Imagine an airplane: wings, fuselage, tail, all this together - a real huge, immense, whole airplane. And you can make a model of an airplane, small, but everything is real, the same wings, etc., but compact. So is the mathematical model. There is a text problem, cumbersome, you can look at it, read it, but not quite understand it, and even more so it is not clear how to solve it. But what if we make a small model of it, a mathematical model, out of a large verbal problem? What does mathematical mean? So, using the rules and laws of mathematical notation, remake the text into a logically correct representation using numbers and arithmetic signs. So, A mathematical model is a representation of a real situation using a mathematical language.
Let's start simple: The number is greater than the number by. We need to write it down without using words, just the language of mathematics. If more by, then it turns out that if we subtract from, then the very difference of these numbers will remain equal. Those. or. Got the gist?
Now it’s more complicated, now there will be a text that you should try to present in the form of a mathematical model, until you read how I will do it, try it yourself! There are four numbers: , and. A product and more products and twice.
What happened?
In the form of a mathematical model, it will look like this:
Those. the product is related to as two to one, but this can be further simplified:
Well, with simple examples, you get the point, I guess. Let's move on to full-fledged tasks in which these mathematical models also need to be solved! Here is the task.
Mathematical model in practice
Task 1
After rain, the water level in the well may rise. The boy measures the time of falling small pebbles into the well and calculates the distance to the water using the formula, where is the distance in meters and is the time of falling in seconds. Before the rain, the time for the fall of the pebbles was s. How much must the water level rise after the rain in order for the measured time to change to s? Express your answer in meters.
Oh God! What formulas, what kind of well, what is happening, what to do? Did I read your mind? Relax, in tasks of this type, conditions are even more terrible, the main thing to remember is that in this task you are interested in formulas and relationships between variables, and what all this means in most cases is not very important. What do you see useful here? I personally see. The principle of solving these problems is as follows: you take all known quantities and substitute them.But sometimes you have to think!
Following my first advice, and substituting all the known ones into the equation, we get:
It was I who substituted the time of the second, and found the height that the stone flew before the rain. And now we need to count after the rain and find the difference!
Now listen to the second advice and think about it, the question specifies "how much the water level must rise after rain in order for the measured time to change by s". You need to figure it out right away, soooo, after the rain the water level rises, which means that the time for the stone to fall to the water level is less, and here the ornate phrase “so that the measured time changes” takes on a specific meaning: the fall time does not increase, but is reduced by the specified seconds. This means that in the case of a throw after the rain, we just need to subtract c from the initial time c, and we get the equation for the height that the stone will fly after the rain:
And finally, in order to find how much the water level should rise after the rain, so that the measured time changes by s, you just need to subtract the second from the first height of the fall!
We get the answer: per meter.
As you can see, there is nothing complicated, most importantly, don’t bother too much where such an incomprehensible and sometimes complex equation came from in the conditions and what everything in it means, take my word for it, most of these equations are taken from physics, and there the jungle is worse than in algebra. It sometimes seems to me that these tasks were invented to intimidate the student at the exam with an abundance of complex formulas and terms, and in most cases they require almost no knowledge. Just carefully read the condition and substitute the known values in the formula!
Here is another problem, no longer in physics, but from the world of economic theory, although knowledge of sciences other than mathematics is again not required here.
Task 2
The dependence of the volume of demand (units per month) for the products of a monopoly enterprise on the price (thousand rubles) is given by the formula
The company's monthly revenue (in thousand rubles) is calculated using the formula. Determine the highest price at which the monthly revenue will be at least a thousand rubles. Give the answer in thousand rubles.
Guess what I'll do now? Yeah, I'll start substituting what we know, but, again, you still have to think a little. Let's go from the end, we need to find at which. So, there is, equal to some, we find what else it is equal to, and it is equal, and we will write it down. As you can see, I don’t particularly bother about the meaning of all these quantities, I just look from the conditions, what is equal to what, that’s what you need to do. Let's return to the task, you already have it, but as you remember, from one equation with two variables, none of them can be found, what to do? Yeah, we still have an unused particle in the condition. Here, there are already two equations and two variables, which means that now both variables can be found - great!
