An example of a mathematical model. Definition, classification and features

In the article, offered to your attention, we offer examples of mathematical models. In addition, we will pay attention to the stages of creating models and analyze some problems associated with mathematical modeling.

Another question that we have is mathematical models in the economy, examples, which we will consider a little later. We suggest starting our conversation from the very concept of "model", briefly review their classification and move on to our main issues.

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 the given object (this particular object can be expressed in different forms: mental, description by means of signs and so on);
• Even under the model is meant the display of a particular situation, vital or administrative;
• The model can be a reduced copy of an object (they are created for more detailed study and analysis, since the model reflects the structure and relationships).

Based on everything that was said before, you can draw a small conclusion: the model allows you to study in detail a complex system or object.

All models can be classified according to a number of characteristics:

• On the field of use (educational, experimental, scientific and technical, game, imitation);
• Dynamics (static and dynamic);
• On the branch of knowledge (physical, chemical, geographic, historical, sociological, economic, mathematical);
• By the mode of presentation (material and information).

Information models, in turn, are divided into sign and verbal. And the iconic ones are for computer and non-computer. Now let us turn to a detailed consideration of examples of a mathematical model.

Mathematical model

As it is not difficult to guess, the mathematical model reflects some features of an object or phenomenon using special mathematical symbols. Mathematics is also needed to model the laws of the surrounding world in its specific language.

The method of mathematical modeling was born long ago, thousands of years ago, together with the advent of this science. However, the impetus for the development of this method of modeling was the emergence of computers (electronic computers).

Now let's proceed to classification. It can also be carried out on some grounds. They are presented in the table below.

Classification by branch of science	Application of mathematical models in physics, sociology, chemistry and so on
The mathematical apparatus used in the modeling process	Models based on differential equations, discrete algebraic transformations, and the like
By the purposes of modeling	According to this principle, distinguish descriptive, optimization, multi-criteria, game and simulation models

We propose to stop and review the latest classification in more detail, since it reflects the general patterns of modeling and the purpose of the models being created.

Descriptive Models

In this chapter, we propose to dwell in more detail on descriptive mathematical models. In order for everything to be clear, an example will be given.

To begin with, this species can be called descriptive. This is due to the fact that we simply do calculations and forecasts, but we can not influence the outcome of the event in any way.

A vivid example of a descriptive mathematical model is the computation of the flight trajectory, speed, distance from the Earth of the comet, which invaded the expanses of our solar system. This model is descriptive, since all the results obtained can only warn us of any danger. To influence the outcome of the event, alas, we can not. However, based on the received calculations, 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 serve different situations. In this case, we are talking about models that help to find the right answer under certain conditions. They necessarily have certain parameters. To become very clear, consider an example from the agrarian part.

We have a granary, but the grain very quickly spoils. In this case, we need to correctly choose the temperature regime and optimize the storage process.

Thus, we can define "optimization model". In the mathematical sense, this is a system of equations (both linear and not), the solution of which helps to find the optimal solution in a specific economic situation. We examined 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.

Let's note one more nuance: models can have a different character (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 uncertain

Multi-criteria models

Now we suggest that you talk a little about the mathematical model of multicriteria optimization. Before that, we gave an example of a mathematical model of process optimization by any one criterion, but what if there are a lot of them?

A vivid example of a multicriteria task is the organization of the right, useful and at the same time economical nutrition for large groups of people. With such tasks are often found in the army, school canteens, summer camps, hospitals and so on.

What criteria are given to us in this task?

1. Meals should be useful.
2. The cost of food should be minimal.

As you can see, these goals do not coincide at all. So, when solving the problem, it is necessary to search for the optimal solution, the balance between the two criteria.

Game Models

Speaking about gaming models, it is necessary to understand the concept of "game theory". Simply put, these models reflect mathematical models of real conflicts. Only it is worthwhile to understand that, unlike a real conflict, the game mathematical model has its own specific rules.

Now will be given a minimum of information from game theory, which will help you understand what a game model is. And so, in the model necessarily there are sides (two or more), which are usually called players.

All models have certain characteristics.

Subjects	Number of Players
Strategy	Possible action options
Payment	The outcome of the conflict (winning or losing).

The game model can be a pair or multiple. If we have two subjects, then the conflict is a pair, if more - multiple. It is also possible to single out an antagonistic game, it is also called a zero-sum game. This is a model in which the gain of one of the participants equals 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 motion of molecules, and so on.

In this case, we are talking about models that are as close to real processes as possible. By and large, they imitate any manifestation in nature. In the first case, for example, we can simulate the dynamics of the number of ants in one colony. In this case, one can observe the fate of each individual individual. In this case, the mathematical description is rarely used, more often there are written conditions:

• In five days the female lays eggs;
• In twenty days the ant dies, and so on.

Thus, simulation models are used to describe a large system. The mathematical conclusion is the processing of the received statistical data.

Requirements

It is very important to know that some requirements are imposed on this type of model, among which are the ones listed in the table below.

Versatility	This property allows you to use the same model when describing the same type of groups of objects. It is important to note that universal mathematical models are completely independent of the physical nature of the object under study
Adequacy	It is important to understand here that this property allows you to reproduce real processes as correctly as possible. In the tasks of exploitation, this property of mathematical modeling is very important. An example of a model can be the process of optimizing the use of the gas system. In this case, the calculated and actual indicators are compared, as a result, the correctness of the 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
Economical	The requirement of profitability, imposed on any mathematical model, is characterized by the costs of implementation. 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 with the help of this mathematical model. If we are talking about computer-aided design, we calculate the time and computer memory

Stages of modeling

In total in mathematical modeling it is customary to distinguish four stages.

1. Formulation of laws linking parts of the model.
2. Research of mathematical problems.
3. Clarifying the coincidence of practical and theoretical results.
4. Analysis and modernization of the model.

Economic-mathematical model

In this section we will briefly discuss the issue of economic and mathematical models. Examples of tasks are:

• The formation of a production program for the production of meat products, which ensures the maximum profit of production;
• Maximizing the profit of the organization by calculating the optimal number of output tables and chairs in the furniture factory, and so on.

Economic-mathematical model reflects economic abstraction, which is expressed by means of mathematical terms and signs.

Computer mathematical model

Examples of a computer mathematical model are:

• Tasks of hydraulics with the help of block diagrams, diagrams, tables, and so on;
• Problems on the mechanics of solids, and so on.

A computer model is an image of an object or system, represented as:

• Tables;
• Flowcharts;
• Diagrams;
• Graphics, and so on.

At the same time, this model reflects the structure and interrelationships of the system.

Construction of the economic-mathematical model

We have already said about what an economic-mathematical model is. An example of the solution of the problem will be considered right now. We need to perform an analysis of the production program to identify a reserve of increased profit in the shift in the range.

We will not fully consider the problem, but only build an economic-mathematical model. The criterion of our task is the maximization of profit. Then the function has the form: A = p1 * x1 + p2 * x2 ..., tending to a 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 constructing a simple mathematical model

A task. The fisherman returned with the following catch:

• 8 fishes - inhabitants of northern seas;
• 20% of the catch - inhabitants of the southern seas;
• From the local river, not a single fish was found.

How many fish did he buy in the store?

Thus, an example of constructing a mathematical model of a given problem is as follows. Denote the total number of fish for x. Following the condition, 0.2x is the number of fish inhabiting the southern latitudes. Now we combine all available information and obtain a mathematical model of the problem: x = 0.2x + 8. Solve the equation and get the answer to the main question: he bought 10 fish in the store.