Lecture No. 1

Introduction. Concept of mathematical models and methods

Section 1. Introduction

2. Methods for constructing mathematical models. Concept of systematic approach. 1

3. Basic concepts mathematical modeling economic systems.. 4

4. Methods of analytical, simulation and full-scale modeling. 5

Test questions.. 6

1. Contents, goals and objectives of the discipline “Modeling Methods”

This discipline is devoted to the study of modeling methods and the practical application of the acquired knowledge. The purpose of the discipline is to train students in general issues of modeling theory, methods for constructing mathematical models and formal descriptions of processes and objects, the use of mathematical models for conducting computational experiments and solving optimization problems using modern computing tools.

The objectives of the discipline include:

To familiarize students with the basic concepts of the theory of mathematical modeling, systems theory, similarity theory, theory of experimental planning and processing of experimental data used to construct mathematical models,

To provide students with skills in the field of setting modeling problems, mathematical descriptions of objects/processes/, numerical methods for implementing mathematical models on a computer and solving optimization problems.

As a result of studying the discipline, the student must master methods of mathematical modeling of processes and objects from problem formulation to implementation of mathematical models on a computer and presentation of the results of model research.

The discipline course consists of 12 lectures and 12 practical works. As a result of studying the discipline, the student must master methods of mathematical modeling from problem formulation to implementation of mathematical models on a computer

2. Methods for constructing mathematical models. The concept of a systems approach

5. Solving the problem.

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

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

4. Methods of analytical, simulation and full-scale modeling

Modeling is a powerful method of scientific knowledge, in which the object under study is replaced by a simpler object called a model. The main types of the modeling process can be considered two types - mathematical and physical modeling. During physical (full-scale) modeling, the system under study is replaced by another corresponding to it material system, which reproduces the properties of the system under study while preserving them physical nature. An example of this type of modeling is a pilot network, with the help of which the fundamental possibility of building a network based on certain computers, communication devices, operating systems and applications is studied.

Possibilities physical modeling quite limited. It allows you to solve individual problems when specifying a small number of combinations of the system parameters under study. Indeed, with full-scale modeling of a computer network, it is almost impossible to check its operation for options using various types of communication devices - routers, switches, etc. Testing in practice is about a dozen different types routing is associated not only with great effort and time costs, but also with considerable material costs.

But even in cases where, during network optimization, it is not the types of devices and operating systems that change, but only their parameters, conducting experiments in real time to huge amount all possible combinations of these parameters are practically impossible in the foreseeable time. Even simply changing the maximum packet size in any protocol requires reconfiguration operating system on hundreds of computers on the network, which requires a lot of work from the network administrator.

Therefore, when optimizing networks, in many cases it is preferable to use mathematical modeling. A mathematical model is a set of relationships (formulas, equations, inequalities, logical conditions), defining the process of changing the state of the system depending on its parameters, input signals, initial conditions and time.

A special class of mathematical models are simulation models. Such models are computer program, which reproduces step by step the events occurring in a real system. In relation to computer networks, their simulation models reproduce the processes of generating messages by applications, breaking messages into packets and frames of certain protocols, delays associated with processing messages, packets and frames within the operating system, the process of a computer gaining access to a shared network environment, the process of processing incoming packets by a router etc. When simulating a network, there is no need to purchase expensive equipment - its operation is simulated by programs that quite accurately reproduce all the main features and parameters of such equipment.

The advantage of simulation models is the ability to replace the process of changing events in the system under study in real time with an accelerated process of changing events at the pace of the program. As a result, in a few minutes it is possible to reproduce the operation of the network for several days, which makes it possible to evaluate the operation of the network in a wide range of varying parameters.

The result of the simulation model is statistical data collected during observation of ongoing events on the most important characteristics of the network: response times, utilization rates of channels and nodes, probability of packet loss, etc.

Exist special languages simulation modeling, which facilitate the process of creating a software model compared to using universal languages programming. Examples of simulation languages ​​include languages ​​such as SIMULA, GPSS, SIMDIS.

There are also simulation modeling systems that focus on a narrow class of systems being studied and allow you to build models without programming.

Control questions

Formulate a definition of the modeling process. What is a model? Simulation properties. Formulate the main stages of building a model using the classical method. Formulate the main stages of building a model using a systems approach. Name the functions of the models. What are the stages in the process of solving economic problems? Main types of modeling process.

Mathematical models

Mathematical model - approximate opithe meaning of the modeling object, expressed usingof mathematical symbolism.