Can you solve such a system?
We solve by substitution, we have already expressed it, which means we will substitute it into the first equation and simplify it.
It turns out here is such a quadratic equation: , we solve, the roots are like this, . In the task, it is required to find the highest price at which all the conditions that we took into account when we compiled the system will be met. Oh, it turns out that was the price. Cool, so we found the prices: and. The highest price, you say? Okay, the largest of them, obviously, we write it in response. Well, is it difficult? I think not, and you don’t need to delve into it too much!
And here's a frightening physics for you, or rather, another problem:
Task 3
To determine the effective temperature of stars, the Stefan–Boltzmann law is used, according to which, where is the radiant power of the star, is a constant, is the surface area of the star, and is the temperature. It is known that the surface area of a certain star is equal, and the power of its radiation is equal to W. Find the temperature of this star in degrees Kelvin.
Where is it clear? Yes, the condition says what is equal to what. Previously, I recommended that all unknowns be immediately substituted, but here it is better to first express the unknown sought. Look how simple everything is: there is a formula and they are known in it, and (this is the Greek letter "sigma". In general, physicists love Greek letters, get used to it). The temperature is unknown. Let's express it in the form of a formula. How to do it, I hope you know? Such assignments for the GIA in grade 9 usually give:
Now it remains to substitute numbers instead of letters on the right side and simplify:
Here's the answer: degrees Kelvin! And what a terrible task it was!
We continue to torment problems in physics.
Task 4
The height above the ground of a ball tossed up changes according to the law, where is the height in meters, is the time in seconds that has elapsed since the throw. How many seconds will the ball be at a height of at least three meters?
Those were all the equations, but here it is necessary to determine how much the ball was at a height of at least three meters, which means at a height. What are we going to make? Inequality, yes! We have a function that describes how the ball flies, where is exactly the same height in meters, we need the height. Means
And now you just solve the inequality, most importantly, do not forget to change the inequality sign from more or equal to less or equal when you multiply by both parts of the inequality in order to get rid of the minus in front.
Here are the roots, we build intervals for inequality:
We are interested in the interval where the sign is minus, since the inequality takes negative values there, this is from to both inclusive. And now we turn on the brain and think carefully: for inequality, we used an equation that describes the flight of the ball, it somehow flies along a parabola, i.e. it takes off, reaches a peak and falls, how to understand how long it will be at a height of at least meters? We found 2 turning points, i.e. the moment when it soars above meters and the moment when it reaches the same mark while falling, these two points are expressed in our form in the form of time, i.e. we know at what second of the flight it entered the zone of interest to us (above meters) and into which it left it (fell below the meter mark). How many seconds was he in this zone? It is logical that we take the time of exit from the zone and subtract from it the time of entry into this zone. Accordingly: - so much he was in the zone above meters, this is the answer.
You are so lucky that most of the examples on this topic can be taken from the category of problems in physics, so catch one more, it is the final one, so push yourself, there is very little left!
Task 5
For a heating element of a certain device, the temperature dependence on the operating time was experimentally obtained:
Where is the time in minutes. It is known that at a temperature of the heating element above the device may deteriorate, so it must be turned off. Find the maximum time after the start of work to turn off the device. Express your answer in minutes.
We act according to a well-established scheme, everything that is given, we first write out:
Now we take the formula and equate it to the temperature value to which the device can be heated as much as possible until it burns out, that is:
Now we substitute numbers instead of letters where they are known:
As you can see, the temperature during operation of the device is described by a quadratic equation, which means that it is distributed along a parabola, i.e. the device heats up to a certain temperature, and then cools down. We received answers and, therefore, during and during minutes of heating, the temperature is critical, but between and minutes it is even higher than the limit!
So, you need to turn off the device after a minute.
MATHEMATICAL MODELS. BRIEFLY ABOUT THE MAIN
Most often, mathematical models are used in physics: after all, you probably had to memorize dozens of physical formulas. And the formula is the mathematical representation of the situation.
