases where the processes occurring in a real object are not amenable to physical modeling, or it can precede other types of modeling. If we introduce a symbol for individual concepts, i.e. signs, as well as certain operations between these signs, then we can implement iconic modeling and using signs to display a set of concepts - to make separate chains of words and sentences. Using the operations of union, intersection and addition of set theory, it is possible to give a description of some real object in separate symbols.

At the core language modeling lies some thesaurus. The latter is formed from a set of incoming concepts, and this set must be fixed. It should be noted that there are fundamental differences between a thesaurus and a regular dictionary. Thesaurus is a dictionary in which only one concept can correspond to each word, although in a regular dictionary one word can correspond to several concepts.

Symbolic modeling is an artificial process of creating a logical object that replaces the real one and expresses the main properties of its relations using a certain system of signs or symbols.

Math modeling. Under mathematical modeling we will understand the process of establishing correspondence to a given real object of some mathematical object, called a mathematical model, and the study of this model, which allows obtaining the characteristics of the real object under consideration. The type of a mathematical moth depends both on the nature of the real object and the tasks of studying the object and the required reliability and accuracy of solving this problem.

For analytical modeling it is characteristic that the processes of functioning of the elements of the system are written in the form of some functional relations (algebraic, integro-differential, finite-difference, etc.) or logical conditions.

Simulation allows you to obtain information about the state of the process at certain points in time from the initial data, making it possible to evaluate the characteristics of the system S.