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

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== Cutting Planes Gruppe 2 ==
 
  
 
== Idea ==
 
== Idea ==
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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.
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The Idea of the Cutting Plane is to add Restrictions (the Cutting Planes) to contract the solution space more and more to get an integer solution. These restrictions cut off the non-integer parts of the solution.
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+
  
 
== Approach ==
 
== Approach ==
 
First of all you should divide your restrictions trough the greates common factor.
 
First of all you should divide your restrictions trough the greates common factor.
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:: Example: 6x1 – 15x2 <= 120
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:: Example: <math>6*x_1 + 15*x_2 <= 120</math> | gcf = 3
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:: <math>\Rightarrow</math> 3 x1 – 5 x2 <= 40
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:: <math>\Rightarrow</math><math>2*x_1 + 5*x_2 <= 40</math>
  
 
:(1) You search for the continous Optimum with the Simplex Algorithm. If your solution is integer you are finished.
 
:(1) You search for the continous Optimum with the Simplex Algorithm. If your solution is integer you are finished.
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:::<math>y_{neu} = g_r - \sum(g_{rj}*NBV_j)-BV_r</math>
 
:::<math>y_{neu} = g_r - \sum(g_{rj}*NBV_j)-BV_r</math>
 
::
 
::
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:<math>\Rightarrow</math><math>\sum(f_{rj}*NBV_j) = f_r + y_{neu}</math>
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:::<math>\Rightarrow</math><math>\sum(f_{rj}*NBV_j) = f_r + y_{neu}</math>
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::Now we have to constract our new formula in the same form like the others. You have to consider, that <math>y_{neu}</math> cannot have a algebraic sign.
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::Now we have to constract our new formula in the same form like the others. You have to consider, that <math>y_{neu}</math> cannot have an algebraic sign.
 
:::<math>y_{neu} - (f_{rj}*NBV_j) = -f_r</math>
 
:::<math>y_{neu} - (f_{rj}*NBV_j) = -f_r</math>
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::This is now our Cutting Plane.
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 +
== Example ==
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:::<math>max G = 2*x_1 + 3*x_2</math>
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::Subject to
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:::<math>6*x_1 + 15*x_2 <= 120</math>
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:::<math>14*x_1 + 6*x_2 <= 141</math>
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::
 +
:(0) if possible: round
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:::<math>6*x_1 + 15*x_2 <= 120</math> |:3
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:::<math>\Rightarrow</math><math>2*x_1 + 5*x_2 <= 40</math>
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::
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:::<math>14*x_1 + 6*x_2 <= 141</math> |:2
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:::<math>\Rightarrow</math><math>7*x_1 + 3*x_2 <= 70</math>
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::
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:(1) Create the Simplex-Tableau and find the continuous optimum with the Simplex-Algorithm:
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 +
:::{| class="wikitable float-right"  style="text-align:center"
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|+ Initial tableau
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|- style="height: 50px;"
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| scope="col" width="50" |  || scope="col" width="50" | <math>x_1</math> || scope="col" width="50" | <math>x_2</math> ||  scope="col" width="50" |
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|- style="height: 50px;"
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| <math>G</math> || <math>-2</math> || <math>-3</math> || <math>0</math>
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|- style="height: 50px;"
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| <math>y_1</math> || <math>2</math> || <math>-5</math> || <math>40</math>
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|- style="height: 50px;"
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| <math>y_2</math> || <math>7</math> || style="background:#fedcba" | <math>3</math> || <math>10</math>
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|}
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 +
:After one iteration we get the first optimal solution. But it's not integer yet.
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:::{| class="wikitable float-right"  style="text-align:center"
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|+ Optimal solution (non integer)
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|- style="height: 50px;"
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| scope="col" width="50" |  || scope="col" width="50" | <math>x_1</math> || scope="col" width="50" | <math>y_2</math> ||  scope="col" width="50" |
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|- style="height: 50px;"
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| <math>G</math> || <math>5</math> || <math>1</math> || <math>70</math>
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|- style="height: 50px;"
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| <math>y_1</math> || <math>\frac{41}{3}</math> || <math>\frac{5}{3}</math> || <math>\frac{470}{3}</math>
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|- style="height: 50px;"
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| <math>x_2</math> || <math>\frac{7}{3}</math> || <math>\frac{1}{3}</math> || <math>\frac{70}{3}</math>
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|}
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 +
:Now we have to insert the Cutting Plane. We choose <math>y_1</math> because it has the greatest non integer part.
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:: Source row:
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::<math>y_1 + \frac{41}{3} * x_1 + \frac{5}{3} * y_2 =\frac{470}{3}</math>
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::Divided:
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::<math>y_1 + (13 + \frac{2}{3}) * x_1 + (1 + \frac{2}{3}) * y_2 =156 + \frac{2}{3}</math>
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::with slack variable:
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::<math>y_1 + \frac{2}{3} * x_1 + \frac{2}{3} * y_2 =\frac{2}{3} + y_3</math>
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::Restriction for the Simplex_Tableau:
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::<math>y_3 - \frac{2}{3} * x_1 - \frac{2}{3} * y_2 = -\frac{2}{3}</math>
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:::{| class="wikitable float-right"  style="text-align:center"
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|+ Extended Simplex Tableau
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|- style="height: 50px;"
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| scope="col" width="50" |  || scope="col" width="50" | <math>x_1</math> || scope="col" width="50" | <math>y_2</math> ||  scope="col" width="50" |
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|- style="height: 50px;"
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| <math>G</math> || <math>5</math> || <math>1</math> || <math>70</math>
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|- style="height: 50px;"
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| <math>y_1</math> || <math>\frac{41}{3}</math> || <math>\frac{5}{3}</math> || <math>\frac{470}{3}</math>
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|- style="height: 50px;"
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| <math>x_2</math> || <math>\frac{7}{3}</math> || <math>\frac{1}{3}</math> || <math>\frac{70}{3}</math>
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|- style="height: 50px;"
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| <math>y_3</math> || <math>-\frac{2}{3}</math> || <math>-\frac{2}{3}</math> || <math>-\frac{2}{3}</math>
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|}
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::After two further iterations we get the optimal integer solution:
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:::{| class="wikitable float-right"  style="text-align:center"
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|+ Optimal Solution (Integer)
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|- style="height: 50px;"
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| scope="col" width="50" |  || scope="col" width="50" | <math>y_3</math> || scope="col" width="50" | <math>x_1</math> ||  scope="col" width="50" |
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|- style="height: 50px;"
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| <math>G</math> || <math>\frac{3}{2}</math> || <math>4</math> || <math>69</math>
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|- style="height: 50px;"
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| <math>y_1</math> || <math>\frac{5}{2}</math> || <math>12</math> || <math>155</math>
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|- style="height: 50px;"
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| <math>x_2</math> || <math>\frac{1}{2}</math> || <math>2</math> || <math>23</math>
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|- style="height: 50px;"
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| <math>y_2</math> || <math>-\frac{3}{2}</math> || <math>1</math> || <math>1</math>
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|}
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::The solution is integer an optimal.
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== References ==
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[http://en.wikipedia.org/wiki/Cutting-plane_method] Wikipedia Article about Cutting Planes

