 | Service for Solving Linear Programming Problemsand other interesting typical problems |
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Welcome
To explain the solution of your typical problems is the main idea of creating this site.
Please keep in mind that the solution may differ from the one made at your educational institution.
To explain the solution of your typical problems is the main idea of creating this site.
Please keep in mind that the solution may differ from the one made at your educational institution.
Graphical Method of Linear Programming Problem
This is the first problem that is solved when studying linear programming.
This method allows solving the linear programming problem for the function of two variables.
The solution is accompanied by detailed comments and a large number of pictures.
Example ¹1. Function has a maximum value at the point
Example ¹2. Function has a minimum value at the point
Example ¹3. Function has a maximum value on the line segment
Example ¹4. Function has a minimum value on the line segment
Example ¹5. Function has a maximum value on the ray
Example ¹6. Function has a minimum value on the ray
Example ¹7. Function increases unlimitedly
Example ¹8. Function decreases unlimitedly
Example ¹9. Region of feasible solutions is a point
Example ¹10. Region of feasible solutions is an empty set
This is the first problem that is solved when studying linear programming.
This method allows solving the linear programming problem for the function of two variables.
The solution is accompanied by detailed comments and a large number of pictures.
Example ¹1. Function has a maximum value at the point
Example ¹2. Function has a minimum value at the point
Example ¹3. Function has a maximum value on the line segment
Example ¹4. Function has a minimum value on the line segment
Example ¹5. Function has a maximum value on the ray
Example ¹6. Function has a minimum value on the ray
Example ¹7. Function increases unlimitedly
Example ¹8. Function decreases unlimitedly
Example ¹9. Region of feasible solutions is a point
Example ¹10. Region of feasible solutions is an empty set
Simplex Method
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.
Example ¹1. Simplex method. Finding a maximum value of the function
Example ¹2. Simplex method. Finding a minimum value of the function
Example ¹3. Simplex method. Finding a maximum value of the function (artificial variables)
Example ¹4. Simplex method. Finding a minimum value of the function (artificial variables)
Example ¹5. Simplex method. Solution is not the only one
Example ¹6. Simplex method. Function increases unlimitedly
Example ¹7. Simplex method. Function decreases unlimitedly
Example ¹8. Simplex method. Region of feasible solutions is an empty set
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.
Example ¹1. Simplex method. Finding a maximum value of the function
Example ¹2. Simplex method. Finding a minimum value of the function
Example ¹3. Simplex method. Finding a maximum value of the function (artificial variables)
Example ¹4. Simplex method. Finding a minimum value of the function (artificial variables)
Example ¹5. Simplex method. Solution is not the only one
Example ¹6. Simplex method. Function increases unlimitedly
Example ¹7. Simplex method. Function decreases unlimitedly
Example ¹8. Simplex method. Region of feasible solutions is an empty set
Transportation Problem of Linear Programming
The transportation problem is a special linear programming problem.
The calculator finds the initial solution by the North-West corner method or the least cost method.
If necessary the initial solution will be improved by the MODI method.
The solution is accompanied by a large number of illustrations.
Example ¹1. Transportation problem by the least cost method (Balanced problem)
Example ¹2. Transportation problem by the least cost method (Unbalanced problem. Fictitious supplier)
Example ¹3. Transportation problem by the least cost method (Unbalanced problem. Fictitious consumer)
Example ¹4. Transportation problem by the North-West corner method (Balanced problem)
Example ¹5. Transportation problem by the North-West corner method (Unbalanced problem. Fictitious supplier)
Example ¹6. Transportation problem by the North-West corner method (Unbalanced problem. Fictitious consumer)
The transportation problem is a special linear programming problem.
The calculator finds the initial solution by the North-West corner method or the least cost method.
If necessary the initial solution will be improved by the MODI method.
The solution is accompanied by a large number of illustrations.
Example ¹1. Transportation problem by the least cost method (Balanced problem)
Example ¹2. Transportation problem by the least cost method (Unbalanced problem. Fictitious supplier)
Example ¹3. Transportation problem by the least cost method (Unbalanced problem. Fictitious consumer)
Example ¹4. Transportation problem by the North-West corner method (Balanced problem)
Example ¹5. Transportation problem by the North-West corner method (Unbalanced problem. Fictitious supplier)
Example ¹6. Transportation problem by the North-West corner method (Unbalanced problem. Fictitious consumer)
Other interesting typical problems:
Matrix Determinant
Each square matrix is associated a number called the matrix determinant. There are rules that allow to calculate determinants.