Mathematical models appeared along with mathematics many centuries ago. The advent of computers gave a huge impetus to the development of mathematical modeling. The use of computers has made it possible to analyze and apply in practice many mathematical models that were previously not amenable to analytical research. Implemented on a computer mathematicallysky model called computer mathematical model, A carrying out targeted calculations using a computer model called computational experiment.

Stages of computer mathematical sciencedivision are shown in the figure. Firststage - defining modeling goals. These goals can be different:

1. a model is needed in order to understand how a specific object works, what its structure is, basic properties, laws of development and interaction
   with the outside world (understanding);
2. a model is needed in order to learn how to manage an object (or process) and determine the best ways management with given goals and criteria (management);
3. the model is needed to predict the direct and indirect consequences of implementation given methods and forms of influence on the object (forecasting).

Let's explain with examples. Let the object of study be the interaction of a flow of liquid or gas with a body that is an obstacle to this flow. Experience shows that the force of resistance to flow on the part of the body increases with increasing flow speed, but at some sufficiently high speed this force decreases abruptly so that with a further increase in speed it increases again. What caused the decrease in resistance force? Mathematical modeling allows us to obtain a clear answer: at the moment of an abrupt decrease in resistance, the vortices formed in the flow of liquid or gas behind the streamlined body begin to break away from it and are carried away by the flow.

An example from a completely different area: populations of two species of individuals that had peacefully coexisted with stable numbers and had a common food supply, “suddenly” begin to change their numbers sharply. And here mathematical modeling allows (with a certain degree of reliability) to establish the cause (or at least refute a certain hypothesis).

Developing a concept for managing an object is another possible goal of modeling. Which aircraft flight mode should I choose to ensure that the flight is safe and most economically profitable? How to schedule hundreds of types of work on the construction of a large facility so that it is completed in the shortest possible time? Many such problems systematically arise before economists, designers, and scientists.

Finally, predicting the consequences of certain impacts on an object can be both relatively simple matter in simple physical systems, and extremely complex - on the verge of feasibility - in biological, economic, social systems. While it is relatively easy to answer the question about changes in the mode of heat distribution in a thin rod due to changes in its constituent alloy, 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: determination of input and output parameters of the model; division of input parameters according to the degree of importance of the influence of their changes on the output. This process is called ranking, or separation by rank (see. "Formalizationtion and modeling").

Third stage: construction of a mathematical model. At this stage, there is a transition from an 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: choosing a method for studying a 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. From the right choice method often depends on the success of the entire modeling process.

Fifth stage: developing an algorithm, compiling and debugging a computer program is a difficult process to formalize. Among the programming languages, many professionals prefer FORTRAN for mathematical modeling: both due to traditions and due to the unsurpassed efficiency of compilers (for calculation work) and the availability of huge, carefully debugged and optimized libraries of standard programs for 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 checked for test task with a known answer. This is just the beginning of a testing procedure that is difficult to describe in a formally comprehensive manner. Typically, testing ends when the user, based on his professional characteristics, considers the program correct.

Seventh stage: the actual computational experiment, during which it is determined 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. You can classify models by branches of science (mathematical models in physics, biology, sociology, etc.). Can be classified according to the mathematical apparatus used (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 problems of modeling in different sciences, regardless of the mathematical apparatus, the following classification is most natural:

• descriptive (descriptive) models;
• optimization models;
• multicriteria models;
• game models.

Let's explain this with examples.

Descriptive (descriptive) models. For example, modeling the motion of a comet that has invaded the solar system is carried out to predict its flight path, the distance at which it will pass from the Earth, etc. In this case, the modeling goals are descriptive in nature, since there is no way to influence the movement of the comet or change anything in it.

Optimization models are used to describe 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, when changing the thermal regime in a granary, you can set the goal of choosing a regime that will achieve maximum grain safety, i.e. optimize the storage process.

Multicriteria models. It is often necessary to optimize a process along several parameters simultaneously, and the goals can be quite contradictory. For example, knowing the prices of food and a person’s need for food, it is necessary to organize nutrition 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, i.e. When modeling, several criteria will be used, between which a balance must be sought.

Game models may relate not only to computer games, but also to very serious things. For example, before a battle, a commander, if there is incomplete information about the opposing army, must develop a plan: in what order to introduce certain units into battle, etc., taking into account the possible reaction of the enemy. There is a special branch of modern mathematics - game theory - that studies methods of decision-making under conditions of incomplete information.

IN school course Informatics, students receive an initial understanding of computer mathematical modeling within the framework of basic course. In high school, mathematical modeling can be studied in depth in a general education course for physics and mathematics classes, as well as as part of a specialized elective course.