In the OGE and the Unified State Examination there are tasks just on this topic. In the USE (profile) this is task number 11 (formerly B12). In the OGE - task number 20.
The solution scheme is obvious:
1) From the text of the condition, it is necessary to “isolate” useful information - what we write in problems in physics under the word “Given”. This useful information is:
- Formula
- Known physical quantities.
That is, each letter from the formula must be assigned a certain number.
2) Take all the known quantities and substitute them into the formula. The unknown value remains as a letter. Now you just need to solve the equation (usually quite simple), and the answer is ready.
Well, the topic is over. If you are reading these lines, then you are very cool.
Because only 5% of people are able to master something on their own. And if you have read to the end, then you are in the 5%!
Now the most important thing.
You've figured out the theory on this topic. And, I repeat, it's ... it's just super! You are already better than the vast majority of your peers.
The problem is that this may not be enough ...
For what?
For the successful passing of the exam, for admission to the institute on the budget and, MOST IMPORTANTLY, for life.
I will not convince you of anything, I will just say one thing ...
People who have received a good education earn much more than those who have not received it. This is statistics.
But this is not the main thing.
The main thing is that they are MORE HAPPY (there are such studies). Perhaps because much more opportunities open up before them and life becomes brighter? Don't know...
But think for yourself...
What does it take to be sure to be better than others on the exam and be ultimately ... happier?
FILL YOUR HAND, SOLVING PROBLEMS ON THIS TOPIC.
On the exam, you will not be asked theory.
You will need solve problems on time.
And, if you haven’t solved them (LOTS!), you will definitely make a stupid mistake somewhere or simply won’t make it in time.
It's like in sports - you need to repeat many times to win for sure.
Find a collection anywhere you want necessarily with solutions, detailed analysis and decide, decide, decide!
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In conclusion...
If you don't like our tasks, find others. Just don't stop with theory.
“Understood” and “I know how to solve” are completely different skills. You need both.
Find problems and solve!
Mathematical models
Mathematical model - approximate opidescription of the object of modeling, expressed usingschyu mathematical symbolism.
Mathematical models appeared along with mathematics many centuries ago. A huge impetus to the development of mathematical modeling was given by the appearance of computers. The use of computers made it possible to analyze and put into practice many mathematical models that had not previously been amenable to analytical research. Computer-implemented mathematicalsky model called computer mathematical model, a carrying out targeted calculations using a computer model called computational experiment.
Stages of computer mathematical modeletion shown in the figure. The firststage - definition of modeling goals. These goals can be different:
- a model is needed in order to understand how a particular object works, what is its structure, basic properties, laws of development and interaction
with the outside world (understanding); - a model is needed in order to learn how to manage an object (or process) and determine the best ways to manage for given goals and criteria (management);
- the model is needed in order to predict the direct and indirect consequences of the implementation of the specified methods and forms of impact on the object (forecasting).
An example from a completely different area: peacefully coexisting with stable populations of two species of individuals with a common food base, “suddenly” begin to dramatically change their numbers. And here mathematical modeling allows (with a certain degree of certainty) to establish the cause (or at least to refute a certain hypothesis).
Development of the concept of object management is another possible goal of modeling. Which aircraft flight mode should be chosen in order for the flight to be safe and most economically advantageous? How to schedule hundreds of types of work on the construction of a large facility so that it ends as soon as possible? Many such problems systematically arise before economists, designers, and scientists.
Finally, predicting the consequences of certain impacts on an object can be both a relatively simple matter in simple physical systems, and extremely complex - on the verge of feasibility - in biological, economic, social systems. If it is relatively easy to answer the question about the change in the mode of heat propagation in a thin rod with changes in its constituent alloy, then it is incomparably more difficult to trace (predict) the environmental and climatic consequences of the construction of a large hydroelectric power station or the social consequences of changes in tax legislation. Perhaps, here, too, mathematical modeling methods will provide more significant assistance in the future.
Second phase: definition of input and output parameters of the model; division of input parameters according to the degree of importance of the impact of their changes on the output. This process is called ranking, or division by rank (see below). "Formalisation and modeling").
