In a large-scale version, the model is a certain image, diagram, map, description, image of a certain phenomenon or process. The phenomenon itself is called the original mathematical or economic model.
What is modeling?
Modeling is the study of a certain 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. During the experiment, a specific phenomenon is replaced by a scheme or a simplified model (copy). In some cases, the current model is assembled in order to understand the mechanism of work by its example, to analyze the economic feasibility of introducing the results of experience into a market economy. One and 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 used in situations where a real experiment involves significant material costs.
Features of mathematical modeling
In science, mathematical models are indispensable, as well as tools for them - mathematical concepts. Over several millennia, they have accumulated, modernized. In modern mathematics, there are universal and powerful research methods. 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, systematize the received information, make a full description of the object.
Mathematical formalization involves operating during the study with special concepts: matrix, function, derivative, antiderivative, numbers. Those relationships and relationships that can be found in the studied object between the constituent elements and details are written down by mathematical relations: equations, inequalities, equalities. The result is a mathematical description of the phenomenon or process, and therefore its mathematical model.
Rules for studying a mathematical model
There is a certain order of simulation stages, which allows you to establish relationships between consequences and causes. The central stage in the design or study of the system is the construction of a full-fledged mathematical model. It is the quality of the actions performed that directly determines the further analysis of this object. The construction of a mathematical or economic model is not a formal procedure. It should be convenient to use, accurate so that there are no distortions 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 the variables used to describe a phenomenon or object.
A similar approach is based on information about the principle of the object. In many cases, the simulated phenomenon has a complex structure; deciphering it requires a lot of time and knowledge. In such situations, such modeling steps are selected that will allow the experiments to be carried out on the original, to process the results obtained without going into the theoretical features of the object. Most often they use statistics and probability theory. The result is a stochastic model. There is a random connection between the variables in it. A huge number of different factors causes a random set of variables with which a phenomenon or object is characterized.
The modern stages of modeling are applied to static and dynamic models. In static types, a description of the relationships between the variables of the created phenomenon does not imply taking into account the time changes of the main parameters. For dynamic models, the description of relationships between variables is carried out taking into account temporary changes.
Varieties of models:
Different stages of mathematical modeling make it possible to describe relations and functions in linear models using the direct connection of variables.
What are the requirements for models?
- Universality. The model should be a complete display of all the properties inherent in a real object.
- Adequacy. Important characteristics of the object should not exceed the specified error value.
- Accuracy. It characterizes the degree of coincidence of the characteristics of an object existing in reality with similar parameters obtained in the study of the model.
- Profitability. The model should be minimal in terms of material costs.
Modeling steps
Consider the main stages of mathematical modeling.
- Choice of a task. The purpose of the study is selected, methods for its implementation are selected, and an experimental strategy is developed. This stage involves serious work. The final simulation result depends on the correctness of the task.
- Analysis of the theoretical foundations, summation of information obtained about the object. Such a stage involves the selection or creation of a theory. In the absence of theoretical knowledge about the object, cause-effect relationships are established between all the variables selected to describe the phenomenon or object. At this stage, determine the initial and final data, put forward a hypothesis.
- Formalization. A system of special notation is selected that will help to record in the form of mathematical expressions the relations between the components of the object under consideration.
Algorithm Additions
After setting the model parameters, a certain method or solution method is chosen.
- Implementation of the created model. After the stages of modeling systems are selected, a program is created that is tested and applied to solve the problem.
- Analysis of the information collected. An analogy is drawn between the task and the solution obtained, the modeling error is determined.
- Verify that the model matches the real object. If there will be a significant difference between them, a new model is being developed. Until the ideal correspondence of the model to its real analogue 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 has increased. Sections such as “mathematical chemistry”, “mathematical linguistics”, and “mathematical economics” dealing with the study of phenomena and objects appeared, and 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 and look for ways to control it. Carrying out a computer experiment is supposed in those cases when it is not possible to conduct a real one. After constructing a mathematical model of the phenomenon under study by 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 the task of an economic plan, natural phenomena, design, 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 the qualitative characteristics are translated into a mathematical language, 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 problems on computer technology, identifying measurement errors.
- Translation of information obtained during research into the language of the field for which the experiment was conducted.
These three stages of mathematical modeling are supplemented by checking the adequacy of the resulting model. The correspondence between the results obtained in the experiment and theoretical knowledge is checked. If necessary, a modification of the created model is carried out. 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 systems of equations. Complex objects are built using graph theory. It involves many points in space or on a plane, partially connected by edges. The main stages of economic modeling involve the selection of resources, their distribution, accounting for transportation, network planning. What action is not a modeling step? It is difficult to answer this question clearly, it all depends on the specific situation. The main stages of the modeling process involve the formulation of the goal and subject of the study, the identification of the main characteristics to achieve the goal, a description of the relationship between the fragments of the model. Next, perform calculations using mathematical formulas.
For example, service theory is a queuing problem. It is important to find a balance between the costs of maintaining the devices and the costs of being in the queue. After constructing a formal description of the model, calculations are carried out using computational and analytical technologies. With high-quality drafting 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 genuine 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, it should be highlighted:
- in a complex system, there are several connections between elements;
- it is difficult to take into account all random factors when analyzing a real system;
- it is difficult to compare the mathematical apparatus with the results that you want to get
Because of the many difficulties that arise in the process of studying multifaceted systems, simulation modeling has been developed. By it is meant 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 using computer technology. What are the advantages of this system? In a similar way, it is possible to achieve greater proximity to a genuine system, which is impossible in the case of a mathematical model. Using the block principle, individual blocks can be analyzed before they are included in a single system. This option allows you to use complex dependencies that cannot be described using ordinary mathematical relationships.
Among the disadvantages of building a simulation system, we highlight the costs of time and resources, as well as the need to use modern computer technology.
The stages of development of modeling are comparable with changes in society. In the field of use, all models are divided into training programs, simulators, educational and visual aids. Experienced models can be reduced copies of real objects (cars). Scientific and technical options are 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. Simulation is seen as a method of error and trial.
There is a subdivision of all models according to the presentation option. Material models are called subject models. Such options are endowed with geometric and physical characteristics of the original itself, they can be translated into reality. Information models cannot be touched. They characterize the state and properties of the studied object, phenomenon, process, and their connection with the real world. Verbal options imply information models, implemented in colloquial or mental form. Sign types are expressed by applying certain signs of a multifaceted mathematical language.
Conclusion
Mathematical modeling in the form of a method of scientific knowledge appeared simultaneously with the foundations of higher mathematics. An important role in such a process was played by I. Newton, R. Descartes, G. Leibniz. Mathematical models were first built by P. Fermat, B. Pascal. Mathematical modeling in production and economics was paid attention to V.V. Leontyev, V.V. Novozhilov, A. L. Lurie. Today, a similar version of the study of an object or phenomenon is used in various fields of activity. Using the designed systems, engineers study such phenomena and processes that cannot be analyzed in real conditions.
Scientific research by modeling was used in ancient times, capturing over time a variety of 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.