The main forms of teaching computer mathematical modeling in high school are lectures, laboratory and test classes. Typically, the work of creating and preparing to study each new model takes 3-4 lessons. During the presentation of the material, problems are set that must be solved by students independently in the future, and ways to solve them are outlined in general terms. Questions are formulated, the answers to which must be obtained when completing tasks. Additional literature is indicated that allows you to obtain auxiliary information for more successful completion of tasks.

The form of organization of classes when studying new material is usually a lecture. After completing 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 completing a task, students choose an appropriate solution method and test the developed program using some well-known private solution. In case of quite possible difficulties when completing tasks, consultation is given, and a proposal is made to study these sections in more detail in literary sources.

Most relevant to the practical part of training computer modeling is the project method. The task is formulated for the student in the form of an educational project and is completed over several lessons, with the main organizational form at the same time are computer laboratory works. Training in modeling using the method educational projects can be implemented on different levels. The first is a problematic presentation of the process of completing the project, 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 for students to independently complete an educational research project.

The results of the work must be presented in numerical form, in the form of graphs and diagrams. If possible, the process is presented on the computer screen in dynamics. Upon completion of the calculations and receipt of the results, they are analyzed and compared with known facts from theory, reliability is confirmed and meaningful interpretation is made, which is subsequently reflected in a written report.

If the results satisfy the student and 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, and a list of references.

When all the reports have been compiled, students present their short messages about the work done, defend their project. This is an effective form of report from the group carrying out the project 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, and making predictions. 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 grades during theory quizzes.

An essential question is what tools to use in a school computer science course for mathematical modeling? Computer implementation of models can be carried out:

• by using table processor(usually MS Excel);
• by creating programs in traditional languages programming (Pascal, BASIC, etc.), as well as their modern versions (Delphi, Visual
  Basic for Application, etc.);
• using special application packages for solving mathematical problems (MathCAD, etc.).

At the basic school level, the first method seems to be more preferable. However, in high school, when programming is, along with modeling, key theme computer science, it is advisable to use it as a modeling tool. During the programming process, 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 specialized computer science course as a supplement to other tools.

Exercise :

• Make a diagram of key concepts.

Mathematical model of a technical object - a set mathematical objects and the relationships between them, which adequately reflects the properties of the object under study that are of interest to the researcher (engineer).

The model can be represented in various ways.

Model presentation forms:

invariant - recording model relationships using traditional mathematical language regardless of the method for solving the model equations;

analytical - recording the model in the form of the result of an analytical solution of the initial equations of the model;

algorithmic - recording the relationships between the model and the selected numerical method solutions in the form of an algorithm.

schematic (graphical) - representation of the model in some graphic language (for example, graph language, equivalent circuits, diagrams, etc.);

physical

analog

The most universal is the mathematical description of processes - mathematical modeling.

The concept of mathematical modeling also includes the process of solving a problem on a computer.

Generalized mathematical model

The mathematical model describes the relationship between the initial data and the desired quantities.

The elements of the generalized mathematical model are (Fig. 1): a set of input data (variables) X,Y;

X is a set of variable variables; Y - independent variables (constants);

mathematical operator L, which defines operations on this data; by which is meant complete system mathematical operations, describing numerical or logical relationships between sets of input and output data (variables);

set of output data (variables) G(X,Y); is a set of criterion functions, including (if necessary) an objective function.

A mathematical model is a mathematical analogue of the designed object. The degree of its adequacy to the object is determined by the formulation and correctness of solutions to the design problem.

The set of varied parameters (variables) X forms the space of varied parameters Rx (search space), which is metric with dimension n equal to the number of varied parameters.

The set of independent variables Y form the metric input data space Ry. In the case when each component of the space Ry is specified by a range of possible values, the set of independent variables is mapped to some limited subspace of the space Ry.

The set of independent variables Y determines the operating environment of the object, i.e. external conditions in which the designed object will operate

It can be:

• - technical parameters of the object that are not subject to change during the design process;
• - physical disturbances of the environment with which the design object interacts;
• - tactical parameters that the design object must achieve.

The output data of the generalized model under consideration forms the metric space of criterion indicators RG.

The diagram for using a mathematical model in a computer-aided design system is shown in Fig. 2.

Requirements for the mathematical model

The main requirements for mathematical models are the requirements of adequacy, versatility and efficiency.

Adequacy. The model is considered adequate if it reflects the specified properties with acceptable accuracy. Accuracy is defined as the degree of agreement between the values ​​of the output parameters of the model and the object.