Third stage: construction of a mathematical model. At this stage, there is a transition from the abstract formulation of the model to a formulation that has a specific mathematical representation. A mathematical model is equations, systems of equations, systems of inequalities, differential equations or systems of such equations, etc.
Fourth stage: choice of method for studying the mathematical model. Most often, numerical methods are used here, which lend themselves well to programming. As a rule, several methods are suitable for solving the same problem, differing in accuracy, stability, etc. The success of the entire modeling process often depends on the correct choice of method.
Fifth stage: the development of an algorithm, the compilation and debugging of a computer program is a process that is difficult to formalize. Of the programming languages, many professionals for mathematical modeling prefer FORTRAN: both due to tradition, and due to the unsurpassed efficiency of compilers (for computational work) and the presence of huge, carefully debugged and optimized libraries of standard programs of mathematical methods written in it. Languages such as PASCAL, BASIC, C are also in use, depending on the nature of the task and the inclinations of the programmer.
Sixth stage: program testing. The operation of the program is tested on a test problem with a known answer. This is just the beginning of a testing procedure that is difficult to describe in a formally exhaustive way. Usually, testing ends when the user, according to his professional characteristics, considers the program correct.
Seventh stage: the actual computational experiment, during which it is found out whether the model corresponds to a real object (process). The model is sufficiently adequate to the real process if some characteristics of the process obtained on a computer coincide with the experimentally obtained characteristics with a given degree of accuracy. If the model does not correspond to the real process, we return to one of the previous stages.
Classification of mathematical models
The classification of mathematical models can be based on various principles. It is possible to classify models by branches of science (mathematical models in physics, biology, sociology, etc.). It can be classified according to the applied mathematical apparatus (models based on the use of ordinary differential equations, partial differential equations, stochastic methods, discrete algebraic transformations, etc.). Finally, if we proceed from the general tasks of modeling in different sciences, regardless of the mathematical apparatus, the following classification is most natural:
- descriptive (descriptive) models;
- optimization models;
- multicriteria models;
- game models.
Let's explain this with examples.
Descriptive (descriptive) models. For example, simulations of the movement of a comet that invades the solar system are made in order to predict the trajectory of its flight, the distance at which it will pass from the Earth, and so on. In this case, the goals of modeling are descriptive, since there is no way to influence the motion of the comet, to change something in it.
Optimization Models are used to describe the processes that can be influenced in an attempt to achieve a given goal. In this case, the model includes one or more parameters that can be influenced. For example, by changing the thermal regime in a granary, one can set a goal to choose such a regime in order to achieve maximum grain preservation, i.e. optimize the storage process.
Multicriteria models. Often it is necessary to optimize the process in several parameters at the same time, and the goals can be very contradictory. For example, knowing food prices and a person's need for food, it is necessary to organize meals for large groups of people (in the army, children's summer camp, etc.) physiologically correctly and, at the same time, as cheaply as possible. It is clear that these goals do not coincide at all; when modeling, several criteria will be used, between which a balance must be sought.
Game models can be related not only to computer games, but also to very serious things. For example, before a battle, in the presence of incomplete information about the opposing army, a commander must develop a plan: in what order to bring certain units into battle, etc., taking into account the possible reaction of the enemy. There is a special section of modern mathematics - game theory - that studies the methods of decision making under conditions of incomplete information.
In the school course of computer science, students receive an initial idea of computer mathematical modeling as part of the basic course. In high school, mathematical modeling can be deeply studied in a general education course for classes in physics and mathematics, as well as within a specialized elective course.
The main forms of teaching computer mathematical modeling in high school are lectures, laboratory and credit classes. Usually, the work on creating and preparing for the study of each new model takes 3-4 lessons. In the course of the presentation of the material, tasks are set, which in the future should be solved by students on their own, in general terms, ways to solve them are outlined. Questions are formulated, the answers to which should be obtained when performing tasks. Additional literature is indicated, which allows obtaining auxiliary information for more successful completion of tasks.
