Linear optimization: Sensibility analysis 1: Unterschied zwischen den Versionen
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== Complete Example == | == Complete Example == | ||
| − | A garden center has got a bed with a developable area of 100 m². | + | A garden center has got a bed with a developable area of 100 m². <br /> |
| − | There is a budget of $720 available to plant roses and carrots in the bed. | + | There is a budget of $720 available to plant roses and carrots in the bed. <br /> |
| − | However, the maximal planting area of carrots shall not exceed 60m². | + | However, the maximal planting area of carrots shall not exceed 60m². <br /> |
| − | The planting shall maximize profit. | + | The planting shall maximize profit. <br /> |
| − | The costs for the seed are as following: | + | |
| − | < | + | The costs for the seed are as following: <br /> |
| − | </ | + | *Roses: <math> 6 \frac{$}{m^{2}} </math> Carrots: <math> 9 \frac{$}{m^{2}} </math> <br /> |
| + | |||
| + | The selling prices of the products are: | ||
| + | *Roses: <math> 7 \frac{$}{m^{2}} </math> Carrots: <math> 11 \frac{$}{m^{2}} </math> <br /> | ||
| + | |||
| + | ==== We formulate the LP: ==== | ||
| + | : <math>Z: x _1 + 2x_2 \xrightarrow {} max! \qquad | x_1 ; x_2 \ge 0 </math> | ||
| + | : <math>y_1: x_2 \le 60 </math> | ||
| + | : <math>y_2: x_1 + x_2 \le 100 </math> | ||
| + | : <math>y_3: 6x_1 + 9x_2 \le 720 </math> | ||
| + | |||
| + | ==== Simplex tableaus: ==== | ||
== Presentation of the Problem == | == Presentation of the Problem == | ||
Version vom 27. Juni 2013, 19:02 Uhr
Besetzt H&S
Inhaltsverzeichnis
Theory
The german term "sensibilitätsanalyse" was derived from the original english term sensitivity analysis by Müller-Merbach [Operations Research] in 1992. Sensibility analysis is the retranslation of that. These two terms are being here used equally.
After having found an optimal solution for a linear optimization problem by the means of the Simplex algorithm, one can use the sensibility analysis to see, how much the initial data may be changed, without changing the structure of the optimal solution. In other words you can see, the robustness of your optimal programm regarding the change of the objective function or constraints.
Non basic variable change
In the following section, all relations concerning dimensions of numbers are absolute values
The smallest negative and the smallest positive quotient of the optimal solution's right hand side and the corresponding column element define, when added or subtracted, the upper and lower endpoint of the interval in which the constraints may vary
So, you can see the robustness of the optimal solution, concerning the change of the restrictions.
Basic variable change
The smallest negative and the smallest positive quotient of the optimal solution's Objective function coefficient and the corresponding row element define, when subtracted or added, the lower andupper endpoint of the interval in which the coefficients of the objective function may vary
By the interval of basic variables, you can see the robustness of the optimal solution, concerning the change of the objective function.
Complete Example
A garden center has got a bed with a developable area of 100 m².
There is a budget of $720 available to plant roses and carrots in the bed.
However, the maximal planting area of carrots shall not exceed 60m².
The planting shall maximize profit.
The costs for the seed are as following:
- Roses: Carrots:
The selling prices of the products are:
- Roses: Carrots:
We formulate the LP:
- Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): Z: x _1 + 2x_2 \xrightarrow {} max! \qquad | x_1 ; x_2 \ge 0
- Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): y_1: x_2 \le 60
- Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): y_2: x_1 + x_2 \le 100
- Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): y_3: 6x_1 + 9x_2 \le 720