Model accuracy varies depending on different conditions functioning of the object. These conditions are characterized by external parameters. In the space of external parameters, select the area of ​​model adequacy where the error is less than the specified maximum permissible error. Determining the range of adequacy of models is a complex procedure that requires large computational costs, which quickly grow with increasing dimension of the space of external parameters. This problem in scope can significantly exceed the problem of parametric optimization of the model itself, and therefore may not be solved for newly designed objects.

Universality is determined mainly by the number and composition of external and output parameters taken into account in the model.

The cost-effectiveness of the model is characterized by the cost of computing resources for its implementation - the cost of computer time and memory.

The contradictory requirements for a model to have a wide range of adequacy, a high degree of versatility and high efficiency determines the use of a number of models for objects of the same type.

Methods for obtaining models

Obtaining models in the general case is an unformalized procedure. Basic decisions regarding the choice of type mathematical relations, the nature of the variables and parameters used, is decided by the designer. At the same time, operations such as calculating numerical values ​​of model parameters, determining areas of adequacy and others are algorithmized and solved on a computer. Therefore, modeling of the elements of the designed system is usually carried out by specialists in specific technical fields using traditional experimental studies.

Methods for obtaining functional models of elements are divided into theoretical and experimental.

Theoretical methods are based on studying the physical laws of the processes occurring in an object, determining the mathematical description corresponding to these laws, justifying and accepting simplifying assumptions, performing the necessary calculations and bringing the result to the accepted form of model representation.

Experimental methods are based on the use of external manifestations of the properties of an object, recorded during the operation of objects of the same type or during targeted experiments.

Despite the heuristic nature of many operations, modeling has a number of provisions and techniques that are common to obtain models of various objects. Enough general character have

macro modeling technique,

mathematical methods for planning experiments,

algorithms for formalized operations for calculating numerical values ​​of parameters and determining areas of adequacy.

Using Mathematical Models

The computing power of modern computers, combined with the provision of all system resources to the user, the possibility of an interactive mode when solving a problem and analyzing the results, allows us to minimize the time required to solve a problem.

When compiling a mathematical model, the researcher is required to:

study the properties of the object under study;

the ability to separate the main properties of an object from the secondary ones;

evaluate the assumptions made.

The model describes the relationship between the initial data and the desired quantities. The sequence of actions that must be performed in order to move from the initial data to the desired values ​​is called an algorithm.

The algorithm for solving the problem on a computer is associated with the choice of a numerical method. Depending on the form of representation of the mathematical model (algebraic or differential form) various numerical methods are used.

The essence of economic and mathematical modeling is to describe socio-economic systems and processes in the form of economic and mathematical models.

Let us consider the issues of classification of economic and mathematical methods. These methods, as noted above, represent a complex of economic and mathematical disciplines, which are an alloy of economics, mathematics and cybernetics.

Therefore, the classification of economic and mathematical methods comes down to the classification of the scientific disciplines that make up them. Although generally accepted classification these disciplines have not yet been developed, with a certain degree of approximation, within the composition of economic and mathematical methods we can distinguish following sections:

• * economic cybernetics: system analysis of economics, theory economic information and theory of control systems;
• * mathematical statistics: economic applications of this discipline -- sampling method, analysis of variance, correlation analysis, regression analysis, multidimensional statistical analysis, factor analysis, index theory, etc.;
• * mathematical economics and econometrics, which studies the same issues from the quantitative side: theory of economic growth, theory of production functions, input balances, national accounts, analysis of demand and consumption, regional and spatial analysis, global modeling, etc.;
• * methods for making optimal decisions, including the study of operations in economics. This is the most voluminous section, including the following disciplines and methods: optimal (mathematical) programming, including branch and bound methods, network methods of planning and management, program-targeted methods of planning and management, theory and methods of inventory management, theory queuing, game theory, theory and methods of decision making, scheduling theory. Optimal (mathematical) programming, in turn, includes linear programming, nonlinear programming, dynamic programming, discrete (integer) programming, fractional linear programming, parametric programming, separable programming, stochastic programming, geometric programming;
• * methods and disciplines specific separately for both a centrally planned economy and a market (competitive) economy. The first include the theory of optimal functioning of the economy, optimal planning, the theory of optimal pricing, models of material and technical supply, etc. The second include methods that allow the development of models of free competition, models of the capitalist cycle, models of monopoly, models of indicative planning, models of the theory of the firm etc.

Many of the methods developed for a centrally planned economy can also be useful in economic and mathematical modeling in a market economy;

* experimental study methods economic phenomena. These usually include mathematical methods of analysis and planning of economic experiments, methods of machine simulation ( simulation), business games. This also includes methods expert assessments, designed to assess phenomena that cannot be directly measured.