The form of organizing classes in the study of new material is usually a lecture. After the completion of the discussion of the next model students have at their disposal the necessary theoretical information and a set of tasks for further work. In preparation for the task, students choose the appropriate solution method, using some known private solution, they test the developed program. In case of quite possible difficulties in the performance of tasks, consultation is given, a proposal is made to work out these sections in more detail in the literature.
The most relevant to the practical part of teaching computer modeling is the method of projects. The task is formulated for the student in the form of an educational project and is carried out over several lessons, and the main organizational form in this case is computer laboratory work. Learning to model using the learning project method can be implemented at different levels. The first is a problem statement of the project implementation process, which is led by the teacher. The second is the implementation of the project by students under the guidance of a teacher. The third is the independent implementation by students of an educational research project.
The results of the work should be presented in numerical form, in the form of graphs, diagrams. If possible, the process is presented on the computer screen in dynamics. Upon completion of the calculations and the receipt of the results, they are analyzed, compared with known facts from the theory, the reliability is confirmed and a meaningful interpretation is carried out, which is subsequently reflected in a written report.
If the results satisfy the student and the teacher, then the work counts completed, and its final stage is the preparation of a report. The report includes brief theoretical information on the topic under study, a mathematical formulation of the problem, a solution algorithm and its justification, a computer program, the results of the program, analysis of the results and conclusions, a list of references.
When all the reports have been drawn up, at the test session, students make brief reports on the work done, defend their project. This is an effective form of report of the project team to the class, including setting the problem, building a formal model, choosing methods for working with the model, implementing the model on a computer, working with the finished model, interpreting the results, forecasting. As a result, students can receive two grades: the first - for the elaboration of the project and the success of its defense, the second - for the program, the optimality of its algorithm, interface, etc. Students also receive marks in the course of surveys on theory.
An essential question is what kind of tools to use in the school informatics course for mathematical modeling? Computer implementation of models can be carried out:
- using a spreadsheet (usually MS Excel);
- by creating programs in traditional programming languages (Pascal, BASIC, etc.), as well as in their modern versions (Delphi, Visual
Basic for Application, etc.); - using special software packages for solving mathematical problems (MathCAD, etc.).
At the elementary school level, the first remedy appears to be the preferred one. However, in high school, when programming is, along with modeling, a key topic of computer science, it is desirable to involve it as a modeling tool. In the process of programming, the details of mathematical procedures become available to students; moreover, they are simply forced to master them, and this also contributes to mathematical education. As for the use of special software packages, this is appropriate in a profile computer science course as a supplement to other tools.
Exercise :
- Outline key concepts.
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 problems associated with mathematical modeling.
Another issue of ours is mathematical models in economics, examples of which we will consider a definition 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 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);
- a model also means a display of any specific situation, life or management;
- a small copy of an object can serve as a model (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, we 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 criteria:
- 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);
- according to the method 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 laws of the 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 are presented in the table below.
We propose to stop and take a closer look at the last classification, since 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 simply 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 expanses of our solar system. This model is descriptive, since all the results obtained can only warn us of some kind of danger. Unfortunately, we cannot influence the outcome of the event. However, based on the calculations obtained, it is possible to take any measures to preserve life on Earth.
Optimization Models
Now we will talk a little about economic and mathematical models, examples of which can be various 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 agrarian 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 one more thing: this type belongs to the class of extreme problems, they help to describe the functioning of the economic system.
We note one more nuance: models can be of a different nature (see the table below).
Multicriteria models
Now we invite you to talk a little about the mathematical 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, healthy and at the same time economical nutrition of large groups of people. Such tasks are often encountered 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 expenses 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 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 is only worth understanding that, unlike a real conflict, a game mathematical model has its own specific rules.
Now I 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.
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 focus on simulation mathematical models. Examples of tasks are:
- model of the dynamics of the number of microorganisms;
- model of molecular motion, 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;
- after twenty days the ant dies, and so on.
Thus, are used to describe a large system. Mathematical conclusion is the processing of the received statistical data.
