Linear optimization: Parametrical objective function 4: Unterschied zwischen den Versionen

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 "${\displaystyle q}$". For simple cases you just summate the new variable "${\displaystyle q}$" multiplicated with a constant vektor " ${\displaystyle \beta }$ " to the objective function. In this case the constant vektor is the value which change the input parameter "${\displaystyle c}$".

The objectiv function is now with a parameter:

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): c(q)=c+q* ${\displaystyle \beta }$

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

${\displaystyle P(q)}$${\displaystyle =}$${\displaystyle (c+}$${\displaystyle q}$${\displaystyle \beta )}$${\displaystyle ^{T}x}$ Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \rightarrow

${\displaystyle max!}$

${\displaystyle Ax=b}$

${\displaystyle x}$ Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \ge

${\displaystyle 0}$

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.

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:

In Tableau 1 and 2 you do the normal simplex iteration. "Ignoring" the variable "${\displaystyle q}$", setting it equal "0".

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

${\displaystyle 0}$ ,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

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