In a scale version, a model is a kind of image, diagram, map, description, image of a certain phenomenon or process. The phenomenon itself is called the original of a mathematical or economic model.
What is modeling?
Modeling is the study of some object, system. For its implementation, a model is built and analyzed.
All stages of modeling involve a scientific experiment, the object of which is an abstract or subject model. When conducting an experiment, a specific phenomenon is replaced by a scheme or a simplified model (copy). In some cases, a working model is assembled in order to understand the mechanism of work using its example, to analyze the economic feasibility of introducing the results of experience into a market economy. The same phenomenon can be considered by different models.
The researcher must choose the necessary stages of modeling, optimally use them. The use of models is relevant in cases where a real object is not available, or experiments with it are associated with serious environmental problems. The current model is also applied in situations where a real experimentinvolves significant material costs.
Features of mathematical modeling
Mathematical models are indispensable in science, as well as tools for them - mathematical concepts. Over the course of several millennia, they accumulated and modernized. In modern mathematics, there are universal and powerful ways of research. Any objects considered by the "queen of sciences" are a mathematical model. For a detailed analysis of the selected object, the stages of mathematical modeling are selected. With their help, details, features, characteristic features are distinguished, the information received is systematized, and a full description of the object is made.
Mathematical formalization involves operating during research with special concepts: matrix, function, derivative, antiderivative, numbers. Those relations and connections that can be found in the object under study between the constituent elements and details are recorded by mathematical relations: equations, inequalities, equalities. As a result, a mathematical description of a phenomenon or process is obtained, and, consequently, its mathematical model.
Rules for studying a mathematical model
There is a certain order of modeling steps that allows you to establish links between effects and causes. The central stage in the design or study of the system is the construction of a full-fledged mathematical model. The further analysis of this object directly depends on the quality of the performed actions. Buildingmathematical or economic model is not a formal procedure. It should be easy to use, accurate, so that there is no distortion in the results of the analysis.
On the classification of mathematical models
There are two varieties: deterministic and stochastic models. Deterministic models involve the establishment of a one-to-one correspondence between variables used to describe a phenomenon or object.
This approach is based on information about the principle of the object's operation. In many cases, the phenomenon being modeled has a complex structure, and it takes a lot of time and knowledge to decipher it. In such situations, such modeling stages are selected that will allow conducting experiments on the original, processing the results obtained, without going into the theoretical features of the object. Most often used statistics and probability theory. The result is a stochastic model. There is a random relationship between the variables. A huge number of different factors causes a random set of variables that characterize a phenomenon or an object.
Modern modeling steps apply to static and dynamic models. In static views, the description of relationships between the variables of the created phenomenon does not imply taking into account the change in time of the main parameters. For dynamic models, the description of relationships between variables is carried out taking into account temporary changes.
Varieties of models:
- continuous;
- discrete;
- mixed
Different stages of mathematical modeling make it possible to describe relationships and functions in linear models using a direct connection of variables.
What are the requirements for models?
- Versatility. The model must be a complete representation of all the properties inherent in the real object.
- Adequacy. Important characteristics of the object must not exceed the specified error value.
- Accuracy. It characterizes the degree of coincidence of the characteristics of an object that exists in reality with similar parameters obtained during the study of the model.
- Economy. The model should be minimal in terms of material costs.
Modeling steps
Let's consider the main stages of mathematical modeling.
Choosing a task. The purpose of the study is chosen, methods for its implementation are selected, and an experiment strategy is developed. This stage involves serious work. The final result of the simulation depends on the correctness of the task
- Analysis of the theoretical foundations, summing up the information received about the object. This stage involves the selection or creation of a theory. In the absence of theoretical knowledge about the object, causal relationships are established between all the variables chosen to describe the phenomenon or object. At this stage, the initial and final data are determined, and a hypothesis is put forward.
- Formalization. Implementedselection of a system of special notation that will help to write in the form of mathematical expressions the relationship between the components of the object in question.
Additions to the algorithm
After setting the model parameters, a certain method or method of solution is chosen.
- Implementation of the created model. After the stages of system modeling are selected, a program is created that is tested and applied to solve the problem.
- Analysis of collected information. An analogy is drawn between the task and the solution obtained, and the modeling error is determined.
- Checking whether the model matches the real object. If there is a significant difference between them, a new model is developed. Until an ideal match of the model with its real counterpart is obtained, refinement and change of details are carried out.
Simulation characteristic
In the middle of the last century, computer technology appeared in the life of a modern person, the relevance of mathematical methods for studying objects and phenomena increased. Such sections as "mathematical chemistry", "mathematical linguistics", "mathematical economics", dealing with the study of phenomena and objects, appeared, the main stages of modeling were created.
