Linear optimization: Parametrical objective function 1

Parametrical objective function is a component of Operation Research. It belongs to the linear optimization and is a speciall part of the parametrical optimization . With this tool you can determine the limits of the simplex-restrictions as the sensitivity analysis. It can be changed the objective function coefficients. Hereby it can be checked how much one of the objective function coefficients depend on the objective function.

Explanation

First of all you define one of the objective function coefficients and give them the value q. The value q is in the first iteration zero. Then you determine the pivot element with the normal simplex rules. So you get the first allowable solution.

So that the table is not the optimal solution, the part with the variable q has to be negative. In case of the part with the variable q is negative you can pivot the table for second time.

Now you have new limits for q and you can do one more iteration step and get other limits. The limits are always a lower bound and an upper bound. You end as soon as q has to be larger than the new limit. In addition the limit must be a lower bound for q and q can not be an upper bound.

Example

First of all we have our beginning tableau with the variable q. The red circle in evry tableau shows the pivot element.

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

                                                                                   Tableau 1

In tableau 1 we set q=0. Then we do the normal simplex-algorithm which give us Tableau 2.

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

                                                                                   Tableau 2

The second tableau is not an optimal solution because the term with q can be negative. Our variable q can be between 0 and 100. So we do a further iteration step with the normal simplex rules and get tableau 3.

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

                                                                                   Tableau 3

After the second iteration we get the final tableau with the optimal solution. The variable q has to be larger than or equal 100. The expression "larger than or equal" give us the hint that we are at the end. Furthermore no value in the objective function we have no negative numbers. Now the objective function depends on our variable q.

Lecture

| Sensitivitätsanalyse und Parametrische Optimierung (English)

Sources

• script of the summer semester 2013 at TU Kaiserslautern by Prof. Dr. Oliver Wendt

• Wikipedia link

• Müller-Merbach: Operation Research

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• Matriculation number 382141 and 380643