Service for Solving Linear Programming Problems

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Simplex Method Calculator

The simplex method is universal. It allows you to solve any linear programming problems.
Òhe solution by the simplex method is not as difficult as it might seem at first glance.
This calculator only finds a general solution when the solution is a straight line segment.
You can solve your problem or see examples of solutions that this calculator has made.
Enter integers or ordinary fractions. For example: 12, -3/4.
Find the     value of the function
F   =   x1 +   x2 +   x3 +   x4 +   x5 +   x6
subject to the constraints:
Çíàê ñèñòåìû x1 + x2 + x3 + x4 + x5 + x6
x1 + x2 + x3 + x4 + x5 + x6
x1 + x2 + x3 + x4 + x5 + x6
x1 + x2 + x3 + x4 + x5 + x6
x1 + x2 + x3 + x4 + x5 + x6

Problem:

Find the maximum value of the function
F = 30 x1 + 50 x2 + 80 x3
subject to the constraints:
Çíàê ñèñòåìû 20 x1 + 20 x2 + 30 x3 ≤ 9500
10 x1 + 10 x2 + 10 x3 ≤ 5500
10 x2 + 10 x3 ≤ 2000
x1 ≥ 0    x2 ≥ 0    x3 ≥ 0    

Solution:

1.This is a necessary condition for solving the problem:
the numbers on the right parts of the constraint system must be non-negative.
This condition is done.

2. This is a necessary condition for solving the problem:
all constraints must be equations.
Çíàê ñèñòåìû 20 x1 + 20 x2 + 30 x3 ≤ 9500
10 x1 + 10 x2 + 10 x3 ≤ 5500
10 x2 + 10 x3 ≤ 2000
Çíàê ñèñòåìû 20 x1 + 20 x2 + 30 x3 + S1 = 9500
10 x1 + 10 x2 + 10 x3 + S2 = 5500
10 x2 + 10 x3 + S3 = 2000
S1 ≥ 0, S2 ≥ 0, S3 ≥ 0.   The entered variables S1, S2, S3, are called slack variables.

3. Finding the initial basis and the value of the function F which corresponds to the found initial basis.

What is a basis?
A variable is called a basic variable for an equation if it enters into this equation with a coefficient of one and does not enter into other equations system (provided that there is a non-negative number on the right side of the equation).
If each equation has a basic variable, then they say that the system has a basis.
Variables that are not basic are called non-basic.

What is the idea of the simplex method?
Each basis is corresponded to one function value. One of them is the maximum value of the function F.
We will move from one basis to another.
The next basis will be chosen in such a way that the value of the function F will be no less than we have now.
Obviously, the number of possible bases for any problem is not very large.
So sooner or later the answer will be received.

How will we move from one basis to another?
It is more convenient to record the solution in the form of tables. Each row of the table is equivalent to a system equation. The highlighted row consists of the coefficients of the function (see the table below). This allows us not to rewrite variables every time. It saves time.
In the highlighted row, select a maximum positive coefficient (we can select any positive coefficient).
This is necessary in order to get a value of the function F no less than we have.
Column is selected.
For the positive coefficients of the selected column, we count the coefficient Θ and select the minimum value.
This is necessary in order to get non-negative numbers in the right part of the equations after moving to another basis.
Row is selected.
An element is found that will be basic. Next, we will need to calculate.

Does our system have a basis?
Çíàê ñèñòåìû 20 x1 + 20 x2 + 30 x3 + S1 = 9500
10 x1 + 10 x2 + 10 x3 + S2 = 5500
10 x2 + 10 x3 + S3 = 2000
There is a basis in our system. We can begin to solve our problem.
F = 30 x1 + 50 x2 + 80 x3
Non-basic variables are zero. In the mind, we can find the values of the basic variables. (see system)
Function F contains only non-basic variables. Therefore, the value of the function F for this basis can be found in the mind.
x1 = 0   x2 = 0   x3 = 0  
S1 = 9500   S2 = 5500   S3 = 2000   		=> F = 0
The initial basis was found. The value of the function F corresponding to the initial basis was found.

4. Finding a maximum value of the function F.

Step ¹1
x1 x2 x3 S1 S2 S3 const. Θ
20 20 30 1 0 0 9500 9500 : 30 ≈ 316,667
10 10 10 0 1 0 5500 5500 : 10 = 550
0 10 10 0 0 1 2000 2000 : 10 = 200
30 50 80 0 0 0 F - 0
20 20 30 1 0 0 9500
10 10 10 0 1 0 5500
0 1 1 0 0 1/10 200
30 50 80 0 0 0 F - 0
20 -10 0 1 0 -3 3500
10 0 0 0 1 -1 3500
0 1 1 0 0 1/10 200
30 -30 0 0 0 -8 F - 16000
Non-basic variables are zero. In the mind, we can find the values of the basic variables. (see table)
Function F contains only non-basic variables. Therefore, the value of the function F for this basis can be found in the mind. (see the highlighted row in the table)
x1 = 0   x2 = 0   S3 = 0  
x3 = 200   S1 = 3500   S2 = 3500   		=> F - 16000 = 0   => F = 16000

Step ¹2
x1 x2 x3 S1 S2 S3 const. Θ
20 -10 0 1 0 -3 3500 3500 : 20 = 175
10 0 0 0 1 -1 3500 3500 : 10 = 350
0 1 1 0 0 1/10 200
30 -30 0 0 0 -8 F - 16000
1 -1/2 0 1/20 0 -3/20 175
10 0 0 0 1 -1 3500
0 1 1 0 0 1/10 200
30 -30 0 0 0 -8 F - 16000
1 -1/2 0 1/20 0 -3/20 175
0 5 0 -1/2 1 1/2 1750
0 1 1 0 0 1/10 200
0 -15 0 -3/2 0 -7/2 F - 21250
Non-basic variables are zero. In the mind, we can find the values of the basic variables. (see table)
Function F contains only non-basic variables. Therefore, the value of the function F for this basis can be found in the mind. (see the highlighted row in the table)
x2 = 0   S1 = 0   S3 = 0  
x1 = 175   x3 = 200   S2 = 1750   		=> F - 21250 = 0   => F = 21250
There are not any positive coefficients in the highlighted row. Therefore, the maximum value of the function F was found.
Result:

x1 = 175   x2 = 0   x3 = 200  

Fmax = 21250


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