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This JavaScript Works Well in Netscape Navigator Version 4 (such as 4.7). If this is not feasible for you, you may download (free-of-charge) a software package that solves Linear Programs models by the Simplex Method and/or the Push-and-Pull Method:
This JavaScript E-labs learning object is intended for finding the optimal solution, and post-optimality analysis of small-size linear programs. It provides the optimal value and the optimal strategy for the decision variables. The necessary tools are produced to perform various sensitivity analyses on the coefficients of the objective function and on the right-hand-side values of the constraints.Other JavaScript learning objects for decision making in this series are categorized under different areas of applications at the MENU section on this page.
In entering your data to move from cell to cell in the>≤ and/or ≥ form. You may enter in the non-negativity conditions, if you wish.
Solve the standard formatted problem, and then substitute these changes back to get the values for the original variables and optimal value.
An Example: Consider the following problem with an equality constraint:
Maximization 3x + 2y + z
subject to:
4x + 2y + 3z = 12
x + z ≥ 1
x, y, and z ≥ 0.
Converting the equality constraints to two inequality constraints, we have the following equivalent problem:
Maximization 3x + 2y + z
subject to:
4x + 2y + 3z ≤ 24
4x + 2y + 3z ≥ 24
x + z ≥ 1
x, y, and z ≥ 0.
Enter your standard LP problem in the following table, then click on the 'Calculate' button.
Linear Optimization For Technical Details on Construction of the Sensitivity Region, Back to:
Construction of the Sensitivity Region for LP Models
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Professor Hossein Arsham
Example (part 2): Graphical method
Solve using the Graphical method the following problem:
| Maximize | Z = f(x,y) = 3x + 2y |
| subject to: | 2x + y ≤ 18 |
| 2x + 3y ≤ 42 | |
| 3x + y ≤ 24 | |
| x ≥ 0 , y ≥ 0 |
- Initially the coordinate system is drawn and each variable is associated to an axis (generally 'x' is associated to the horizontal axis and 'y' to the vertical one), as shown in figure 1.
- A numerical scale is marked in axis, appropriate to the values that variables can take according to the problem constraints. In order to do this, for each variable corresponding to an axis, all variables are set to zero except the variable associated to the studied axis in each constraint.
- The following step is to represent the restrictions. Beginning with the first, the line obtained by considering the constraint as an equality is drawn. In the example, this line is the segment connecting A and B points, and the region delimiting this restriction is indicated by the color YELLOW. This process is repeated with the other restrictions, BLUE and RED regions correspond to the second and third constraint respectively.
- The feasible region is the intersection of the regions defined by the set of constraints and the coordinate axis (conditions of non-negativity of variables). This feasible region is represented by the O-F-H-G-C polygon in PURPLE color.
- As a feasible region exists, extreme values (or polygon vertices) are calculated. These vertices are the points candidate as optimal solutions. In the example, these points are O, F, H, G, and C, as shown in the figure.
- Finally, the objective function (3x + 2y) is evaluated in each of these points (results are shown in the tableau below). Since G-point provides the greatest value to the Z-function and the objective is to maximize, this point is the optimal solution: Z = 33 with x = 3 and y = 12.
| Extreme point | Coordinates (x,y) | Objective value (Z) |
|---|---|---|
| O | (0,0) | 0 |
| C | (0,14) | 28 |
| G | (3,12) | 33 |
| H | (6,6) | 30 |
| F | (8,0) | 24 |
Graphical method and Simplex method comparison
Successive constructed tableaux in the Simplex method will provide the value of the objective function at the vertices of the feasible region, adjusting simultaneously, the coefficients of initial and slack variables.
In the initial tableau the value of the objective function at the O-vertex is calculated, the coordinates (0,0) correspond to the value which have the basic variables, being the result 0.
| Tableau I . 1st iteration. | |||||||
|---|---|---|---|---|---|---|---|
| 3 | 2 | 0 | 0 | 0 | |||
| Base | Cb | P0 | P1 | P2 | P3 | P4 | P5 |
| P3 | 0 | 18 | 2 | 1 | 1 | 0 | 0 |
| P4 | 0 | 42 | 2 | 3 | 0 | 1 | 0 |
| P5 | 0 | 24 | 3 | 1 | 0 | 0 | 1 |
| Z | 0 | -3 | -2 | 0 | 0 | 0 | |
The input base variable in the Simplex method determines towards what new vertex is performed the displacement. In this example, as P1 (corresponding to 'x') enters, the displacement is carried out by the OF-edge to reach the F-vertex, where the Z-function value is calculated. This step occurs in the second iteration of the Simplex method, as shown in tableau II. The corresponding value to F-vertex is calculated in it, and Z = 24 is the obtained value for the function.
| Tableau II . 2nd iteration. | |||||||
|---|---|---|---|---|---|---|---|
| 3 | 2 | 0 | 0 | 0 | |||
| Base | Cb | P0 | P1 | P2 | P3 | P4 | P5 |
| P3 | 0 | 2 | 0 | 1/3 | 1 | 0 | -2/3 |
| P4 | 0 | 26 | 0 | 7/3 | 0 | 1 | -2/3 |
| P1 | 3 | 8 | 1 | 1/3 | 0 | 0 | 1/3 |
| Z | 24 | 0 | -1 | 0 | 0 | 1 | |
Graphic Lp Optimizer Free
A new displacement by the FH-edge is made, up to H-vertex (data in Table III). In the third iteration, the value of the function at the H-vertex is calculated to obtain Z = 30.
| Tableau III . 3rd iteration. | |||||||
|---|---|---|---|---|---|---|---|
| 3 | 2 | 0 | 0 | 0 | |||
| Base | Cb | P0 | P1 | P2 | P3 | P4 | P5 |
| P2 | 2 | 6 | 0 | 1 | 3 | 0 | -2 |
| P4 | 0 | 12 | 0 | 0 | -7 | 1 | 4 |
| P1 | 3 | 6 | 1 | 0 | -1 | 0 | 1 |
| Z | 30 | 0 | 0 | 3 | 0 | -1 | |
The process goes on through the HG-edge up to G-vertex, obtained data are shown in tableau IV. At this point, the process ends, being able to check that the solution does not improve moving along GC-edge up to C-vertex (the current value of the Z-function is not increased).
| Tableau IV . 4th iteration. | |||||||
|---|---|---|---|---|---|---|---|
| 3 | 2 | 0 | 0 | 0 | |||
| Base | Cb | P0 | P1 | P2 | P3 | P4 | P5 |
| P2 | 2 | 12 | 0 | 1 | -1/2 | 1/2 | 0 |
| P5 | 0 | 3 | 0 | 0 | -7/4 | 1/4 | 1 |
| P1 | 3 | 3 | 1 | 0 | 3/4 | -1/4 | 0 |
| Z | 33 | 0 | 0 | 5/4 | 1/4 | 0 | |
The maximum value of the objective function is 33, and it corresponds to the values x = 3 and y = 12 (G-vertex coordinates).
In Graphical method is necessary to calculate the value of the objective function at each vertex of feasible region, while the Simplex method ends when the optimum value is found.
Graphic Lp Optimizer Download
Glp Graphic Lp Optimizer Descargar
Solve with PHPSimplex: Simplex method.
Solve with PHPSimplex: Graphical method.