In the article brought to your attention, we offer examples of mathematical models. In addition, we will pay attention to the stages of creating models and analyze some of the tasks associated with mathematical modeling.
One more of our questions is about mathematical models in the economy, examples of which we will consider a definition a bit later. We propose to start our conversation with the very concept of “model”, briefly consider their classification and move on to our main questions.
The concept of "model"
We often hear the word "model". What is it? This term has many definitions, here are just three of them:
- a specific object that is created to receive and store information, reflecting some properties or characteristics, and so on, of the original of this object (this specific object can be expressed in different forms: mental, description using signs, and so on);
- model also means the display of any specific situation, life ormanagerial;
- model can serve as a reduced copy of any object (they are created for more detailed study and analysis, as the model reflects the structure and relationships).
Based on everything that was said earlier, we can draw a small conclusion: the model allows you to study a complex system or object in detail.
All models can be classified according to a number of features:
- by area of use (educational, experimental, scientific and technical, gaming, simulation);
- by dynamics (static and dynamic);
- by branch of knowledge (physical, chemical, geographical, historical, sociological, economic, mathematical);
- by way of presentation (material and informational).
Information models, in turn, are divided into sign and verbal. And iconic - on computer and non-computer. Now let's move on to a detailed consideration of examples of a mathematical model.
Mathematical model
As you might guess, a mathematical model reflects some features of an object or phenomenon using special mathematical symbols. Mathematics is needed in order to model the patterns of the surrounding world in its own specific language.
The method of mathematical modeling originated quite a long time ago, thousands of years ago, along with the advent of this science. However, the impetus for the development of this modeling method was given by the appearance of computers (electronic computers).
Now let's move on to classification. It can also be carried out according to some signs. They areare presented in the table below.
| Classification by branch of science | Application of mathematical models in physics, sociology, chemistry and so on |
| According to the mathematical apparatus used in the modeling process | Models based on differential equations, discrete algebraic transformations and the like |
| By modeling goals | According to this principle, there are descriptive, optimization, multi-criteria, game and simulation models |
We propose to stop and take a closer look at the last classification, as it reflects the general patterns of modeling and the goals of the models being created.
Descriptive models
In this chapter, we propose to dwell in more detail on descriptive mathematical models. In order to make everything very clear, an example will be given.
To begin with, this view can be called descriptive. This is due to the fact that we just make calculations and forecasts, but we cannot influence the outcome of the event in any way.
A striking example of a descriptive mathematical model is the calculation of the flight path, speed, distance from the Earth of a comet that invaded the vastness of our solar system. This model is descriptive, since all the results obtained can only warn us of some kind of danger. Influence the outcome of the event, alas, we do notCan. However, based on the calculations obtained, it is possible to take any measures to save life on Earth.
Optimization models
Now we will talk a little about economic and mathematical models, examples of which can be different situations. In this case, we are talking about models that help to find the right answer in certain conditions. They must have some parameters. To make it very clear, consider an example from the agricultural part.
We have a granary, but the grain spoils very quickly. In this case, we need to choose the right temperature regime and optimize the storage process.
Thus, we can define the concept of "optimization model". In a mathematical sense, this is a system of equations (both linear and not), the solution of which helps to find the optimal solution in a particular economic situation. We have considered an example of a mathematical model (optimization), but I would like to add: this type belongs to the class of extreme problems, they help to describe the functioning of the economic system.
Note one more nuance: models can be of different nature (see the table below).
| deterministic | In this case, the result depends on the input data |
| stochastic | Description of random processes. In this case, the result remains undefined |
Multicriteria models
Now we invite you to talk a little aboutmathematical model of multiobjective optimization. Before that, we gave an example of a mathematical model for optimizing a process according to any one criterion, but what if there are a lot of them?
A striking example of a multicriteria task is the organization of proper, he althy and at the same time economical nutrition for large groups of people. Such tasks are often found in the army, school canteens, summer camps, hospitals and so on.
What criteria are we given in this problem?
- Food should be he althy.
- Spending on food should be kept to a minimum.
As you can see, these goals do not coincide at all. This means that when solving a problem, it is necessary to look for the optimal solution, a balance between two criteria.