Requirements
It is very important to know 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 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, 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 obtain when calculating the mathematical model and the input parameters of our real object |
Economy | The requirement of economy 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 |
Modeling steps
In total, it is customary to distinguish four stages in mathematical modeling.
- Formulation of laws linking parts of the model.
- Study of mathematical problems.
- Finding out 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. Examples of tasks can be:
- 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:
- hydraulics tasks using flowcharts, diagrams, tables, and so on;
- problems on solid mechanics, and so on.
A computer model is an image of an object or system, presented as:
- tables;
- block diagrams;
- diagrams;
- graphics, and so on.
At the same time, this model reflects the structure and interconnections of the system.
Building an economic and mathematical model
We have already talked about what an 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 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=р1*х1+р2*х2… 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
A 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 a 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.
The basis for solving economic problems are mathematical models.
Drawing up a mathematical model includes:mathematical model problem is a set of mathematical relationships that describe the essence of the problem.
- task variable selection
- drawing up a system of restrictions
- choice of objective function
Task variables are called quantities X1, X2, Xn, which fully characterize the economic process. Usually they are written as a vector: X=(X 1 , X 2 ,...,X n).
The system of restrictions tasks are a set of equations and inequalities that describe the limited resources in the problem under consideration.
target function task is called a function of task variables that characterizes the quality of the task and the extremum of which is required to be found.
In general, a linear programming problem can be written as follows:
This entry means the following: find the extremum of the objective function (1) and the corresponding variables X=(X 1 , X 2 ,...,X n) provided that these variables satisfy the system of constraints (2) and non-negativity conditions (3) .
Acceptable Solution(plan) of a linear programming problem is any n-dimensional vector X=(X 1 , X 2 ,...,X n) that satisfies the system of constraints and non-negativity conditions.
The set of feasible solutions (plans) of the problem forms range of feasible solutions(ODR).
The optimal solution(plan) of a linear programming problem is such a feasible solution (plan) of the problem, in which the objective function reaches an extremum.
An example of compiling a mathematical model
The task of using resources (raw materials)
Condition: For the manufacture of n types of products, m types of resources are used. Make a mathematical model.
Known:
- b i (i = 1,2,3,...,m) are the reserves of each i-th type of resource;
- a ij (i = 1,2,3,...,m; j=1,2,3,...,n) are the costs of each i-th type of resource for the production of a unit volume of the j-th type of product;
- c j (j = 1,2,3,...,n) is the profit from the sale of a unit volume of the j-th type of product.
It is required to draw up a plan for the production of products that provides maximum profit with given restrictions on resources (raw materials).
Solution:
We introduce a vector of variables X=(X 1 , X 2 ,...,X n), where x j (j = 1,2,...,n) is the volume of production of the j-th type of product.
The costs of the i-th type of resource for the production of a given volume x j of products are equal to a ij x j , therefore, the restriction on the use of resources for the production of all types of products has the form:
The profit from the sale of the j-th type of product is equal to c j x j , so the objective function is equal to:
Answer- The mathematical model looks like:
Canonical form of a linear programming problem
In the general case, a linear programming problem is written in such a way that both equations and inequalities are constraints, and variables can be either non-negative or arbitrarily changing.
In the case when all constraints are equations and all variables satisfy the non-negativity condition, the linear programming problem is called canonical.
It can be represented in coordinate, vector and matrix notation.
The canonical linear programming problem in coordinate notation has the form:
The canonical linear programming problem in matrix notation has the form:
- A is the matrix of coefficients of the system of equations
- X is a column matrix of task variables
- Ao is the matrix-column of the right parts of the constraint system
Often, linear programming problems are used, called symmetric ones, which in matrix notation have the form:
Reduction of a general linear programming problem to canonical form
In most methods for solving linear programming problems, it is assumed that the system of constraints consists of equations and natural conditions for the non-negativity of variables. However, when compiling models of economic problems, constraints are mainly formed in the form of a system of inequalities, so it is necessary to be able to move from a system of inequalities to a system of equations.
