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Version vom 31. Mai 2013, 17:18 Uhr

Cutting-plane method

Cutting plane method is an approach to the solution of integer linear programming problems.First of all we do not consider that variables ${\displaystyle x_{i}}$ is an integer. But by adding linear constraints (so called cutting plane) the original feasible region will be cut.That part which was cut off contains only non-integer solution.Any integer feasible solution is kept in the ”modified” feasible region. This method is to point out how to find the right kinds of cutting planes (not necessarily only one to find), so that after cutting the feasible region finally we actually the get integer optimal solution. This method is proposed by R. E. Gomory, and it is also known as Gomory's cutting plane method.

Example

Consider the integer optimization problem

Maximize ${\displaystyle Z=4x_{1}+3x_{2}}$

Subject to

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 4x_1+5x_2 \le 20

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 2x_1+x_2 \le 6

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_i \geq 0,x_i \mbox{ an integer for } i=1,2

Introduce slack variables ${\displaystyle y_{A},y_{B}}$ to produce the standard form

Maximize ${\displaystyle Z=4x_{1}+3x_{2}}$

Subject to

${\displaystyle y_{A}+4x_{1}+5x_{2}=20}$

${\displaystyle y_{B}+2x_{1}+x_{2}=6}$

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_i \geq 0,x_i \mbox{ an integer for } i=1,2

We use simplex method to get the optimal solution.(See "First optimal solution")

We.........