Let us now move on to the issues of classification of economic and mathematical models, in other words, mathematical models of socio-economic systems and processes.

At present, there is also no unified classification system for such models, however, more than ten main characteristics of their classification, or classification headings, are usually identified. Let's look at some of these headings.

According to their general purpose, economic and mathematical models are divided into theoretical and analytical models used in the study general properties and patterns of economic processes, and applied ones, used in solving specific economic problems of analysis, forecasting and management. Various types applied economic and mathematical models are discussed in this textbook.

Based on the degree of aggregation of modeling objects, models are divided into macroeconomic and microeconomic. Although there is no clear distinction between them, the first of them include models that reflect the functioning of the economy as a whole, while microeconomic models are associated, as a rule, with such parts of the economy as enterprises and firms.

According to the specific purpose, i.e., according to the purpose of creation and use, balance models are distinguished that express the requirement for correspondence between the availability of resources and their use; trend models, in which the development of the modeled economic system is reflected through the trend (long-term trend) of its main indicators; optimization models designed to select the best option from a certain number of production, distribution or consumption options; simulation models intended for use in the process of machine simulation of the systems or processes being studied, etc.

Based on the type of information used in the model, economic-mathematical models are divided into analytical, built on a priori information, and identifiable, built on a posteriori information.

Based on the time factor, models are divided into static, in which all dependencies are related to one point in time, and dynamic, describing economic systems in development.

Taking into account the uncertainty factor, models fall into deterministic ones, if their output results are uniquely determined by control actions, and stochastic (probabilistic) ones, if when specifying a certain set of values ​​at the model input, different results can be obtained at its output depending on the action of a random factor.

Economic-mathematical models can also be classified according to the characteristics of the mathematical objects included in the model, in other words, by type mathematical apparatus, used in the model. Based on this feature, matrix models, linear and nonlinear programming models, correlation-regression models,

Basic concepts of mathematical modeling of queuing theory models, models network planning and control, game theory models, etc.

Finally, according to the type of approach to the socio-economic systems being studied, descriptive and normative models are distinguished. With the descriptive approach, models are obtained that are intended to describe and explain actually observed phenomena or to predict these phenomena; As an example of descriptive models, we can cite the previously mentioned balance and trend models. With the normative approach, one is not interested in how the economic system is structured and develops, but how it should be structured and how it should operate in the sense of certain criteria. In particular, all optimization models are of the normative type; Another example is normative models of living standards.

Let us consider, as an example, the economic-mathematical model of the input-output balance (EMM IOB). Taking into account the above classification headings, this is an applied, macroeconomic, analytical, descriptive, deterministic, balance sheet, matrix model; there are both static and dynamic methods

Linear programming is a special branch of optimal programming. In turn, optimal (mathematical) programming - section applied mathematics, studying problems conditional optimization. In economics, such problems arise during the practical implementation of the principle of optimality in planning and management.

A necessary condition for using an optimal approach to planning and management (the principle of optimality) is flexibility and alternativeness of production and economic situations under which planning and management decisions have to be made. It is precisely such situations that, as a rule, constitute the daily practice of an economic entity (choice production program, attachment to suppliers, routing, cutting materials, preparing mixtures, etc.).

The essence of the optimality principle is the desire to choose such a planning and management solution X = (xi, X2 xn), where Xy, (y = 1. i) are its components, which the best way would take into account the internal capabilities and external conditions of the production activity of an economic entity.

The words “best” here mean the choice of some optimality criterion, i.e. some economic indicator that allows you to compare the effectiveness of certain planning and management decisions. Traditional optimality criteria: “maximum profit”, “minimum costs”, “maximum profitability”, etc. The words “would take into account internal capabilities and external conditions of production activity” mean that a number of conditions are imposed on the choice of planning and management decision (behavior), i.e. .e. the choice of X is carried out from a certain region of possible (admissible) solutions D; this area is also called the problem definition area. common task optimal (mathematical) programming, otherwise - a mathematical model of the optimal programming problem, the construction (development) of which is based on the principles of optimality and consistency.

A vector X (a set of control variables Xj, j = 1, n) is called an admissible solution, or plan, of an optimal programming problem if it satisfies the system of constraints. And that plan X (admissible solution) that delivers the maximum or minimum objective function f(xi, *2, ..., xn) is called the optimal plan ( optimal behavior, or simply solving) an optimal programming problem.

Thus, the choice of optimal managerial behavior in a specific production situation is associated with carrying out from the standpoint of consistency and optimality economic-mathematical modeling and solving the optimal programming problem. Optimal programming problems in the most general form are classified according to the following criteria.

