In the article you have submitted, wewe offer examples of mathematical models. In addition, we will pay attention to the stages of creating models and discuss some problems associated with mathematical modeling.
Another of our questions is the mathematical models ineconomy, examples, the definition of which we will consider a little later. We propose to start our conversation from the very concept of "model", briefly review their classification and move on to our main issues.
We often hear the word "model". What is it? This term has many definitions, here are just three of them:
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:
Information models, in turn, are divided into sign and verbal. A landmark - for computer and non-computer. Now we turn to a detailed examination of examples of a mathematical model.
As it is not difficult to guess, the mathematical modelreflects any features of an object or phenomenon using special mathematical symbols. Mathematics is also needed in order to model the laws of the surrounding world in its specific language.
The method of mathematical modeling was bornlong enough, 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 the 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, multicriteria, gaming 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.
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 prime example of a descriptive mathematical modelis the calculation of the trajectory of flight, 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.
Now we will talk a little abouteconomic-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 spoils very quickly. In this case, we need to correctly choose the temperature regime and optimize the storage process.
Thus, we can define the concept"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 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 |
Now we suggest you talk a little aboutmathematical model of multicriteria optimization. Prior to this, we gave an example of a mathematical model of process optimization by any one criterion, but what if there are many?
A vivid example of a multicriteria problem isthe 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?
As you can see, these goals do not coincide at all. So, in solving the problem, it is necessary to search for the optimal solution, the balance between the two criteria.
Speaking of gaming models, you need to understandthe concept of "game theory". To put it simply, 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 the minimum of information fromThe theory of games, 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 (win or loss). |
The game model can be a pair ormultiple. 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.
In this section we will pay attention to simulation mathematical models. Examples of tasks are:
In this case, we are talking about models thatas close 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, one can observe the fate of each individual individual. In this case, the mathematical description is rarely used, more often there are written conditions:
Thus, simulation models are used to describe a large system. The mathematical conclusion is the processing of the received statistical data.
It is very important to know that there are some requirements for this type of model, among which are the ones listed in the table below.
Versatility | This property allows you to use the samemodel in 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 | Here it is important to understand that this propertyallows you to properly reproduce real processes. In the tasks of exploitation, this property of mathematical modeling is very important. An example of the 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 compiled model is checked |
Accuracy | This requirement implies the coincidence of the values that we get when calculating the mathematical model and input parameters of our real object |
Cost-effective | The requirement of profitability, imposed on anymathematical model, characterized by the cost 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 |
In total in mathematical modeling it is customary to distinguish four stages.
In this section we will briefly discuss the issue of economic and mathematical models. Examples of tasks are:
The economic-mathematical model reflects economic abstraction, which is expressed through mathematical terms and signs.
Examples of a computer mathematical model are:
A computer model is an image of an object or system, represented as:
At the same time, this model reflects the structure and interrelationships of the system.
We have already said about what iseconomic-mathematical model. 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 completely consider the problem, butjust 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.
A task. The fisherman returned with the following catch:
How many fish did he buy in the store?
So, an example of constructing a mathematical modelof this 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 an answer to the main question: he bought 10 fish in the store.
</ p>
