Intelligent Tutoring System of Linear Programming

There is a growing technological development in intelligent teaching systems. This field has become interesting to many researchers. In this paper, we present an intelligent tutoring system for teaching mathematics that helps students un-derstand the basics of linear programming using Linear Program Solver and Service for Solving Linear Programming Problems, through which students will be able to solve economic problems. It comes down to determining the minimum or maximum value of a linear function, which is called the objective function, according to pre-set limiting conditions expressed by linear equations and inequalities. The goal function and the limiting conditions represent a mathematical model of the observed problem. Working as a professor of mathematics in high school, I felt the need for one such work and dealing with the study of linear programming as an integral part of mathematics. There are a number of papers in this regard, but exclusively related to traditional ways of working, as stated in the introductory part of the paper. The center of work as well as the final part deals with the study of linear programming using programs that deal with this topic.


Introduction
Many problems from everyday life can be formulated as the minimum or maximum of a target function given limited resources and mutual constraints. If it is possible to define an objective function as a linear function of certain variables then we get the problem of linear programming.
The beginnings of the problem of linear programming can be found in Fouri-

Convex Figures of Points in a Plane
The notion of convex figures as a set of points is very present in the theory of linear programming.   Figure 2).

Linear inequalities with two unknowns
To make it easier to show the way in which the problem of linear programming is solved, we must touch on linear inequalities with two unknowns. The most appropriate way to solve such linear inequalities with two unknowns is to represent the solution using graphs. Take  ( ) where instead of the sign > the signs ≤, ≥, < can also appear.
where instead of the sign > inequality signs can also appear.
The procedure for solving the inequalities 0 ax by c + + ≥ and 0 ax by c + + ≤ is reduced to examining the sign of the linear trinomial ax by c + + for all possible values ( ) , x y . It is known that the set of points whose coordinates satisfy the equation 0 ax by c + + =, represents a straight line in the coordinate plane xOy, which we will denote by l. The line l divides the coordinate plane into two half-planes. The set of all points on the same side of the line l represents an open half-plane whose boundary is the line l. The line divides the plane into two half-planes. The vertical direction divides the plane into the left half-plane and the right half-plane (see Figure 4), the oblique direction divides the plane into the upper half-plane and the lower half-plane (see Figure 5).

The Basic Problem of Linear Programming
The basic problem of linear programming is to determine the maximum or minimum of the goal function:

( )
, F x y ax by c = + + (2) under restriction The function of Goal (2) and Constraint (3) represent a mathematical model of the linear programming problem.
The optimal solution to the problem of linear programming with a mathematical model defined by Conditions (1) and (2), are the coordinates of these points of a convex figure determined by (2), for which the objective function

Intelligent Tutoring Systems
Intelligent Tutoring Systems (ITS) have been researched in AI for decades. With Advances in Linear Algebra & Matrix Theory the huge development and increasing availability of programs, the application of intelligent learning systems is becoming more probable and precise, and the research of intelligent characteristics has received more attention than before. As a result, a number of new ITS, programs and websites have been established in recent years [1].
There are many ITS developed for: ITS for learning Java objects, evaluation of Java expression, linear programming, ITS for teaching Mongo database, ITS efficiency of e-learning, efficiency of CPP-Tutor, teaching AI search algorithms, teaching database, ITS for teaching 7 characteristics for mercury beings, ITS for teaching proper pronunciation in reciting the Holy Qur'an, ITS for health problems related to video game addiction, ITS for teaching advanced topics in information security, Oracle Intelligent Tutoring System (OITS), ITS for learning computer theory, e-learning system, ADO-Tutor: Intelligent tutoring system to rely on ADO.NET, ITS to pass parameters in Java programming, and predict student performance using NT and ITS, CPP-Tutor for C++ programming language, comparative study between animated intelligent teaching systems (AITS) and video-based Intelligent Tutoring Systems (VITS), ITS for Stomach Diseases Intelligent Teaching System, ITS for dia betes, computer networks, DSETutor for teaching DES information security algorithm.

