Linear optimization: Sensibility analysis 1: Unterschied zwischen den Versionen
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''In the following section, all relations concerning dimensions of numbers are [http://en.wikipedia.org/wiki/Absolute_value absolute values]'' | ''In the following section, all relations concerning dimensions of numbers are [http://en.wikipedia.org/wiki/Absolute_value absolute values]'' | ||
| − | The smallest negative and the smallest positive quotient of the optimal solution's '''right hand side''' and the corresponding '''column element''' of the optimal solution define, when added or subtracted to the '''right hand Side of the initial solution''', the '''upper''' and '''lower endpoint''' of the interval in which the constraints may vary | + | The smallest negative and the smallest positive quotient of the optimal solution's '''right hand side''' and the corresponding '''column element''' of the optimal solution define, when added or subtracted to the '''right hand Side of the initial solution''', 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. | So, you can see the robustness of the optimal solution, concerning the change of the restrictions. | ||
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=== Basic variable change === | === Basic variable change === | ||
| − | The smallest negative and the smallest positive quotient of the optimal solution's '''Objective function coefficient''' and the corresponding '''row element''' of the optimal solution define, when subtracted or added to the '''coefficients of the initial objective function''', the '''lower''' and '''upper endpoint''' of the interval in which the coefficients of the objective function may vary | + | The smallest negative and the smallest positive quotient of the optimal solution's '''Objective function coefficient''' and the corresponding '''row element''' of the optimal solution define, when subtracted or added to the '''coefficients of the initial objective function''', the '''lower''' and '''upper 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. | By the interval of basic variables, you can see the robustness of the optimal solution, concerning the change of the objective function. | ||
Version vom 28. Juni 2013, 00:28 Uhr
Besetzt H&S
Inhaltsverzeichnis
Theory
Terminology
The german term "Sensibilitätsanalyse" was derived from the original english term "sensitivity analysis" by Müller-Merbach [Operations Research] in 1992. The term sensibility analysis, that is used in this Article, is used synonymous to that.
Use
After finding 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 of the optimal solution define, when added or subtracted to the right hand Side of the initial solution, 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 of the optimal solution define, when subtracted or added to the coefficients of the initial objective function, the lower and upper 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
Text
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:
Formal LP
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): OF: x _1 + 2x_2 \xrightarrow {} max! \qquad | x_1 ; x_2 \ge 0
1) Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_2+ y_1 \le 60
2) Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_1 + x_2 + y_2 \le 100
3) Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 6x_1 + 9x_2 + y_3 \le 720
Simplex tableaus
Initial Tableau