This can be done like this:Take a linear inequality a 1 x 1 +a 2 x 2 +...+a n x n ≤b and add to its left side some value x n+1 such that the inequality becomes the equality a 1 x 1 +a 2 x 2 + ...+a n x n +x n+1 =b. Moreover, this value x n+1 is non-negative.
Let's consider everything with an example.
Example 26.1Reduce the linear programming problem to canonical form:
Solution:
Let's move on to the problem of finding the maximum of the objective function.
To do this, we change the signs of the coefficients of the objective function.
To convert the second and third inequalities of the constraint system into equations, we introduce non-negative additional variables x 4 x 5 (this operation is marked with the letter D on the mathematical model).
The variable x 4 is entered on the left side of the second inequality with a "+" sign, since the inequality has the form "≤".
The variable x 5 is entered on the left side of the third inequality with the "-" sign, since the inequality has the form "≥".
Variables x 4 x 5 are entered into the objective function with a coefficient. equal to zero.
We write the problem in canonical form.
Example 1.5.1.
Let some economic region produce several (n) types of products exclusively on its own and only for the population of this region. It is assumed that the technological process has been worked out, and the demand of the population for these goods has been studied. It is necessary to determine the annual volume of output of products, taking into account the fact that this volume must provide both final and industrial consumption.
Let's make a mathematical model of this problem. According to its condition, the following are given: types of products, demand for them and the technological process; find the volume of output for each type of product.
Let us denote the known quantities:
c i- public demand for i-th product ( i=1,...,n); a ij- amount i-th product required to produce a unit of the j -th product using this technology ( i=1,...,n ; j=1,...,n);
X i - volume of output i-th product ( i=1,...,n); totality With =(c 1 ,..., c n ) is called the demand vector, the numbers a ij– technological coefficients, and the set X =(X 1 ,..., X n ) is the release vector.
By the condition of the problem, the vector X is divided into two parts: for final consumption (vector With ) and reproduction (vector x-s ). Calculate that part of the vector X which goes to reproduction. According to our designations for production X j quantity of the j-th product goes a ij · X j quantities i-th product.
Then the sum a i1 · X 1 +...+ a in · X n shows the value i-th product, which is needed for the entire output X =(X 1 ,..., X n ).
Therefore, the equality must hold:
Extending this reasoning to all types of products, we arrive at the desired model:
Solving this system of n linear equations with respect to X 1 ,...,X n and find the required output vector.
In order to write this model in a more compact (vector) form, we introduce the notation:
Square ( 
) -matrix BUT called the technology matrix. It is easy to check that our model will now be written like this: x-s=Ah or
(1.6)
We got the classic model " Input - Output ”, the author of which is the famous American economist V. Leontiev.
Example 1.5.2.
An oil refinery has two grades of oil: grade BUT in the amount of 10 units, grade AT- 15 units. When processing oil, two materials are obtained: gasoline (we denote B) and fuel oil ( M). There are three options for the processing technology:
I: 1 unit BUT+ 2 units AT gives 3 units. B+ 2 units M
II: 2 units BUT+ 1 unit AT gives 1 unit. B+ 5 units M
III: 2 units BUT+ 2 units AT gives 1 unit. B+ 2 units M
The price of gasoline is $10 per unit, fuel oil is $1 per unit.
It is required to determine the most advantageous combination of technological processes for processing the available amount of oil.
Before modeling, we clarify the following points. It follows from the conditions of the problem that the “profitability” of the technological process for the plant should be understood in the sense of obtaining the maximum income from the sale of its finished products (gasoline and fuel oil). In this regard, it is clear that the "choice (making) decision" of the plant is to determine which technology and how many times to apply. Obviously, there are many such possibilities.
Let us denote the unknown quantities:
X i- amount of use i-th technological process (i=1,2,3). Other parameters of the model (reserves of oil grades, prices of gasoline and fuel oil) known.
