Linear optimization: Upper and lower bounds 1
Linear Optimization
Linear Optimization is a mathematical method for improving your outcome by using restrictions.
Normally you add contraints to your Simplex system. The Problem is that you will get a great simplex tableau and to many iteration steps. Therefore we will show a way to consider upper and lower bounds much easier by modifying the standard simplex sheet.
How to integrate lower bounds like Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_{j}\geq l_{j}
?
Mathematically you just need to transform the old variable into new one .
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): {\color{Red} {x}'_{j}}= x_{j}-l_{j}
Now the lower bound changes to a non-negative-constraint.
By changing to we also need to replace the right site of our simplex tableau.
The standard linear projection looks like
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): y_{i}+\sum a_{ij}\cdot x_{j}= b_{i}
Our modification leads us to the following formula.
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): y_{i}+\sum a_{ij}\cdot ({x}'_{j}+l_{j})= b_{i}
Now we just change the composition a bit to get our standard structure of the linear projection.
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): y_{i}+\sum a_{ij}\cdot {x}'_{j}= b_{i}-\sum a_{ij}\cdot l_{j}
It is important to replace back to for getting the right results to the processed exercise.
Graphically you need to shift the coordination system like pictured below.
Here you see that the whole coordination system was moved up by the respective lower bound. was moved up by and and by .
Example:
We also want to solve an example for you as you can see below.