• 1. By the nature of the relationship between the variables -
• a) linear,
• b) nonlinear.

In case a) all functional connections in the system of constraints and the goal function are linear functions; the presence of nonlinearity in at least one of the mentioned elements leads to case b).

• 2. By the nature of the change in variables --
• a) continuous,
• b) discrete.

In case a) the values ​​of each of the control variables can completely fill a certain area real numbers; in case b) all or at least one variable can take only integer values.

• 3. Taking into account the time factor --
• a) static,
• b) dynamic.

In problems a) modeling and decision-making are carried out under the assumption that the elements of the model are independent of time during the period of time for which the planning and management decision is made. In case b) such an assumption cannot be accepted with sufficient reasoning and it is necessary to take into account the time factor.

• 4. Based on the availability of information about variables --
• a) tasks under conditions of complete certainty (deterministic),
• b) tasks under conditions of incomplete information,
• c) tasks under conditions of uncertainty.

In problems b) individual elements are probabilistic quantities, but their distribution laws are known or can be established by additional statistical studies. In case c) it is possible to make an assumption about the possible outcomes of random elements, but there is no way to make a conclusion about the probabilities of the outcomes.

• 5. According to the number of criteria for evaluating alternatives --
• a) simple, single-criteria tasks,
• b) complex, multi-criteria tasks.

In problems a) it is economically acceptable to use one optimality criterion or can be achieved using special procedures (for example, “weighing priorities”)

PREFACE

The purpose of the course on modeling hoisting and transport systems is to teach the basics of modeling hoisting and transport machines (HTM), which includes the compilation of mathematical models of HTM, software implementation of models on a computer, as well as obtaining, processing and analysis of modeling results.

For independent familiarization with the listed issues, the following literature is recommended: Braude V.I., Ter-Mkhitarov M.S. “System methods for calculating lifting machines”, Ignatiev N.B., Ilyevsky B.Z., Klaus L.P. “Modeling machine systems", Rachkov E.V., Silikov Yu.V. "Lifting and transport machines and mechanisms", as well as reference books and teaching aids on numerical methods of computational mathematics and the use of the mathematical editor MathCad.

§1. MAIN GOALS, DEFINITIONS AND PRINCIPLES OF MATHEMATICAL MODELING, TYPES OF MODELS

1.1 Basic definitions

Modeling is a theoretical and experimental method of cognitive activity; it is a method of studying and explaining phenomena, processes and systems (original objects) based on the creation of new objects - models.

Modeling is the replacement of the object under study (original) with its conventional image or another object (model) and the study of the properties of the original by studying the properties of the model.

Depending on the method of implementation, all models can be divided into 4 groups: physical, mathematical, subject-mathematical and combined [, ].

A physical model is a real embodiment of those properties of the original that interest the researcher. Physical models are also called layouts, so physical modeling is called prototyping.

A mathematical model is a formalized description of a system (or process) using some abstract language (mathematically), for example, in the form of graphs, equations, algorithms, mathematical correspondences, etc.

Subject-mathematical models are analog, i.e. in this case, for modeling, the principle of the same mathematical description of processes, real and occurring in the model, is used.

Combined models are a combination of a mathematical or subject-mathematical and physical model. They are used when the mathematical description of one of the elements of the system under study is unknown or difficult, and also, according to the modeling conditions, it is necessary to introduce a physical model (for example, a simulator) as an element.

Mathematical modeling is the replacement of the original with a mathematical model and the study of the properties of the original using this model.

A system is a combination of several objects (elements) interconnected, forming a certain integrity.

An element is a relatively independent part of the system, considered at this level analysis as a single whole, designed to implement a certain function.

The system has the following, so-called "system" properties:

structure, i.e. a strictly defined order of combining elements into groups;

purposefulness or functionality, i.e. the presence of a purpose for which the system was created;

efficiency, the ability to achieve goals with the least expenditure of resources;

stability, the ability to maintain the characteristics of its properties unchanged within certain limits when external conditions change.

Currently, in technology, the concept of “human-machine system” (HMS) is used to study the operation of machine complexes and machines, i.e. mixed system, an integral part of which, along with technical objects, is the human operator [, ]. In addition, HMS interacts with the environment. Thus, to model the PTS, it is necessary to consider the Man-Machine-Environment system, which can be displayed by the following graph (Fig. 1).

R
is. 1 Graph of the Man-Machine-Environment system.

The arrows on the graph depict the flows of energy, matter and information that are exchanged between the elements of the system.

The processes occurring in technical systems are formed by a set of simple operations. Operations are transformations of input physical quantities into output ones in a low-level element of the system (Fig. 2).