What Is an Intelligent Tutoring System?
The terms tutor and mentor refer to someone who provides professional assistance in a particular field; to someone who helps to acquire certain skills and knowledge in the learning process. Transferred to the context of information technology and artificial intelligence, Intelligent Tutoring Systems (ITS) is software that simulates the behavior of tutors/teachers/mentors, i.e. learning assistant. More specifically, this software provides individual support and assistance in learning by asking students questions, analyzing their answers and offering customized instructions and feedback that are in the function of learning and understanding the content.
There are several programs that I think we can use when solving tasks from linear programming, which apply an intelligent system in such a way that each task solving procedure is available and elaborated in the right way and easily accessible to students.

Graphing Method Calculator for Linear Programming
This is the first problem to be solved when studying linear programming. This method allows solving linear programming problems for a two-variable function. The solution is accompanied by detailed comments and a large number of images. You can solve your problem or see all possible solutions to this problem. The program is available on the website (see Application of the program Example 3.1.
Find the maximum value of the function By clicking on Solve we get the solution of the whole task.
Solution: see Figure 9   The points whose coordinates satisfy all the inequalities of the constraint system are called the area of feasible solutions.
Any inequality of the constraint system needs to be addressed in order to find an area of feasible solutions to this problem (see Step 1 and Step 2).
The last two steps are necessary to get an answer (see Step 3 and Step 4).
This is a standard solution plan. If the area of feasible solutions is a point or an empty set then the solution will be shorter.
Step 1 Let us solve 1. The inequality of the constraint system. We can now draw a straight line (1) through the two points found.
Let's go back to inequality.
We need to transform the inequality so that only 2 x is on the left. Therefore, we must consider the points below the straight line (1).
Let's combine this result with the previous picture.
We now have an area of feasible solutions shown in Figure 10.  Step 2 Let us solve 2 inequalities of the constraint system. We can now draw a straight line (2) through the two points found.
Let's go back to inequality.
We need to transform the inequality so that only 2 x is on the left. Here is another example that, thanks to the program, can solve the following task with an explanation. Example 3.2.
Find the maximum value of the function Figure 11) By clicking on Solve we get the solution of the whole task.
Solution: Figure 12 The points whose coordinates satisfy all the inequalities of the constraint system are called the area of feasible solutions.
Any inequality of the constraint system needs to be addressed in order to find an area of feasible solutions to this problem (see Step 1 and Step 2).
The last two steps are necessary to get an answer (see Step 3 and Step 4).
This is a standard solution plan. If the area of feasible solutions is a point or an empty set then the solution will be shorter.
See the picture for a plan to solve this problem According to the condition of the task: We now have an area of feasible solutions shown in Figure 13.   Step 1 Let us solve 1. The inequality of the constraint system.
We need to draw a straight line: Let it be 2 Two points were found: ( ) 0,9 i ( ) 3, 0 .
We can now draw a straight line (1) through the two points found.
Let's go back to inequality We need to transform the inequality so that only 2 x is on the left. Therefore, we must consider the points below the straight line (1).
Let's combine this result with the previous picture.
We now have an area of feasible solutions shown in Figure 14.
Step 2 Let us solve 2 inequalities of the constraint system. Let it be 2 Two points were found: ( ) 0, 4 i ( ) 7, 0 . We can now draw a straight line (2) through the two points found. Let's go back to inequality.
We need to transform the inequality so that only 2 x is on the left.
The sign of the inequality is ≥. Therefore, we must consider the points above the line (2). Let's combine this result with the previous picture.
Step 3 We now have an area of feasible solutions shown in Figure 15. We need to draw a vector ( )
Step 4 We will move the "red" straight line perpendicular to the vector C from the upper left corner to the lower right corner.
The function F has a minimum value at the point where the "red" straight line crosses the range of feasible solutions for the first time.
The function F has a maximum value at the point where the "red" straight line last crosses the range of feasible solutions.
The function F has a maximum value at point A (see Figure 16).   The given example is by setting the maximum value of the function. We will show the same example by specifying the minimum value of the function.
Find the minimum value of the function Figure 17) By clicking on Solve we get the solution of the whole task. Solution: The points whose coordinates satisfy all the inequalities of the constraint system are called the area of feasible solutions (see Figure 18).
Any inequality of the constraint system needs to be addressed in order to find an area of feasible solutions to this problem (see Step 1 and Step 2).
The last two steps are necessary to get an answer (see Step 3 and Step 4).
This is a standard solution plan. If the area of feasible solutions is a point or an empty set then the solution will be shorter.
See the picture for a plan to solve this problem.
According to the condition of the task: Step 1 Let us solve 1. The inequality of the constraint system (see Figure 19).  A. Hasic, S. Jukic Two points were found: ( ) We can now draw a straight line (1) through the two points found.
Let's go back to inequality.
We need to transform the inequality so that only 2 x is on the left.
The sign of the inequality is ≥.
Therefore, we must consider the points above the line (1).
Let's combine this result with the previous picture.
We now have an area of feasible solutions shown in the figure.
Step 2 Let us solve 2 inequalities of the constraint system (see Figure 20).  Let it be 2 Two points were found: ( ) 0, 4 i ( ) We can now draw a straight line (2) through the two points found.
Let's go back to inequality.
We need to transform the inequality so that only 2 x is on the left. Therefore, we must consider the points above the line (2).
Let's combine this result with the previous picture.
We now have an area of feasible solutions shown in Figure 21.
Step 3. (see Figure 22) We need to draw a vector ( ) 3, 2 3 C = − , whose coordinates are the coefficients of the function F.
We will move the "red" straight line perpendicular to the vector C from the upper left corner to the lower right corner.
The function F has a minimum value at the point where the "red" straight line crosses the range of feasible solutions for the first time.