Their main goal was the prediction of planned observations, the study of certain objects. In addition, with the help of modeling, you can learn about the world around you, look for ways to control it. A computer experiment is supposed to be carried out in those cases whenthe real one doesn't work. After constructing a mathematical model of the phenomenon under study, using computer graphics, one can study nuclear explosions, plague epidemics, etc.
Specialists distinguish three stages of mathematical modeling, and each has its own characteristics:
- Building a model. This stage involves setting an economic plan, natural phenomena, construction, production process. It is difficult to clearly describe the situation in this case. First you need to identify the specifics of the phenomenon, to determine the relationship between it and other objects. Then all qualitative characteristics are translated into mathematical language, and a mathematical model is built. This stage is the most difficult in the entire modeling process.
- The stage of solving a mathematical problem associated with the development of algorithms, methods for solving a problem on computer technology, identifying measurement errors.
- Translating information obtained during research into the language of the area for which the experiment was conducted.
These three stages of mathematical modeling are supplemented by checking the adequacy of the resulting model. A check is made of the correspondence between the results obtained in the experiment with theoretical knowledge. If necessary, modify the created model. It is complicated or simplified, depending on the results obtained.
Features of economic modeling
3 stages of mathematical modeling involve the use of algebraic, differential systemsequations. Complex objects are built using graph theory. It involves a set of points in space or on a plane, partially connected by edges. The main stages of economic modeling involve the choice of resources, their distribution, accounting for transportation, network planning. Which action is not a modeling step? It is difficult to answer this question unambiguously, it all depends on the specific situation. The main stages of the modeling process involve the formulation of the goal and subject of research, the identification of the main characteristics to achieve the goal, and the description of the relationship between model fragments. Next, perform calculations using mathematical formulas.
For example, service theory is the queuing problem. It is important to find a balance between the cost of maintaining devices and the cost of being in line. After constructing a formal description of the model, calculations are carried out using computational and analytical technologies. With a qualitative compilation of the model, you can find answers to all questions. If the model is bad, it is impossible to understand which action is not a modeling step.
Practicality is a true criterion for assessing the adequacy of a phenomenon or model. Multicriteria models, including optimization options, involve goal setting. But the way to achieve this goal is different. Among the difficulties that are possible in the process, we should highlight:
- in a complex system, there are sever alties;
- it is difficult to account for all random factors when analyzing a real system;
- it is problematic to compare the mathematical apparatus with the results you want to get
Due to the many complexities that arise in the process of studying multifaceted systems, simulation modeling has been developed. It is understood as a set of special programs for computer technology, which describes the operation of individual elements of the system and the relationship between them. The use of random variables involves repeated repetition of experiments, statistical processing of the results. Working with a simulation system is an experiment that is carried out with the help of computer technology. What are the advantages of this system? In this way it is possible to achieve greater proximity to the original system, which is impossible in the case of a mathematical model. Using the block principle, you can analyze individual blocks before they are included in a single system. This option allows you to use complex relationships that cannot be described using ordinary mathematical relationships.
Among the disadvantages of building a simulation system, we highlight the cost of time and resources, as well as the need to use modern computer technology.
The stages of development of modeling are comparable to the changes taking place in society. According to the area of use, all models are divided into training programs, simulators, teaching and visual aids. Experimental models can be reduced copies of real objects (cars). Scientific and technical optionsare stands created for the analysis of electronic equipment. Simulation models not only reflect the real reality, they involve testing on laboratory mice, experiments in the education system. Imitation is seen as a method of trial and error.
There is a division of all models according to the presentation variant. Material models are called subject. Such options are endowed with the geometric and physical characteristics of the original itself, they can be translated into reality. Information models cannot be touched by hands. They characterize the state and properties of the studied object, phenomenon, process, and their connection with the real world. Verbal options involve information models that are implemented in a colloquial or mental form. Signed types are expressed by applying certain signs of a polyhedral mathematical language.
Conclusion
Mathematical modeling as a method of scientific knowledge appeared simultaneously with the foundations of higher mathematics. An important role in this process was played by I. Newton, R. Descartes, G. Leibniz. Mathematical models were first built by P. Fermat, B. Pascal. V. V. Leontiev, V. V. Novozhilov, A. L. Lurie paid attention to mathematical modeling in production and economics. Nowadays, a similar option for studying an object or phenomenon is used in various fields of activity. With the help of designed systems, engineers explore such phenomena and processes that cannot be analyzed in real conditions.
Scientific researchby modeling, they were used in ancient times, capturing over time various types of scientific knowledge: architecture, design, chemistry, construction, physics, biology, ecology, geography, as well as social sciences. In any modeling process, three components are used: subject, object, model. Of course, the study of an object or phenomenon is not limited to modeling, there are other ways to obtain the necessary information.