In each element of the system (E i), the transformation of input influences (X i) into output influences (Y i) occurs, and the output influences of one element can be the input of the next. The connection of elements into a structural diagram according to the nature of the transfer of influences occurs sequentially or in parallel.

Rice. 2 Block diagram of the system.

Lifting and transport systems (HTS), studied in this course, will be called systems that include a person, the environment and hoisting and transport machines (HTM).

PTMs are machines designed to move cargo over relatively short distances without processing it. PTMs are used to facilitate, speed up, and increase the efficiency of reloading operations.

1.2 Principles and types of mathematical modeling

Mathematical models must have the following properties:

adequacy, property of correspondence between the model and the object of research;

reliability, ensuring the specified probability of modeling results falling into the confidence interval,

accuracy, insignificant (within the permissible error) discrepancy between the simulation results and the indicators of real objects (processes);

stability, the property of correspondence of small changes in output parameters to small changes in input parameters;

efficiency, the ability to achieve a goal with low expenditure of resources;

adaptability, the ability to easily adapt to solve various problems.

To achieve these properties, there are some principles (rules) of mathematical modeling, a number of which are given below.

The principle of purposefulness is that the model must ensure the achievement of strictly defined goals and, first of all, reflect those properties of the original that are necessary to achieve the goal.

The principle of information sufficiency consists in limiting the amount of information about an object when creating its model and searching for the optimum between the input information and the modeling results. It can be illustrated by the following diagram.

All possible simulation cases are located in column 2.

Feasibility principle is that the model must ensure achievement of the set goal with a probability close to 1 and in a finite time. This principle can be expressed in two terms

And
, (1)

Where
- probability of achieving the goal, - time to achieve the goal,
and - acceptable values ​​of the probability and time of achieving the goal.

Aggregation principle is that the model should consist of 1st level subsystems, which, in turn, consist of 2nd level subsystems, etc. Subsystems must be designed as separate independent blocks. Such a model construction allows the use of standard calculation procedures, and also makes it easier to adapt the model to solve various problems.

Parameterization principle consists in replacing, when modeling, certain parameters of subsystems described by functions corresponding to numerical characteristics.

The modeling process using these rules consists of the following 5 steps (stages).

Define modeling goals.

Development conceptual model(calculation scheme).

Formalization.

Implementation of the model.

Analysis and interpretation of simulation results.

Significant differences in the implementation of stages 3-5 suggest two approaches to building a model.

Analytical Modeling is the use of a mathematical model in the form of equations supplemented by a system of constraints that connect input variables with output parameters. Analytical modeling is used if there is a complete formulation of the research problem and it is necessary to obtain one final result corresponding to it.

Simulation modeling is the use of a mathematical model to describe the functioning of a system over time under various combinations of system parameters and various external influences. Simulation modeling is used if there is no final formulation of the problem and it is necessary to study the processes occurring in the system. Simulation modeling assumes compliance with a time scale. Those. events on the model occur at time intervals proportional to events on the original with a constant proportionality coefficient.

Based on the use of tools to implement the model, one more type of modeling can be distinguished, computer modeling. Computer modeling is mathematical modeling using computer technology.

1.3 Classification of mathematical models

All mathematical models can be divided into several groups according to the following classification criteria.

Depending on the type of system being modeled, models can be static or dynamic. Static models are used to study static systems, dynamic models are used to study dynamic ones. Dynamic systems are characterized by having multiple states that change over time.

According to the purposes of modeling, models are divided into load, managerial and functional. Load models are used to determine the loads acting on the elements of the system, management models are used to determine the kinematic parameters of the system under study, which include speeds and movements of system elements, functional models are used to determine the coordinates of the model in the space of possible functional states systems.

Based on the degree of discretization, models are divided into discrete, mixed and continuous. Discrete models contain interconnected elements whose characteristics are concentrated at points. These can be masses, volumes, forces and other influences concentrated at points. Continuum models contain elements whose parameters are distributed along the length, area or volume of the entire element. Mixed models contain elements of both types.

Model (from Latin modulus - measure) and modeling are general scientific concepts. Modeling from a general scientific point of view acts as a way of cognition through the construction of special objects, systems - models of the studied objects, phenomena or processes. In this case, one or another object is called a model when it is used to obtain information regarding another object - a prototype of the model.

The modeling method is used in virtually all sciences without exception and at all stages of scientific research. The heuristic power of this method is determined by the fact that with the help of the modeling method it is possible to reduce the study of the complex to the simple, the invisible and intangible, the visible and tangible, etc.

