Integer linear optimization: Cutting Planes 2: Unterschied zwischen den Versionen

Version vom 20. Juni 2013, 12:10 Uhr

Cutting Planes Gruppe 2

Idea

The Idea of the Cutting Plane is to add Restrictions (the Cutting Planes) to contract the solution space more and more to get integers. These restrictions cut of the non-integer parts of the solution.

Approach

First of all you should divide your restrictions trough the greates common factor.

Example: 6x1 – 15x2 <= 120

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

3 x1 – 5 x2 <= 40

(1) You search for the continous Optimum with the Simplex Algorithm. If your solution is integer you are finished.

(2) Otherwise you have to insert Cutting Planes

Your source row is the one with the greates non integer part.

Row r:

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): BV_r+\sum (a_{ij}*NBV_j) = b_r

Converting r to: You have to break all ${\displaystyle a_{rj}}$ and ${\displaystyle b_{j}}$ into the integer part ${\displaystyle g_{rj}}$ and the rest (non integer) ${\displaystyle f_{j}}$.

${\displaystyle a_{ij}=g_{ij}+f_{ij}}$ ${\displaystyle \qquad }$ (Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 0 \leq f_{ij} < 1

)

${\displaystyle b_{i}=g_{i}+f_{i}}$ ${\displaystyle \qquad }$ (Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 0 \leq f_i < 1

)