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=Linear optimization: Parametrical objective function=
 
  
 
'''Parametrical objective function''' is a special part  of linear optimization.
 
'''Parametrical objective function''' is a special part  of linear optimization.
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== Basic  Knowledge ==
  
=Basic  Knowledge=
 
  
To include large changes in the input data you have to add a new variable "q". For simple cases you just summate the new variable "q" multiplicated with a constant vektor "ß" to the objective function. In this case the constant vektor is the value which change the input parameter "c".
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To include large changes in the input data you have to add a new variable "<math>q</math>". For simple cases you just summate the new variable "<math>q</math>" multiplicated with a constant vektor " <math>\beta</math> " to the objective function. In this case the constant vektor is the value which change the input parameter "<math>c</math>".
  
 
The objectiv function is now with a parameter:
 
The objectiv function is now with a parameter:
  
c(q)=c+q*ß
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<math>c(q)=c+q*</math><math>\beta</math>
  
 
Thereby there is a new optimization problem which can be solved.
 
Thereby there is a new optimization problem which can be solved.
  
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<math>P(q)</math><math>=</math><math>(c+</math><math>q</math><math>\beta)</math><math>^T x</math>  <math>\rightarrow</math> <math>max!</math>
  
P(q)=(c+q*ß)^T x --> max!
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<math>Ax=b</math>
Ax=b
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<math>x=>0</math>
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<math>x</math> <math>\ge</math> <math>0</math>
  
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== Exemplification ==
  
=Exemplification=
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In the follow we are going to show an example of "Linear Optimization of Parametrical Objective Function".
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We want to find an optimal combination of product 1 and product 2 to minimize the costs.
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Diffrent from "normal" simplex are the parameters in the objective function.
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This parameters express a modification in the objective funktion, it is usefull for changing the input data without calculating the whole simplex again.
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[[Datei:Unbenannt.PNG]]
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From this tableau you can read out the profit function  with a variable inside:
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<math>G(q)= q*x_1+500x_2-36000</math>
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Transformed into the Simplex-Tableau:
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[[Datei:Tabelle_1.PNG]]  (Tableau 1)
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In Tableau 1 and 2 you do the normal simplex iteration. "Ignoring" the variable "<math>q</math>", setting it equal "0".
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[[Datei:Tabelle_2.PNG]]  (Tableau 2)
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[[Datei:Tabbelle_3.PNG]]  (Tableau 3)
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In Tableau 3 you have an optimal solution if you define <math>q</math>  <math>\le</math> <math>0</math> ,because your head row gets positive <math>\rightarrow</math> acceptable and optimal solution
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== Origin ==
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-Skript OperationResearch_2012SS_Wendt
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- http://www.orklaert.de/parametrische-lineare-optimierung
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- https://bisor.wiwi.uni-kl.de/orwiki/Parametrische_Optimierung
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==Made By==
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Matrknr.: 381935; 380535

Aktuelle Version vom 28. Juli 2013, 21:43 Uhr

Parametrical objective function is a special part of linear optimization. The foundation of parametrical optimization is the sensitivity analysis. Compared to the sensitivity analysis the Parametrical objective function makes a statement about large changes in the input data.


Basic Knowledge

To include large changes in the input data you have to add a new variable "". For simple cases you just summate the new variable "" multiplicated with a constant vektor " " to the objective function. In this case the constant vektor is the value which change the input parameter "".

The objectiv function is now with a parameter:

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): c(q)=c+q*

Thereby there is a new optimization problem which can be solved.

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \rightarrow


Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \ge


Exemplification

In the follow we are going to show an example of "Linear Optimization of Parametrical Objective Function". We want to find an optimal combination of product 1 and product 2 to minimize the costs. Diffrent from "normal" simplex are the parameters in the objective function. This parameters express a modification in the objective funktion, it is usefull for changing the input data without calculating the whole simplex again.

Fehler beim Erstellen des Vorschaubildes: Die Miniaturansicht konnte nicht am vorgesehenen Ort gespeichert werden


From this tableau you can read out the profit function with a variable inside:


Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): G(q)= q*x_1+500x_2-36000



Transformed into the Simplex-Tableau:


Fehler beim Erstellen des Vorschaubildes: Die Miniaturansicht konnte nicht am vorgesehenen Ort gespeichert werden
(Tableau 1)

In Tableau 1 and 2 you do the normal simplex iteration. "Ignoring" the variable "", setting it equal "0".


Fehler beim Erstellen des Vorschaubildes: Die Miniaturansicht konnte nicht am vorgesehenen Ort gespeichert werden
(Tableau 2)



Fehler beim Erstellen des Vorschaubildes: Die Miniaturansicht konnte nicht am vorgesehenen Ort gespeichert werden
(Tableau 3)


In Tableau 3 you have an optimal solution if you define Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \le

 ,because your head row gets positive Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \rightarrow
acceptable and optimal solution

Origin

-Skript OperationResearch_2012SS_Wendt

- http://www.orklaert.de/parametrische-lineare-optimierung

- https://bisor.wiwi.uni-kl.de/orwiki/Parametrische_Optimierung


Made By

Matrknr.: 381935; 380535