Aktuelle Version vom 21. Juni 2013, 17:32 Uhr

Idea

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

Approach

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

Example: Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 6*x_1 + 15*x_2 <= 120
| gcf = 3
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.): 2*x_1 + 5*x_2 <= 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 and into the integer part and the rest (non integer) .
(Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 0 \leq f_{ij} < 1

)

(Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 0 \leq f_i < 1

)

We convert r to:
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): BV_r+\sum ((f_{rj} + g_{rj})*NBV_j) = g_r + f_r
Take the integer parts on the right side of the formula
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \sum(f_{rj}*NBV_j) = g_r + f_r - \sum(g_{rj}*NBV_j)-BV_r
Merge the integer parts into the slack variable
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): y_{neu} = g_r - \sum(g_{rj}*NBV_j)-BV_r
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.): \sum(f_{rj}*NBV_j) = f_r + y_{neu}

Now we have to constract our new formula in the same form like the others. You have to consider, that cannot have an algebraic sign.
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): y_{neu} - (f_{rj}*NBV_j) = -f_r
This is now our Cutting Plane.

Example

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): max G = 2*x_1 + 3*x_2
Subject to
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 6*x_1 + 15*x_2 <= 120
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 14*x_1 + 6*x_2 <= 141
(0) if possible: round
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 6*x_1 + 15*x_2 <= 120
|:3
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.): 2*x_1 + 5*x_2 <= 40

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 14*x_1 + 6*x_2 <= 141
|:2
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.): 7*x_1 + 3*x_2 <= 70

(1) Create the Simplex-Tableau and find the continuous optimum with the Simplex-Algorithm:
Initial tableau
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): -2 Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): -3
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): -5
After one iteration we get the first optimal solution. But it's not integer yet.
Optimal solution (non integer)
Now we have to insert the Cutting Plane. We choose because it has the greatest non integer part.
Source row:
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): y_1 + \frac{41}{3} * x_1 + \frac{5}{3} * y_2 =\frac{470}{3}
Divided:
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): y_1 + (13 + \frac{2}{3}) * x_1 + (1 + \frac{2}{3}) * y_2 =156 + \frac{2}{3}
with slack variable:
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): y_1 + \frac{2}{3} * x_1 + \frac{2}{3} * y_2 =\frac{2}{3} + y_3
Restriction for the Simplex_Tableau:
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): y_3 - \frac{2}{3} * x_1 - \frac{2}{3} * y_2 = -\frac{2}{3}


Extended Simplex Tableau
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): -\frac{2}{3} Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): -\frac{2}{3} Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): -\frac{2}{3}
After two further iterations we get the optimal integer solution:
Optimal Solution (Integer)
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): -\frac{3}{2}
The solution is integer an optimal.

References

[1] Wikipedia Article about Cutting Planes

Von „https://www.wiwi.uni-kl.de/bisor-orwiki/index.php?title=Integer_linear_optimization:_Cutting_Planes_2&oldid=3699“