Game Models
Speaking of game models, it is necessary to understand the concept of "game theory". Simply put, these models reflect mathematical models of real conflicts. Just be aware that, unlike a real conflict, the game mathematical model has its own specific rules.
Now there will be a minimum of information from game theory that will help you understand what a game model is. And so, in the model there are necessarily parties (two or more), which are usually called players.
All models have some characteristics.
| Subjects | Number of players |
| Strategy | Options for possible actions |
| Payment | Outcome of the conflict (win or lose). |
The game model can be paired or multiple. If we have two subjects, then the conflict is paired, if more - multiple. An antagonistic game can also be distinguished, it is also called a zero-sum game. This is a model in which the gain of one of the participants is equal to the loss of the other.
Simulation models
In this section, we will pay attention to simulation mathematical models. Examples of tasks are:
- model of the dynamics of the number of microorganisms;
- model of the movement of molecules, and so on.
In this case, we are talking about models that are as close as possible to real processes. By and large, they imitate any manifestation in nature. In the first case, for example, we can model the dynamics of the number of ants in one colony. In this case, you can observe the fate of each individual. In this case, the mathematical description is rarely used, more often there are written conditions:
- after five days the female lays eggs;
- 20 days later the ant dies, and so on.
Thus, simulation models are used to describe a large system. Mathematical conclusion is the processing of the received statistical data.
Requirements
Very importantbe aware that there are some requirements for this type of model, among which are those given in the table below.
| Versatility | This property allows you to use the same model when describing groups of objects of the same type. It is important to note that universal mathematical models are completely independent of the physical nature of the object under study |
| Adequacy | Here it is important to understand that this property allows the most correct reproduction of real processes. In operational problems, this property of mathematical modeling is very important. An example of a model is the process of optimizing the use of a gas system. In this case, calculated and actual indicators are compared, as a result, the correctness of the compiled model is checked |
| Accuracy | This requirement implies the coincidence of the values that we get when calculating the mathematical model and the input parameters of our real object |
| Economy | The cost-effectiveness requirement for any mathematical model is characterized by implementation costs. If the work with the model is carried out manually, then it is necessary to calculate how much time it will take to solve one problem using this mathematical model. If we are talking about computer-aided design, then indicators of time and computer memory are calculated |
Stagesmodeling
In total, it is customary to distinguish four stages in mathematical modeling.
- Formulate the laws that link the parts of the model.
- Research of mathematical problems.
- Clarifying the coincidence of practical and theoretical results.
- Analysis and modernization of the model.
Economic and mathematical model
In this section, we will briefly highlight the issue of economic and mathematical models. Examples of tasks are:
- formation of a production program for the production of meat products, ensuring the maximum profit of production;
- maximizing the profit of the organization by calculating the optimal number of tables and chairs to be produced in a furniture factory, and so on.
The economic-mathematical model displays an economic abstraction, which is expressed using mathematical terms and signs.
Computer mathematical model
Examples of a computer mathematical model are:
- problems of hydraulics using flowcharts, diagrams, tables, and so on;
- problems on solid mechanics, and so on.
Computer model is an image of an object or system presented as:
- tables;
- flowcharts;
- diagrams;
- graphics, and so on.
At the same time, this model reflects the structure and interconnections of the system.
Building an economic-mathematical model
We have already talked about what economicmathematical model. An example of solving the problem will be considered right now. We need to analyze the production program to identify the reserve for increasing profits with a shift in the assortment.
We will not fully consider the problem, but only build an economic and mathematical model. The criterion of our task is profit maximization. Then the function has the form: Л=р1х1+р2х2… tending to the maximum. In this model, p is the profit per unit, x is the number of units produced. Further, based on the constructed model, it is necessary to make calculations and summarize.
An example of building a simple mathematical model
Task. The fisherman returned with the following catch:
- 8 fish - inhabitants of the northern seas;
- 20% of the catch - the inhabitants of the southern seas;
- not a single fish was found from the local river.
How many fish did he buy at the store?
So, an example of constructing a mathematical model of this problem is as follows. We denote the total number of fish as x. Following the condition, 0.2x is the number of fish living in southern latitudes. Now we combine all the available information and get the mathematical model of the problem: x=0, 2x+8. We solve the equation and get the answer to the main question: he bought 10 fish in the store.