Now one specific decision of the plant is reduced to the choice of one vector X =(x 1 ,X 2 ,X 3 ) , for which the plant's revenue is equal to (32x 1 +15x 2 +12x 3 ) dollars. Here, 32 dollars is the income received from one application of the first technological process (10 dollars 3 units. B+ $1 2 units M= $32). Coefficients 15 and 12 have a similar meaning for the second and third technological processes, respectively. Accounting for the oil reserve leads to the following conditions:
for variety BUT:
for variety AT:,
where in the first inequality the coefficients 1, 2, 2 are the consumption rates of grade A oil for a one-time application of technological processes I,II,III respectively. The coefficients of the second inequality have a similar meaning for grade B oil.
The mathematical model as a whole has the form:
Find such a vector x = (x 1 ,X 2 ,X 3 ) to maximize
f(x) = 32x 1 +15x 2 +12x 3
when the conditions are met:
The abbreviated form of this entry is as follows:
under restrictions
(1.7)
We got the so-called linear programming problem.
Model (1.7.) is an example of an optimization model of a deterministic type (with well-defined elements).
Example 1.5.3.
The investor needs to determine the best set of stocks, bonds and other securities to purchase them for a certain amount in order to obtain a certain profit with minimal risk to himself. Return on every dollar invested in a security j-th type, characterized by two indicators: expected profit and actual profit. For an investor, it is desirable that the expected profit per dollar of investments for the entire set of securities is not lower than a given value b.
Note that for the correct modeling of this problem, a mathematician requires certain basic knowledge in the field of portfolio theory of securities.
Let us denote the known parameters of the problem:
n- the number of types of securities; a j– actual profit (random number) from the j-th type of security; 
is the expected profit from j th type of security.
Denote the unknown quantities :
y j - funds allocated for the purchase of securities of the type j.
In our notation, the total amount invested is expressed as 
. To simplify the model, we introduce new quantities
.
In this way, X i- this is the share of all funds allocated for the purchase of securities of the type j.
It's clear that 
It can be seen from the condition of the problem that the goal of the investor is to achieve a certain level of profit with minimal risk. Essentially, risk is a measure of deviation of actual profit from expected one. Therefore, it can be identified with the profit covariance for securities of type i and type j. Here M is the designation of the mathematical expectation.
The mathematical model of the original problem has the form:
under restrictions
,
,
,
. (1.8)
We have obtained the well-known Markowitz model for optimizing the structure of a securities portfolio.
Model (1.8.) is an example of an optimization model of a stochastic type (with elements of randomness).
Example 1.5.4.
On the basis of a trade organization, there are n types of one of the products of the assortment minimum. Only one of the types of this product must be delivered to the store. It is required to choose the type of goods that it is advisable to bring to the store. If the product type j will be in demand, then the store will profit from its sale R j, if it is not in demand - a loss q j .
Before modeling, we will discuss some fundamental points. In this problem, the decision maker (DM) is the store. However, the outcome (getting the maximum profit) depends not only on his decision, but also on whether the imported goods will be in demand, i.e. whether they will be bought out by the population (it is assumed that for some reason the store does not have the opportunity to study the demand of the population ). Therefore, the population can be considered as the second decision maker, choosing the type of goods according to their preferences. The worst "decision" of the population for the store is: "the imported goods are not in demand." So, in order to take into account all kinds of situations, the store needs to consider the population as its “opponent” (conditionally), pursuing the opposite goal - to minimize the store’s profit.
So, we have a decision problem with two participants pursuing opposite goals. Let us clarify that the store chooses one of the types of goods for sale (there are n solutions), and the population chooses one of the types of goods that is in the greatest demand ( n solution options).
To compile a mathematical model, we draw a table with n lines and n columns (total n 2 cells) and agree that the rows correspond to the choice of the store, and the columns correspond to the choice of the population. Then the cage (i, j) corresponds to the situation when the store chooses i-th type of goods ( i-th line), and the population chooses j-th type of goods ( j- th column). In each cell, we write a numerical assessment (profit or loss) of the corresponding situation from the point of view of the store:
Numbers q i written with a minus to reflect the loss of the store; in each situation, the “payoff” of the population is (conditionally) equal to the “payoff” of the store, taken with the opposite sign.
An abbreviated view of this model is as follows:
(1.9)
We got the so-called matrix game. Model (1.9.) is an example of game decision making models.