Linear Program Solver (LIPS)
Linear Program Solver (LIPS) [2] is an optimization package designed to solve the problems of linear, integer and target programming.
The main characteristics of LIPS are: LIPS solver is based on efficient implementation of a modified simplex method.
LIPS provides not only the answer, but also a detailed solution process as a series of simplex tables, so you can use it in the study (teaching) of linear programming.  LIPS provides sensitivity analysis procedures that allow us to study the behavior of the model when you change its parameters, including: analysis of changes on the right side of the constraint, analysis of changes in the coefficients of the target function, analysis of changes in the column/row technology matrix. Such information can be extremely useful in the practical application of LP models.
LIPS provides goal programming methods, including lexicographic and weighted GP methods. Goal programming methods are designed to solve multi-purpose optimization problems.
Main components of the program: 1) Client area of the main window, contains child windows; 2) Toolbar contains buttons for quick access to key program functions 3) Main menu bar allows access to all program functions; 4) Status bar contains simple notes on the purpose of the main menu items and toolbar items; 5) Sub-window contains the definition of the model or the results of the program (report).
Use the LIPS >> Solve Model menu to start the solution process. The Solver model has two modes of operation: basic and advanced. In the basic mode (which is suitable for learning the simplex method) LIPS provides not only the answer, but also a detailed solution process as a series of simplex tables. At each iteration, the output includes: the corresponding simplex table, the variable to be made basic, the variable outside the basic set, etc. The form of the computer solution simulates the process of manual problem solving (artificial basis, intelligent selection of the initial basis, fractions, etc.).
In advanced mode [3], LIPS provides a set of methods for solving large-scale problems: a modified primal and dual simplex method based on LU-decomposition, branching, and constraint method for MILP.
Sensitivity analysis allows us to study the behavior of the model when you change its parameters.
LIPS provides the following types of sensitivity analysis:  Analysis of changes on the right side of the constraint; The company that develops the Linear Program Solver is konobey. The latest version released by its developer is 1.11.0.
The next step is to get the next solution, that is LIPS report or LIPS Linear programming report which looks as follows. We will show it in several Figure   25 and Figure 26.
From the above few pictures, we have shown the way of solving in a few steps through the program.

Discussion and Conclusions
The main goal of this paper was to show how much difference there is between the existence of advanced technology and intelligent system and the traditional way of application that existed until a few years ago. With the development of technological aids, our life has become much easier and with much less time, an increasing number of programs are available that can solve the most demanding problems in the best way.
We are witnessing the ruble of the economic crisis, the work that occurred in Ukraine, and the fear of all that there is a lack of food primarily wheat and other foodstuffs, this assessment of the study through such programs can be of great benefit to us as we have already in the introductory part, they stated that the very way of applying linear programming arose from the need in the age of economic crisis and war events many years ago.
In addition, I would like to mention that technological advances have made it easier in terms of mathematical problems that were not available and which required a lot of strength and will, and desire to reach the goal, I hope that this small contribution will help pupils and students to take advantage of our recommendation and to use intelligent systems as much as possible and apply them in everyday practice and life.