When studying an object (process or phenomenon) using the modeling method, as a model you can select those properties that we this moment interested. Scientific research any object is always relative. IN case study it is impossible to consider an object in all its diversity. Consequently, the same object can have many different models, and none of them can be said to be the only true model of this object.

It is customary to distinguish four main properties models:

· simplification in comparison with the object being studied;

· ability to reflect or reproduce the object of study;

· the ability to replace the object of research at certain stages of its cognition;

· opportunity to receive new information about the object being studied.

Study of various phenomena or processes mathematical methods carried out using a mathematical model. Mathematical model is a formalized description in the language of mathematics of the object under study. Such a formalized description can be a system of linear, nonlinear or differential equations, a system of inequalities, definite integral, a polynomial with unknown coefficients, etc. The mathematical model must cover the most important characteristics of the object under study and reflect the connections between them.

Before creating a mathematical model of an object (process or phenomenon), it is studied for a long time various methods: observation, specially organized experiments, theoretical analysis etc., that is, they study the qualitative side of the phenomenon quite well, identify the relationships in which the elements of the object are located. Then the object is simplified, and the most essential ones are singled out from the variety of its inherent properties. If necessary, assumptions are made about existing connections with the outside world.

As stated earlier, any model is not identical to the phenomenon itself; it only provides some approximation to reality. But the model lists all the assumptions that underlie it. These assumptions may be crude and yet provide a completely satisfactory approximation to reality. Several models, including mathematical ones, can be built for the same phenomenon. For example, you can describe the movement of the planets of the solar system using:

8 Kepler's model, which consists of three laws, including mathematical formulas (ellipse equation);

8 of Newton's model, which consists of one formula, but nevertheless it is more general and accurate.

In optics, several models of light were considered: corpuscular, wave and electromagnetic. Numerous patterns have been derived for them quantitative nature. Each of these models required its own mathematical approach and corresponding mathematical tools. Corpuscular optics used the means of Euclidean geometry and came to the conclusion of the laws of reflection and refraction of light. Ox new model The theory of light required new mathematical ideas and, purely computationally, new facts were discovered related to the phenomena of diffraction and interference of light, which had not previously been observed. Geometric optics, associated with the corpuscular model, turned out to be powerless here.

The constructed model must be such that it can replace an object (process or phenomenon) in research and must have similar features with it. Similarity is achieved either through similarity in structure (isomorphism) or analogy in behavior or functioning (isofunctionality). Based on the similarity of structure or function between the model and the original in modern technology check, calculate and design highly complex systems, machines and structures.

As mentioned above, many different models can be built for the same object, process or phenomenon. Some of them (not necessarily all) may be isomorphic. For example, in analytical geometry a curve on a plane is used as a model corresponding equation with two variables. In this case, the model (curve) and the prototype (equation) are isomorphic to systems (points lying on the curve and corresponding pairs of numbers satisfying the equation),

In the book “Mathematics Conducts an Experiment,” academician N.N. Moiseev writes that any mathematical model can arise in three ways:

· As a result of direct study and understanding of an object (process or phenomenon) (phenomenological) (example - equations describing the dynamics of the atmosphere, ocean),

· As a result of some process of deduction, when a new model is obtained as special case a more general model (asymptomatic) (example - equations of hydro-thermodynamics of the atmosphere),

· As a result of some process of induction, when the new model is a natural generalization of “elementary” models (ensemble model or generalized model).

The process of developing mathematical models consists of the following stages:

· formulation of the problem;

· determination of the purpose of modeling;

· organization and conduct of research subject area(research of the properties of the modeling object);

· model development;

· checking its accuracy and compliance with reality;

· practical use, i.e. transfer of knowledge obtained using the model to the object or process under study.

Special meaning modeling as a way of understanding the laws and phenomena of nature acquires in the study of objects that are not fully accessible to direct observation or experimentation. These include social systems, only possible way the study of which is often done by modeling.

There are no general methods for constructing mathematical models. In each specific case, it is necessary to proceed from the available data, target orientation, take into account the objectives of the study, and also balance the accuracy and detail of the model. It should reflect the most important features of the phenomenon, essential factors on which the success of the modeling mainly depends.

When developing models, you must adhere to the following basic principles: methodological principles modeling social phenomena:

· the principle of problematicity, which implies a movement not from ready-made “universal” mathematical models to problems, but from real, actual problems - to the search and development of special models;

· the principle of systematicity, which considers all the relationships of the modeled phenomenon in terms of the elements of the system and its environment;

· the principle of variability in the formalization of management processes associated with specific differences in the laws of development of nature and society. To explain it, it is necessary to reveal the fundamental difference between models of social processes and models describing natural phenomena.