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Example ¹1. Solving a Linear Programming Problem Using the Simplex Method.
Finding a Maximum Value of the Function
This solution was made using the calculator presented on the site.
Example ¹2. Finding a minimum value of the function
Example ¹3. Finding a maximum value of the function (artificial variables)
Example ¹4. Finding a minimum value of the function (artificial variables)
Example ¹5. Solution is not the only one
Example ¹6. Function increases unlimitedly
Example ¹7. Function decreases unlimitedly
Example ¹8. Region of feasible solutions is an empty set
Example ¹3. Finding a maximum value of the function (artificial variables)
Example ¹4. Finding a minimum value of the function (artificial variables)
Example ¹5. Solution is not the only one
Example ¹6. Function increases unlimitedly
Example ¹7. Function decreases unlimitedly
Example ¹8. Region of feasible solutions is an empty set
Problem:
Find the maximum value of the function
| F | = | 3 | x1 | + | 4 | x2 |
subject to the constraints:
 | x1 | + | x2 | ≤ | 55 | |||
| 2 | x1 | + | 3 | x2 | ≤ | 120 | ||
| 12 | x1 | + | 30 | x2 | ≤ | 960 |
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.
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.
all constraints must be equations.
| | x1 | + | x2 | ≤ | 55 | |||
| 2 | x1 | + | 3 | x2 | ≤ | 120 | ||
| 12 | x1 | + | 30 | x2 | ≤ | 960 |
| | x1 | + | x2 | + | S1 | = | 55 | ||||||||||
| 2 | x1 | + | 3 | x2 | + | S2 | = | 120 | |||||||||
| 12 | x1 | + | 30 | x2 | + | S3 | = | 960 |
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?
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?
| | x1 | + | x2 | + | S1 | = | 55 | ||||||||||
| 2 | x1 | + | 3 | x2 | + | S2 | = | 120 | |||||||||
| 12 | x1 | + | 30 | x2 | + | S3 | = | 960 |
There is a basis in our system. We can begin to solve our problem.
| F | = | 3 | x1 | + | 4 | x2 |
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.
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 S1 = 55 S2 = 120 S3 = 960 |
=> 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 | S1 | S2 | S3 | const. | Θ |
| 1 | 1 | 1 | 0 | 0 | 55 | 55 : 1 = 55 |
| 2 | 3 | 0 | 1 | 0 | 120 | 120 : 2 = 60 |
| 12 | 30 | 0 | 0 | 1 | 960 | 960 : 12 = 80 |
| 3 | 4 | 0 | 0 | 0 | F - 0 | |
| 1 | 1 | 1 | 0 | 0 | 55 | |
| 0 | 1 | -2 | 1 | 0 | 10 | |
| 0 | 18 | -12 | 0 | 1 | 300 | |
| 0 | 1 | -3 | 0 | 0 | F - 165 |
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)
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 x1 = 55 S2 = 10 S3 = 300 |
=> F - 165 = 0 => F = 165 |
Step ¹2
| x1 | x2 | S1 | S2 | S3 | const. | Θ |
| 1 | 1 | 1 | 0 | 0 | 55 | 55 : 1 = 55 |
| 0 | 1 | -2 | 1 | 0 | 10 | 10 : 1 = 10 |
| 0 | 18 | -12 | 0 | 1 | 300 | 300 : 18 ≈ 16,67 |
| 0 | 1 | -3 | 0 | 0 | F - 165 | |
| 1 | 0 | 3 | -1 | 0 | 45 | |
| 0 | 1 | -2 | 1 | 0 | 10 | |
| 0 | 0 | 24 | -18 | 1 | 120 | |
| 0 | 0 | -1 | -1 | 0 | F - 175 |
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)
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)
| S1 = 0 S2 = 0 x1 = 45 x2 = 10 S3 = 120 |
=> F - 175 = 0 => F = 175 |
There are not any positive coefficients in the highlighted row. Therefore, the maximum value of the function F was found.
Result:x1 = 45 x2 = 10
Fmax = 175
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