The calculator algorithm is able to use elementary transformations of the determinant.
Elementary transformations makes it easier to calculate the determinant, but this is possible only for simple problems.
The solution is accompanied by a large number of illustrations.
Example ¹1. Finding the determinant of a 3x3 matrix
Example ¹2. Finding the determinant of a 4x4 matrix
Example ¹3. Finding the determinant of a 5x5 matrix
Each square matrix is associated a number called the matrix determinant. There are rules that allow to calculate determinants.
The calculator algorithm is able to use elementary transformations of the determinant.
Elementary transformations makes it easier to calculate the determinant, but this is possible only for simple problems.
The solution is accompanied by a large number of illustrations.
Example ¹1. Finding the determinant of a 3x3 matrix
Example ¹2. Finding the determinant of a 4x4 matrix
Example ¹3. Finding the determinant of a 5x5 matrix
Gaussian Elimination and Gauss-Jordan Elimination
This calculator solves systems of linear equations using Gaussian elimination or Gauss-Jordan elimination.
These methods differ only in the second part of the solution.
The calculator's algorithm "tries" to count without using fractions.
The solution is accompanied by a large number of illustrations.
Example ¹1. Solving of a system of linear equations by the Gauss elimination (one solution)
Example ¹2. Solving of a system of linear equations by the Gauss elimination (many solutions)
Example ¹3. Solving of a system of linear equations by the Gauss elimination (no solution)
Example ¹4. Solving of a system of linear equations by the Gauss-Jordan elimination (one solution)
Example ¹5. Solving of a system of linear equations by the Gauss-Jordan elimination (many solutions)
This calculator solves systems of linear equations using Gaussian elimination or Gauss-Jordan elimination.
These methods differ only in the second part of the solution.
The calculator's algorithm "tries" to count without using fractions.
The solution is accompanied by a large number of illustrations.
Example ¹1. Solving of a system of linear equations by the Gauss elimination (one solution)
Example ¹2. Solving of a system of linear equations by the Gauss elimination (many solutions)
Example ¹3. Solving of a system of linear equations by the Gauss elimination (no solution)
Example ¹4. Solving of a system of linear equations by the Gauss-Jordan elimination (one solution)
Example ¹5. Solving of a system of linear equations by the Gauss-Jordan elimination (many solutions)
Cramer's Rule
Cramer’s rule uses determinants to solve systems of linear equations.
The calculator algorithm is able to use elementary transformations of the determinant.
The solution is accompanied by a large number of illustrations.
Example ¹1. Solving of a 2x2 system of linear equations by the Cramer's rule
Example ¹2. Solving of a 3x3 system of linear equations by the Cramer's rule
Cramer’s rule uses determinants to solve systems of linear equations.
The calculator algorithm is able to use elementary transformations of the determinant.
The solution is accompanied by a large number of illustrations.
Example ¹1. Solving of a 2x2 system of linear equations by the Cramer's rule
Example ¹2. Solving of a 3x3 system of linear equations by the Cramer's rule
Inverse Matrix
The inverse matrix can be calculated only for square matrices, but not every square matrix has an inverse matrix.
If the found matrix A-1 is inverse for the given matrix A, then A-1 * A = A * A-1 = E.
This calculator uses the algebraic additions to calculate the inverse matrix.
The solution is accompanied by a large number of illustrations.
Example ¹1. Finding the inverse of a 2x2 matrix
Example ¹2. Finding the inverse of a 3x3 matrix
The inverse matrix can be calculated only for square matrices, but not every square matrix has an inverse matrix.
If the found matrix A-1 is inverse for the given matrix A, then A-1 * A = A * A-1 = E.
This calculator uses the algebraic additions to calculate the inverse matrix.
The solution is accompanied by a large number of illustrations.
Example ¹1. Finding the inverse of a 2x2 matrix
Example ¹2. Finding the inverse of a 3x3 matrix
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