In this article, 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 our question is mathematical models in economics, examples, the definition of which we will consider a little later. We propose to start our conversation with the very concept of “model”, briefly consider 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 that reflects some properties or characteristics and so on of the original of this object (this specific object can be expressed in different forms: mental, description with the help of signs and so on);
- even under the model is meant the display of any particular situation, life or management;
- a model can be a reduced copy of an object (they are created for a more detailed study and analysis, since the model reflects the structure and relationships).
Based on everything that was said earlier, a small conclusion can be made: the model allows you to study in detail a complex system or object.
All models can be classified according to a number of signs:
- in the field of use (educational, experimental, scientific and technical, game, imitation);
- by dynamics (static and dynamic);
- in the field of knowledge (physical, chemical, geographical, historical, sociological, economic, mathematical);
- by the method of presentation (material and informational).
Information models, in turn, are divided into sign and verbal. And the iconic ones are for computer and non-computer ones. Now we turn to a detailed discussion of examples of a mathematical model.
Mathematical model
As you might guess, a mathematical model reflects some features of an object or phenomenon with the help of special mathematical symbols. Mathematics is needed in order to model the laws of the world in its own specific language.
The method of mathematical modeling originated a long time ago, thousands of years ago, together 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 the classification. It can also be carried out on some grounds. They are presented in the table below.
Classification by branch of science | The use 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, descriptive, optimization, multi-criteria, game and simulation models are distinguished |
We propose to stop and take a closer look at the latest classification, since it reflects the general laws of modeling and the goals of the created models.
Descriptive Models
In this chapter, we propose to dwell on descriptive mathematical models. In order to make everything extremely clear, an example will be given.
To begin with, this species can be called descriptive. This is due to the fact that we simply make calculations and forecasts, but in no way can affect the outcome of the event.
A striking example of a descriptive mathematical model is the calculation of the flight path, speed, distance from the Earth of a comet that has invaded the vastness of our solar system. This model is descriptive, since all the results obtained can only warn us of any danger. Alas, we cannot influence the outcome of the event. However, based on the obtained calculations, any measures can be taken 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 find the right answer in certain conditions. They necessarily have certain parameters. To make it extremely clear, consider an example from the agricultural sector.
We have a granary, but the grain spoils very quickly. In this case, we need to choose the right temperature and optimize the storage process.
Thus, we can define the term “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 considered an example of a mathematical model (optimization), but I would like to add: this type belongs to the class of extreme tasks, they help to describe the functioning of the economic system.
We note one more nuance: models can be of a different nature (see table below).
deterministic | In this case, the result depends on the input. |
stochastic | Description of random processes. In this case, the result remains uncertain. |
Multi-criteria models
Now we suggest you talk a little about the mathematical model of multi-criteria optimization. Prior to this, 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 striking example of a multi-criteria task is the organization of proper, healthy and at the same time economical nutrition of large groups of people. 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?
- Food should be healthy.
- Food costs should be minimal.
As you can see, these goals do not coincide at all. So, when solving a problem, it is necessary to search for the optimal solution, a balance between the two criteria.
Game models
Speaking about game models, it is necessary to understand the concept of “game theory”. Simply put, these models reflect mathematical models of real conflicts. It’s only worth understanding that, unlike a real conflict, a game mathematical model has its own specific rules.
Now we will give a minimum of information from game theory, which 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 certain characteristics.
Subjects | Number of players |
Strategy | Options for action |
Payment | The outcome of the conflict (win or lose). |
The game model may be a pair or multiple. If we have two subjects, then the conflict is paired, if more - multiple. You can also highlight the 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 include:
- model of the dynamics of the number of microorganisms;
- molecular motion model, 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 simulate the dynamics of the number of ants in one colony. In this case, you can observe the fate of each individual. In this case, a mathematical description is rarely used, written conditions are more often present:
- five days later, the female lays eggs;
- twenty days later, the ant dies, and so on.
Thus, simulation models are used to describe a large system. The mathematical conclusion is the processing of the obtained statistical data.
Requirements
It is very important to know that some requirements are imposed on this type of model, among which are those listed in the table below.
Versatility | This property allows you to use the same model when describing similar groups of objects. It is important to note that universal mathematical models are completely independent of the physical nature of the studied object |
Adequacy | It is important to understand here that this property allows reproducing real processes as correctly as possible. In operation 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, the 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 |
Cost-effectiveness | The requirement of profitability presented to any mathematical model is characterized by the cost of implementation. If working 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 the indicators of the cost of time and computer memory are calculated |
Modeling steps
In total, it is customary to distinguish four stages in mathematical modeling.
- The formulation of laws linking parts of the model.
- The study of mathematical problems.
- Clarification of the coincidence of practical and theoretical results.
- Analysis and modernization of the model.
Economic and mathematical model
This section briefly highlights the issue of economic-mathematical models. Examples of tasks include:
- the formation of a production program for the production of meat products, providing maximum production profit;
- maximizing the organization’s profits by calculating the optimal number of tables and chairs produced at a furniture factory, and so on.
The economic-mathematical model reflects an economic abstraction, which is expressed using mathematical terms and signs.
Computer mathematical model
Examples of computer mathematical models are:
- hydraulics tasks using flowcharts, diagrams, tables, and so on;
- tasks on solid mechanics, and so on.
A computer model is an image of an object or system, presented in the form:
- Tables
- flowcharts;
- Charts
- graphics, and so on.
Moreover, this model reflects the structure and relationships of the system.
Building an economic-mathematical model
We have already said what the economic-mathematical model is. 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 during 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: A = p1 * x1 + p2 * x2 ..., 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 are 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. Denote the total number of fish per x. Following the condition, 0.2x is the number of fish that live in the southern latitudes. Now we unite 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.